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

@article{Fyodorov2004ComplexityOR, 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}, year={2004}, volume={92 24}, pages={ 240601 } }

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…

## 153 Citations

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

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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…

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### Critical behavior of the number of minima of a random landscape at the glass transition point and the Tracy-Widom distribution.

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### Random Matrices and Complexity of Spin Glasses

- Computer Science
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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.

### Replica Symmetry Breaking Condition Exposed by Random Matrix Calculation of Landscape Complexity

- Physics
- 2007

Abstract
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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…

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### Hessian spectrum at the global minimum of high-dimensional random landscapes

- MathematicsJournal of Physics A: Mathematical and Theoretical
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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…

### Density of stationary points in a high dimensional random energy landscape and the onset of glassy behavior

- Physics
- 2007

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…

### Complex Energy Landscapes in Spiked-Tensor and Simple Glassy Models: Ruggedness, Arrangements of Local Minima, and Phase Transitions

- Computer SciencePhysical Review X
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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.

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