Probability theory

A point process is a random locally finite configuration of points, usually represented as a random counting measure. A finite eigenvalue configuration of a random matrix is an example. Multiplicities and whether the process is simple should be specified.

A point process is determinantal with kernel $K$ relative to a specified reference measure if every correlation function of a point process is the indicated determinant. A rank-$m$ finite-rank projection kernel gives an exactly $m$-point process, whose symmetric joint density is $\det [K(x_i,x_j)]/m!$.

The $k$th correlation function is the density of the $k$th factorial moment measure: integrating a nonnegative test function against $R_k$ gives the expectation of its sum over ordered $k$-tuples of distinct points. For exactly $m$ exchangeably labelled points with symmetric joint density $f_m$, it is $m!/(m-k)!$ times the marginal density of $k$ labels. It is not normalized to have total mass one.

A random matrix is a matrix whose entries are random variables. Its eigenvalues form a random configuration, and its empirical spectral measure records their distribution with equal mass on each eigenvalue. Entry assumptions and normalization determine the spectral regime.

For an $N$-by-$N$ Hermitian matrix, the empirical spectral measure puts mass $1/N$ at each eigenvalue, counted with multiplicity. It is a probability measure. Integrating a test function against it gives the average of that function evaluated at the eigenvalues.

A real Wigner matrix is a real symmetric matrix whose upper off-diagonal entries are independent centered copies of a fixed random variable, scaled to have variance of order $1/N$. Diagonal conventions may vary and should be specified separately. A common version has variance-one unscaled off-diagonal entries and either zero diagonal entries or independent diagonal entries with a bounded second moment.

Truncate the unscaled entries of a Wigner matrix at a fixed threshold and subtract their expected values. The difference between original and truncated entries is $N^{-1/2}(Z_C-\mathbb{E}{Z}_C)$ with $Z_C=Y{1}_{|Y|\ge C}$, so its variance is bounded by $N^{-1}\mathbb{E}[Y^2{1}_{|Y|\ge C}]$. The centering term cannot be discarded for an asymmetric entry law.

For a fixed mean-zero variance-one real entry law, centered truncation of a Wigner matrix satisfies $\mathbb{E}[N^{-1}\mathrm{Tr}(X-\hat{X}{)}^2]\le t(C)$, where $t(C)=\mathbb{E}[Y^2{1}_{|Y|\ge C}]\to 0$. The spectral Lipschitz bound from Frobenius distance and Markov inequality give probability at most $t(C)/{\epsilon }^2$ for a spectral test-function discrepancy exceeding $\epsilon$. Thus $t(C)<{\epsilon }^3$ works uniformly in matrix size. A varying entry-law family requires uniform second-moment tail decay instead.

A complex Wishart matrix is the Gram matrix of an $n$-by-$m$ matrix of independent centered complex Gaussian random variables with an indicated variance convention. It is a positive semidefinite matrix. In the standard convention with joint entry density proportional to $e^{-\mathrm{Tr}{X}^{\ast }X}$ and $n\ge m$, its eigenvalue density contains the weight $x^{n-m}{e}^{-x}$ and a squared Vandermonde determinant.

The Laguerre unitary ensemble is the positive-eigenvalue ensemble with joint density proportional to ${\prod }_{i<j}(x_j-x_i{)}^2{\prod }_i{x}_i^a{e}^{-x_i}$, for $a>-1$. Integer $a=n-m\ge 0$ is realized by a complex Wishart matrix. The Laguerre projection kernel expresses its correlation functions of a point process as determinants.

For a metric space with a probability measure $\mu$, the concentration function is $\alpha (\epsilon )={\sup }_{\mu (A)\ge 1/2}(1-\mu (A_{\epsilon }))$, where $A_{\epsilon }$ is the metric neighbourhood of radius $\epsilon$. A sequence exhibits concentration of measure if these functions tend to zero for every fixed positive radius. The metric scale is part of this definition.

A Lévy family of graphs consists of finite connected graphs with their uniform probability measures and graph distances divided by their diameters, whose concentration of measure functions vanish at every fixed positive normalized radius. Equivalently, every family occupying at least half the vertices has a radius-$\lfloor \epsilon D_n\rfloor$ neighbourhood occupying a proportion tending to one, uniformly over such families. Harper theorem and binomial distribution tail bounds prove this for the hypercube graphs.

