Off-Criticality and the Massive Brownian Loop Soup

TL;DR

This paper introduces a massive version of the Brownian loop soup that exhibits conformal covariance and decay, linking it to near-critical models and conjecturing its relation to the massive Gaussian free field.

Contribution

It defines a natural massive Brownian loop soup, relates it to near-critical scaling limits, and explores its potential connection to the massive Gaussian free field.

Findings

Massive Brownian loop soup displays conformal covariance.

It arises as the near-critical limit of a random walk loop soup with killing.

Conjectured to describe zero level lines of the massive Gaussian free field.

Abstract

We introduce a natural "massive" version of the Brownian loop soup of Lawler and Werner which displays conformal covariance and exponential decay. We show that this massive Brownian loop soup arises as the near-critical scaling limit of a random walk loop soup with killing and is related to the massive SLE(2) identified by Makarov and Smirnov as the near-critical scaling limit of a loop-erased random walk with killing. We conjecture that the massive Brownian loop soup describes the zero level lines of the massive Gaussian free field, and discuss possible relations to other models, such as Ising, in the off-critical regime.