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.

[1] S. Adams. Complete equivalence of the Gibbs ensembles for one-dimensional Markov systems. J. Statist. Phys. 105 (2001) 879-908. | MR | Zbl

[2] S. Adams. Large deviations for empirical path measures in cycles of integer partitions, preprint. Available at arXiV:math.PR/0702053, 2007.

[3] S. Adams. Interacting Brownian bridges and probabilistic interpretation of Bose-Einstein condensation, Habilitation thesis, University of Leipzig, 2008.

[4] S. Adams, J. B. Bru and W. König. Large deviations for trapped interacting Brownian particles and paths. Ann. Probab. 34 (2006) 1340-1422. | MR | Zbl

[5] S. Adams, W. König and J. B. Bru. Large systems of path-repellent Brownian motions in a trap at positive temperature. Electron. J. Probab. 11 (2006) 460-485. | MR | Zbl

[6] S. Adams and W. König. Large deviations for many Brownian bridges with symmetrised initial-terminal condition. Probab. Theory Related Fields 142 (2008) 79-124. | MR | Zbl

[7] G. Benfatto, M. Cassandro, I. Merola and E. Presutti. Limit theorems for statistics of combinatorial partitions with applications to mean field Bose gas. J. Math. Phys. 46 (2005) 033303. | MR | Zbl

[8] W. Cegla, J. T. Lewis and G. A. Raggio. The free energy of quantum spin systems and large deviations. Comm. Math. Phys. 118 (1988) 337-354. | MR | Zbl

[9] D. A. Dawson and J. Gärtner. Multilevel large deviations and interacting diffusions. Probab. Theory Related Fields 98 (1994) 423-487. | MR | Zbl

[10] T. Dorlas. A non-commutative central limit theorem. J. Math. Phys. 37 (1996) 4662-4682. | MR | Zbl

[11] T. Dorlas. Probabilistic derivation of a noncommutative version of Varadhan's Theorem. Proceedings of the Royal Irish Academy, 2007. To appear. | MR | Zbl

[12] J.-D. Deuschel and D. W. Stroock. Large Deviations. AMS Chelsea Publishing, Amer. Math. Soc., 2001.

[13] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time, I-IV. Comm. Pure Appl. Math. 28 (1975) 1-47, 279-301, 29 (1979) 389-461, 36 (1983) 183-212. | Zbl

[14] I. H. Dinwoodie and S. L. Zabell. Large deviations for exchangeable random vectors. Ann. Probab. 20 (1992) 1147-1166. | MR | Zbl

[15] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, Berlin, 1998. | MR | Zbl

[16] R. P. Feynman. Atomic theory of the λ transition in Helium. Phys. Rev. 91 (1953) 1291-1301. | Zbl

[17] H. Föllmer. Random fields and diffusion processes. Ecole d'Eté de Saint Flour XV-XVII 101-203. Lecture Notes in Math. 1362. Springer, 1988. | MR | Zbl

[18] J. Ginibre. Some applications of functional integration in statistical mechanics, and field theory. C. de Witt and R. Storaeds (Eds). Gordon and Breach, New York, 1970.

[19] F. Den Hollander. Large Deviations. Amer. Math. Soc., Providence, RI, 2000. | MR | Zbl

[20] O. Kallenberg. Foundations of Modern Probability. Springer, New York, 2001. | MR | Zbl

[21] E. H. Lieb, R. Seiringer, J. P. Solovej and Y. Yngvason. The Mathematics of the Bose Gas and Its Condensation. Birkhäuser, Basel, 2005. | MR | Zbl

[22] D. Petz, G. A. Raggio and A. Verbeure. Asymptotics of Varadhan-type and the Gibbs variational principle. Comm. Math. Phys. 121 (1989) 271-282. | MR | Zbl

[23] K. R. Parthasarathy. Probability Measures on Metric Spaces. Academic Press, New York, 1967. | MR | Zbl

[24] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1999. | MR | Zbl

[25] E. Schrödinger. Über die Umkehrung der Naturgesetze. Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1931 (1931) 144-153. | Zbl

[26] E. Seneta. Non-negative Matrices and Markov Chains. Springer, New York, 1981. | MR | Zbl

[27] A. Sütö. Percolation transition in the Bose gas: II. J. Phys. A: Math. Gen. 35 (2002) 6995-7002. | MR | Zbl

[28] B. Tóth. Phase transition in an interacting bose system. An application of the theory of Ventsel' and Freidlin, J. Statist. Phys. 61 (1990) 749-764. | MR

[29] J. Trashorras. Large deviations for a triangular array of exchangeable random variables. Ann. Inst. H. Poincaré Probab. Statist. 35 (2002) 649-680. | EuDML | Numdam | MR | Zbl