Large deviations for empirical path measures in cycles of integer partitions
| dc.creator | Adams, Stefan | |
| dc.date | 2007-02-02 | |
| dc.date | 2007-02-05 | |
| dc.date.accessioned | 2026-07-07T07:44:31Z | |
| dc.date.available | 2026-07-07T07:44:31Z | |
| dc.description | Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ on some fixed time interval $[0,β]$ 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 $σ(i)$-th motion, where $σ$ 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 $ Λ\uparrow\mathbb{R}^d $ and $ N/|Λ|\toρ$. 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 $ ρ$, 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,β] $. For dimensions $ d=1,2 $, and for small densities and small time horizon $ [0,β] $ 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. | |
| dc.identifier | https://arxiv.org/abs/math/0702053 | |
| dc.identifier | http://arxiv.org/abs/math/0702053 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/123226 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.subject | 60F10; 60J65; 82B10; 82B26 | |
| dc.title | Large deviations for empirical path measures in cycles of integer partitions | |
| dc.type | text |