We consider percolation on $\mathbb{Z}^d$ and on the $d$-dimensional discrete torus, in dimensions $d \ge 11$ for the nearest-neighbour model and in dimensions $d>6$ for spread-out models. For $\mathbb{Z}^d$, we employ a wide range of techniques and previous results to prove that there exist positive constants $c$ and $C$ such that the slightly subcritical two-point function and one-arm probabilities satisfy \[ \mathbb{P}_{p_c-\varepsilon}(0 \leftrightarrow x) \leq \frac{C}{\|x\|^{d-2}} e^{-c\varepsilon^{1/2} \|x\|} \quad \text{ and } \quad \frac{c}{r^{2}} e^{-C \varepsilon^{1/2}r} \leq \mathbb{P}_{p_c-\varepsilon}\Bigl(0 \leftrightarrow \partial [-r,r]^d \Bigr) \leq \frac{C}{r^2} e^{-c \varepsilon^{1/2}r}. \] Using this, we prove that throughout the critical window the torus two-point function has a "plateau," meaning that it decays for small $x$ as $\|x\|^{-(d-2)}$ but for large $x$ is essentially constant and of order $V^{-2/3}$ where $V$ is the volume of the torus. The plateau for the two-point function leads immediately to a proof of the torus triangle condition, which is known to have many implications for the critical behaviour on the torus, and also leads to a proof that the critical values on the torus and on $\mathbb{Z}^d$ are separated by a multiple of $V^{-1/3}$. The torus triangle condition and the size of the separation of critical points have been proved previously, but our proofs are different and are direct consequences of the bound on the $\mathbb{Z}^d$ two-point function. In particular, we use results derived from the lace expansion on $\mathbb{Z}^d$, but in contrast to previous work on high-dimensional torus percolation we do not need or use a separate torus lace expansion.
We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the critical point, and prove an upper bound $|x|^{-(d-2)}\exp[-c|x|/ξ]$, where the correlation length $ξ$ has a square root divergence at the critical point. As an application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions $d>4$ has a "plateau." We also discuss the significance and consequences of the plateau for the analysis of critical behaviour on…
For $d \geq 3$ we obtain an approximation of the zero-average Gaussian free field on the discrete $d$-dimensional torus of large side length $N$ by the Gaussian free field on $\mathbb Z^d$, valid in boxes of roughly side length $N - N^δ$ with $δ\in (\frac12,1)$. As an implication, the level sets of the zero-average Gaussian free field on the torus can be approximated by the level sets of the Gaussian free field on $\mathbb Z^d$. This leads to a series of applications related to level-set percolation. We show that level sets of the zero-average Gaussian free field on the torus for levels $h > h…
This paper analyzes various questions pertaining to bootstrap percolation on the $d$-dimensional Hamming torus where each node is open with probability $p$ and the percolation threshold is 2. For each $d'<d$ we find the critical exponent for the event that a $d'$-dimensional subtorus becomes open and compute the limiting value of its probability under the critical scaling. For even $d'$, we use the Chen-Stein method to show that the number of $d'$-dimensional subtori that become open can be approximated by a Poisson random variable.
我们调查的最大的临界渗流集团在一个大的d维环面的标度，在高维的最近邻渗流，或当d>6足够的扩散渗流。我们使用一个相对简单的耦合参数表明，这个最大的关键集群是，具有很高的概率，由一个大的常数倍$V^{2/3}$以上和以下由一个小的常数倍$V^{2/3}(log V)^{-4/3}$，其中V是体积的环面。我们还根据Z^d上的亚临界渗流两点函数给出了一个简单的判据，在该判据下，下界可改进为小常数倍$V^{2/3}$，即，我们证明了随机图的最大临界集群的高维环面上的渐近。这建立了Aizenman（1997）的一个猜想，除了对数修正。我们讨论了这些结果的影响对高维渗流的边界条件的依赖。我们的方法主要基于Borgs、Chayes、van der Hofstad、Slade和Spencer（2005 a，2005 b）的结果，其中证明了$V^{2/3}$标度服从一个适当定义的临界窗口包含Z^d上的渗流阈值的假设。我们还强烈依赖于Hara（1990，2005）、Hara和Slade（1990）以及Hara，van der Hofstad和Slade（2003）证明的Z^d上渗流的平均场结果。
For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further…
In this note we study the geometry of the largest component C_1 of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. There it is shown that this component is of size n^{2/3}, and here we show that its diameter is n^{1/3} and that the simple random walk takes n steps to mix on it. Our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus Z_n^d (with d large and n tending to infinity) and the Hamming cube {0,1}^n.
