Large deviations for empirical path measures in cycles of integer partitions

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.