One goal of this paper is to prove that dynamical critical site percolation on the planar triangular lattice has exceptional times at which percolation occurs. In doing so, new quantitative noise sensitivity results for percolation are obtained. The latter is based on a novel method for controlling the "level k" Fourier coefficients via the construction of a randomized algorithm which looks at random bits, outputs the value of a particular function but looks at any fixed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff dimension of the set of percolating times. We then study the problem of exceptional times for certain "k-arm" events on wedges and cones. As a corollary of this analysis, we prove, among other things, that there are no times at which there are two infinite "white" clusters, obtain an upper bound on the Hausdorff dimension of the set of times at which there are both an infinite white cluster and an infinite black cluster and prove that for dynamical critical bond percolation on the square grid there are no exceptional times at which three disjoint infinite clusters are present.
In dynamical critical site percolation on the triangular lattice or bond percolation on \Z^2, we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster. We show that at a typical time with respect to this measure, the percolation configuration has the law of Kesten's Incipient Infinite Cluster. In the most technical result of this paper, we show that, on the other hand, at the first exceptional time, the law of the configuration is different. We also study the collapse of the infinite cluster near typical exceptional times, and establish a …
We analyse in this paper a conservative analogue of the celebrated model of dynamical percolation introduced by Häggström, Peres and Steif in [HPS97]. It is simply defined as follows: start with an initial percolation configuration $ω(t=0)$. Let this configuration evolve in time according to a simple exclusion process with symmetric kernel $K(x,y)$. We start with a general investigation (following [HPS97]) of this dynamical process $t \mapsto ω_K(t)$ which we call $K$-exclusion dynamical percolation. We then proceed with a detailed analysis of the planar case at the critical point (both for th…
In first-passage percolation (FPP), we let $(τ_v)$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If $F$ is the distribution function of $τ_v$, there are different regimes: if $F(0)$ is small, this weight typically grows like a linear function of the distance, and when $F(0)$ is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. We study a dynamical version of critical FPP on the triangular lattice where vertices resample their…
Benjamini and Kesten introduced in 1995 the problem of embedding infinite binary sequences into a Bernoulli percolation configuration, known as "percolation of words". We give a positive answer to their Open Problem 2: almost surely, all words are seen for site percolation on Z^3 with parameter p = 1/2. We also extend this result in various directions, proving the same result for any dimension d at least three and for any value p in the interval (p_c(Z^d), 1 - p_c(Z^d)), and for restrictions to slabs. Finally, we provide an explicit estimate on the probability to find all words starting from a…
We study a percolation model on the square lattice, where clusters "freeze" (stop growing) as soon as their volume (i.e. the number of sites they contain) gets larger than N, the parameter of the model. A model where clusters freeze when they reach diameter at least N was studied in earlier papers. Using volume as a way to measure the size of a cluster - instead of diameter - leads, for large N, to a quite different behavior (contrary to what happens on the binary tree, where the volume model and the diameter model are "asymptotically the same"). In particular, we show the existence of a seque…
Consider the indicator function $f$ of a two-dimensional percolation crossing event. In this paper, the Fourier transform of $f$ is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension 31/36 a.s., and the corresponding dimension in the half-plane is 5/9. It is also proved that critical bond percolation on the square grid has exceptional times a.s. Also, the…
Ahlberg, Griffiths, Morris and Tassion have proved that, asymptotically almost surely, the quenched crossing probabilities for critical planar Voronoi percolation do not depend on the environment. We prove an analogous result for arm events. In particular, we prove that the variance of the quenched probability of an arm event is at most a constant times the square of the annealed probability. The fact that the arm events are degenerate and non-monotonic add two major difficulties. As an application, we prove that there exists $ε> 0$ such that the following holds for the annealed percolation fu…
We study the heat kernel and the Green's function on the infinite supercritical percolation cluster in dimension $d \geq 2$ and prove a quantitative homogenization theorem for these functions with an almost optimal rate of convergence. These results are a quantitative version of the local central limit theorem proved by Barlow and Hambly. The proof relies on a structure of renormalization for the infinite percolation cluster introduced by Armstrong and the first author, Gaussian bounds on the heat kernel established by Barlow and tools of the theory of quantitative stochastic homogenization. A…
Häggström, Peres, and Steif (1997) have introduced a dynamical version of percolation on a graph $G$. When $G$ is a tree they derived a necessary and sufficient condition for percolation to exist at some time $t$. In the case that $G$ is a spherically symmetric tree, Häggström, Peres, and Steif (1997) derived a necessary and sufficient condition for percolation to exist at some time $t$ in a given target set $D$. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time $t\in D$, in the case that the underlying tree is not neces…
We prove optimal quantitative estimates on the first-order correctors on supercritical percolation clusters: we show that they are bounded in $d\geq 3$ and have logarithmic growth in $d = 2$, in the sense of stretched exponential moments. The main ingredients are a renormalization scheme of the supercritical percolation cluster, following the works of Pisztora and Barlow; large-scale regularity estimates developed in the previous paper; and a nonlinear concentration inequality of Efron-Stein type which is used to transfer quantitative information from the environment to the correctors.
