PMID: 15245072Jul 13, 2004Paper

Complexity of random energy landscapes, glass transition, and absolute value of the spectral determinant of random matrices

Physical Review Letters
Yan V Fyodorov

Abstract

Finding the mean of the total number N(tot) of stationary points for N-dimensional random energy landscapes is reduced to averaging the absolute value of the characteristic polynomial of the corresponding Hessian. For any finite N we provide the exact solution to the problem for a class of landscapes corresponding to the "toy model" of manifolds in a random environment. For N>1 our asymptotic analysis reveals a phase transition at some critical value mu(c) of a control parameter mu from a phase with a finite landscape complexity: N(tot) approximately e(N Sigma), Sigma(mu<mu(c))>0 to the phase with vanishing complexity: Sigma(mu>mu(c))=0. Finally, we discuss a method of dealing with the modulus of the spectral determinant applicable to a broad class of problems.

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Citations

Feb 18, 2006·The Journal of Chemical Physics·Tomás S Grigera
Jun 9, 2016·Proceedings of the National Academy of Sciences of the United States of America·Yan V Fyodorov, Boris A Khoruzhenko
May 16, 2007·Physical Review Letters·Alan J Bray, David S Dean
Mar 29, 2020·Physical Review Letters·Simon BeckerAlpha A Lee
Mar 15, 2015·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Dhagash MehtaDaniel A Stariolo
Mar 15, 2020·Physical Review. E·Yan V Fyodorov, Pierre Le Doussal
Oct 20, 2015·Extremes·Dan Cheng, Armin Schwartzman
Jun 4, 2008·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·David S Dean, Satya N Majumdar
Jul 16, 2014·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Gernot AkemannPierpaolo Vivo
Jul 16, 2014·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Ofer M ShirHerschel Rabitz
Mar 15, 2013·Physical Review Letters·Gilles Wainrib, Jonathan Touboul
Mar 24, 2005·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Gerardo G Naumis
Dec 13, 2006·Physical Review Letters·David S Dean, Satya N Majumdar
Feb 1, 2008·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Z BurdaO C Martin
Feb 7, 2007·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Satya N Majumdar, Olivier C Martin
Aug 11, 2005·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Gerardo G Naumis
Sep 13, 2019·Bernoulli : Official Journal of the Bernoulli Society for Mathematical Statistics and Probability·Dan Cheng, Armin Schwartzman
Apr 1, 2017·Annals of Statistics·Dan Cheng, Armin Schwartzman
Mar 20, 2021·Physical Review. E·S Belga FedeliJ R Ipsen
Aug 22, 2021·Proceedings of the National Academy of Sciences of the United States of America·Gérard Ben ArousBoris A Khoruzhenko

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