# Large deviations for empirical path measures in cycles of integer
partitions

Research paper by **Stefan Adams**

Indexed on: **05 Feb '07**Published on: **05 Feb '07**Published in: **Mathematics - Probability**

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#### 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.