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nowraplinks"><tbody><tr><td class="sidebar-pretitle" style="padding-bottom:0.15em;">Part of <a href="/wiki/Category:Network_science" title="Category:Network science">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle" style="font-size:175%;"><a href="/wiki/Network_science" title="Network science">Network science</a></th></tr><tr><td class="sidebar-image"><div class="center"><div class="center"> <div style="width: 250px; height: 250px; overflow: hidden;"> <div style="position: relative; top: -0px; left: -0px; width: 250px"><div class="noresize"><span typeof="mw:File"><a href="/wiki/File:Internet_map_1024.jpg" class="mw-file-description"><img alt="Internet_map_1024.jpg" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Internet_map_1024.jpg/250px-Internet_map_1024.jpg" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Internet_map_1024.jpg/375px-Internet_map_1024.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Internet_map_1024.jpg/500px-Internet_map_1024.jpg 2x" data-file-width="1280" data-file-height="1280" /></a></span></div></div> </div> </div></div></td></tr><tr><th class="sidebar-heading"> <div class="hlist"><ul><li><a href="/wiki/Network_theory" title="Network theory">Theory</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <ul><li><a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">Graph</a></li> <li><a href="/wiki/Complex_network" title="Complex network">Complex network</a></li> <li><a href="/wiki/Complex_contagion" title="Complex contagion">Contagion</a></li> <li><a href="/wiki/Small-world_network" title="Small-world network">Small-world</a></li> <li><a href="/wiki/Scale-free_network" title="Scale-free network">Scale-free</a></li> <li><a href="/wiki/Community_structure" title="Community structure">Community structure</a></li> <li><a class="mw-selflink selflink">Percolation</a></li> <li><a href="/wiki/Evolving_networks" class="mw-redirect" title="Evolving networks">Evolution</a></li> <li><a href="/wiki/Network_controllability" title="Network controllability">Controllability</a></li> <li><a href="/wiki/Graph_drawing" title="Graph drawing">Graph drawing</a></li> <li><a href="/wiki/Social_capital" title="Social capital">Social capital</a></li> <li><a href="/wiki/Link_analysis" title="Link analysis">Link analysis</a></li> <li><a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">Optimization</a></li> <li><a href="/wiki/Reciprocity_(network_science)" title="Reciprocity (network science)">Reciprocity</a></li> <li><a href="/wiki/Triadic_closure" title="Triadic closure">Closure</a></li> <li><a href="/wiki/Homophily" title="Homophily">Homophily</a></li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitivity</a></li> <li><a href="/wiki/Preferential_attachment" title="Preferential attachment">Preferential attachment</a></li> <li><a href="/wiki/Balance_theory" title="Balance theory">Balance theory</a></li> <li><a href="/wiki/Network_effect" title="Network effect">Network effect</a></li> <li><a href="/wiki/Social_influence" title="Social influence">Social influence</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Network types</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <ul><li><a href="/wiki/Computer_network" title="Computer network">Informational (computing)</a></li> <li><a href="/wiki/Telecommunications_network" title="Telecommunications network">Telecommunication</a></li> <li><a href="/wiki/Transport_network" class="mw-redirect" title="Transport network">Transport</a></li> <li><a href="/wiki/Social_network" title="Social network">Social</a></li> <li><a href="/wiki/Scientific_collaboration_network" title="Scientific collaboration network">Scientific collaboration</a></li> <li><a href="/wiki/Biological_network" title="Biological network">Biological</a></li> <li><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural</a></li> <li><a href="/wiki/Interdependent_networks" title="Interdependent networks">Interdependent</a></li> <li><a href="/wiki/Semantic_network" title="Semantic network">Semantic</a></li> <li><a href="/wiki/Spatial_network" title="Spatial network">Spatial</a></li> <li><a href="/wiki/Dependency_network" title="Dependency network">Dependency</a></li> <li><a href="/wiki/Flow_network" title="Flow network">Flow</a></li> <li><a href="/wiki/Network_on_a_chip" title="Network on a chip">on-Chip</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">Graphs</a></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Features</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Clique_(graph_theory)" title="Clique (graph theory)">Clique</a></li> <li><a href="/wiki/Connected_component_(graph_theory)" class="mw-redirect" title="Connected component (graph theory)">Component</a></li> <li><a href="/wiki/Cut_(graph_theory)" title="Cut (graph theory)">Cut</a></li> <li><a href="/wiki/Cycle_(graph_theory)" title="Cycle (graph theory)">Cycle</a></li> <li><a href="/wiki/Graph_(abstract_data_type)" title="Graph (abstract data type)">Data structure</a></li> <li><a href="/wiki/Edge_(graph_theory)" class="mw-redirect" title="Edge (graph theory)">Edge</a></li> <li><a href="/wiki/Loop_(graph_theory)" title="Loop (graph theory)">Loop</a></li> <li><a href="/wiki/Neighbourhood_(graph_theory)" title="Neighbourhood (graph theory)">Neighborhood</a></li> <li><a href="/wiki/Path_(graph_theory)" title="Path (graph theory)">Path</a></li> <li><a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">Vertex</a></li> <li><span class="nowrap"><a href="/wiki/Adjacency_list" title="Adjacency list">Adjacency list</a> / <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">matrix</a></span></li> <li><span class="nowrap"><a href="/wiki/Incidence_list" class="mw-redirect" title="Incidence list">Incidence list</a> / <a href="/wiki/Incidence_matrix" title="Incidence matrix">matrix</a></span></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Types</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Bipartite_graph" title="Bipartite graph">Bipartite</a></li> <li><a href="/wiki/Complete_graph" title="Complete graph">Complete</a></li> <li><a href="/wiki/Directed_graph" title="Directed graph">Directed</a></li> <li><a href="/wiki/Hypergraph" title="Hypergraph">Hyper</a></li> <li><a href="/wiki/Labeled_graph" class="mw-redirect" title="Labeled graph">Labeled</a></li> <li><a href="/wiki/Multigraph" title="Multigraph">Multi</a></li> <li><a href="/wiki/Random_graph" title="Random graph">Random</a></li> <li><a href="/wiki/Weighted_graph" class="mw-redirect" title="Weighted graph">Weighted</a></li></ul></td> </tr></tbody></table></td> </tr><tr><th class="sidebar-heading"> <div class="hlist"><ul><li><a href="/wiki/Metrics_(networking)" title="Metrics (networking)">Metrics</a></li><li><a href="/wiki/List_of_algorithms#Networking" title="List of algorithms">Algorithms</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <ul><li><a href="/wiki/Centrality" title="Centrality">Centrality</a></li> <li><a href="/wiki/Degree_(graph_theory)" title="Degree (graph theory)">Degree</a></li> <li><a href="/wiki/Network_motif" title="Network motif">Motif</a></li> <li><a href="/wiki/Clustering_coefficient" title="Clustering coefficient">Clustering</a></li> <li><a href="/wiki/Degree_distribution" title="Degree distribution">Degree distribution</a></li> <li><a href="/wiki/Assortativity" title="Assortativity">Assortativity</a></li> <li><a href="/wiki/Distance_(graph_theory)" title="Distance (graph theory)">Distance</a></li> <li><a href="/wiki/Modularity_(networks)" title="Modularity (networks)">Modularity</a></li> <li><a href="/wiki/Efficiency_(network_science)" title="Efficiency (network science)">Efficiency</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Models</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Topology</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Random_graph" title="Random graph">Random graph</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi</a></li> <li><a