Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N
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Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N Grégory Schehr · Satya N. Majumdar · Alain Comtet · Peter J. Forrester
Received: 19 July 2012 / Accepted: 6 October 2012 / Published online: 23 October 2012 © Springer Science+Business Media New York 2012
Abstract 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 (called “watermelons” with a wall) 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 analyze 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 O( N ) and L ≥ O( 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 = Lc (N ) ∼ 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 ∼ Lc (N ), is described in terms of the TracyWidom 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 Lc (N )] and the left [L Lc (N )] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical L ∼ Lc (N ) and atypical fluctuations in the right tail L Lc (N ) is rather unconventional, compared to the usual behavior found for the distribution of the largest eigenvalue of GUE random matrices. This
G. Schehr () · S.N. Majumdar · A. Comtet Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, Bât. 100, 91405 Orsay Cedex, France e-mail: [email protected] A. Comtet Université Pierre et Marie Curie-Paris 6, 75005, Paris, France P.J. Forrester Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
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G. Schehr et al.
paper is an extended version of (Schehr et al. in Phys. Rev. Lett. 101:150601, 2008) and (Forrester et al. in Nucl. Phys. B 844:500–526, 2011). Keywords Vicious walkers · Random matrices · Tracy-Widom · Extreme statistics
1 Introduction Non-intersecting random walkers, first introduced by de Gennes [1], followed by Fisher [2] (who called them “vicious walkers”), have been studied extensively in statistical physics as they appear in a variety of physical contexts ranging from wetting an
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