Abstract. -We study the joint probability distribution function (pdf) Pt(M, XM ) of the maximum M of the height and its position XM of a curved growing interface belonging to the universality class described by the Kardar-Parisi-Zhang equation in 1 + 1 dimensions, in the long time t limit. We obtain exact results for the related problem of p non-intersecting Brownian bridges where we compute the joint pdf Pp(M, τM ), for any finite p, where τM is the time at which the maximal height M is reached. This yields an approximation of Pt(M, XM ) for the interface problem, whose accuracy is systematically improved as p is increased, becoming exact for p → ∞. We show that our results, for moderate values of p ∼ 10, describe accurately our numerical data of a prototype of these systems, the polynuclear growth model in droplet geometry. We also discuss applications of our results to the ground state configuration of the directed polymer in a random medium with one fixed endpoint.
We study the extreme statistics of N nonintersecting Brownian motions (vicious walkers) over a unit time interval in one dimension. Using path-integral techniques we compute exactly the joint distribution of the maximum M and of the time τ(M) at which this maximum is reached. We focus in particular on nonintersecting Brownian bridges ("watermelons without wall") and nonintersecting Brownian excursions ("watermelons with a wall"). We discuss in detail the relationships between such vicious walkers models in watermelon configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium (DPRM) with one free end point on the other hand. We also check our results using numerical simulations of Dyson's Brownian motion and confront them with numerical simulations of the polynuclear growth model (PNG) and of a model of DPRM on a discrete lattice. Some of the results presented here were announced in a recent letter [J. Rambeau and G. Schehr, Europhys. Lett. 91, 60006 (2010)].
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