Functional renormalization group for multilinear disordered Langevin dynamics II: Revisiting the $p=2\,$ spin dynamics for Wigner and Wishart ensembles

TL;DR

This paper develops a nonperturbative renormalization group approach for analyzing large-time behavior in modified p=2 disordered Langevin dynamics, focusing on Wigner and Wishart ensembles, with implications for signal detection.

Contribution

It introduces a local-in-time renormalization group formalism for disordered Langevin dynamics with Wigner and Wishart disorder, avoiding non-locality of previous methods.

Findings

Constructed a nonperturbative RG formalism valid at large N

Applied the approach to models with Wigner and Wishart disorder

Provided insights into equilibrium states and signal detection paradigms

Abstract

In this paper, we investigate the large-time behavior for a slightly modified version of the standard p=2 soft spins dynamics model, including a quartic or higher potential. The equilibrium states of such a model correspond to an effective field theory, which has been recently considered as a novel paradigm for signal detection in data science based on the renormalization group argument. We consider a Langevin-like equation, including a disorder term that leaves in the Wigner or Wishart ensemble. Then we construct a nonperturbative renormalization group formalism valid in the large N limit, where eigenvalues distributions for the disorder can be replaced by their analytic limits, namely the Wigner and Marchenko-Pastur laws. One of the main advantages of this approach is that the interactions remain local in time, avoiding the non-locality arising from the approaches that integrate out the disorder at the partition function level.