Rémy Sanchis

Analyticity of the d-Dimensional Bond Percolation Probability Around p=1

Let θ(p) be the percolation probability of a d-dimensional bond percolation process on Zd. We prove that 1−θ(p) can be written as an absolutely convergent series in powers of (1−p)/p, provided that |(1−p)/p| is sufficiently small. This implies that θ(p) is an analytic function of the complex variable p, around p=1.

Percolation on Infinite Graphs and Isoperimetric Inequalities

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 inequality if the graph has a bi-infinite geodesic, or two isoperimetric inequalities if the graph has not a bi-infinite geodesic. This new criterion extends previous criteria and brings together a large class of amenable graphs (such as regular lattices) and non-amenable graphs (such trees). We also study the finite connectivity in graphs satisfying the new general criterion and show that graphs in this class with a bi-infinite geodesic always have finite connectivity functions with exponential decay when p is sufficiently close to one. On the other hand, we show that there are graphs in the same class with no bi-infinite geodesic for which the finite connectivity decays sub-exponentially (down to polynomially) in the highly supercritical phase even for p arbitrarily close to one.

Percolation of words on Z d with long range connections

Consider an independent site percolation model on

A Remark on Monotonicity in Bernoulli Bond Percolation

\Z^d

Diffusive–Ballistic Transition in Random Walks with Long-Range Self-Repulsion

, with parameter

Decay Properties of the Connectivity for Mixed Long Range Percolation Models on ℤd

p \in (0,1)

Percolation of words on Zd with long-range connections

, where all long range connections in the axes directions are allowed. In this work we show that given any parameter

Ornstein-Zernike behavior for the Bernoulli bond percolation on

p

\mathbb Z^d

, there exists and integer

in the supercritical regime

K(p)