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

  title={Complexity of random energy landscapes, glass transition, and absolute value of the spectral determinant of random matrices.},
  author={Yan V. Fyodorov},
  journal={Physical review letters},
  volume={92 24},
  • Y. Fyodorov
  • Published 16 January 2004
  • Mathematics
  • Physical review letters
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… 

Manifolds in a high-dimensional random landscape: Complexity of stationary points and depinning.

We obtain explicit expressions for the annealed complexities associated, respectively, with the total number of (i) stationary points and (ii) local minima of the energy landscape for an elastic

The distribution of vacua in random landscape potentials

Landscape cosmology posits the existence of a convoluted, multidimensional, scalar potential—the “landscape”—with vast numbers of metastable minima. Random matrices and random functions in many

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We exploit a relation between the mean number N(m) of minima of random Gaussian surfaces and extreme eigenvalues of random matrices to understand the critical behavior of N(m) in the simplest

Random Matrices and Complexity of Spin Glasses

This study enables detailed information about the bottom of the energy landscape, including the absolute minimum, and the other local minima, and describes an interesting layered structure of the low critical values for the Hamiltonians of these models.

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Abstract We start with a rather detailed, general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random N(≫1)-dimensional

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Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in [Formula: see text] obtained by

High-Dimensional Random Fields and Random Matrix Theory

The important role of the GOE "edge scaling" spectral region and the Tracy-Widom distribution of the maximal eigenvalue of GOE matrices for providing an accurate quantitative description of the universal features of the topology trivialization scenario are revealed.

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Using the replica method we calculate the mean spectral density of the Hessian matrix at the global minimum of a random dimensional isotropic, translationally invariant Gaussian random landscape

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The density of stationary points and minima of a N ≫ 1 dimensional Gaussian energy landscape has been calculated. It is used to show that the point of zero-temperature replica symmetry breaking in

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This work develops a framework based on the Kac-Rice method that allows to compute the complexity of the landscape, i.e. the logarithm of the typical number of stationary points and their Hessian, and discusses its advantages with respect to previous frameworks.



The Geometry of Random Fields

Preface to the Classics edition Preface Corrections and comments 1. Random fields and excursion sets 2. Homogeneous fields and their spectra 3. Sample function regularity 4. Geometry and excursion

Random Matrices

The elementary properties of random matrices are reviewed and widely used mathematical methods for both hermitian and nonhermitian random matrix ensembles are discussed.

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