A lower bound on the critical parameter of interlacement percolation in high dimension

TL;DR

This paper establishes a lower bound on the critical parameter for percolation in high-dimensional random interlacement models, showing it grows logarithmically with the dimension, aligning with heuristic predictions.

Contribution

The paper provides the first rigorous lower bound on the critical parameter u_* for interlacement percolation in high dimensions, matching heuristic expectations.

Findings

Lower bound on u_* grows as log(d) for large d

Results align with heuristics from interlacements on regular trees

Advances understanding of high-dimensional percolation thresholds

Abstract

We investigate the percolative properties of the vacant set left by random interlacements on Z^d, when d is large. A non-negative parameter u controls the density of random interlacements on Z^d. It is known from arXiv:0704.2560, and arXiv:0808.3344, that there is a non-degenerate critical value u_*, such that the vacant set at level u percolates when u < u_*, and does not percolate when u > u_*. Little is known about u_*, however for large d, random interlacements on Z^d, ought to exhibit similarities to random interlacements on a (2d)-regular tree, for which the corresponding critical parameter can be explicitly computed, see arXiv:0907.0316. We prove in this article a lower bound on u_*, which is equivalent to log(d) as d goes to infinity. This lower bound is in agreement with the above mentioned heuristics.