A random vector $X$ is positively associated if $\mathrm{Cov}(f(X),g(X))\ge 0$ for every pair of coordinatewise nondecreasing bounded functions $f,g$. The same inequality holds when both are nonincreasing, by negating them. Repeatedly applying it to products of nonnegative increasing indicators gives $\mathbb{P}({\bigcap }_i{E}_i)\ge {\prod }_i\mathbb{P}(E_i)$ for increasing events; the same is true for decreasing events. Thus associated rejection indicators have familywise error rate no greater than independent indicators with the same marginals. Pairwise positive correlation coefficients alone do not give this stronger property. The classical definition is in Esary, Proschan and Walkup (1967).

Extreme value theory studies limiting probability distributions of normalized sample maxima and related extremes. The standard limits for maxima include the Fréchet distribution, negative Weibull distribution, and Gumbel distribution.

For $1\le k<n$ and a positive threshold $t=X_{(n-k)}$, the Hill estimator averages the logarithmic ratios of the top $k$ order statistics to $t$. For a Fréchet distribution domain with shape $\alpha$, it estimates the reciprocal tail index $1/\alpha$. At an untied threshold it equals the empirical distribution function plug-in tail integral ${\int }_t^{\infty }(1-F_n(x))/(1-F_n(t))dx/x$. At a tied threshold the plug-in denominator counts strictly larger observations and the identity needs adjustment. Fixed $k$ does not by itself yield a consistent tail-index estimate.

Every non-degenerate limit in convergence in distribution of positively scaled and centered sample maxima is a max-stable distribution. Up to positive affine transformations its law is the Gumbel distribution, a Fréchet distribution with positive shape parameter, or a negative Weibull distribution with positive shape parameter. This classifies possible limits, without asserting that every distribution function admits a limiting normalization.

A non-degenerate distribution function is max-stable when every sample maximum of $n$ independent and identically distributed random variables can be positively scaled and centered to have the original law. Thus for every integer $n\ge 1$ there are $a_n>0,b_n\in \mathbb{R}$ with $F(a_{n}x+b_n{)}^n=F(x)$. The extremal types theorem classifies these laws, up to affine changes, as the Gumbel distribution, Fréchet distribution, and negative Weibull distribution.

For independent repeated time blocks, an event with exceedance probability $p$ has mean waiting time $1/p$ blocks by the geometric distribution. The return period is a long-run average waiting time, not a deterministic recurrence interval. For annual exceedance events, it is measured in years.

A $T$-year return level has annual exceedance probability $1/T$ under the assumed stationary model. It is the corresponding quantile of annual maxima. Daily exceedance probabilities must be converted to annual event probabilities before using the annual return period definition.

A return level plot shows estimated return levels against their return periods, often on a logarithmic horizontal axis. Confidence curves describe uncertainty in estimated return levels. They are not prediction bands for individual extreme observations, and uncertainty typically increases when extrapolating far beyond the observation window.

For scale $\sigma >0$ and shape $\xi$, the generalized Pareto distribution on nonnegative excesses has cumulative distribution functionThe case $\xi =0$ is the continuous limit $1-e^{-y/\sigma }$. Positive $\xi$ gives a power-law tail, zero gives an exponential distribution, and negative $\xi$ gives a finite upper endpoint $-\sigma /\xi$. Its role in modelling high-threshold excesses is explained by the Pickands-Balkema-de Haan theorem.

The peaks-over-threshold method fits a generalized Pareto distribution to exceedances of a high threshold and also estimates their frequency. Together these determine tail probabilities and return levels. For dependent observations, threshold exceedances may occur in clusters; declustering of extremes can separate extreme events before fitting a model for independent events.

Declustering groups dependent threshold exceedances into extreme-event clusters and retains a representative, commonly the cluster maximum. It aims to make the events used in a peaks-over-threshold method approximately independent. The event rate must be estimated consistently with the retained clusters when computing return levels.

For a distribution in an extreme-value domain of attraction, the conditional distribution of the excess above a high threshold is asymptotically approximated by a generalized Pareto distribution with a threshold-dependent scale and a limiting shape parameter. This theorem justifies the peaks-over-threshold method. Threshold selection must balance approximation error against the reduced number of excess observations.

The standard Gumbel distribution for maxima has distribution function $\Lambda (x)=e^{-e^{-x}}$ for every real $x$. The maximum domain of attraction is characterized by an exponential tail ratio on the scale of a Gumbel auxiliary function.

For the Gamma distribution with fixed positive integer shape $m$ and rate one, the survival function is $e^{-t}{\sum }_{j=0}^{m-1}{t}^j/j!$. Its sample maximum centered by $\log n+(m-1)\log \log n-\log ((m-1)!)$ has convergence in distribution to the standard Gumbel distribution, without an additional scale factor.

A positive Gumbel auxiliary function makes $\overline{F}(t+a(t)x)/\overline{F}(t)\to e^{-x}$ as $t$ approaches the right endpoint of a distribution, for every real $x$. Under the Von Mises conditions for extreme values, the reciprocal hazard function is a valid choice.

