Kesten's incipient infinite cluster for the three-dimensional, metric-graph Gaussian free field, from critical level-set percolation, and for the Villain model, from random cluster geometries and a Swendsen-Wang type algorithm

TL;DR

This paper proves the existence of the incipient infinite cluster (IIC) for the three-dimensional metric-graph Gaussian free field, extending classical percolation results and addressing open problems for the Villain model.

Contribution

It introduces a novel approach to construct the IIC for the 3D metric-graph GFF and the Villain model, building on and streamlining previous percolation and crossing probability techniques.

Findings

Established the IIC for the 3D metric-graph GFF.

Constructed the IIC for the Villain model.

Extended percolation theory to new models in three dimensions.

Abstract

We address one open problem in a recent work due to Ding and Wirth, the first version of which was available in $2019$, relating to level-set percolation on metric-graphs for the Gaussian free field in three dimensions, in which it was shown that a percolation estimate that the authors employ for studying connectivity properties of different heights of the metric graph Gaussian free field is bounded above poly-logarithmically. In three dimensions, in order to construct Kesten's incipient infinite cluster which was first seminally introduced for Bernoulli percolation in two dimensions, in $1986$, through the equality of two probabilistic quantities, we make use of a streamlined version of the $1986$ argument due to Basu and Sapozhnikov, which was first made available in $2016$, that introduces properties of crossing probabilities for demonstrating that the IIC exists for Bernoulli percolation on an infinite connected, bounded degree graph. To make use of such arguments for demonstrating the existence of the metric-graph GFF IIC in three dimensions, we also address another open problem raised in a recent work, from October $2022$, due to Dubedat and Falconet, which expresses an open problem pertaining to the construction of an IIC-type limit for the Villain model.