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

Adams, Stefan; Dorlas, Tony

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 5, pp. 837-875.

Nous étudions les principes de grandes déviations pour $N$ processus aléatoires sur réseaux ${\mathbb{Z}}^d$ pour des temps $\left[0,\beta \right]$ finis et sous la condition que la mesure correspondante soit symétrisée, c’est-à-dire que tous les points initiaux et finaux soient uniformément moyennés par rapport aux perturbations aléatoires. Plus précisément, cela signifie que, pour toute permutation $\sigma$ de $N$ éléments et pour tout vecteur $(x_1,...,x_N)$ de $N$ points initiaux, le processus aléatoire peut se terminer aux points $(x_{\sigma \left(1\right)},...,x_{\sigma \left(N\right)})$ et nous sommons donc ensuite sur toutes les permutations possibles ainsi que sur tous les points initiaux avec, pour poids respectif, une distribution initiale. Nous démontrons le principe de grandes déviations de niveau deux pour la valeur moyenne de la mesure des chemins empiriques, pour la valeur moyenne des chemins, ainsi que pour la valeur moyenne de la mesure empirique sur l’espace des chemins via la mesure symétrisée. Nous donnons également quelques applications de ces résultats en mécanique statistique quantique via la formule de Feynman-Kac représentant la trace de certains opérateurs. En particulier, nous montrons un lemme de Varadhan non commutatif pour des systèmes de spins quantiques définis via la statistique de Bose-Einstein et avec une interaction de champ moyen. Un cas spécial de notre principe de grandes déviations pour la valeur moyenne des temps locaux d’occupation de $N$ marches aléatoires montre que la fonction de taux est celle de Donsker-Varadhan dans la limite $N\to \infty$ mais pour un temps $\beta$ fini. Nous donnons une interprétation en mécanique statistique quantique de ce résultat surprenant.

We study large deviations principles for $N$ random processes on the lattice ${\mathbb{Z}}^d$ with finite time horizon $\left[0,\beta \right]$ under a symmetrised measure where all initial and terminal points are uniformly averaged over random permutations. That is, given a permutation $\sigma$ of $N$ elements and a vector $(x_1,...,x_N)$ of $N$ initial points we let the random processes terminate in the points $(x_{\sigma \left(1\right)},...,x_{\sigma \left(N\right)})$ 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 $N\to \infty$ but for finite time $\beta$. We give an interpretation in quantum statistical mechanics for this surprising result.

Classification : 60F10, 60J65, 82B10, 82B26
Mots clés : 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

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