href="/wiki/Barab%C3%A1si%E2%80%93Albert_model" title="Barabási–Albert model">Barabási–Albert</a></li> <li><a href="/wiki/Bianconi%E2%80%93Barab%C3%A1si_model" title="Bianconi–Barabási model">Bianconi–Barabási</a></li> <li><a href="/wiki/Fitness_model_(network_theory)" title="Fitness model (network theory)">Fitness model</a></li> <li><a href="/wiki/Watts%E2%80%93Strogatz_model" title="Watts–Strogatz model">Watts–Strogatz</a></li> <li><a href="/wiki/Exponential_random_graph_models" class="mw-redirect" title="Exponential random graph models">Exponential random (ERGM)</a></li> <li><a href="/wiki/Random_geometric_graph" title="Random geometric graph">Random geometric (RGG)</a></li> <li><a href="/wiki/Hyperbolic_geometric_graph" title="Hyperbolic geometric graph">Hyperbolic (HGN)</a></li> <li><a href="/wiki/Hierarchical_network_model" title="Hierarchical network model">Hierarchical</a></li> <li><a href="/wiki/Stochastic_block_model" title="Stochastic block model">Stochastic block</a></li> <li><a href="/wiki/Blockmodeling" title="Blockmodeling">Blockmodeling</a></li> <li><a href="/wiki/Maximum-entropy_random_graph_model" title="Maximum-entropy random graph model">Maximum entropy</a></li> <li><a href="/wiki/Soft_configuration_model" title="Soft configuration model">Soft configuration</a></li> <li><a href="/wiki/Lancichinetti%E2%80%93Fortunato%E2%80%93Radicchi_benchmark" title="Lancichinetti–Fortunato–Radicchi benchmark">LFR Benchmark</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Dynamics</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Boolean_network" title="Boolean network">Boolean network</a></li> <li><a href="/wiki/Agent-based_model" title="Agent-based model">agent based</a></li> <li><a href="/wiki/Epidemic_model" class="mw-redirect" title="Epidemic model">Epidemic</a>/<a href="/wiki/SIR_model" class="mw-redirect" title="SIR model">SIR</a></li></ul></td> </tr></tbody></table></td> </tr><tr><th class="sidebar-heading"> <div class="hlist"><ul><li>Lists</li><li>Categories</li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <ul><li><a href="/wiki/List_of_network_theory_topics" title="List of network theory topics">Topics</a></li> <li><a href="/wiki/Social_network_analysis_software" title="Social network analysis software">Software</a></li> <li><a href="/wiki/List_of_network_scientists" title="List of network scientists">Network scientists</a></li></ul> <ul><li><a href="/wiki/Category:Network_theory" title="Category:Network theory">Category:Network theory</a></li> <li><a href="/wiki/Category:Graph_theory" title="Category:Graph theory">Category:Graph theory</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Network_science" title="Template:Network science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Network_science" title="Template talk:Network science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Network_science" title="Special:EditPage/Template:Network science"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Statistical_physics" class="mw-redirect" title="Statistical physics">statistical physics</a> and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>percolation theory</b> describes the behavior of a network when nodes or links are added. This is a geometric type of <a href="/wiki/Phase_transition" title="Phase transition">phase transition</a>, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger <a href="/wiki/Glossary_of_graph_theory" title="Glossary of graph theory">connected</a>, so-called spanning clusters. The applications of percolation theory to <a href="/wiki/Materials_science" title="Materials science">materials science</a> and in many other disciplines are discussed here and in the articles <a href="/wiki/Network_theory" title="Network theory">Network theory</a> and <a href="/wiki/Percolation_(cognitive_psychology)" title="Percolation (cognitive psychology)">Percolation (cognitive psychology)</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Perc-wiki.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Perc-wiki.png/220px-Perc-wiki.png" decoding="async" width="220" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Perc-wiki.png/330px-Perc-wiki.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Perc-wiki.png/440px-Perc-wiki.png 2x" data-file-width="515" data-file-height="480" /></a><figcaption>A three-dimensional site percolation graph</figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Transition_de_percolation_2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Transition_de_percolation_2.gif/220px-Transition_de_percolation_2.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Transition_de_percolation_2.gif/330px-Transition_de_percolation_2.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Transition_de_percolation_2.gif/440px-Transition_de_percolation_2.gif 2x" data-file-width="500" data-file-height="500" /></a><figcaption>Bond percolation in a square lattice from p=0.3 to p=0.52</figcaption></figure> <p>A representative question (and the <a href="/wiki/Etymology" title="Etymology">source</a> of the name) is as follows. Assume that some liquid is poured on top of some <a href="/wiki/Porosity" title="Porosity">porous</a> material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is <a href="/wiki/Mathematical_model" title="Mathematical model">modelled</a> mathematically as a <a href="/wiki/Grid_graph" class="mw-redirect" title="Grid graph">three-dimensional network</a> of <span class="texhtml"><i>n</i> × <i>n</i> × <i>n</i></span> <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">vertices</a>, usually called "sites", in which the <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">edge</a> or "bonds" between each two neighbors may be open (allowing the liquid through) with probability <span class="texhtml"><i>p</i></span>, or closed with probability <span class="texhtml">1 – <i>p</i></span>, and they are assumed to be independent. Therefore, for a given <span class="texhtml"><i>p</i></span>, what is the probability that an open path (meaning a path, each of whose links is an "open" bond) exists from the top to the bottom? The behavior for large <span class="texhtml"><i>n</i></span> is of primary interest. This problem, called now <b>bond percolation</b>, was introduced in the mathematics literature by <a href="#CITEREFBroadbentHammersley1957">Broadbent & Hammersley (1957)</a>,<sup id="cite_ref-BroadbentHammersley1957_1-0" class="reference"><a href="#cite_note-BroadbentHammersley1957-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and has been studied intensively by mathematicians and physicists since then. </p><p>In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability <span class="texhtml"><i>p</i></span> or "empty" (in which case its edges are removed) with probability <span class="texhtml">1 – <i>p</i></span>; the corresponding problem is called <b>site percolation</b>. The question is the same: for a given <i>p</i>, what is the probability that a path exists between top and bottom? Similarly, one can ask, given a connected graph at what fraction <span class="texhtml">1 – <i>p</i></span> of failures the graph will become disconnected (no large component). </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tube_Network_Percolation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Tube_Network_Percolation.gif/220px-Tube_Network_Percolation.gif" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Tube_Network_Percolation.gif/330px-Tube_Network_Percolation.