Issue |
EPL
Volume 91, Number 6, September 2010
|
|
---|---|---|
Article Number | 60006 | |
Number of page(s) | 6 | |
Section | General | |
DOI | https://doi.org/10.1209/0295-5075/91/60006 | |
Published online | 15 October 2010 |
Extremal statistics of curved growing interfaces in 1+1 dimensions
Laboratoire de Physique Théorique (UMR du CNRS 8627), Université de Paris-Sud 11 91405 Orsay Cedex, France, EU
a
rambeau@th.u-psud.fr
b
schehr@th.u-psud.fr
Received:
8
July
2010
Accepted:
15
September
2010
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.
PACS: 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion / 02.50.-r – Probability theory, stochastic processes, and statistics / 75.10.Nr – Spin-glass and other random models
© EPLA, 2010
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