Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks

Stefan Adams; Tony Dorlas

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 5, page 837-875
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Adams, Stefan, and Dorlas, Tony. "Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks." Annales de l'I.H.P. Probabilités et statistiques 44.5 (2008): 837-875. <http://eudml.org/doc/77994>.

@article{Adams2008,
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.},
author = {Adams, Stefan, Dorlas, Tony},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {large deviations; large systems of random processes with symmetrised initial-terminal conditions; Feynman–Kac formula; Bose–Einstein statistics; non-commutative Varadhan lemma; quantum spin systems; Donsker–Varadhan function; Feynman-Kac formula; Bose-Einstein statistics; non-commutative Varadhan Lemma; quantum Spin systems; Donsker-Varadhan function},
language = {eng},
number = {5},
pages = {837-875},
publisher = {Gauthier-Villars},
title = {Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks},
url = {http://eudml.org/doc/77994},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Adams, Stefan
AU - Dorlas, Tony
TI - Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 5
SP - 837
EP - 875
AB - 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.
LA - eng
KW - large deviations; large systems of random processes with symmetrised initial-terminal conditions; Feynman–Kac formula; Bose–Einstein statistics; non-commutative Varadhan lemma; quantum spin systems; Donsker–Varadhan function; Feynman-Kac formula; Bose-Einstein statistics; non-commutative Varadhan Lemma; quantum Spin systems; Donsker-Varadhan function
UR - http://eudml.org/doc/77994
ER -

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