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Tube_Network_Percolation.gif/440px-Tube_Network_Percolation.gif 2x" data-file-width="560" data-file-height="420" /></a><figcaption>A 3D tube network percolation determination</figcaption></figure> <p>The same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine <a href="/wiki/Infinite_graph" class="mw-redirect" title="Infinite graph">infinite</a> networks than just large ones. In this case the corresponding question is: does an infinite open cluster exist? That is, is there a path of connected points of infinite length "through" the network? By <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law">Kolmogorov's zero–one law</a>, for any given <span class="texhtml"><i>p</i></span>, the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of <span class="texhtml"><i>p</i></span> (proof via <a href="/wiki/Coupling_(probability)" title="Coupling (probability)">coupling</a> argument), there must be a <b>critical</b> <span class="texhtml"><i>p</i></span> (denoted by <span class="texhtml"><i>p</i><sub>c</sub></span>) below which the probability is always 0 and above which the probability is always 1. In practice, this criticality is very easy to observe. Even for <span class="texhtml"><i>n</i></span> as small as 100, the probability of an open path from the top to the bottom increases sharply from very close to zero to very close to one in a short span of values of <span class="texhtml"><i>p</i></span>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bond_percolation_p_51.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Bond_percolation_p_51.png/220px-Bond_percolation_p_51.png" decoding="async" width="220" height="223" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Bond_percolation_p_51.png/330px-Bond_percolation_p_51.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/7b/Bond_percolation_p_51.png 2x" data-file-width="402" data-file-height="407" /></a><figcaption>Detail of a bond percolation on the square lattice in two dimensions with percolation probability <span class="texhtml"><i>p</i> = 0.51</span></figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Flory%E2%80%93Stockmayer_theory" title="Flory–Stockmayer theory">Flory–Stockmayer theory</a> was the first theory investigating percolation processes.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The history of the percolation model as we know it has its root in the coal industry. Since the industrial revolution, the economical importance of this source of energy fostered many scientific studies to understand its composition and optimize its use. During the 1930s and 1940s, the qualitative analysis by organic chemistry left more and more room to more quantitative studies. <sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In this context, the <a href="/wiki/British_Coal_Utilisation_Research_Association" title="British Coal Utilisation Research Association">British Coal Utilisation Research Association</a> (BCURA) was created in 1938. It was a research association funded by the coal mines owners. In 1942, <a href="/wiki/Rosalind_Franklin" title="Rosalind Franklin">Rosalind Franklin</a>, who then recently graduated in chemistry from the university of Cambridge, joined the BCURA. She started research on the density and porosity of coal. During the Second World War, coal was an important strategic resource. It was used as a source of energy, but also was the main constituent of gas masks. </p><p>Coal is a porous medium. To measure its 'real' density, one was to sink it in a liquid or a gas whose molecules are small enough to fill its microscopic pores. While trying to measure the density of coal using several gases (helium, methanol, hexane, benzene), and as she found different values depending on the gas used, Rosalind Franklin showed that the pores of coal are made of microstructures of various lengths that act as a microscopic sieve to discriminate the gases. She also discovered that the size of these structures depends on the temperature of carbonation during the coal production. With this research, she obtained a PhD degree and left the BCURA in 1946. <sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In the mid fifties, Simon Broadbent worked in the BCURA as a statistician. Among other interests, he studied the use of coal in gas masks. One question is to understand how a fluid can diffuse in the coal pores, modeled as a random maze of open or closed tunnels. In 1954, during a symposium on <a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo methods</a>, he asks questions to <a href="/wiki/John_Hammersley" title="John Hammersley">John Hammersley</a> on the use of numerical methods to analyze this model. <sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Broadbent and Hammersley introduced in their article of 1957 a mathematical model to model this phenomenon, that is percolation. </p> <div class="mw-heading mw-heading2"><h2 id="Computation_of_the_critical_parameter">Computation of the critical parameter</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=3" title="Edit section: Computation of the critical parameter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For most infinite lattice graphs, <span class="texhtml"><i>p</i><sub>c</sub></span> cannot be calculated exactly, though in some cases <span class="texhtml"><i>p</i><sub>c</sub></span> there is an exact value. For example: </p> <ul><li>for the <a href="/wiki/Square_lattice" title="Square lattice">square lattice</a> <span class="texhtml"><b>ℤ</b><sup>2</sup></span> in two dimensions, <span class="texhtml"><i>p</i><sub>c</sub> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by <a href="/wiki/Harry_Kesten" title="Harry Kesten">Harry Kesten</a> in the early 1980s,<sup id="cite_ref-BollobásRiordan2006_6-0" class="reference"><a href="#cite_note-BollobásRiordan2006-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> see <a href="#CITEREFKesten1982">Kesten (1982)</a>. For site percolation on the square lattice, the value of <span class="texhtml"><i>p</i><sub>c</sub></span> is not known from analytic derivation but only via simulations of large lattices which provide the estimate <span class="texhtml"><i>p</i><sub>c</sub> = </span> 0.59274621 ± 0.00000013.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>  </li> <li>A limit case for lattices in high dimensions is given by the <a href="/wiki/Bethe_lattice" title="Bethe lattice">Bethe lattice</a>, whose threshold is at <span class="texhtml"><i>p</i><sub>c</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>z</i> − 1</span></span>⁠</span></span> for a <a href="/wiki/Coordination_number" title="Coordination number">coordination number</a> <span class="texhtml"><i>z</i></span>. In other words: for the regular <a href="/wiki/Tree_(graph_theory)" title="Tree (graph theory)">tree</a> of degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/836639c3805ca867b1ff24dc6db7a6b24fc69158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.203ex; height:2.009ex;" alt="{\displaystyle p_{c}}"></span> is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/(z-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/(z-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c46599bf2ed06d9f97f74a51ef33bac5f7a0ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.225ex; height:2.843ex;" alt="{\displaystyle 1/(z-1)}"></span>.