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

@article{Fyodorov2007ReplicaSB, title={Replica Symmetry Breaking Condition Exposed by Random Matrix Calculation of Landscape Complexity}, author={Yan V. Fyodorov and Ian Williams}, journal={Journal of Statistical Physics}, year={2007}, volume={129}, pages={1081-1116} }

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 Gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity. Then we employ random matrix methods to calculate the density of stationary points, as well as minima, of the associated energy surface. This is used to show that for a generic smooth, concave confining…

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## References

SHOWING 1-10 OF 63 REFERENCES

### Exact solutions for the statistics of extrema of some random 1D landscapes, application to the equilibrium and the dynamics of the toy model

- Physics, Mathematics
- 2002

### 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…

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

- MathematicsPhysical review letters
- 2004

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…

### Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

- Mathematics
- 2003

Abstract: By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick…

### Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2001

Applications to Dirac fermions in random magnetic fields at criticality reveal a peculiar "quasilocalized" regime (corresponding to the glass phase for the particle), where eigenfunctions are concentrated over a finite number of distant regions, and allow us to recover the multifractal spectrum in the delocalized regime.

### Large time nonequilibrium dynamics of a particle in a random potential.

- Physics, MathematicsPhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 1996

The existence of two asymptotic time regimes: stationary dynamics, slow aging dynamics with violation of equilibrium theorems are demonstrated and an analytical solution of these equations is obtained.

### Large deviations of extreme eigenvalues of random matrices.

- MathematicsPhysical review letters
- 2006

The average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number zeta is calculated, thus generalizing the celebrated Wigner semicircle law.

### Free-energy landscapes, dynamics, and the edge of chaos in mean-field models of spin glasses

- Physics
- 2006

Metastable states in Ising spin-glass models are studied by finding iterative solutions of mean-field equations for the local magnetizations. Two different equations are studied: the TAP equations…