We establish quantitative homogenization, large-scale regularity and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is constant on the cluster. The argument passes through a series of renormalization steps: first, we use standard percolation results to find a large scale above which the geometry of the percolation cluster behaves (in a sense made precise) like that of Euclidean space. Then, following the work of Barlow, we find a succession of larger scales on which certain f…
We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first for which the critical probability p_c \ne 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete mod…
We show that planar Bargmann-Fock percolation is noise sensitive under the Ornstein-Ulhenbeck process. The proof is based on the randomized algorithm approach introduced by Schramm and Steif and gives quantitative polynomial bounds on the noise sensitivity of crossing events for Bargmann-Fock. A rather counter-intuitive consequence is as follows. Let $F$ be a Bargmann-Fock Gaussian field in $\mathbb{R}^3$ and consider two horizontal planes $P_1,P_2$ at small distance $\varepsilon$ from each other. Even though $F$ is a.s. analytic, the above noise sensitivity statement implies that the full res…
Consider a Markov process ω_t at equilibrium and some event C (a subset of the state-space of the process). A natural measure of correlations in the process is the pairwise correlation \Pr[ω_0,ω_t \in C] - \Pr[ω_0 \in C]^2. A second natural measure is the probability of the continual occurrence event \{ω_s \in C, \forall s\in [0,t]\}. We show that for reversible Markov chains, and any event C, pairwise decorrelation of the event C implies a decay of the probability of the continual occurrence event \{ω_s \in C, \forall s \in [0,t]\} as t\to\infty. We provide examples showing that our results a…
In this paper, we introduce and study the annealed spectral sample of Voronoi percolation, which is a continuous and finite point process in $\mathbb{R}^2$ whose definition is mostly inspired by the spectral sample of Bernoulli percolation introduced in [GPS10] by Garban, Pete and Schramm. We show a clustering effect as well as estimates on the full lower tail of this spectral object. Our main motivation is the study of two models of dynamical critical Voronoi percolation in the plane. In the first model, the Voronoi tiling does not evolve in time while the colors of the cells are resampled at…
We continue our study of the chemical (graph) distance inside large critical percolation clusters in dimension two. We prove new estimates, which involve the three-arm probability, for the point-to-surface and point-to-point distances. We show that the point-to-point distance in $\mathbb{Z}^2$ between two points in the same critical percolation cluster has infinite second moment. We also give quantitative versions of our previous results comparing the length of the shortest crossing to that of the lowest crossing of a box.
For directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times on the vertices, let $T_{n}$ denote the last passage time from $(0,0)$ to $(n,n)$. We consider asymptotic two point correlation functions of the sequence $T_{n}$. In particular we consider ${\rm Corr}(T_{n}, T_{r})$ for $r\le n$ where $r,n\to \infty$ with $r\ll n$ or $n-r \ll n$. We show that in the former case ${\rm Corr}(T_{n}, T_{r})=Θ((\frac{r}{n})^{1/3})$ whereas in the latter case $1-{\rm Corr}(T_{n}, T_{r})=Θ((\frac{n-r}{n})^{2/3})$. The argument revolves around finer understanding of polymer geometry…
Consider the long-range percolation model on the integer lattice $\mathbb{Z}^d$ in which all nearest-neighbour edges are present and otherwise $x$ and $y$ are connected with probability $q_{x,y}:=1-\exp(-|x-y|^{-s})$, independently of the state of other edges. Throughout the regime where the model yields a locally-finite graph, (i.e.\ for $s>d$,) we determine the spectral dimension of the associated simple random walk, apart from at the exceptional value $d=1$, $s=2$, where the spectral dimension is discontinuous. Towards this end, we present various on-diagonal heat kernel bounds, a number of…
We study long-range Bernoulli percolation on $\mathbb{Z}^d$ in which each two vertices $x$ and $y$ are connected by an edge with probability $1-\exp(-β\|x-y\|^{-d-α})$. It is a theorem of Noam Berger (CMP, 2002) that if $0<α<d$ then there is no infinite cluster at the critical parameter $β_c$. We give a new, quantitative proof of this theorem establishing the power-law upper bound \[ \mathbf{P}_{β_c}\bigl(|K|\geq n\bigr) \leq C n^{-(d-α)/(2d+α)} \] for every $n\geq 1$, where $K$ is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli perco…
