On the $C^1$-property of the percolation function of random interlacements and a related variational problem

TL;DR

This paper proves the $C^1$-smoothness of the percolation probability function for random interlacements on $bZ^d$, and explores a related variational problem to understand decay rates of certain large deviation events.

Contribution

It establishes the $C^1$ regularity of the percolation function and analyzes a variational problem related to large deviations in the vacant set of random interlacements.

Findings

Percolation function is $C^1$ on an interval $[0,\hat{u})$.

Identifies a regime of 'small excess' where minimizers stay below a natural threshold.

Provides insights into the exponential decay rate of large deviation probabilities.

Abstract

We consider random interlacements on $\mathbb{Z}^d$, $d \ge 3$. We show that the percolation function that to each $u \ge 0$ attaches the probability that the origin does not belong to an infinite cluster of the vacant set at level $u$, is $C^1$ on an interval $[0,\^u)$, where $\^u$ is positive and plausibly coincides with the critical level $u_*$ for the percolation of the vacant set. We apply this finding to a constrained minimization problem that conjecturally expresses the exponential rate of decay of the probability that a large box contains an excessive proportion $\nu$ of sites that do not belong to an infinite cluster of the vacant set. When $u$ is smaller than $\^u$, we describe a regime of "small excess" for $\nu$ where all minimizers of the constrained minimization problem remain strictly below the natural threshold value $\sqrt{u}_* - \sqrt{u}$ for the variational problem.