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Front_de_percolation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Front_de_percolation.png/220px-Front_de_percolation.png" decoding="async" width="220" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Front_de_percolation.png/330px-Front_de_percolation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Front_de_percolation.png/440px-Front_de_percolation.png 2x" data-file-width="986" data-file-height="698" /></a><figcaption>Percolation front</figcaption></figure> <ul><li>For a random <a href="/wiki/Tree_(graph_theory)" title="Tree (graph theory)">tree-like</a> network without degree-degree correlation, it can be shown that such network can have a <a href="/wiki/Giant_component" title="Giant component">giant component</a>, and the <a href="/wiki/Percolation_threshold" title="Percolation threshold">percolation threshold</a> (transmission probability) is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}={\frac {1}{g_{1}'(1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{c}={\frac {1}{g_{1}'(1)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a33d3711e5f6dcbc78265dae665dacd6c4a08aa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-left: -0.089ex; width:11.273ex; height:6.176ex;" alt="{\displaystyle p_{c}={\frac {1}{g_{1}'(1)}}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a782ef55bcea8427cd547699721d77a474f90f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.061ex; height:2.843ex;" alt="{\displaystyle g_{1}(z)}"></span> is the <a href="/wiki/Degree_distribution#Generating_functions_method" title="Degree distribution">generating function</a> corresponding to the <a href="/wiki/Degree_distribution#Generating_functions_method" title="Degree distribution">excess degree distribution</a>. So, for random <a href="/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi networks</a> of average degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle k\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle k\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79c8e81fd47c64b42f310aa18c5197183dcbb0d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.021ex; height:2.843ex;" alt="{\displaystyle \langle k\rangle }"></span>, <span class="texhtml"><i>p</i><sub>c</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">⟨k⟩</span></span>⁠</span></span>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li>In networks with low <a href="/wiki/Clustering_coefficient" title="Clustering coefficient">clustering</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<C\ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>C</mi> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<C\ll 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8e3468e975da630ec89bec17eee9a3ca9cf1e6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.804ex; height:2.176ex;" alt="{\displaystyle 0<C\ll 1}"></span>, the critical point gets scaled by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-C)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>C</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-C)^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc0279b4d3e97896d9c093c8164997d3aae32e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.911ex; height:3.176ex;" alt="{\displaystyle (1-C)^{-1}}"></span> such that:</li></ul> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}={\frac {1}{1-C}}{\frac {1}{g_{1}'(1)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>C</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{c}={\frac {1}{1-C}}{\frac {1}{g_{1}'(1)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/465dda7f5a10632157ae7f5ab0063d76dc64f9b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-left: -0.089ex; width:18.525ex; height:6.176ex;" alt="{\displaystyle p_{c}={\frac {1}{1-C}}{\frac {1}{g_{1}'(1)}}.}"></span><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>This indicates that for a given degree distribution, the clustering leads to a larger percolation threshold, mainly because for a fixed number of links, the clustering structure reinforces the core of the network with the price of diluting the global connections. For networks with high clustering, strong clustering could induce the core–periphery structure, in which the core and periphery might percolate at different critical points, and the above approximate treatment is not applicable.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Universality">Universality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=4" title="Edit section: Universality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Universality_(dynamical_systems)" title="Universality (dynamical systems)">universality principle</a> states that the numerical value of <span class="texhtml"><i>p</i><sub>c</sub></span> is determined by the local structure of the graph, whereas the behavior near the critical threshold, <span class="texhtml"><i>p</i><sub>c</sub></span>, is characterized by universal <a href="/wiki/Critical_exponents" class="mw-redirect" title="Critical exponents">critical exponents</a>. For example the distribution of the size of clusters at criticality decays as a power law with the same exponent for all 2d lattices. This universality means that for a given dimension, the various critical exponents, the <a href="/wiki/Fractal_dimension" title="Fractal dimension">fractal dimension</a> of the clusters at <span class="texhtml"><i>p</i><sub>c</sub></span> is independent of the lattice type and percolation type (e.g., bond or site). However, recently percolation has been performed on a <a href="/wiki/Weighted_planar_stochastic_lattice_(WPSL)" class="mw-redirect" title="Weighted planar stochastic lattice (WPSL)">weighted planar stochastic lattice (WPSL)</a> and found that although the dimension of the WPSL coincides with the dimension of the space where it is embedded, its universality class is different from that of all the known planar lattices.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Phases">Phases</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=5" title="Edit section: Phases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Subcritical_and_supercritical">Subcritical and supercritical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=6" title="Edit section: Subcritical and supercritical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The main fact in the subcritical phase is "exponential decay". That is, when <span class="texhtml"><i>p</i> < <i>p</i><sub>c</sub></span>, the probability that a specific point (for example, the origin) is contained in an open cluster (meaning a maximal connected set of "open" edges of the graph) of size <span class="texhtml"><i>r</i></span> decays to zero <a href="/wiki/Big_O_notation#Orders_of_common_functions" title="Big O notation">exponentially</a> in <span class="texhtml"><i>r</i></span>. This was proved for percolation in three and more dimensions by <a href="#CITEREFMenshikov1986">Menshikov (1986)</a> and independently by <a href="#CITEREFAizenmanBarsky1987">Aizenman & Barsky (1987)</a>. In two dimensions, it formed part of Kesten's proof that <span class="texhtml"><i>p</i><sub>c</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>.<sup id="cite_ref-Kesten1982_15-0" class="reference"><a href="#cite_note-Kesten1982-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Dual_graph" title="Dual graph">dual graph</a> of the square lattice <span class="texhtml"><b>ℤ</b><sup>2</sup></span> is also the square lattice. It follows that, in two dimensions, the supercritical phase is dual to a subcritical percolation process. This provides essentially full information about the supercritical model with <span class="texhtml"><i>d</i> = 2</span>. The main result for the supercritical phase in three and more dimensions is that, for sufficiently large <span class="texhtml"><i>N</i></span>, there is almost certainly an infinite open cluster in the two-dimensional slab <span class="texhtml"><b>ℤ</b><sup>2</sup> × [0, <i>N</i>]<sup><i>d</i> − 2</sup></span>. This was proved by <a href="#CITEREFGrimmettMarstrand1990">Grimmett & Marstrand (1990)</a>.<sup id="cite_ref-GrimmettMarstrand1990_16-0" class="reference"><a href="#cite_note-GrimmettMarstrand1990-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>In two dimensions with <span class="texhtml"><i>p</i> < <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, there is with probability one a unique infinite closed cluster (a closed cluster is a maximal connected set of "closed" edges of the graph). Thus the subcritical phase may be described as finite open islands in an infinite closed ocean. When <span class="texhtml"><i>p</i> > <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> just the opposite occurs, with finite closed islands in an infinite open ocean. The picture is more complicated when <span class="texhtml"><i>d</i> ≥ 3</span> since <span class="texhtml"><i>p</i><sub>c</sub> < <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, and there is coexistence of infinite open and closed clusters for <span class="texhtml"><i>p</i></span> between <span class="texhtml"><i>p</i><sub>c</sub></span> and <span class="texhtml">1 − <i>p</i><sub>c</sub></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Criticality">Criticality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=7" title="Edit section: Criticality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Percolation_zoom.