For $\alpha >0$, the standard negative Weibull distribution has distribution function ${\Psi }_{\alpha }(x)=e^{-(-x{)}^{\alpha }}$ for $x<0$ and one for $x\ge 0$. This is the probability distribution of the negative of a standard positive Weibull distribution variable.

For $\alpha >0$, the standard Fréchet distribution has distribution function ${\Phi }_{\alpha }(x)=e^{-x^{-\alpha }}$ for $x>0$ and zero for $x\le 0$. Its maximum domain of attraction consists of distribution functions with an infinite right endpoint of a distribution and a survival function with regular variation of index $-\alpha$.

A distribution function $F$ lies in the maximum domain of attraction of $G$ if $F(a_{n}x+b_n{)}^n\to G(x)$ at every continuity point of $G$ for some $a_n>0$ and real $b_n$. This expresses convergence in distribution of normalized sample maxima.

For an absolutely continuous distribution function with positive probability density function near its right endpoint of a distribution, write $\eta =f/(1-F)$ for its hazard function. The limits $t\eta (t)\to \alpha$ at an infinite endpoint or $(x_F-t)\eta (t)\to \alpha$ at a finite endpoint imply attraction to the Fréchet distribution or negative Weibull distribution, respectively. A continuously differentiable reciprocal hazard function with $(1/\eta {)}^{\prime }\to 0$ implies attraction to the Gumbel distribution, provided the survival function tends to zero at the endpoint.

The probability of an event is its value under a probability measure, a real number between zero and one that quantifies how likely the event is.

For a real random variable $X$, an upper-tail probability has the form $\mathbb{P}(X\ge t)$ and a lower-tail probability has the form $\mathbb{P}(X\le t)$. A concentration inequality bounds such probabilities.

Percolation theory studies random subgraphs formed by independently or dependently declaring vertices or edges open.

For critical site percolation on the triangular lattice, scaling-limit crossing probabilities transform naturally under conformal maps between planar domains. This rigorously established property supports descriptions of interfaces by SLE and calculations of planar percolation critical exponents. Transferring it to a different lattice or to bond percolation requires a separate universality argument.

An iterative expansion of long-range connectivity functions that separates an approximately random-walk contribution from corrections caused by loops and intersections. In suitable high-dimensional percolation models it controls these corrections and proves mean-field percolation exponents. It also applies to other models, including self-avoiding walks; its use always requires model-specific estimates.

In bond percolation with parameter $p$, every edge of a graph is independently open with probability $p$ and closed with probability $1-p$.

A path in the underlying graph all of whose edges are open. Connectivity of open paths determines the percolation clusters. A ray in a graph with all edges open is an infinite ray in a graph.

If independent edges of a locally finite graph receive fair orientations, exploring outward from a root tests each fresh reached-to-unreached boundary edge with success probability $1/2$. Deferred decisions give exactly the exploration law of the root cluster in bond percolation of density $1/2$. Thus the two reachable graph vertex sets have the same distribution. An infinite out-cluster contains an infinite simple directed ray by the König infinity lemma. Geometrically biased orientations do not generally give the same uniform edge-success law.

Here $\tau (x,y)=P_{p_c}(x\leftrightarrow y)$ is critical two-point connectivity. The condition controls the intersection effects that spoil a branching approximation. It is a sufficient criterion for important mean-field percolation exponents, including the order-parameter, percolation susceptibility and cluster-tail exponents. Establishing detailed spatial asymptotics is a further task, commonly addressed by lace expansion.

The two-point percolation two-point connection probability ${\tau }_p(x,y)={\mathbb{P}}_p(x\leftrightarrow y)$ is the probability that two fixed graph vertices belong to the same percolation cluster. On a translation-invariant lattice it depends on their displacement. Restricting the permitted connecting graph paths to a subgraph gives the corresponding confined percolation two-point connection probability. The Harris-FKG inequality implies ${\tau }_p(x,z)\ge {\tau }_p(x,y){\tau }_p(y,z)$.

For bond percolation on the square lattice, write $h_n={\mathbb{P}}_p(0\leftrightarrow (n,0))$ and $g_k={\mathbb{P}}_p(0\leftrightarrow \partial [-k,k{]}^2)$. A maximal-probability boundary graph vertex can be placed at $(k,j)$ by symmetry. Reflection about $x=k$ identifies its connection probability from $0$ with that from $(2k,0)$. The Harris-FKG inequality yields $(g_k/(8k){)}^2\le h_{2k}\le g_{2k}$ for $k\ge 1$.

