Journal article

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

JR Ipsen, PJ Forrester

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL | IOP PUBLISHING LTD | Published : 2018

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 result is universal in the sense that it does not depend on finer details of the correlation..

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Grants

Awarded by ARC


Funding Acknowledgements

We would like to thank Yan Fyodorov for sharing a draft of [5]. The work is part of a research program supported by the Australian Research Council (ARC) through the ARC Centre of Excellence for Mathematical and Statistical frontiers (ACEMS). PJF also acknowledge partial support from ARC grant DP170102028.