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Percolation_zoom.gif/220px-Percolation_zoom.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Percolation_zoom.gif/330px-Percolation_zoom.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Percolation_zoom.gif/440px-Percolation_zoom.gif 2x" data-file-width="500" data-file-height="500" /></a><figcaption>Zoom in a critical percolation cluster (Click to animate)</figcaption></figure> <p>Percolation has a <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularity</a> at the critical point <span class="texhtml"><i>p</i> = <i>p</i><sub>c</sub></span> and many properties behave as of a power-law with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p-p_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p-p_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a798d16ed9486c3d0d3e45543669b11d6e822987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.213ex; height:2.343ex;" alt="{\displaystyle p-p_{c}}"></span>, near <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/836639c3805ca867b1ff24dc6db7a6b24fc69158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.203ex; height:2.009ex;" alt="{\displaystyle p_{c}}"></span>. <a href="/wiki/Critical_scaling" class="mw-redirect" title="Critical scaling">Scaling theory</a> predicts the existence of <a href="/wiki/Critical_exponents" class="mw-redirect" title="Critical exponents">critical exponents</a>, depending on the number <i>d</i> of dimensions, that determine the class of the singularity. When <span class="texhtml"><i>d</i> = 2</span> these predictions are backed up by arguments from <a href="/wiki/Conformal_field_theory" title="Conformal field theory">conformal field theory</a> and <a href="/wiki/Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a>, and include predicted numerical values for the exponents. Most of these predictions are conjectural except when the number <span class="texhtml"><i>d</i></span> of dimensions satisfies either <span class="texhtml"><i>d</i> = 2</span> or <span class="texhtml"><i>d</i> ≥ 6</span>. They include: </p> <ul><li>There are no infinite clusters (open or closed)</li> <li>The probability that there is an open path from some fixed point (say the origin) to a distance of <span class="texhtml"><i>r</i></span> decreases <i>polynomially</i>, i.e. is <a href="/wiki/Big_O_notation" title="Big O notation">on the order of</a> <span class="texhtml"><i>r</i><sup><i>α</i></sup></span> for some <span class="texhtml"><i>α</i></span> <ul><li><span class="texhtml"><i>α</i></span> does not depend on the particular lattice chosen, or on other local parameters. It depends only on the dimension <span class="texhtml"><i>d</i></span> (this is an instance of the <a href="/wiki/Universality_(dynamical_systems)" title="Universality (dynamical systems)">universality</a> principle).</li> <li><span class="texhtml"><i>α<sub>d</sub></i></span> decreases from <span class="texhtml"><i>d</i> = 2</span> until <span class="texhtml"><i>d</i> = 6</span> and then stays fixed.</li> <li><span class="texhtml"><i>α</i><sub>2</sub> = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">48</span></span>⁠</span></span></li> <li><span class="texhtml"><i>α</i><sub>6</sub> = −1</span>.</li></ul></li> <li>The shape of a large cluster in two dimensions is <a href="/wiki/Conformal_map" title="Conformal map">conformally invariant</a>.</li></ul> <p>See <a href="#CITEREFGrimmett1999">Grimmett (1999)</a>.<sup id="cite_ref-Grimmett1999_17-0" class="reference"><a href="#cite_note-Grimmett1999-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> In 11 or more dimensions, these facts are largely proved using a technique known as the <a href="/w/index.php?title=Lace_expansion&action=edit&redlink=1" class="new" title="Lace expansion (page does not exist)">lace expansion</a>. It is believed that a version of the lace expansion should be valid for 7 or more dimensions, perhaps with implications also for the threshold case of 6 dimensions. The connection of percolation to the lace expansion is found in <a href="#CITEREFHaraSlade1990">Hara & Slade (1990)</a>.<sup id="cite_ref-HaraSlade1990_18-0" class="reference"><a href="#cite_note-HaraSlade1990-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>In two dimensions, the first fact ("no percolation in the critical phase") is proved for many lattices, using duality. Substantial progress has been made on two-dimensional percolation through the conjecture of <a href="/wiki/Oded_Schramm" title="Oded Schramm">Oded Schramm</a> that the <a href="/wiki/Scaling_limit" class="mw-redirect" title="Scaling limit">scaling limit</a> of a large cluster may be described in terms of a <a href="/wiki/Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a>. This conjecture was proved by <a href="#CITEREFSmirnov2001">Smirnov (2001)</a><sup id="cite_ref-Smirnov2001_19-0" class="reference"><a href="#cite_note-Smirnov2001-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> in the special case of site percolation on the triangular lattice. </p> <div class="mw-heading mw-heading2"><h2 id="Different_models">Different models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=8" title="Edit section: Different models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Directed_percolation" title="Directed percolation">Directed percolation</a> that models the effect of <a href="/wiki/Gravity" title="Gravity">gravitational forces acting on the liquid</a> was also introduced in <a href="#CITEREFBroadbentHammersley1957">Broadbent & Hammersley (1957)</a>,<sup id="cite_ref-BroadbentHammersley1957_1-1" class="reference"><a href="#cite_note-BroadbentHammersley1957-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and has connections with the <a href="/wiki/Contact_process_(mathematics)" title="Contact process (mathematics)">contact process</a>.</li> <li>The first model studied was Bernoulli percolation. In this model all bonds are independent. This model is called bond percolation by physicists.</li> <li>A generalization was next introduced as the <a href="/wiki/Random_cluster_model" title="Random cluster model">Fortuin–Kasteleyn random cluster model</a>, which has many connections with the <a href="/wiki/Ising_model" title="Ising model">Ising model</a> and other <a href="/wiki/Potts_model" title="Potts model">Potts models</a>.</li> <li>Bernoulli (bond) percolation on <a href="/wiki/Complete_graph" title="Complete graph">complete graphs</a> is an example of a <a href="/wiki/Random_graph" title="Random graph">random graph</a>. The critical probability is <span class="texhtml"><i>p</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>N</i></span></span>⁠</span></span>, where <span class="texhtml"><i>N</i></span> is the number of vertices (sites) of the graph.</li> <li><a href="/wiki/Bootstrap_percolation" title="Bootstrap percolation">Bootstrap percolation</a> removes active cells from clusters when they have too few active neighbors, and looks at the connectivity of the remaining cells.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/First_passage_percolation" title="First passage percolation">First passage percolation</a>.</li> <li><a href="/wiki/Invasion_percolation" title="Invasion percolation">Invasion percolation</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=9" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="In_biology,_biochemistry,_and_physical_virology"><span id="In_biology.2C_biochemistry.2C_and_physical_virology"></span>In biology, biochemistry, and physical virology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=10" title="Edit section: In biology, biochemistry, and physical virology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Percolation theory has been used to successfully predict the fragmentation of biological virus shells (capsids),<sup id="cite_ref-Brunk_Twarock_p._21-0" class="reference"><a href="#cite_note-Brunk_Twarock_p.