Independent Bond Percolation Research Articles

A new proof of the bunkbed conjecture in the p↑1 limit

For a finite simple graph G, the bunkbed graph G± is defined to be the product graph G□K2. We will label the two copies of a vertex v∈V(G) as v− and v+. The bunkbed conjecture, posed by Kasteleyn, states that for independent bond percolation on G±, percolation from u− to v− is at least as likely as percolation from u− to v+, for any u,v∈V(G). Despite the plausibility of this conjecture, so far the problem in full generality remains open. Recently, Hutchcroft, Nizić-Nikolac, and Kent gave a proof of the conjecture in the p↑1 limit. Here we present a new proof of the bunkbed conjecture in this limit, working in the more general setting of allowing different probabilities on different edges of G±.

Barak–Erdős graphs and the infinite-bin model

A Barak-Erd\H{o}s graph is a directed acyclic version of the Erd\H{o}s-R\'enyi random graph. It is obtained by performing independent bond percolation with parameter $p$ on the complete graph with vertices $\{1,...,n\}$, in which the edge between two vertices $i<j$ is directed from $i$ to $j$. The length of the longest path in this graph grows linearly with the number of vertices, at rate $C(p)$. In this article, we use a coupling between Barak-Erd\H{o}s graphs and infinite-bin models to provide explicit estimates on $C(p)$. More precisely, we prove that the front of an infinite-bin model grows at linear speed, and that this speed can be obtained as the sum of a series. Using these results, we prove the analyticity of $C$ for $p >1/2$, and compute its power series expansion. We also obtain the first two terms of the asymptotic expansion of $C$ as $p \to 0$, using a coupling with branching random walks.

Percolation of Finite Clusters and Shielded Paths

In independent bond percolation on $${\mathbb {Z}}^d$$ with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite connected component? Grimmett-Holroyd-Kozma used the triangle condition to show that for $$d \ge 19$$, the set of such p contains values strictly larger than the percolation threshold $$p_c$$. With the work of Fitzner-van der Hofstad, this has been reduced to $$d \ge 11$$. We improve this result by showing that for $$d \ge 10$$ and some $$p>p_c$$, there are infinite paths consisting of “shielded” vertices—vertices all whose adjacent edges are closed—which must be in the complement of the infinite cluster. Using values of $$p_c$$ obtained from computer simulations, this bound can be reduced to $$d \ge 7$$. Our methods are elementary and do not require the triangle condition.

The bunkbed conjecture on the complete graph

The bunkbed conjecture was first posed by Kasteleyn. If G=(V,E) is a finite graph and H some subset of V, then the bunkbed of the pair (G,H) is the graph G×{1,2} plus |H| extra edges to connect for every v∈H the vertices (v,1) and (v,2). The conjecture asserts that (v,1) is more likely to connect with (w,1) than with (w,2) in the independent bond percolation model for any v,w∈V. This is intuitive because (v,1) is in some sense closer to (w,1) than it is to (w,2). The conjecture has however resisted several attempts of proof. This paper settles the conjecture in the case of a constant percolation parameter and G the complete graph.

Bridges in the random-cluster model

The random-cluster model, a correlated bond percolation model, unifies a range of important models of statistical mechanics in one description, including independent bond percolation, the Potts model and uniform spanning trees. By introducing a classification of edges based on their relevance to the connectivity we study the stability of clusters in this model. We prove several exact relations for general graphs that allow us to derive unambiguously the finite-size scaling behavior of the density of bridges and non-bridges. For percolation, we are also able to characterize the point for which clusters become maximally fragile and show that it is connected to the concept of the bridge load. Combining our exact treatment with further results from conformal field theory, we uncover a surprising behavior of the (normalized) variance of the number of (non-)bridges, showing that it diverges in two dimensions below the value 4cos2⁡(π/3)=0.2315891⋯ of the cluster coupling q. Finally, we show that a partial or complete pruning of bridges from clusters enables estimates of the backbone fractal dimension that are much less encumbered by finite-size corrections than more conventional approaches.

Leaf-excluded percolation in two and three dimensions.

We introduce the leaf-excluded percolation model, which corresponds to independent bond percolation conditioned on the absence of leaves (vertices of degree one). We study the leaf-excluded model on the square and simple-cubic lattices via Monte Carlo simulation, using a worm-like algorithm. By studying wrapping probabilities, we precisely estimate the critical thresholds to be 0.3552475(8) (square) and 0.185022(3) (simple-cubic). Our estimates for the thermal and magnetic exponents are consistent with those for percolation, implying that the phase transition of the leaf-excluded model belongs to the standard percolation universality class.

Bounding basic characteristics of spatial epidemics with a new percolation model

We introduce a new 1-dependent percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are independent and identically distributed, but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but, conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above by the corresponding quantities for independent bond percolation with a certain density; this generalizes a result of Kuulasmaa (1982). Many models in the literature are special cases of our general model.

Bounding basic characteristics of spatial epidemics with a new percolation model

We introduce a new 1-dependent percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are independent and identically distributed, but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but, conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above by the corresponding quantities for independent bond percolation with a certain density; this generalizes a result of Kuulasmaa (1982). Many models in the literature are special cases of our general model.

