Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N

Autor: Grégory Schehr, Satya N. Majumdar, Peter J. Forrester, Alain Comtet
Přispěvatelé: Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Institut Henri Poincaré (IHP), Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Melbourne], University of Melbourne, Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)
Rok vydání: 2012
Předmět:
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]
FOS: Physical sciences
01 natural sciences
Combinatorics
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]
Saddle point
0103 physical sciences
FOS: Mathematics
Continuum (set theory)
[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech]
0101 mathematics
010306 general physics
Condensed Matter - Statistical Mechanics
Mathematical Physics
Eigenvalues and eigenvectors
Brownian motion
Physics
Statistical Mechanics (cond-mat.stat-mech)
Probability (math.PR)
010102 general mathematics
Statistical and Nonlinear Physics
Mathematical Physics (math-ph)
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Distribution (mathematics)
Path integral formulation
Large deviations theory
Random matrix
Mathematics - Probability
Zdroj: Journal of Statistical Physics
Journal of Statistical Physics, 2013, 150 (3), pp.491-530. ⟨10.1007/s10955-012-0614-7⟩
Journal of Statistical Physics, Springer Verlag, 2013, 150, pp.491-530
ISSN: 1572-9613
0022-4715
DOI: 10.1007/s10955-012-0614-7
Popis: We consider three different models of N non-intersecting Brownian motions on a line segment [0,L] with absorbing (model A), periodic (model B) and reflecting (model C) boundary conditions. In these three cases we study a properly normalized reunion probability, which, in model A, can also be interpreted as the maximal height of N non-intersecting Brownian excursions on the unit time interval. We provide a detailed derivation of the exact formula for these reunion probabilities for finite N using a Fermionic path integral technique. We then analyse the asymptotic behavior of this reunion probability for large N using two complementary techniques: (i) a saddle point analysis of the underlying Coulomb gas and (ii) orthogonal polynomial method. These two methods are complementary in the sense that they work in two different regimes, respectively for L\ll O(\sqrt{N}) and L\geq O(\sqrt{N}). A striking feature of the large N limit of the reunion probability in the three models is that it exhibits a third-order phase transition when the system size L crosses a critical value L=L_c(N)\sim \sqrt{N}. This transition is akin to the Douglas-Kazakov transition in two-dimensional continuum Yang-Mills theory. While the central part of the reunion probability, for L \sim L_c(N), is described in terms of the Tracy-Widom distributions (associated to GOE and GUE depending on the model), the emphasis of the present study is on the large deviations of these reunion probabilities, both in the right [L \gg L_c(N)] and the left [L \ll L_c(N)] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical L \sim L_c(N) and atypical fluctuations in the right tail L \gg L_c(N) is rather unconventional, compared to the usual behavior found for the distribution of the largest eigenvalue of GUE random matrices.
Comment: 41 pages, 4 figures, to appear in Journal of Statisical Physics (special issue in honor of M. Fisher, J. Percus and B. Widom)
Databáze: OpenAIRE
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