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

Abstract

We study the landscape complexity of the Hamiltonian $X_N(x)+\frac{\mu }{2}\Vert x{\Vert }^2,$ where $X_N$ is a smooth Gaussian process with isotropic increments on ${\mathbb{R}}^N$. 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.

arXiv commentSubmission arXiv:2007.07668 was updated and split into two articles. This submission corresponds to the second part of arXiv:2007.07668v2. 50 pages

References in corpus (12)

• The statistics of critical points of Gaussian fields on large-dimensional spaces
• Complexity of random smooth functions on the high-dimensional sphere
• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• The spherical $2+p$ spin glass model: an exactly solvable model for glass to spin-glass transition
• Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian
• Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
• Hessian spectrum at the global minimum of high-dimensional random landscapes
• Exponential growth of random determinants beyond invariance
• Hessian eigenvalue distribution in a random Gaussian landscape
• Manifolds in high dimensional random landscape: complexity of stationary points and depinning
• Complexity of Gaussian random fields with isotropic increments: critical points with given indices
• Glass--like transition described by toppling of stability hierarchy

Cited by in corpus (1)

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