On the gap between cluster dimensions of loop soups on $\mathbb{R}^3$ and the metric graph of $\mathbb{Z}^3$

TL;DR

This paper investigates the scaling limits of loop soup clusters on metric graphs in three dimensions, revealing that the metric graph clusters are strictly larger than the continuum Brownian loop soup clusters, with differing box counting dimensions.

Contribution

It demonstrates that in three dimensions, the scaling limits of metric graph clusters exceed the size of continuum Brownian loop soup clusters, highlighting a new subtlety in scaling limit behavior.

Findings

Metric graph clusters have box counting dimension 5/2.

Continuum Brownian loop soup clusters have box counting dimension less than 5/2.

Scaling limits of metric graph clusters are strictly larger than continuum clusters.

Abstract

The question of understanding the scaling limit of metric graph critical loop soup clusters and its relation to loop soups in the continuum appears to be one of the subtle cases that reveal interesting new scenarios about scaling limits, with a mixture of macroscopic and microscopic randomness. In the present paper, we show that in three dimensions, scaling limits of the metric graph clusters are strictly larger than the clusters of the limiting continuum Brownian loop soup. We actually show that the upper box counting dimension of the latter clusters is strictly smaller than $5/2$, while that of the former is $5/2$.