Concentration and fluctuations of sine-Gordon measure around topological multi-soliton manifold

TL;DR

This paper analyzes the sine-Gordon measure's behavior around multi-soliton configurations, showing measure concentration, fluctuation patterns, and soliton distribution properties in the low-temperature, infinite-volume limit.

Contribution

It demonstrates measure concentration and fluctuation behavior near multi-soliton manifolds, and characterizes soliton positions as ordered Beta-distributed variables.

Findings

Gibbs measure concentrates near multi-soliton manifold

Soliton positions follow Beta distribution as ordered statistics

Soliton collisions are shown to be unlikely events

Abstract

We study the sine-Gordon measure defined on each homotopy class. The energy space decomposes into infinitely many such classes indexed by the topological degree $Q \in \mathbf{Z}$. Even though the sine-Gordon action admits no minimizer in homotopy classes with $|Q| \ge 2$, we prove that the Gibbs measure on each class nevertheless concentrates and exhibits Ornstein-Uhlenbeck fluctuations near the multi-soliton manifold in the joint low-temperature and infinite-volume limit. Moreover, we show that soliton collisions are unlikely events, so that typical states consist of solitons separated at an appropriate scale. Finally, we identify the joint distribution of the multi-soliton centers as the ordered statistics of independent uniform random variables, so that each soliton's location follows a Beta distribution.