On the chemical distance exponent for the two-sided level-set of the 2D Gaussian free field

TL;DR

This paper introduces the two-sided level-set for the 2D Gaussian free field and investigates its chemical distance, revealing that the distance dimension exceeds 1, thus opening new questions in the field.

Contribution

It defines the two-sided level-set for the 2D Gaussian free field and analyzes its chemical distance, providing new insights into its geometric properties.

Findings

Chemical distance dimension exceeds 1

Introduces the two-sided level-set concept

Stimulates further questions about level-set properties

Abstract

In this paper we introduce the two-sided level-set for the two-dimensional discrete Gaussian free field. Then we investigate the chemical distance for the two-sided level-set percolation. Our result shows that the chemical distance should have dimension strictly larger than $1$, which in turn stimulates some tempting questions about the two-sided level-set.