An equivalence between gauge-twisted and topologically conditioned scalar Gaussian free fields

TL;DR

This paper establishes a connection between gauge-twisted and topologically conditioned scalar Gaussian free fields on metric graphs, expressing probabilities of topological events as ratios of Laplacian determinants, and conjectures a high-dimensional scaling limit involving Brownian loop soups.

Contribution

It demonstrates that gauge-twisted GFFs can be obtained by conditioning the usual GFF on a topological event and relates these probabilities to Laplacian determinants, providing new insights into the structure of sign clusters.

Findings

Gauge-twisted GFFs are equivalent to conditioned usual GFFs up to a deterministic transformation.

Probability of topological events expressed as ratios of Laplacian determinants.

Conjecture that high-dimensional sign clusters converge to a Brownian loop soup with doubled intensity.

Abstract

We study on the metric graphs two types of scalar Gaussian free fields (GFF), the usual one and the one twisted by a $\{-1,1\}$-valued gauge field. We show that the latter can be obtained, up to an additional deterministic transformation, by conditioning the first on a topological event. This event is that all the sign clusters of the field should be trivial for the gauge field, that is to say should not contain loops with holonomy $-1$. We also express the probability of this topological event as a ratio of two determinants of Laplacians to the power $1/2$, the usual Laplacian and the gauge-twisted Laplacian. As an example, this gives on annular planar domains the probability that no sign cluster of the metric graph GFF surrounds the inner hole of the domain. Based on our result on the metric graph, and on previous works by Werner and Cai-Ding on the clusters of the metric graph GFF in high dimension, we formulate an intensity doubling conjecture. According to it, if the space dimension is high enough, the cycles in the sign clusters of the metric graph GFF converge in the scaling limit to a Brownian loop soup of intensity parameter $\alpha = 1 = 2\times \dfrac{1}{2}$, which is the double of the intensity parameter appearing in isomorphism theorems.