The dynamical discrete web (DyDW),introduced in recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical time parameter τ. The evolution is by independent updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed τ. In this paper, we study the existence of exceptional (random) values of τwhere the paths of the web do not behave like usual random walks and the Hausdorff dimension of the set of exceptional such τ. Our results are motivated b…
Let $\mathcal{C}^n$ be the largest open cluster for supercritical Bernoulli bond percolation in $[-n, n]^d \cap \mathbb{Z}^d$ with $d \ge 2$. We obtain a sharp estimate for the effective resistance on $\mathcal{C}^n$. As an application we show that the cover time for the simple random walk on $\mathcal{C}^n$ is comparable to $n^d (\log n)^2$. Noting that the cover time for the simple random walk on $[-n, n]^d \cap \mathbb{Z}^d$ is of order $n^d \log n$ for $d \ge 3$ (and of order $n^2 (\log n)^2$ for $d = 2$), this gives a quantitative difference between the two random walks for $d \ge 3$.
Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell, Saks, Schramm and Servedio (2005)) and the Schramm-Steif inequality for the Fourier-Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm-Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions - expressed in terms of randomized stopping sets - yielding sharp …
We consider directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times on the vertices. A topic of great interest is the coupling structure of the weights of geodesics as the endpoints are varied spatially and temporally. A particular specialization is when one considers geodesics to points varying in the time direction starting from a given initial data. This paper considers the flat initial condition which corresponds to line-to-point last passage times. Settling a conjecture by Ferrari and Spohn (SIGMA, 2016), we show that for the passage times from the line $x+y=0$ …
We consider the Constrained-degree percolation model on the hypercubic lattice, $\mathbb L^d=(\mathbb Z^d,\mathbb E^d)$ for $d\geq 3$. It is a continuous time percolation model defined by a sequence, $(U_e)_{e\in\mathbb E^d}$, of i.i.d. uniform random variables in $[0,1]$ and a positive integer (constraint) $κ$. Each bond $e\in\mathbb E^d$ tries to open at time $U_e$; it succeeds if and only if both its end-vertices belong to at most $κ-1$ open bonds at that time. Our main results are quantitative upper bounds on the critical time, characterising a phase transition for all $d\geq 3$ and most n…
We consider translationally-invariant percolation models on $\mathbb{Z}^d$ satisfying the finite energy and the FKG properties. We provide explicit upper bounds on the probability of having two distinct clusters going from the endpoints of an edge to distance $n$ (this corresponds to a finite size version of the celebrated Burton-Keane [Comm. Math. Phys. 121 (1989) 501-505] argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincaré inequality proved in Chatterjee and Sen (2013). As a consequence, we obtain upper bounds on the probabilit…
In dynamical percolation, the status of every bond is refreshed according to an independent Poisson clock. For graphs which do not percolate at criticality, the dynamical sensitivity of this property was analyzed extensively in the last decade. Here we focus on graphs which percolate at criticality, and investigate the dynamical sensitivity of the infinite cluster. We first give two examples of bounded degree graphs, one which percolates for all times at criticality and one which has exceptional times of nonpercolation. We then make a nearly complete analysis of this question for spherically s…
We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension $1/2$ almost surely, and give bounds on the rate at which the walk diverges at such times. We also show noise sensitivity of the event that our random walk is positive after $n$ steps. In fact this event is maximall…
Geodesic Trees and Exceptional Directions in FPP on Hyperbolic Groups
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究所）；School of Mathematics, Tata Institute of Fundamental Research（数学学院，塔塔基础研究所）
We prove a quantitative Russo-Seymour-Welsh (RSW) type result for random walks on two natural examples of random planar graphs: the supercritical percolation cluster in the square lattice and the Poisson Voronoi triangulation in the plane. More precisely, we prove that the probability that a simple random walk crosses a rectangle in the hard direction with uniformly positive probability is stretched exponentially likely in the size of the rectangle. As an application we prove a near optimal decorrelation result for uniform spanning trees for such graphs. This is the key missing step in this se…