A Note on Anisotropic Percolation

We consider an anisotropic independent bond percolation model on $$\mathbb{Z}^{2}_{+}$$ , i.e. we suppose that the vertical edges of $$\mathbb{Z}^{2}_{+}$$ are open with probability p and closed with probability 1−p, while the horizontal edges of $$\mathbb{Z}^{2}_{+}$$ are open with probability α p and closed with probability 1−α p, with 0 < p, α < 1. Let $$x = (x_{1},x_{2})\in \mathbb{Z}^{2}_{+}$$ , with x1 < x2, and $$x^{\prime} = (x_{2}, x_{1}) \in \mathbb{Z}^{2}_{+}$$ . It is natural to ask how the two point connectivity function Pp,α({0↔ x}) behaves, and whether anisotropy in percolation probabilities implies the strict inequality Pp,α({0↔ x})> Pp,α({0↔ x′}). In this note we give affirmative answer at least for some regions of the parameters involved.

On near-critical and dynamical percolation in the tree case

Consider independent bond percolation with retention probability p on a spherically symmetric tree Γ. Write θΓ(p) for the probability that the root is in an infinite open cluster, and define the critical value pc=inf{p : θΓ(p)>0}. If θΓ(pc)=0, then the root may still percolate in the corresponding dynamical percolation process at the critical value pc, as demonstrated recently by Haggstrom, Peres, and Steif. Here we relate this phenomenon to the near-critical behavior of θΓ(p) by showing that the root percolates in the dynamical percolation process if and only if ∫(θΓ(p))−1 dp<∞. The “only if” direction extends to general trees, whereas the “if” direction fails in this generality. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 311–318, 1999

On near‐critical and dynamical percolation in the tree case

Consider independent bond percolation with retention probability p on a spherically symmetric tree Gamma. Write theta_Gamma(p) for the probability that the root is in an infinite open cluster, and define the critical value p_c=inf{p:theta_Gamma(p)>0}. If theta_Gamma(p_c)=0, then the root may still percolate in the corresponding dynamical percolation process at the critical value p_c, as demonstrated recently by Haggstrom, Peres and Steif. Here we relate this phenomenon to the near-critical behaviour of theta_Gamma(p) by showing that the root percolates in the dynamical percolation process if and only if int_{p_c}^1 (theta_Gamma(p))^{-1}dp<infty. The ``only if'' direction extends to general trees, whereas the ``if'' direction fails in this generality.

The incipient infinite cluster in high-dimensional percolation

We announce our recent proof that, for independent bond percolation in high dimensions, the scaling limits of the incipient infinite cluster’s two-point and three-point functions are those of integrated super-Brownian excursion (ISE). The proof uses an extension of the lace expansion for percolation.

A Note on Inhomogeneous Percolation

Consider a special independent bond percolation model on $Z^2$, in which all bonds with vertices in the $X$ axis are open with probability $\delta$ and closed with probability $1 - \delta$, and all other bonds are open with probability $p$ and closed with probability $1 - p$. In this paper we show that no percolation occurs at $p_c$ for any $\delta < 1$. The method allows us also to show no percolation at $p_c$ in a more general inhomogeneous case.

First Passage Percolation for Random Colorings of $\mathbb{Z}^d$

Random colorings (independent or dependent) of $\mathbb{Z}^d$ give rise to dependent first-passage percolation in which the passage time along a path is the number of color changes. Under certain conditions, we prove strict positivity of the time constant (and a corresponding asymptotic shape result) by means of a theorem of Cox, Gandolfi, Griffin and Kesten about "greedy" lattice animals. Of particular interest are i.i.d. colorings and the $d = 2$ Ising model. We also apply the greedy lattice animal theorem to prove a result on the omnipresence of the infinite cluster in high density independent bond percolation.

Mean-field critical behaviour for correlation length for percolation in high dimensions

Extending the method of [27], we prove that the corrlation length ξ of independent bond percolation models exhibits mean-field type critical behaviour (i.e. ξ(p∼(p c −p)−1/2 asp↗p c ) in two situations: i) for nearest-neighbour independent bond percolation models on ad-dimensional hypercubic lattice ℤ d , withd sufficiently large, and ii) for a class of “spread-out” independent bond percolation models, which are believed to belong to the same universality class as the nearest-neighbour model, in more than six dimensions. The proof is based on, and extends, a method developed in [27], where it was used to prove the triangle condition and hence mean-field behaviour of the critical exponents γ, β, δ, Δ and ν2 for the above two cases.

Mean-field critical behaviour for percolation in high dimensions

The triangle condition for percolation states that\(\sum\limits_{x,y} {\tau (0,x)\tau (0,y) \cdot \tau (y,0)} \) is finite at the critical point, where τ(x, y) is the probability that the sitesx andy are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on thed-dimensional hypercubic lattice, ifd is sufficiently large, and (ii) in more than six dimensions for a class of “spread-out” models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number, and is zero otherwise. In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values\((\gamma = \beta = 1,\delta = \Delta _t = 2, t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 2)\) and that the percolation density is continuous at the critical point. We also prove thatv 2 in (i) and (ii), wherev 2 is the critical exponent for the correlation length.

One dimensional 1/|j ? i| S percolation models: The existence of a transition forS?2

Consider a one-dimensional independent bond percolation model withpj denoting the probability of an occupied bond between integer sitesi andi±j,j≧1. Ifpj is fixed forj≧2 and\(\mathop {\lim }\limits_{j \to \infty }\)j2pj>1, then (unoriented) percolation occurs forp1 sufficiently close to 1. This result, analogous to the existence of spontaneous magnetization in long range one-dimensional Ising models, is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size. Oriented percolation is shown to occur forp1 close to 1 if\(\mathop {\lim }\limits_{j \to \infty }\)jspj>0 for somes<2. Analogous results are valid for one-dimensional site-bond percolation models.