On the number of equilibria with a given number of unstable directions
Xavier Garcia
TL;DR
This paper analyzes the asymptotic behavior of the average number of equilibria with a specific number of unstable directions in high-dimensional Gaussian ODEs on a sphere, exploring the influence of the Lagrange multiplier.
Contribution
It provides the first large-dimensional asymptotic formulas for the count of equilibria with fixed unstable directions in Gaussian ODEs, addressing a problem posed by Fyodorov.
Findings
Derived asymptotic formulas for equilibrium counts
Identified the impact of the Lagrange multiplier on equilibria
Enhanced understanding of high-dimensional Gaussian dynamical systems
Abstract
We compute the large-dimensional asymptotics for the average number of equilibria with a fixed number of unstable directions for random Gaussian ODEs on a sphere. We also discuss the effects that the value of the Lagrange multiplier of the vector field has on the number of such equilibria. This was a problem posed by Fyodorov.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Theoretical and Computational Physics · Mathematical Dynamics and Fractals