In first-passage percolation, one assigns i.i.d. nonnegative weights $(t_e)$ to the edges of $\mathbb{Z}^d$ and studies the induced distance (passage time) $T(x,y)$ between vertices $x$ and $y$. It is known that for $d=2$, the fluctuations of $T(x,y)$ are at least order $\sqrt{\log |x-y|}$ under mild assumptions on $t_e$. We study the question of fluctuation lower bounds for $T_n$, the minimal passage time between two opposite sides of an $n$ by $n$ square. The main result is that, under a curvature assumption, this quantity has fluctuations at least of order $n^{1/8-ε}$ for any $ε>0$ when the…
The Hamming torus of dimension $d$ is the graph with vertices $\{1,\dots,n\}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $θ$ starts with a random set of open vertices, to which every vertex belongs independently with probability $p$, and at each time step the open set grows by adjoining every vertex with at least $θ$ open neighbors. We assume that $n$ is large and that $p$ scales as $n^{-α}$ for some $α>1$, and study the probability that an $i$-dimensional subgraph ever becomes open. For large $θ$, we prove that the critical …
We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder with base a d-1-dimensional discrete torus of side-length N, or the set of points visited by simple random walk on the d-dimensional discrete torus of side-length N by times of order uN^d. We study the percolative properties of the …
We study the percolation time of the $r$-neighbour bootstrap percolation model on the discrete torus $(\Z/n\Z)^d$. For $t$ at most a polylog function of $n$ and initial infection probabilities within certain ranges depending on $t$, we prove that the percolation time of a random subset of the torus is exactly equal to $t$ with high probability as $n$ tends to infinity. Our proof rests crucially on three new extremal theorems that together establish an almost complete understanding of the geometric behaviour of the $r$-neighbour bootstrap process in the dense setting. The special case $d-r=0$ o…
In this paper we introduce and study a higher-dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant k-dimensional cycles (with 0-cycles being connected components). Considering a continuum percolation model in the flat d-dimensional torus, we show that all the giant k-cycles (k=1,...,d-1) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant k-cycles are increasing in k and are tightly related to the critical values in continuum percolation. …
We consider random walk on dynamical percolation on the discrete torus $\mathbb{Z}_n^d$. In previous work, mixing times of this process for $p<p_c(\mathbb{Z}^d)$ were obtained in the annealed setting where one averages over the dynamical percolation environment. Here we study exit times in the quenched setting, where we condition on a typical dynamical percolation environment. We obtain an upper bound for all $p$ which for $p<p_c$ matches the known lower bound.
Consider a graph $G$ and an initial random configuration, where each node is black with probability $p$ and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least $r$ black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a $d$-dimensional torus and identify the threshold values.
We consider dynamical percolation on the $d$-dimensional discrete torus of side length $n$, $\mathbb{Z}_n^d$, where each edge refreshes its status at rate $μ=μ_n\le 1/2$ to be open with probability $p$. We study random walk on the torus, where the walker moves at rate $1/(2d)$ along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case $p<p_c(\mathbb{Z}^d)$, the (annealed) mixing time of the walk is $Θ(n^2/μ)$, and it was conjectured that in the supercritical case $p>p_c(\mathbb{Z}^d)$, the mixing time is $Θ(n^2+1/μ)$; here the implie…
In this paper a random graph model $G_{\mathbb{Z}^2_N,p_d}$ is introduced, which is a combination of fixed torus grid edges in $(\mathbb{Z}/N \mathbb{Z})^2$ and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices $u,v\in(\mathbb{Z}/N \mathbb{Z})^2$ with graph distance $d$ on the torus grid is $p_d=c/Nd$, where $c$ is some constant. We show that, {\em whp}, the diameter $D(G_{\mathbb{Z}^2_N,p_d})=Θ(\log N)$. Moreover, we consider non-monotonous bootstrap percolation on $G_{\mathbb{Z}^2_N,p_d}$. We prove the presence of phase …
In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or Erdos-Renyi random graph, raising the question whether the scaling limits of the largest connected components, as identified by Aldous (1997), are also equal. In this paper, we investigate the cycle structure of the largest critical components for high-dimensional percolation on the torus (Z/rZ)^d. While percolation …
We show that the six-vertex model with parameter $c\in[\sqrt 3, 2]$ on a square lattice torus has an ergodic infinite-volume limit as the size of the torus grows to infinity. Moreover we prove that for $c\in[\sqrt{2+\sqrt 2}, 2]$, the associated height function on $\mathbb Z^2$ has unbounded variance. The proof relies on an extension of the Baxter-Kelland-Wu representation of the six-vertex model to multi-point correlation functions of the associated spin model. Other crucial ingredients are the uniqueness and percolation properties of the critical random cluster measure for $q\in[1,4]$, and r…
Assign to each vertex of the one-dimensional torus i.i.d. weights with a heavy-tail of index $τ-1>0$. Connect then each couple of vertices with probability roughly proportional to the product of their weights and that decays polynomially with exponent $α>0$ in their distance. The resulting graph is called scale-free percolation. The goal of this work is to study the mixing time of the simple random walk on this structure. We depict a rich phase diagram in $α$ and $τ$. In particular we prove that the presence of hubs can speed up the mixing of the chain. We use different techniques for each pha…
Define the scale-free Gilbert graph based on a Boolean model with heavy-tailed radius distribution on the $d$-dimensional torus by connecting two centers of balls by an edge if at least one of the balls contains the center of the other. We investigate two asymptotic properties of this graph as the size of the torus tends to infinity. First, we determine the tail index associated with the asymptotic distribution of the sum of all power-weighted incoming and outgoing edge lengths at a randomly chosen vertex. Second, we study the behavior of chemical distances on scale-free Gilbert graphs and sho…
The model of random interlacements is a one-parameter family $\mathcal I^u,$ $u \ge 0,$ of random subsets of $\mathbb{Z}^d,$ which locally describes the trace of simple random walk on a $d$-dimensional torus run up to time $u$ times its volume. Its complement, the so-called vacant set $\mathcal V^u$, has been shown to undergo a non-trivial percolation phase-transition in $u;$ i.e., there exists $u_*(d) \in (0, \infty)$ such that for $u \in [0, u_*(d))$ the vacant set $\mathcal V^u$ contains a unique infinite connected component $\mathcal V_\infty^u,$ while for $u > u_*(d)$ it consists of finit…