-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> with the fragmentation threshold of <a href="/wiki/Hepatitis_B" title="Hepatitis B">Hepatitis B</a> virus <a href="/wiki/Capsid" title="Capsid">capsid</a> predicted and detected experimentally.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> When a critical number of subunits has been randomly removed from the nanoscopic shell, it fragments and this fragmentation may be detected using Charge Detection Mass Spectroscopy (CDMS) among other single-particle techniques. This is a molecular analog to the common board game <a href="/wiki/Jenga" title="Jenga">Jenga</a>, and has relevance to the broader study of virus disassembly. More stable viral particles (tilings with greater fragmentation thresholds) are found in greater abundance in nature.<sup id="cite_ref-Brunk_Twarock_p._21-1" class="reference"><a href="#cite_note-Brunk_Twarock_p.-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="In_ecology">In ecology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=11" title="Edit section: In ecology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Percolation theory has been applied to studies of how environment fragmentation impacts animal habitats<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> and models of how the plague bacterium <i><a href="/wiki/Yersinia_pestis" title="Yersinia pestis">Yersinia pestis</a></i> spreads.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Bunkbed_conjecture" title="Bunkbed conjecture">Bunkbed conjecture</a> – Conjecture in probabilistic combinatorics</li> <li><a href="/wiki/Continuum_percolation_theory" title="Continuum percolation theory">Continuum percolation theory</a></li> <li><a href="/wiki/Critical_exponent" title="Critical exponent">Critical exponent</a> – Parameter describing physics near critical points</li> <li><a href="/wiki/Directed_percolation" title="Directed percolation">Directed percolation</a> – Physical models of filtering under forces such as gravity</li> <li><a href="/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi model</a> – Two closely related models for generating random graphs</li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a> – Infinitely detailed mathematical structure</li> <li><a href="/wiki/Giant_component" title="Giant component">Giant component</a> – Large connected component of a random graph</li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a> – Area of discrete mathematics</li> <li><a href="/wiki/Interdependent_networks" title="Interdependent networks">Interdependent networks</a> – Subfield of network science</li> <li><a href="/wiki/Invasion_percolation" title="Invasion percolation">Invasion percolation</a></li> <li><a href="/wiki/Kahn%E2%80%93Kalai_conjecture" title="Kahn–Kalai conjecture">Kahn–Kalai conjecture</a> – Mathematical proposition</li> <li><a href="/wiki/Network_theory" title="Network theory">Network theory</a> – Study of graphs as a representation of relations between discrete objects</li> <li><a href="/wiki/Network_science" title="Network science">Network science</a> – Academic field</li> <li><a href="/wiki/Percolation_threshold" title="Percolation threshold">Percolation threshold</a> – Threshold of percolation theory models</li> <li><a href="/wiki/Percolation_critical_exponents" title="Percolation critical exponents">Percolation critical exponents</a> – Mathematical parameter in percolation theory</li> <li><a href="/wiki/Scale-free_network" title="Scale-free network">Scale-free network</a> – Network whose degree distribution follows a power law</li> <li><a href="/wiki/Shortest_path_problem" title="Shortest path problem">Shortest path problem</a> – Computational problem of graph theory</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-BroadbentHammersley1957-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-BroadbentHammersley1957_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BroadbentHammersley1957_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBroadbentHammersley1957" class="citation journal cs1">Broadbent, Simon; <a href="/wiki/John_Hammersley" title="John Hammersley">Hammersley, John</a> (1957). 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Barsky, David (1987), <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.cmp/1104116538">"Sharpness of the phase transition in percolation models"</a>, <i>Communications in Mathematical Physics</i>, <b>108</b> (3): <span class="nowrap">489–</span>526, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1987CMaPh.108..489A">1987CMaPh.108..489A</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01212322">10.1007/BF01212322</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:35592821">35592821</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Mathematical+Physics&rft.atitle=Sharpness+of+the+phase+transition+in+percolation+models&rft.volume=108&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E489-%3C%2Fspan%3E526&rft.date=1987&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A35592821%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01212322&rft_id=info%3Abibcode%2F1987CMaPh.108..489A&rft.aulast=Aizenman&rft.aufirst=Michael&rft.au=Barsky%2C+David&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.cmp%2F1104116538&rfr_id=info%3Asid%2Fen.wikipedia.org%3APercolation+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMenshikov1986" class="citation cs2"><a href="/wiki/Mikhail_Vasiliyevich_Menshikov" class="mw-redirect" title="Mikhail Vasiliyevich Menshikov">Menshikov, Mikhail</a> (1986), "Coincidence of critical points in percolation problems", <i>Soviet Mathematics - Doklady</i>, <b>33</b>: <span class="nowrap">856–</span>859</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Soviet+Mathematics+-+Doklady&rft.atitle=Coincidence+of+critical+points+in+percolation+problems&rft.volume=33&rft.pages=%3Cspan+class%3D%22nowrap%22%3E856-%3C%2Fspan%3E859&rft.date=1986&rft.aulast=Menshikov&rft.aufirst=Mikhail&rfr_id=info%3Asid%2Fen.wikipedia.org%3APercolation+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Further_reading">Further reading</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=14" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMalthe-Sørenssen2024" class="citation book cs1">Malthe-Sørenssen, Anders (2024). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-031-59900-2"><i>Percolation Theory Using Python</i></a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-59900-2" title="Special:BookSources/978-3-031-59900-2"><bdi>978-3-031-59900-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Percolation+Theory+Using+Python&rft.date=2024&rft.isbn=978-3-031-59900-2&rft.aulast=Malthe-S%C3%B8renssen&rft.aufirst=Anders&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-031-59900-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3APercolation+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAustin2008" class="citation web cs1">Austin, David (July 2008). <a rel="nofollow" class="external text" href="https://www.ams.org/featurecolumn/archive/percolation.html">"Percolation: Slipping through the Cracks"</a>. American Mathematical Society. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091113083050/http://www.ams.org/featurecolumn/archive/percolation.html">Archived</a> from the original on 2009-11-13<span class="reference-accessdate">. Retrieved <span class="nowrap">2021-04-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Percolation%3A+Slipping+through+the+Cracks&rft.pub=American+Mathematical+Society&rft.date=2008-07&rft.aulast=Austin&rft.aufirst=David&rft_id=https%3A%2F%2Fwww.ams.org%2Ffeaturecolumn%2Farchive%2Fpercolation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APercolation+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKesten2006" class="citation journal cs1">Kesten, Harry (May 2006). