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

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.

arXiv comment20 pages, 2 figures

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• Manifolds pinned by a high-dimensional random landscape: Hessian at the global energy minimum
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• Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Parametric Determinants in the Orthogonal Case