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

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 $d \to \infty$.