Large deviations for empirical path measures in cycles of integer partitions

TL;DR

This paper studies the large-system behavior of symmetrized Brownian motions, revealing phase transitions in empirical path measures related to Bose-Einstein condensation phenomena.

Contribution

It provides a variational formula for the rate function of the empirical path measure and describes phase transition behavior depending on dimension and density.

Findings

Phase transition in empirical path measure depending on dimension and density

Support of empirical path measure varies with parameters

Connection to Bose-Einstein condensation phenomena

Abstract

Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ on some fixed time interval $[0,\beta]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $\sigma(i)$-th motion, where $\sigma$ is a uniformly distributed random permutation of $1,...,N$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the $N$ paths) when $ \Lambda\uparrow\mathbb{R}^d $ and $ N/|\Lambda|\to\rho $. The rate function is given as a variational formula involving a certain entropy functional and a Fenchel-Legendre transform. Depending on the dimension and the density $ \rho $, there is phase transition behaviour for the empirical path measure. For certain parameters (high density, large time horizon) and dimensions $ d\ge 3 $ the empirical path measure is not supported on all paths $ [0,\infty)\to\mathbb{R}^d $ which contain a bridge path of any finite multiple of the time horizon $ [0,\beta] $. For dimensions $ d=1,2 $, and for small densities and small time horizon $ [0,\beta] $ in dimensions $ d\ge 3$, the empirical path measure is supported on those paths. In the first regime a finite fraction of the motions lives in cycles of infinite length. We outline that this transition leads to an empirical path measure interpretation of {\it Bose-Einstein condensation}, known for systems of Bosons.