Energy Landscape of the Finite-Size Mean-field 2-Spin Spherical Model and Topology Trivialization

Abstract

Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite size $2$-spin spherical model using both numerical polynomial homotopy continuation and a reformulation via non-hermitian matrices. The continuation approach computes all of the complex stationary points of this model while the matrix approach computes the real stationary points. Using these methods, we compute the average number of stationary points while changing the topology of the PEL as well as the variance. Histograms of these stationary points are presented along with an analysis regarding the complex stationary points. This work connects topology trivialization to two different branches of mathematics: algebraic geometry and catastrophe theory, which is fertile ground for further interdisciplinary research.

arXiv comment9 pages, 17 figures

References in corpus (22)

• Random matrices: Universality of local spectral statistics of non-Hermitian matrices
• Phase transitions and configuration space topology
• Topological and Dynamical Complexity of Random Neural Networks
• Finding All the Stationary Points of a Potential Energy Landscape via Numerical Polynomial Homotopy Continuation Method
• Phase transitions detached from stationary points of the energy landscape
• Tumbling through a landscape: Evidence of instabilities in high-dimensional moduli spaces
• Topology trivialization and large deviations for the minimum in the simplest random optimization
• Energy landscape analysis of the two-dimensional nearest-neighbor ϕ^4 model
• Enumerating Gribov copies on the lattice
• High-Dimensional Random Fields and Random Matrix Theory
• Numerical Polynomial Homotopy Continuation Method and String Vacua
• Spectral density asymptotics for Gaussian and Laguerre $\beta$-ensembles in the exponentially small region
• Energy Landscape of the Finite-Size Mean-field 3-Spin Spherical Model
• Exploring the energy landscape of XY models
• Communication: Newton homotopies for sampling stationary points of potential energy landscapes
• Certifying the Potential Energy Landscape
• On a microcanonical relation between continuous and discrete spin models
• Certification and the Potential Energy Landscape
• Density of states of continuous and discrete spin models: a case study
• Critical point correlations in random gaussian fields
• Gauge-fixing on the Lattice via Orbifolding
• Experiments on the zeros of harmonic polynomials using certified counting

Cited by in corpus (6)

• Perspective: Energy Landscapes for Machine Learning
• Topology trivialization transition in random non-gradient autonomous ODE's on a sphere
• The Loss Surface of XOR Artificial Neural Networks
• Statistics of Stationary Points of Random Finite Polynomial Potentials
• Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass
• An index-resolved fixed-point homotopy and potential energy landscapes