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200605/what-is-kesten.pdf">"What Is ... Percolation?"</a> <span class="cs1-format">(PDF)</span>. <i>Notices of the American Mathematical Society</i>. <b>53</b> (5): <span class="nowrap">572–</span>573. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1088-9477">1088-9477</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210502033025/http://www.ams.org/notices/200605/what-is-kesten.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2021-05-02<span class="reference-accessdate">. Retrieved <span class="nowrap">2021-04-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=What+Is+...+Percolation%3F&rft.volume=53&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E572-%3C%2Fspan%3E573&rft.date=2006-05&rft.issn=1088-9477&rft.aulast=Kesten&rft.aufirst=Harry&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F200605%2Fwhat-is-kesten.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APercolation+theory" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Percolation_theory&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://danlarremore.com/PercoVIS.html">PercoVIS: a macOS program to visualize percolation on networks in real time</a></li> <li><a rel="nofollow" class="external text" href="http://ibiblio.org/e-notes/Perc/contents.htm">Interactive Percolation</a></li> <li><a rel="nofollow" class="external text" href="http://nanohub.org/resources/5660">Nanohub online course on <i>Percolation Theory</i></a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Stochastic_processes" title="Template:Stochastic processes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Stochastic_processes" title="Template talk:Stochastic processes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Stochastic_processes" title="Special:EditPage/Template:Stochastic processes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li> <li><a href="/wiki/Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li> <li><a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Moran_process" title="Moran process">Moran process</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a> <ul><li><a href="/wiki/Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li> <li><a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li> <li><a href="/wiki/Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li> <li><a href="/wiki/Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Additive_process" title="Additive process">Additive process</a></li> <li><a href="/wiki/Bessel_process" title="Bessel process">Bessel process</a></li> <li><a href="/wiki/Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a> <ul><li><a href="/wiki/Birth_process" title="Birth process">pure birth</a></li></ul></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a> <ul><li><a href="/wiki/Brownian_bridge" title="Brownian bridge">Bridge</a></li> <li><a href="/wiki/Brownian_excursion" title="Brownian excursion">Excursion</a></li> <li><a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li> <li><a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li> <li><a href="/wiki/Brownian_meander" title="Brownian meander">Meander</a></li></ul></li> <li><a href="/wiki/Cauchy_process" title="Cauchy process">Cauchy process</a></li> <li><a href="/wiki/Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li> <li><a href="/wiki/Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li> <li><a href="/wiki/Cox_process" title="Cox process">Cox process</a></li> <li><a href="/wiki/Diffusion_process" title="Diffusion process">Diffusion process</a></li> <li><a href="/wiki/Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson Brownian motion</a></li> <li><a href="/wiki/Empirical_process" title="Empirical process">Empirical process</a></li> <li><a href="/wiki/Feller_process" title="Feller process">Feller process</a></li> <li><a href="/wiki/Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li> <li><a href="/wiki/Gamma_process" title="Gamma process">Gamma process</a></li> <li><a href="/wiki/Geometric_process" title="Geometric process">Geometric process</a></li> <li><a href="/wiki/Hawkes_process" title="Hawkes process">Hawkes process</a></li> <li><a href="/wiki/Hunt_process" title="Hunt process">Hunt process</a></li> <li><a href="/wiki/Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li> <li><a href="/wiki/It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li> <li><a href="/wiki/It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li> <li><a href="/wiki/Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li> <li><a href="/wiki/Jump_process" title="Jump process">Jump process</a></li> <li><a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy process</a></li> <li><a href="/wiki/Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li> <li><a href="/wiki/Markov_additive_process" title="Markov additive process">Markov additive process</a></li> <li><a href="/wiki/McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li> <li><a href="/wiki/Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson process</a> <ul><li><a href="/wiki/Compound_Poisson_process" title="Compound Poisson process">Compound</a></li> <li><a href="/wiki/Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li> <li><a href="/wiki/Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li> <li><a href="/wiki/Semimartingale" title="Semimartingale">Semimartingale</a></li> <li><a href="/wiki/Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li> <li><a href="/wiki/Stable_process" title="Stable process">Stable process</a></li> <li><a href="/wiki/Superprocess" title="Superprocess">Superprocess</a></li> <li><a href="/wiki/Telegraph_process" title="Telegraph process">Telegraph process</a></li> <li><a href="/wiki/Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a></li> <li><a href="/wiki/Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian process</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li> <li><a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li> <li><a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a> <ul><li><a href="/wiki/Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li> <li><a href="/wiki/Local_martingale" title="Local martingale">Local</a></li> <li><a href="/wiki/Submartingale" class="mw-redirect" title="Submartingale">Sub-</a></li> <li><a href="/wiki/Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li> <li><a href="/wiki/Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li> <li><a href="/wiki/Regenerative_process" title="Regenerative process">Regenerative process</a></li> <li><a href="/wiki/Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li> <li><a href="/wiki/Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li> <li><a href="/wiki/White_noise" title="White noise">White noise</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li> <li><a href="/wiki/Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li> <li><a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li> <li><a href="/wiki/Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li> <li><a href="/wiki/Ising_model" title="Ising model">Ising model</a> <ul><li><a href="/wiki/Potts_model" title="Potts model">Potts model</a></li> <li><a href="/wiki/Boolean_network" title="Boolean network">Boolean network</a></li></ul></li> <li><a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a></li> <li><a class="mw-selflink selflink">Percolation</a></li> <li><a href="/wiki/Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li> <li><a href="/wiki/Point_process" title="Point process">Point process</a> <ul><li><a href="/wiki/Point_process#Cox_point_process" title="Point process">Cox</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li> <li><a href="/wiki/Random_field" title="Random field">Random field</a></li> <li><a