High-Dimensional Random Fields and Random Matrix Theory

Abstract

Our goal is to discuss in detail the calculation of the mean number of stationary points and minima for random isotropic Gaussian fields on a sphere as well as for stationary Gaussian random fields in a background parabolic confinement. After developing the general formalism based on the high-dimensional Kac-Rice formulae we combine it with the Random Matrix Theory (RMT) techniques to perform analysis of the random energy landscape of $p-$spin spherical spinglasses and a related glass model, both displaying a zero-temperature one-step replica symmetry breaking glass transition as a function of control parameters (e.g. a magnetic field or curvature of the confining potential). A particular emphasis of the presented analysis is on understanding in detail the picture of "topology trivialization" (in the sense of drastic reduction of the number of stationary points) of the landscape which takes place in the vicinity of the zero-temperature glass transition in both models. We will reveal 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.

40 pages; 2 figures; In this version the original lecture notes are converted to an article format, new Eqs. (82)-(85) and Appendix about anisotropic fields added, noticed misprints corrected, references updated. references updated

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Cited by in corpus (22)

• Nonlinear Analogue of the May-Wigner Instability Transition
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• On the number of connected components of random algebraic hypersurfaces
• Exponential number of equilibria and depinning threshold for a directed polymer in a random potential
• Explorations on high dimensional landscapes
• Rethinking generalization requires revisiting old ideas: statistical mechanics approaches and complex learning behavior
• Dynamical Instantons and Activated Processes in Mean-Field Glass Models
• Triviality of the geometry of mixed $p$-spin spherical Hamiltonians with external field
• Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates
• Nonlinearity-generated Resilience in Large Complex Systems
• Consequences of Dale's law on the stability-complexity relationship of random neural networks
• Manifolds in high dimensional random landscape: complexity of stationary points and depinning
• The number of saddles of the spherical $p$-spin model
• Energy Landscape of the Finite-Size Mean-field 2-Spin Spherical Model and Topology Trivialization
• When you can't count, sample! Computable entropies beyond equilibrium from basin volumes
• Sharp complexity asymptotics and topological trivialization for the (p, k) spiked tensor model
• Statistics of Stationary Points of Random Finite Polynomial Potentials
• Glass--like transition described by toppling of stability hierarchy
• Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity
• Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses
• On critical points of Gaussian random fields under diffeomorphic transformations