Regularity of the time constant for a supercritical Bernoulli percolation

TL;DR

This paper investigates how the chemical distance in supercritical Bernoulli percolation on Z^d varies smoothly with the open edge probability p, revealing regularity properties of the associated time constant.

Contribution

It analyzes the regularity of the time constant map p → μ_p in supercritical Bernoulli percolation, extending understanding of its continuity and potential differentiability.

Findings

The map p → μ_p is continuous in the supercritical regime.

The study provides insights into the regularity properties of the chemical distance.

It relates the problem to a special case of first passage percolation with a specific distribution.

Abstract

We consider an i.i.d. supercritical bond percolation on Z^d , every edge is open with a probability p > p\_c (d), where p\_c (d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite open cluster C\_p [11]. We are interested in the regularity properties of the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y $\in$ C\_p corresponds to the length of the shortest path in C\_p joining the two points. The chemical distance between 0 and nx grows asymptotically like n$\mu$\_p (x). We aim to study the regularity properties of the map p $\rightarrow$ $\mu$\_p in the supercritical regime. This may be seen as a special case of first passage percolation where the distribution of the passage time is G\_p = p$\delta$\_1 + (1 -- p)$\delta$\_$\infty$ , p > p c (d). It is already known that the map p $\rightarrow$ $\mu$\_p is continuous (see [10]).