href="/wiki/Random_graph" title="Random graph">Random graph</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li> <li><a href="/wiki/Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li> <li><a href="/wiki/Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li> <li><a href="/wiki/Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li> <li><a href="/wiki/Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li> <li><a href="/wiki/Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li> <li><a href="/wiki/Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li> <li><a href="/wiki/Chen_model" title="Chen model">Chen</a></li> <li><a href="/wiki/Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li> <li><a href="/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li> <li><a href="/wiki/Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li> <li><a href="/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li> <li><a href="/wiki/Heston_model" title="Heston model">Heston</a></li> <li><a href="/wiki/Ho%E2%80%93Lee_model" title="Ho–Lee model">Ho–Lee</a></li> <li><a href="/wiki/Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li> <li><a href="/wiki/Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li> <li><a href="/wiki/LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li> <li><a href="/wiki/Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li> <li><a href="/wiki/SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li> <li><a href="/wiki/Vasicek_model" title="Vasicek model">Vašíček</a></li> <li><a href="/wiki/Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li> <li><a href="/wiki/Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li> <li><a href="/wiki/Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li> <li><a href="/wiki/Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bulk_queue" title="Bulk queue">Bulk</a></li> <li><a href="/wiki/Fluid_queue" title="Fluid queue">Fluid</a></li> <li><a href="/wiki/G-network" title="G-network">Generalized queueing network</a></li> <li><a href="/wiki/M/G/1_queue" title="M/G/1 queue">M/G/1</a></li> <li><a href="/wiki/M/M/1_queue" title="M/M/1 queue">M/M/1</a></li> <li><a href="/wiki/M/M/c_queue" title="M/M/c queue">M/M/c</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li> <li><a href="/wiki/Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li> <li><a href="/wiki/Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li> <li><a href="/wiki/Ergodicity" title="Ergodicity">Ergodic</a></li> <li><a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li> <li><a href="/wiki/Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li> <li><a href="/wiki/Markov_property" title="Markov property">Markov</a></li> <li><a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li> <li><a href="/wiki/Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li> <li><a href="/wiki/Predictable_process" title="Predictable process">Predictable</a></li> <li><a href="/wiki/Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li> <li><a href="/wiki/Self-similar_process" title="Self-similar process">Self-similar</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationary</a></li> <li><a href="/wiki/Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Donsker%27s_theorem" title="Donsker's theorem">Donsker's theorem</a></li> <li><a href="/wiki/Doob%27s_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li> <li><a href="/wiki/Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li> <li><a href="/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li> <li><a href="/wiki/Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li> <li><a href="/wiki/Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li> <li><a href="/wiki/Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li> <li><a href="/wiki/Sanov%27s_theorem" title="Sanov's theorem">Sanov's theorem</a></li> <li><a href="/wiki/Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="/wiki/Blumenthal%27s_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="/wiki/Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="/wiki/Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="/wiki/Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="/wiki/L%C3%A9vy%27s_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li> <li><a href="/wiki/Doob%27s_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li> <li><a href="/wiki/Doob%27s_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li> <li><a href="/wiki/Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li> <li><a href="/wiki/Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li> <li><a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li> <li><a href="/wiki/Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li> <li><a href="/wiki/Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li> <li><a href="/wiki/Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li> <li><a href="/wiki/Doob%27s_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li> <li><a href="/wiki/Dynkin%27s_formula" title="Dynkin's formula">Dynkin's formula</a></li> <li><a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li> <li><a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li> <li><a href="/wiki/Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li> <li><a href="/wiki/Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li> <li><a href="/wiki/It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li> <li><a href="/wiki/It%C3%B4%27s_lemma" title="Itô's lemma">Itô's lemma</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li> <li><a href="/wiki/Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li> <li><a href="/wiki/Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li> <li><a href="/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li> <li><a href="/wiki/Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li> <li><a href="/wiki/Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li> <li><a href="/wiki/Prokhorov%27s_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li> <li><a href="/wiki/Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li> <li><a href="/wiki/Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li> <li><a href="/wiki/Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li> <li><a href="/wiki/Skorokhod%27s_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li> <li><a href="/wiki/Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li> <li><a href="/wiki/Snell_envelope" title="Snell envelope">Snell envelope</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a> <ul><li><a href="/wiki/Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li> <li><a href="/wiki/Stopping_time" title="Stopping time">Stopping time</a></li> <li><a href="/wiki/Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li> <li><a href="/wiki/Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li> <li><a href="/wiki/Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li> <li><a href="/wiki/Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical</a></li> <li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li> <li><a href="/wiki/Stochastic_control" title="Stochastic control">Control theory</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li> <li><a href="/wiki/Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li> <li><a href="/wiki/Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a></li> <li><a href="/wiki/Renewal_theory" title="Renewal theory">Renewal theory</a></li> <li><a href="/wiki/Ruin_theory" title="Ruin theory">Ruin theory</a></li> <li><a href="/wiki/Signal_processing" title="Signal processing">Signal processing</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li> <li><a href="/wiki/Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li> <li><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li> <li><a href="/wiki/Category:Stochastic_processes" title="Category:Stochastic processes">Category</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div 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