相关论文: Quantitative noise sensitivity and exceptional tim…
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…
We analyse in this paper a conservative analogue of the celebrated model of dynamical percolation introduced by H\"aggstr\"om, Peres and Steif in [HPS97]. It is simply defined as follows: start with an initial percolation configuration…
Consider critical Bernoulli percolation in the plane. We give a new proof of the sharp noise sensitivity theorem shown by Garban, Pete and Schramm. Contrary to the previous approaches, we do not use any spectral tool. We rather study…
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…
We prove absence of infinite clusters and contours in a class of critical constrained percolation models on the square lattice. The percolation configuration is assumed to satisfy certain hard local constraints, but only weak symmetry and…
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…
We consider the constrained-degree percolation (CDP) model on the hypercubic lattice. This is a continuous-time percolation model defined by a sequence $(U_e)_{e\in\mathcal{E}^d}$ of i.i.d. uniform random variables and a positive integer…
We study site percolation on Angel & Schramm's uniform infinite planar triangulation. We compute several critical and near-critical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes…
We study the autocorrelation time of the size of the cluster at the origin in discrete-time dynamical percolation. We focus on binary trees and high-dimensional tori, and show in both cases that this autocorrelation time is linear in the…
We prove Tsirelson's conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential…
How does the percolation transition behave in the absence of quenched randomness? To address this question, we study two nonrandom self-dual quasiperiodic models of square-lattice bond percolation. In both models, the critical point has…
This work is the first in a series of papers devoted to the construction and study of scaling limits of dynamical and near-critical planar percolation and related objects like invasion percolation and the Minimal Spanning Tree. We show here…
In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievements is the proof of the Cardy-Smirnov formula. This theorem, together with the introduction of…
Three-dimensional quantum percolation problems are studied by analyzing energy level statistics of electrons on maximally connected percolating clusters. The quantum percolation threshold $\pq$, which is larger than the classical…
We consider the random cluster model with parameter $q<1$, for which the FKG inequalities are not valid. On the square lattice, stochastic comparison with Bernoulli percolation implies that the model is subcritical (respectively…
Consider percolation on $T\times \mathbb{Z}^d$, the product of a regular tree of degree $k\geq 3$ with the hypercubic lattice $\mathbb{Z}^d$. It is known that this graph has $0<p_c<p_u<1$, so that there are non-trivial regimes in which…
We study percolation on the hierarchical lattice of order $N$ where the probability of connection between two points separated by distance $k$ is of the form $c_k/N^{k(1+\delta)},\; \delta >-1$. Since the distance is an ultrametric, there…
We study constrained percolation models on planar lattices including the $[m,4,n,4]$ lattice and the square tilings of the hyperbolic plane, satisfying certain local constraints on faces of degree 4, and investigate the existence of…
We consider two-dimensional dependent dynamical site percolation where sites perform majority dynamics. We introduce the critical percolation function at time t as the infimum density with which one needs to begin in order to obtain an…
We consider percolation on the discrete torus $\mathbb{Z}_n^d$ at $p_c(\mathbb{Z}^d)$, the critical value for percolation on the corresponding infinite lattice $\mathbb{Z}^d$, and within the scaling window around it. We assume that $d$ is a…