For nearest-neighbour bond percolation with $0<p<1$ on the strip $T_k=\mathbb{Z}\times \{-k,\dots ,k\}$, put $q_k(n)={\mathbb{P}}_p((0,0)\leftrightarrow (n,0)\text{ in }{T}_k)$. The Harris-FKG inequality and translation invariance imply $q_k(n+m)\ge q_k(n)q_k(m)$. Thus $-\log q_k(n)$ is a subadditive sequence, and the Fekete lemma givesThe direct horizontal graph path gives $0\le f_k(p)\le -\log p$. Enlarging the strip increases every percolation two-point connection probability, so $f_k(p)$ is nonincreasing in $k$ and has a nonnegative limit of a sequence.

Let $q(n)$ be the unrestricted planar two-point percolation two-point connection probability. Every finite connecting graph path is contained in some strip, so $q_k(n)\uparrow q(n)$ for each fixed $n$. ThereforeThe equality uses commuting infima and monotonicity, not an unjustified interchange of two general limits. The right-hand side is the whole-plane connection decay rate by the Fekete lemma.

A vertical cut between consecutive columns of $T_k$ contains $2k+1$ edges. All are closed with probability $(1-p{)}^{2k+1}$. Cuts in different columns use disjoint edges, so their closed-cut events are independent. A connection from column zero to column $n$ must cross each of the intervening $n$ cuts. ConsequentlyThis finite-width obstruction persists even when the unrestricted planar percolation is supercritical.

The independent edge law of bond percolation on ${\mathbb{Z}}^d$ is ergodic under lattice translations: every translation-invariant event has probability zero or one.

In bond percolation, a percolation cluster is a connected component of a graph formed by the open edges, with every graph vertex retained. In site percolation, it is a connected component of a graph of the subgraph induced by the open graph vertices. Write $C(x)$ for the cluster containing $x$; for site percolation set $C(x)=\varnothing$ when $x$ is closed.

Suppose the one-arm probability on the cubic lattice satisfies ${\beta }_r=\exp (-\lambda r+o(r))$ with $\lambda >0$. Define $M_n$ as the largest maximum-norm radius of any cluster rooted in $[-n,n{]}^d$. Then $M_n/\log n\to d/\lambda$ convergence in probability. An upper bound is a union bound over $O(n^d)$ roots. For a lower bound, pack $\Theta ((n/\log n{)}^d)$ disjoint radius-$O(\log n)$ boxes and use the independent first-hit connection events inside them. Clusters are finite almost surely under the positive-rate hypothesis.

An infinite percolation cluster is an open connected component of a graph with infinitely many graph vertices. On a countable translation-invariant lattice, a zero root percolation probability implies that no infinite percolation cluster exists almost surely, by the countable union over all graph vertices. A positive root probability implies almost-sure existence by translation ergodicity of Bernoulli percolation; uniqueness is an additional theorem.

For independent site percolation or bond percolation on the cubic lattice ${\mathbb{Z}}^d$, there is almost surely at most one infinite percolation cluster at every fixed parameter. Above the percolation critical probability, there is exactly one almost surely. This theorem is called the Burton-Keane theorem. Existence uses translation ergodicity together with positive percolation probability; uniqueness is a separate conclusion. At $p=1$ uniqueness is deterministic.

Independent bond percolation on ${\mathbb{Z}}^2$ has at most one infinite percolation cluster at every parameter, including a possible critical parameter. The number of infinite percolation clusters is constant almost surely by translation ergodicity of Bernoulli percolation. A finite constant larger than one is impossible: a box meeting two clusters can be made entirely open, joining them and decreasing that number with positive probability by finite modification of Bernoulli percolation.

If infinitely many clusters existed, a box would meet three with positive probability. Preserve one infinite exterior arm from each and replace the finitely many interior and boundary edges by a three-armed tree joining them, closing the remaining edges. Its branch graph vertex becomes a trifurcation vertex in percolation. Translation invariance therefore gives a positive density $q$. But the trifurcation boundary-counting lemma gives $q|W|\le |{\partial }^{+}W|$ for every large box, contradicting vanishing boundary-to-volume ratio. This proves uniqueness without assuming absence of an infinite percolation cluster at criticality.

A trifurcation graph vertex is a graph vertex of an infinite percolation cluster whose deletion separates that cluster into at least three infinite connected components of a graph. In a finite modification argument one may make it have exactly three open incident edges leading to three different infinite components. On a translation-invariant lattice, a positive probability of trifurcation at one graph vertex gives a positive expected density of trifurcations.

