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Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks
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Adams, S. (Stefan) and Dorlas, Tony (2008) Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks. Annales de l'Institut Henri Poincare (B) Probability and Statistics, Vol.44 (No.5). pp. 837-875. doi:10.1214/07-AIHP132 ISSN 0246-0203.
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Official URL: http://dx.doi.org/10.1214/07-AIHP132
Abstract
We study large deviations principles for N random processes on the lattice ℤd with finite time horizon [0, β] under a symmetrised measure where all initial and terminal points are uniformly averaged over random permutations. That is, given a permutation σ of N elements and a vector (x1, …, xN) of N initial points we let the random processes terminate in the points (xσ(1), …, xσ(N)) and then sum over all possible permutations and initial points, weighted with an initial distribution. We prove level-two large deviations principles for the mean of empirical path measures, for the mean of paths and for the mean of occupation local times under this symmetrised measure. The symmetrised measure cannot be written as a product of single random process distributions. We show a couple of important applications of these results in quantum statistical mechanics using the Feynman–Kac formulae representing traces of certain trace class operators. In particular we prove a non-commutative Varadhan lemma for quantum spin systems with Bose–Einstein statistics and mean field interactions.
A special case of our large deviations principle for the mean of occupation local times of N simple random walks has the Donsker–Varadhan rate function as the rate function for the limit N→∞ but for finite time β. We give an interpretation in quantum statistical mechanics for this surprising result.
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
| Journal or Publication Title: | Annales de l'Institut Henri Poincare (B) Probability and Statistics | ||||
| Publisher: | Institute of Mathematical Statistics | ||||
| ISSN: | 0246-0203 | ||||
| Official Date: | October 2008 | ||||
| Dates: |
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| Volume: | Vol.44 | ||||
| Number: | No.5 | ||||
| Page Range: | pp. 837-875 | ||||
| DOI: | 10.1214/07-AIHP132 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access |
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