Preprint 13/2007

Large deviations for empirical path measures in cycles of integer partitions

Stefan Adams

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Submission date: 02. Feb. 2007
Pages: 27
paper prepared for: Stochastic Processes and their Applications
MSC-Numbers: 60F10, 60J65, 82B10, 82B26
Keywords and phrases: large deviations, integer partitions, Brownian bridges, path measure, symmetrised distribution, Bose-Einstein condensation
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Abstract:
Consider a large system of N Brownian motions in formula14 on some fixed time interval formula16 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 formula22-th motion, where formula24 is a uniformly distributed random permutation of formula26. We integrate over all initial points confined in boxes formula28 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 formula30.

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 formula36 and formula38. 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 formula42, there is phase transition behaviour for the empirical path measure. For certain parameters (high density, large time horizon) and dimensions formula44 the empirical path measure is not supported on all paths formula46 which contain a bridge path of any finite multiple of the time horizon formula16. For dimensions d=1,2 , and for small densities and small time horizon formula16 in dimensions formula44, 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.

19.04.2013, 01:41