For a finite graph vertex set $W$ in a locally finite graph, let ${\partial }^{+}W$ be its exterior graph neighbours. The number of trifurcation vertices in percolation lying in $W$ is at most $|{\partial }^{+}W|$.

For each cluster meeting $W$, contract its connected components of a graph outside $W$ to terminal graph vertices, keeping only those adjacent to $W$. The resulting incidence graph is finite and connected. Each trifurcation graph vertex in $W$ separates at least three groups of terminals, because every infinite branch must leave the finite set $W$. Take a minimal subtree connecting all terminals. Each such trifurcation is unavoidable in that subtree and has degree of a vertex at least three. All leaves are terminals. The tree identity ${\sum }_v(\mathrm{deg}(v)-2)=-2$ bounds the number of these branching graph vertices by the number of leaves minus two, hence by the number of terminals. Different terminals, even across different clusters, can be assigned different exterior graph neighbours. Summing proves the bound. For square boxes in ${\mathbb{Z}}^2$, the boundary has order $n$ and the volume has order $n^2$.

Delete a finite set of degree-three trifurcation vertices in percolation in each infinite percolation cluster and contract the remaining components to nodes. The incidence graph is a tree: a cycle would bypass an original trifurcation vertex in percolation. All leaf components are infinite, otherwise a graph vertex would have a finite branch after deletion. The tree degree identity gives at least two more infinite leaves than degree-three graph vertices. Distinct infinite components escape a containing box through distinct outer boundary graph vertices. Hence the number of counted trifurcations is at most the outer boundary size.

The percolation susceptibility is the expected size of the open cluster containing a specified root:

The percolation susceptibility is $\chi (p)=1+{\sum }_{m\ge 1}{\sum }_{y\in \partial {\Lambda }_m}{\tau }_p(0,y)$. Finiteness gives a boundary sum $a<1$. The weighted BK boundary-splitting estimate then gives $g_k\le a^{\lfloor k/m\rfloor }$. The strict bound $g_1=1-(1-p{)}^{2d}<1$ absorbs the finitely many smaller radii into a positive exponential rate with unit prefactor. At $p=0$, all positive-radius connection probabilities are zero.

At critical planar percolation, the probability of an open crossing of a rectangle of any fixed aspect ratio stays bounded away from zero and one uniformly over its scale.

For independent bond percolation on the square lattice, a positive uniform lower bound for square crossings implies a positive uniform lower bound for rectangle crossings of every fixed aspect ratio $\rho >1$. The RSW reflection extension lemma first extends a square to aspect ratio $3/2$. Overlapping rectangles and transverse overlap-square crossings then glue using the Harris-FKG inequality. The required constant depends on $\delta$ and $\rho$, not on scale. At a self-dual parameter, dual crossings also give a uniform upper bound below one.

For independent bond percolation on the square lattice at $p=1/2$, square crossing probabilities bounded below by self-duality extend to a uniform positive bound for rectangles of aspect ratio $3/2$. Expose an extremal vertical square crossing, reflect its geometry, and attach it to an exterior side through an unexposed independent region. Reflection symmetry bounds the attachment probability below. Two attachments and a horizontal square crossing join by the Harris-FKG inequality. Overlapping rectangles and transverse overlap-square crossings then give all fixed aspect ratios. Reflecting a geometrical graph path does not require its reflected edges to be open.

The one-arm probability $\pi (r)$ is the probability that a specified vertex has an open path to distance $r$.

For independent bond percolation on the cubic lattice, define ${\tau }_p(x,y)={\mathbb{P}}_p(x\leftrightarrow y)$ and $g_n={\mathbb{P}}_p(0\leftrightarrow \partial {\Lambda }_n)$, with $g_0=1$. Split an open self-avoiding walk at its first radius-$m$ boundary vertex $y$. The remaining segment reaches distance at least $n-m$ from $y$, using disjoint edges. The BK inequality, translation invariance and the union bound yield the displayed estimate. Keeping the individual connection weights can be stronger than replacing their sum by $|\partial {\Lambda }_m|g_m$.

At $p=1/2$ for independent bond percolation on the square lattice, there exist $C<\infty$ and $\alpha >0$ with ${\mathbb{P}}_{1/2}(0\leftrightarrow \partial [-n,n{]}^2)\le C{n}^{-\alpha }$. The Russo-Seymour-Welsh theorem and the Harris-FKG inequality give a uniform positive probability of a closed dual graph cycle around each geometrically spaced square ring. The independent annular barriers for percolation give the polynomial bound and imply zero critical percolation probability. This does not determine the exact critical exponent.

