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We compute the joint probability density function (jpdf) P_N(M, \tau_M) ofthe maximum M and its position \tau_M for N non-intersecting Brownianexcursions, 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 typicalfluctuations 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 theTracy-Widom distribution for the Gaussian Orthogonal Ensemble (GOE) of RandomMatrix Theory and a psi-function for the Hastings-McLeod solution to thePainlev\'e II equation. Our result yields, up to a rescaling of the randomvariables s and w, an expression for the jpdf of the maximum and its positionfor the Airy_2 process minus a parabola. This latter describes the fluctuationsin many different physical systems belonging to the Kardar-Parisi-Zhang (KPZ)universality class in 1+1 dimensions. In particular, the marginal probabilitydensity function (pdf) P(w) yields, up to a model dependent length scale, thedistribution of the endpoint of the directed polymer in a random medium withone free end, at zero temperature. In the large w limit one shows theasymptotic behavior \log P(w) \sim - w^3/12.

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