Extremes of N Vicious Walkers for Large N: Application to the Directed Polymer and KPZ Interfaces

We compute the joint probability density function (jpdf) PN(M,τM) of the maximum M and its position τM for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For N→∞, this jpdf is peaked around $M = \sqrt{2N}$ and τM=1/2, while the typical fluctuations behave for large N like $M - \sqrt{2N} \propto s N^{-1/6}$ and τM−1/2∝wN−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 Painlevé 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 Airy2 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 logP(w)∼−w3/12.