How to count in hierarchical landscapes: a 'full' solution to mean-field complexity

TL;DR

This paper provides a comprehensive solution for counting stationary points in mean-field complex landscapes, integrating Parisi's ground state solution, and applies it to models with both multi-step and full replica symmetry breaking.

Contribution

It introduces a general method for counting stationary points in mean-field landscapes that includes Parisi's solution and applies it to complex models.

Findings

Derived the general solution for stationary points counting.

Successfully applied the solution to models with different symmetry breaking.

Validated the approach with models exhibiting full replica symmetry breaking.

Abstract

We derive the general solution for counting the stationary points of mean-field complex landscapes. It incorporates Parisi's solution for the ground state, as it should. Using this solution, we count the stationary points of two models: one with multi-step replica symmetry breaking, and one with full replica symmetry breaking.