Closed dual graph cycles in disjoint square rings depend on disjoint primal edge sets. In independent bond percolation their barrier events are independent. If each event has probability at least $\delta >0$, an open graph path from the origin across $k$ rings has probability at most $(1-\delta {)}^k$. A geometric sequence of radii turns this estimate into the polynomial critical one-arm upper bound.

For bond percolation on the square lattice, put $g_n={\mathbb{P}}_p(0\leftrightarrow \partial [-n,n{]}^2)$. Splitting an open self-avoiding walk at its first visit to the radius-$m$ boundary gives disjoint certificates for reaching a boundary graph vertex and then reaching radius $n$ about that graph vertex. The van den Berg-Kesten inequality and the union bound give $g_{m+n}\le 8m{g}_m{g}_n$ for positive $m,n$. With $g_0=1$ and $|\partial {\Lambda }_0|=1$, the boundary-cardinality form holds for zero indices as well.

For bond percolation on the square lattice, the one-arm probability $g_n={\mathbb{P}}_p(0\leftrightarrow \partial [-n,n{]}^2)$ has a root limit of a sequence $\gamma (p)\in [p,1]$. The BK boundary-splitting estimate implies that $32n^2{g}_n$ is submultiplicative for positive integers. Applying the Fekete lemma to its logarithm proves existence when $p>0$; at $p=0$ the rate is zero. The same rate is the root limit of the percolation two-point connection probability along a coordinate axis, by the reflection lower bound for two-point percolation.

For independent bond percolation of density $0<p<1$ on the cubic lattice in dimension $d\ge 2$, let ${\beta }_n$ be the one-arm probability for reaching maximum-norm distance $n$. The weighted BK boundary-splitting estimate gives ${\beta }_{m+n}\le |\partial {\Lambda }_n|{\beta }_m{\beta }_n$. Since the logarithm of this boundary size is $o(n)$, the asymmetrically almost-subadditive sequence argument applies to $\log {\beta }_n$. The rate exists and lies in $[0,-\log p]$; positive rate gives exponential decay.

The Russo-Seymour-Welsh theorem and the Harris-FKG inequality imply $\pi (2r)\ge c\pi (r)$ for a scale-independent constant $c>0$.

RSW rectangle crossings and positive association join two separated one-arm events with probability bounded below uniformly over scale.

For increasing events $A,B$ in a product percolation measure, their disjoint occurrence satisfies $\mathbb{P}(A\square B)\le \mathbb{P}(A)\mathbb{P}(B)$.

A finite open coordinate set witnesses an increasing event when prescribing those coordinates open forces the event, regardless of the remaining configuration. The disjoint occurrence of increasing events $A\square B$ means that $A$ and $B$ have disjoint finite witnesses. For finite-coordinate events this is the usual disjoint-occurrence definition. The van den Berg-Kesten inequality bounds its probability by $\mathbb{P}(A)\mathbb{P}(B)$ under an independent Bernoulli distribution product measure.

The percolation probability $\theta (p)$ is the probability that a specified root belongs to an infinite open cluster at parameter $p$.

If an infinite percolation cluster exists almost surely, the probability that a large square box intersects one tends to one. The four rotationally symmetric exterior-side arm events are increasing. By the Harris-FKG inequality, their common probability is at least $1-(1-P(D_N){)}^{1/4}$, where $D_N$ is their union. Thus a pair of specified opposite sides both has an infinite exterior arm with probability at least $1-2(1-P(D_N){)}^{1/4}$, tending to one. The rays start at a boundary graph vertex and immediately leave the graph vertex box.

Suppose a planar bond percolation model has critical one-arm probability at most $C{n}^{-\alpha }$ with $\alpha >0$, and a finite-radius connection event depends on $O(n^2)$ edges. The finite-box comparison for percolation parameters gives $\theta (p_c+\epsilon )\le C{n}^{-\alpha }+C^{\prime }\epsilon n^2$. Balancing at $n\asymp {\epsilon }^{-1/(\alpha +2)}$ gives $\theta (p_c+\epsilon )\le A{\epsilon }^{\alpha /(\alpha +2)}$. For the square lattice use $p_c=1/2$ and the polynomial critical one-arm upper bound.

For independent site percolation or bond percolation on a rooted locally finite graph, $\theta (p)$ is right continuous on $[0,1)$. Finite-radius connection events have polynomial probabilities in $p$ and decrease to infinite connection. Thus $\theta$ is their infimum by continuity from above of a measure. This infimum of continuous functions is upper semicontinuous; parameter monotonicity from the monotone coupling of Bernoulli percolation then gives right continuity.

