Counting Stationary Points of Random Landscapes as a Random Matrix Problem
Abstract
Finding the mean of the total number Ntot of stationary points for N-dimensional random Gaussian landscapes can be reduced to averaging the absolute value of characteristic polynomial of the corresponding Hessian. First such a reduction is illustrated for a class of models describing energy landscapes of elastic manifolds in random environment, and a general method of attacking the problem analytically is suggested. Then the exact solution to the problem (Y.V. Fyodorov, Phys. Rev. Lett. 93, 149901(E) ( 2004) ) for a class of landscapes corresponding to the simplest, yet nontrivial ``toy model'' with N degrees of freedom is described. For N gg 1 our asymptotic analysis reveals a phase transition at some critical value mu c of a control parameter mu from a phase with finite landscape complexity: Ntot sim eN{Sigma }, {Sigma }(mu <mu c)>0 to the phase with vanishing complexity: {Sigma }(mu >mu c) = 0. This is interpreted as a transition to a glass-like state of the matter.
- Publication:
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Acta Physica Polonica B
- Pub Date:
- September 2005
- DOI:
- 10.48550/arXiv.cond-mat/0507059
- arXiv:
- arXiv:cond-mat/0507059
- Bibcode:
- 2005AcPPB..36.2699F
- Keywords:
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- Condensed Matter - Disordered Systems and Neural Networks
- E-Print:
- Presentation at "Applications of Random Matrices to Economy and Other Complex Systems", May 26-28, 2005, Krakow