Related papers: A note about critical percolation on finite graphs

We consider self-avoiding walk on finite graphs with large girth. We study a few aspects of the model originally considered by Lawler, Schramm and Werner on finite balls in Z^d. The expected length of a random self avoiding path is…

We study the distribution of finite clusters in slightly supercritical ($p \downarrow p_c$) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if $G$ is a transitive nonamenable graph satisfying the…

This study delves into first-passage percolation on random geometric graphs in the supercritical regime, where the graphs exhibit a unique infinite connected component. We investigate properties such as geodesic paths, moderate deviations,…

The binomial random bipartite graph $G(n,n,p)$ is the random graph formed by taking two partition classes of size $n$ and including each edge between them independently with probability $p$. It is known that this model exhibits a similar…

The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a_1 n X a_2 n X ... X a_d n box in R^d (for constants a_1, ..., a_d > 0), and whose edges connect all vertices within Hamming distance one.…

A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states of the edges in $S_1$ are independent of the…

We present a unifying, consistent, finite-size-scaling picture for percolation theory bringing it into the framework of a general, renormalization-group-based, scaling scheme for systems above their upper critical dimensions $d_c$.…

In this paper, we introduce the generic circular triangle-free graph $\mathbb C_3$ and propose a finite axiomatization of its first order theory. In particular, our main results show that a countable graph $G$ embeds into $\mathbb C_3$ if…

We consider a synchronous process of particles moving on the vertices of a graph $G$, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, $M$ particles are placed on a vertex of $G$. In subsequent time steps, all…

We consider the Bernoulli bond percolation process (with parameter $p$) on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non-trivial percolation threshold based either on a single isoperimetric…

The largest connected component in duplication-divergence growing graphs with symmetric coupled divergence is studied. Finite-size scaling reveals a phase transition occurring at a divergence rate $\delta_c$. The $\delta_c$ found stands…

The largest components of the critical Erd\H{o}s-R\'enyi graph, $G(n,p)$ with $p=1/n$, have size of order $n^{2/3}$ with high probability. We give detailed asymptotics for the probability that there is an unusually large component, i.e. of…

We describe a probabilistic methodology, based on random walk estimates, to obtain exponential upper bounds for the probability of observing unusually small maximal components in two classical (near-)critical random graph models. More…

In the random $r$-neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get…

A uniformly random graph on $n$ vertices with a fixed degree sequence, obeying a $\gamma$ subpower law, is studied. It is shown that, for $\gamma>3$, in a subcritical phase with high probability the largest component size does not exceed…

Percolation is a model for random damage to a network. It is one of the simplest models that displays a phase transition: when the network is severely damaged, it falls apart in many small connected components, while if the damage is light,…

We show that there exists a connected graph G with subexponential volume growth such that critical percolation on the product of G with the line has infinitely many infinite clusters. We also give some conditions under which this cannot…

Consider balls $\Lambda_n$ of growing volumes in the $d$-dimensional hierarchical lattice, and place edges independently between each pair of vertices $x\neq y\in\Lambda_n$ with probability $1-\exp(-\beta J(x, y) )$ where $J(x, y) \asymp \|…

In $H$-percolation, we start with an Erd\H{o}s--R\'enyi graph ${\mathcal G}_{n,p}$ and then iteratively add edges that complete copies of $H$. The process percolates if all edges missing from ${\mathcal G}_{n,p}$ are eventually added. We…

We prove that critical percolation has no infinite clusters almost surely on any unimodular quasi-transitive graph satisfying a return probability upper bound of the form $p_n(v,v) \leq \exp\left[-\Omega(n^\gamma)\right]$ for some…