Extremal statistics of curved growing interfaces in 1+1 dimensions

TL;DR

This paper derives exact statistical properties of the maximum height and its position for curved interfaces in the KPZ universality class, linking Brownian bridges, growth models, and directed polymers.

Contribution

It provides exact joint probability distributions for the maximum and its position in curved KPZ interfaces, connecting Brownian bridges and growth models.

Findings

Exact joint pdf for maximum height and position in Brownian bridges.

Analytical results match numerical data for the polynuclear growth model.

Results applicable to the ground state of directed polymers in random potentials.

Abstract

We study the joint probability distribution function (pdf) of the maximum M of the height and its position X_M 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 P_p(M,\tau_M) where \tau_M is there the time at which the maximal height M is reached. Our analytical results, in the limit p \to \infty, become exact for the interface problem in the growth regime. We show that our results, for moderate values of p \sim 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.