Independent site percolation and bond percolation on the cubic lattice have continuous percolation probability on $(p_c,1]$. Fix $p>p_c$ and an intermediate $q\in (p_c,p)$. By uniqueness of the infinite percolation cluster, every origin in the infinite $p$-cluster has a finite $p$-open graph path to the infinite $q$-cluster. In the uniform-label monotone coupling of Bernoulli percolation, the finitely many labels on this graph path are strictly below fixed $p$ almost surely, so it persists at some parameter below $p$. The coupled onset parameter for origin percolation has no attained atom there, giving left continuity, including at $1$. Combine this with right continuity of percolation probability.

The percolation critical probability is

Independent bond percolation on the cubic lattice has exponential connection decay at every $p<p_c$ and positive percolation probability at every $p>p_c$. The finite-set criterion for percolation sharpness gives decay below its auxiliary threshold. The Margulis–Russo formula and exploration from a finite box boundary give $g_n^{\prime }(p)\ge (1-g_n(p)){\inf }_{S\ni 0}{\phi }_p(S)/(p(1-p))$, with the infimum over interior finite sets. Above the auxiliary threshold the infimum is at least one; integrating gives positive percolation probability and identifies the two thresholds. This theorem does not identify the value of $p_c$ for a particular lattice.

For independent bond percolation on a transitive locally finite lattice, let $c_n$ count rooted self-avoiding walks and let $\mu =\lim c_n^{1/n}$ be the connective constant. An infinite open percolation cluster supplies an open walk of every length. The union bound gives $\theta (p)\le c_n{p}^n$, which tends to zero when $p\mu <1$. Thus $p_c\ge 1/\mu$.

The percolation critical probability of independent bond percolation on the square lattice is $1/2$.

At parameter $1/2$, primal and dual bond percolation on the square lattice have the same probability distribution. Suppose an infinite percolation cluster exists. Large boxes meet it with probability tending to one. The square-root trick for positively associated events and quarter-turn symmetry imply that each side has an infinite exterior open arm with probability tending to one. The same holds for dual arms. A union bound then gives positive probability of primal top and bottom arms together with dual left and right arms.

Use a primal box $[-r,r{]}^2$ and a dual box with sides at $\pm (r+1/2)$. Open only primal edges whose endpoints both lie in the primal box, preserving all outward and exterior arm edges. The two primal arms are joined inside. Planarity prevents the two alternating dual arms from joining: a proposed dual connection together with the inner boundary arc separates one primal infinite arm from infinity. Thus the modified configuration has two infinite dual clusters. Its positive probability by finite modification of Bernoulli percolation contradicts uniqueness of the infinite percolation cluster. Consequently the percolation probability at $1/2$ is zero. This is often called Zhang's argument.

Four infinite exterior arms entering a square in alternating primal-open and dual-open order force the two primal arms to be disjoint outside the square. If they met, their finite connecting crosscut together with a boundary arc would enclose one dual arm, preventing its escape to infinity. Connecting the two primal arms through the square creates a proper doubly infinite open path, separating the two remaining dual infinite tails. finite-energy property of Bernoulli percolation can then turn this configuration into two infinite dual clusters, contradicting uniqueness. Care is needed to preserve the exterior dual tails even if the finite modification destroys their original boundary attachments.

Assign every site or edge $v$ an independent $U_v\sim \mathrm{Unif}[0,1]$ and declare it open at parameter $p$ exactly when $U_v\le p$. The resulting configurations have the correct Bernoulli laws and increase with $p$.

If an increasing event $A$ depends on $m$ sites or edges, the uniform-label monotone coupling of Bernoulli percolation and the union bound give $0\le {\mathbb{P}}_q(A)-{\mathbb{P}}_p(A)\le m(q-p)$ for $p\le q$. A change of the event requires a coordinate with label in $(p,q]$. For bond percolation from the origin to the boundary of $[-n,n{]}^2$, first-hit graph paths use the $4n(2n+1)$ internal edges, so the same count bounds the parameter sensitivity.

In the uniform-label monotone coupling of Bernoulli percolation, let $I_p$ mean that the origin belongs to an infinite percolation cluster, and let $M=\inf \{p:I_p\}$. The occurrence parameters form an upper interval. Therefore $\{M<p\}={\bigcup }_{r<p,r\in \mathbb{Q}}{I}_r$ and $\theta (p-)=\mathbb{P}(M<p)$. Since $\{M<p\}\subseteq I_p\subseteq \{M\le p\}$, the jump is $\theta (p)-\theta (p-)=\mathbb{P}(I_p\cap \{M=p\})$. Occurrence at the infimum is not presumed.

For $p<q$, first sample Bernoulli percolation at parameter $p$, then independently open each remaining closed coordinate with probability $(q-p)/(1-p)$. Every coordinate is then open with probability $q$, so the union has the parameter-$q$ law. This coupling separates existing $p$-clusters from the additional sprinkled connections between them.

