• Corpus ID: 250089150

Complexity of Gaussian random fields with isotropic increments: critical points with given indices

  title={Complexity of Gaussian random fields with isotropic increments: critical points with given indices},
  author={Antonio Auffinger and Qiang Zeng},
We study the landscape complexity of the Hamiltonian X N ( x )+ µ 2 k x k 2 , where X N is a smooth Gaussian process with isotropic increments on R N . This model describes a single particle on a random potential in statistical physics. We derive asymptotic formulas for the mean number of critical points of index k with critical values in an open set as the dimension N goes to infinity. In a companion paper, we provide the same analysis without the index constraint. 



High-dimensional Gaussian fields with isotropic increments seen through spin glasses

We study the free energy of a particle in (arbitrary) high-dimensional Gaussian random potentials with isotropic increments. We prove a computable saddle point variational representation in terms of

Expected Number and Height Distribution of Critical Points of Smooth Isotropic Gaussian Random Fields.

  • Dan ChengA. Schwartzman
  • Mathematics
    Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
  • 2018
Formulae are obtained for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant matrices.

Statistics of critical points of Gaussian fields on large-dimensional spaces.

The results give a complete picture of the organization of critical points and are of relevance to glassy and disordered systems and landscape scenarios coming from the anthropic approach to string theory.

Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite-dimensional Euclidean spaces

We construct an N-dimensional Gaussian landscape with multiscale, translation invariant, logarithmic correlations and investigate the statistical mechanics of a single particle in this environment.

Random Matrices and Complexity of Spin Glasses

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

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

Exponential growth of random determinants beyond invariance

We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance.

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

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

Replica field theory for random manifolds

We consider the field theory formulation for manifolds in random media using the replica method. We use a variational (Hartree-Fock like) method which shows that replica symmetry is spontaneously