Large deviations for the isoperimetric constant in 2D percolation

Abstract

Isoperimetric profile describes the minimal boundary size of a set with a prescribed volume. Itai Benjamini conjectured that the isoperimetric profile of the giant component in supercritical percolation experiences an averaging effect and satisfies the law of large numbers. This conjecture was settled by Biskup-Louidor-Procaccia-Rosenthal for 2D percolation [6], and later resolved by Gold for higher-dimensional lattices [31]. However, more refined properties of the isoperimetric profile, such as fluctuations and large deviations, remain unknown. In this paper, we precisely analyze a large deviation behavior of the "extrinsic" notion of isoperimetry in 2D supercritical percolation. Notably, while the large deviation probability is of surface order throughout the entire upper tail regime, we observe a phase transition in the lower tail regime. This transition unveils large deviations of both surface and volume orders.