# Solidification of porous interfaces and disconnection

## TL;DR

This paper establishes uniform estimates on Brownian motion absorption by porous interfaces around compact sets and applies these to improve large deviation bounds for random walk and interlacements disconnection probabilities in high dimensions.

## Contribution

It introduces new uniform estimates for Brownian motion absorption and strengthens large deviation results for disconnection events without convexity assumptions.

## Key findings

- Uniform estimates on Brownian motion absorption by porous interfaces.
- Large deviation upper bounds for disconnection probabilities in Z^d.
- No convexity assumption required on the compact set.

## Abstract

In this article we obtain uniform estimates on the absorption of Brownian motion by porous interfaces surrounding a compact set. An important ingredient is the construction of certain resonance sets, which are hard to avoid for Brownian motion starting in the compact set. As an application of our results, we substantially strengthen the results of arXiv:1412.3960, and obtain when $d \ge 3$, large deviation upper bounds on the probability that simple random walk in $Z^d$, or random interlacements in $Z^d$, when their vacant set is in a strongly percolative regime, disconnect the discrete blow-up of a regular compact set from the boundary of the discrete blow-up of a box containing the compact set in its interior. Importantly, we make no convexity assumption on the compact set. It is plausible, although open at the moment, that the upper bounds that we derive in this work match in principal order the lower bounds of Xinyi Li and the second author (see arXiv:1310.2177) in the case of random interlacements, and of Xinyi Li (see arXiv:1412.3959) for the simple random walk.