Kac-Rice fixed point analysis for single- and multi-layered complex systems

TL;DR

This paper introduces a null model for complex systems using Gaussian maps and employs Kac-Rice formalism and Random Matrix Theory to analyze fixed points, revealing a universal phase transition in high dimensions.

Contribution

It develops a novel analytical framework combining Kac-Rice and Random Matrix Theory to study fixed points in multi-layered Gaussian systems, identifying a universal phase transition.

Findings

Mean number of fixed points linked to the expectation of characteristic polynomial

Identifies a third-order phase transition in high-dimensional limit

Transition between single and exponentially many fixed points

Abstract

We present a null model for single- and multi-layered complex systems constructed using homogeneous and isotropic random Gaussian maps. By means of a Kac-Rice formalism, we show that the mean number of fixed points can be calculated as the expectation of the absolute value of the characteristic polynomial for a product of independent Gaussian (Ginibre) matrices. Furthermore, using techniques from Random Matrix Theory, we show that the high-dimensional limit of our system has a third-order phase transition between a phase with a single fixed point and a phase with exponentially many fixed points. This is result is universal in the sense that it does not depend on finer details of the correlations for the random maps.