Phase transition and level-set percolation for the Gaussian free field

TL;DR

This paper investigates the phase transition in level-set percolation of the Gaussian free field on Z^d, establishing finiteness of critical levels and decay properties of connectivity functions, with implications for high-dimensional cases.

Contribution

It proves the finiteness of the critical level h_* and introduces a second critical parameter h_**, advancing understanding of percolation thresholds in Gaussian free fields.

Findings

h_* is finite for all d ≥ 3

h_** is finite and connectivity decays stretched exponentially for h > h_**

h_* > 0 in high dimensions

Abstract

We consider level-set percolation for the Gaussian free field on Z^d, with d bigger or equal to 3, and prove that there is a non-trivial critical level h_* such that for h > h_*, the excursion set above level h does not percolate, and for h < h_*, the excursion set does percolate. It is known from the work of Bricmont-Lebowitz-Maes that h_* is non-negative for all d bigger or equal to 3, and finite, when d=3. We prove here that h_* is finite for all d bigger or equal to 3. In fact, we introduce a second critical parameter h_**, which is bigger or equal to h_*. We show that h_** is finite for all d bigger or equal to 3, and that the connectivity function of the excursion set above level h has stretched exponential decay for all h > h_**. Finally we prove that h_* > 0 in high dimension. It remains open whether h_* and h_** actually coincide, and whether h_* > 0 for all d bigger or equal to 3.