Nonintersecting Brownian interfaces and Wishart random matrices

Céline Nadal and Satya N. Majumdar
Phys. Rev. E 79, 061117 – Published 22 June 2009

Abstract

We study a system of N nonintersecting (1+1)-dimensional fluctuating elastic interfaces (“vicious bridges”) at thermal equilibrium, each subject to periodic boundary condition in the longitudinal direction and in presence of a substrate that induces an external confining potential for each interface. We show that, for a large sytem and with an appropriate choice of the external confining potential, the joint distribution of the heights of the N nonintersecting interfaces at a fixed point on the substrate can be mapped to the joint distribution of the eigenvalues of a Wishart matrix of size N with complex entries (Dyson index β=2), thus providing a physical realization of the Wishart matrix. Exploiting this analogy to random matrix, we calculate analytically (i) the average density of states of the interfaces, (ii) the height distribution of the uppermost and lowermost interfaces (extrema), and (iii) the asymptotic (large-N) distribution of the center of mass of the interfaces. In the last case, we show that the probability density of the center of mass has an essential singularity around its peak, which is shown to be a direct consequence of a phase transition in an associated Coulomb gas problem.

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  • Received 5 March 2009
  • Publisher error corrected 28 July 2009

DOI:https://doi.org/10.1103/PhysRevE.79.061117

©2009 American Physical Society

Corrections

28 July 2009

Erratum

Authors & Affiliations

Céline Nadal and Satya N. Majumdar

  • Laboratoire de Physique Théorique et Modèles Statistiques (UMR 8626 du CNRS), Université Paris-Sud, Bâtiment 100 91405 Orsay Cedex, France

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Issue

Vol. 79, Iss. 6 — June 2009

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