MiS Preprint Repository

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 $1,\dots ,N$. We integrate over all initial points confined in boxes $\mathrm{\Lambda }\subset {\mathbb{R}}^d$ with respect to the Lebesgue measure, and we divide by an appropriate normalisation (partition function).

Such systems play an important role in quantum physics in the description of Boson systems at positive temperature $1/\beta$.

In this article, we describe the large-$N$ behaviour of the empirical path measure (the mean of the Dirac measures in the $N$ paths) when $\mathrm{\Lambda }\uparrow {\mathbb{R}}^d$ and $N/|\mathrm{\Lambda }|\to \rho$. The rate function is given as a variational formula involving a certain entropy functional and a Fenchel-Legendre transform. The entropy term governs the large-$N$ behaviour of discrete shape measures of integer partitions. Any integer partition determines a conjugacy class of permutations of certain cycle structure.

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 $[0,\mathrm{\infty })\to {\mathbb{R}}^d$ 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 Bose-Einstein condensation, known for systems of Bosons.