Extremal statistics of curved growing interfaces in 1+1 dimensions
Abstract
We study the joint probability distribution function (pdf) 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. We obtain exact results for the closely related problem of p non-intersecting Brownian bridges where we compute the joint pdf Pp(M,τM) where τM is there the time at which the maximal height M is reached. Our analytical results, in the limit p ∞, become exact for the interface problem in the growth regime. 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 potential with one fixed endpoint.