Chemical Distance for the Level Sets of the Gaussian Free Field

TL;DR

This paper investigates the connectivity properties of level sets of the Gaussian free field in high dimensions, establishing bounds on the chemical distance compared to Euclidean distance using renormalization and bootstrap techniques.

Contribution

It provides the first rigorous bounds on the chemical distance in the percolating level sets of the Gaussian free field for $d \u2265 3$, advancing understanding of their geometric structure.

Findings

Bounds on the probability that chemical distance exceeds Euclidean distance significantly

Application of renormalization and bootstrap methods to Gaussian free field level sets

Insights into the geometric complexity of high-dimensional Gaussian free field percolation

Abstract

We consider the Gaussian free field $\varphi$ on $\mathbb{Z}^d$ for $d \geq 3$ and study the level sets $\{\varphi \geq h \}$ in the percolating regime. We prove upper and lower bounds for the probability that the chemical distance is much larger than Euclidean distance. Our proof uses a renormalization scheme combined with a bootstrap argument.