In site percolation, vertices are declared open or closed and one studies connected components of the induced open subgraph.

For independent site percolation on the triangular lattice, the critical probability is $p_c=1/2$. Self-duality, crossing estimates, and exclusion of an infinite critical cluster identify the threshold.

Conditioned on the sites revealed by an exploration interface, every unrevealed site retains its independent Bernoulli law. The explored boundary supplies boundary conditions for subsequent crossing events.

Dependent percolation allows the open states of different vertices or edges to be statistically dependent.

On a finite graph, the random-cluster model assigns an edge configuration $\omega$ probability proportional towhere $o,c,k$ count open edges, closed edges, and open connected components.

The multiplicative factor $q$ assigned to each connected component of a graph in a finite random-cluster measure. At fixed edge probability, the weight of a configuration with $k$ components includes $q^k$. For $q=1$ this reduces to independent bond percolation; increasing $q$ at fixed $p$, for $q\ge 1$, suppresses increasing events by random-cluster weight monotonicity.

The threshold for the origin to have an infinite percolation cluster under a consistently chosen infinite-volume random-cluster measure, conventionally the wired measure: $p_c(q)=\inf \{p:{\theta }^{\mathrm{w}}(p,q)>0\}$. The definition alone makes no assertion about percolation exactly at the threshold.

The finite-volume random-cluster weight monotonicity passes to consistently wired or free infinite-volume measures for finite increasing connection events. Decreasing these events to origin-to-infinity connectivity yields $\theta (p,q_1)\ge \theta (p,q_2)$ for $q_1\le q_2$. The corresponding supercritical parameter sets are nested, and hence $p_c(q_1)\le p_c(q_2)$.

For fixed $p\in (0,1)$ and $1\le q_1\le q_2$, the random-cluster measure at $q_1$ stochastically dominates that at $q_2$. In Holley's condition, Bernoulli factors cancel. Put $a=k(S)-k(S\cup T)$ and $b=k(S\cap T)-k(T)$; supermodularity of graph component count gives $b\ge a\ge 0$, and the weight ratio is $(q_2/q_1{)}^a{q}_2^{b-a}\ge 1$. The comparison also holds with the same boundary wiring in both measures.

Planar random-cluster duality relates edge odds by $[p/(1-p)][p^{\ast }/(1-p^{\ast })]=q$. The fixed point $p=p^{\ast }$ has odds $\sqrt{q}$, giving the displayed parameter. A self-dual parameter does not by itself establish a critical threshold on an arbitrary planar graph. At this parameter the symmetric cluster weight at the self-dual parameter makes the primal-dual cluster counts explicit.

The planar cluster-count identity rewrites $o(\omega )+2k(\omega )$ as $|V|-1+k(\omega )+k({\bar{\omega }}^{\ast })$. At the self-dual parameter of the random-cluster model, $p/(1-p)=\sqrt{q}$, so its weight is $(\sqrt{q}{)}^{o(\omega )+2k(\omega )}$ up to a configuration-independent factor. Absorbing $(\sqrt{q}{)}^{|V|-1}$ into normalization proves the symmetric weight, with the unbounded dual face counted.

For an edge with endpoints already connected without that edge, including any boundary wiring, the conditional open probability is $p$. Otherwise opening the edge merges two components, so the conditional probability is $p/(p+q(1-p))$. For $q\ge 1$, this conditional probability increases when more other edges are open or more boundary vertices are identified. This is the local mechanism of boundary monotonicity of the random-cluster measure.

A boundary condition for a finite-box random-cluster measure is a partition of the boundary vertices. All vertices in one block are identified before counting open components; the identifications carry no Bernoulli edge factors. The weight is $p^{o(\omega )}(1-p{)}^{|E|-o(\omega )}{q}^{k_{\xi }(\omega )}$. A coarser partition makes more identifications. General partitions need not have a planar realization.

Here $\xi ⪯\zeta$ means each block of $\xi$ lies in a block of $\zeta$, and the measure order is stochastic domination of probability measures. The random-cluster single-edge conditional probability is increasing in both exterior open edges and wiring for $q\ge 1$. Couple two heat-bath Markov chains by identical update edges and uniform random variables, starting from ordered states. Order persists, and convergence of the finite chains to their stationary laws proves the displayed inequality. At $q=1$, the law does not depend on the boundary partition.

All boundary vertices are identified into one vertex for component counting. This is the most wired random-cluster boundary condition on a fixed boundary, and maximizes increasing-event probabilities when $q\ge 1$ by boundary monotonicity of the random-cluster measure.