Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces

Abstract

We compute the joint probability density function (jpdf) P_N(M, \tau_M) of the maximum M and its position \tau_M for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For N \to \infty, this jpdf is peaked around M = \sqrt{2N} and \tau_M = 1/2, while the typical fluctuations behave for large N like M - \sqrt{2N} \propto s N^{-1/6} and \tau_M - 1/2 \propto w N^{-1/3} where s and w are correlated random variables. One obtains an explicit expression of the limiting jpdf P(s,w) in terms of the Tracy-Widom distribution for the Gaussian Orthogonal Ensemble (GOE) of Random Matrix Theory and a psi-function for the Hastings-McLeod solution to the Painlev\'e II equation. Our result yields, up to a rescaling of the random variables s and w, an expression for the jpdf of the maximum and its position for the Airy_2 process minus a parabola. This latter describes the fluctuations in many different physical systems belonging to the Kardar-Parisi-Zhang (KPZ) universality class in 1+1 dimensions. In particular, the marginal probability density function (pdf) P(w) yields, up to a model dependent length scale, the distribution of the endpoint of the directed polymer in a random medium with one free end, at zero temperature. In the large w limit one shows the asymptotic behavior \log P(w) \sim - w^3/12.