Heterochromatic two-arm probabilities for metric graph Gaussian free fields

TL;DR

This paper analyzes the probability that two points in a Gaussian free field on a metric graph are in opposite-sign clusters with large diameters, revealing asymptotic decay rates and cluster volume growth behavior.

Contribution

It establishes the asymptotic behavior of heterochromatic two-arm probabilities for Gaussian free fields on metric graphs in dimensions other than the critical dimension 6.

Findings

Two-arm probability decays as N^{-[ (d/2)+1 ]∧4} for d≠6.

Cluster volume within a box scales as M^{(d/2)+1} or M^4 depending on dimension.

Conditioned clusters exhibit typical volume growth, similar to unconditioned clusters.

Abstract

For the Gaussian free field on the metric graph of $\mathbb{Z}^d$ ($d\ge 3$), we consider the heterochromatic two-arm probability, i.e., the probability that two points $v$ and $v'$ are contained in distinct clusters of opposite signs with diameters at least $N$. For all $d\ge 3$ except the critical dimension $d_c=6$, we prove that this probability is asymptotically proportional to $N^{-[(\frac{d}{2}+1)\land 4]}$. Furthermore, we prove that conditioned on this two-arm event, the volume growth of each involved cluster is comparable to that of a typical (unconditioned) cluster; precisely, each cluster has a volume of order $M^{(\frac{d}{2}+1)\land 4}$ within a box of size $M$.