Extremal statistics of curved growing interfaces in 1+1 dimensions

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