# Large deviations for many Brownian bridges with symmetrised
initial-terminal condition

Research paper by **Stefan Adams, Wolfgang König**

Indexed on: **30 Mar '06**Published on: **30 Mar '06**Published in: **Mathematics - Probability**

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#### Abstract

Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ with some
non-degenerate initial measure 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 $1,...,N$. Such systems play an important role in quantum
physics in the description of Boson systems at positive temperature $1/\beta$.
In this paper, we describe the large-N behaviour of the empirical path
measure (the mean of the Dirac measures in the $N$ paths) and of the mean of
the normalised occupation measures of the $N$ motions in terms of large
deviations principles. The rate functions are given as variational formulas
involving certain entropies and Fenchel-Legendre transforms. Consequences are
drawn for asymptotic independence statements and laws of large numbers.
In the special case related to quantum physics, our rate function for the
occupation measures turns out to be equal to the well-known Donsker-Varadhan
rate function for the occupation measures of one motion in the limit of
diverging time. This enables us to prove a simple formula for the large-N
asymptotic of the symmetrised trace of ${\rm e}^{-\beta \mathcal{H}_N}$, where
$\mathcal{H}_N$ is an $N$-particle Hamilton operator in a trap.