Multipleminima in glassy random-matrix models

Abstract

Certain simple models of structural glasses (Cugliandolo L F,Kurchan J, Parisi G and Ritort F 1995 Phys. Rev. Lett. 74 1012, Parisi G 1997 Statistical properties of random matricesand the replica method Preprint cond-mat/9701032) map ontorandom-matrix models. These random-matrix models have gaps in theireigenvalue distributions. It turns out that matrix models with gapsin their eigenvalue distributions have the unusual property ofmultiple solutions or minima of the free energy at the same point inphase space. I present evidence for the presence of multiplesolutions in these models both analytically and numerically. Themultiple solutions have different free energies and observablecorrelation functions, the differences arising at higher order in{1/N}. The system can get trapped into different minima dependingupon the path traversed in phase space to reach a particular point.The thermodynamic limit also depends upon the sequence by which Nis taken to infinity (e.g. odd or even N), which is reminiscent ofthe structure discussed for another model for glasses (Marinari E,Parisi G and Ritort F 1994 J. Phys. A: Math. Gen. 27 7615). Hence it would be of interest to study the landscape ofthese multiple solutions and determine whether it corresponds to asupercooled liquid or glass.