Percolation of thick points of the log-correlated Gaussian field in high dimensions

TL;DR

This paper demonstrates that in high dimensions, the set of thick points of the log-correlated Gaussian field contains unbounded paths, contrasting with the totally disconnected nature in two dimensions, and explores implications for the exponential metric.

Contribution

It establishes the existence of unbounded paths in the thick points set in high dimensions and analyzes the phase transition behavior of the exponential metric.

Findings

Thick points form unbounded paths in high dimensions.

Set-to-set distance exponent becomes negative for large parameters.

Critical probability for fractal percolation tends to zero as dimension increases.

Abstract

We prove that the set of thick points of the log-correlated Gaussian field contains an unbounded path in sufficiently high dimensions. This contrasts with the two-dimensional case, where Aru, Papon, and Powell (2023) showed that the set of thick points is totally disconnected. This result has an interesting implication for the exponential metric of the log-correlated Gaussian field: in sufficiently high dimensions, when the parameter $\xi$ is large, the set-to-set distance exponent (if it exists) is negative. This suggests that a new phase may emerge for the exponential metric, which does not appear in two dimensions. In addition, we establish similar results for the set of thick points of the branching random walk. As an intermediate result, we also prove that the critical probability for fractal percolation converges to 0 as $d \to \infty$.