Random Matrix Theory and Networks

Multiply connected microwave (quantum) graphs form a complex network for wave propagation on extended bonds, which must also satisfy boundary conditions at the nodes. Classical trajectories on these networks are sufficiently complex to render the finite-wavelength properties of the graphs wave chaotic. Random Matrix Theory (RMT) methods have been utilized to study the statistical properties of both closed and open versions of these microwave graphs. Here we study the properties of microwave graphs connected to the outside world through a limited number of channels. In particular, we examine the time delay for wave excitations to go into the graph, and the re-emerge in any of the scattering channels. We generalize time delay into a complex quantity that describes both the delay and distortion of the wave packets,[1] and study its statistics over many realizations of the graph structure and dimensions. Experimental statistical properties, such as the distributions of the real and imaginary parts of complex time delay as either uniform attenuation or lumped loss are varied, are compared to RMT predictions. We find detailed agreement between experiment and RMT for many statistical quantities, while some show clear deviations. Analytical expressions for the complex time delay show that it is directly related to the zeros and poles of the graph scattering (S) matrix in the complex energy plane. This understanding allows us to measure and manipulate the S-matrix zero and pole locations so that the phenomenon of coherent perfect absorption (divergence of the delay time) can be achieved. Complex time delay enables a new level of control over the complicated wave scattering properties of networks, and will likely find further applications in the future. [1] Lei Chen, Steven M. Anlage, and Yan V. Fyodorov, “Generalization of Wigner Time Delay to Sub-Unitary Scattering Systems,” Phys. Rev. E 103, L050203 (2021). https://doi.org/10.1103/PhysRevE.103.L050203 Acknowledgements: This work is supported by AFOSR COE Grant No. FA9550-15-1-0171, NSF/DMR 2004386, and ONR Grants N000141912481 and N000141912480.

Biological systems need to react to stimuli over a broad spectrum of timescales. If and how this ability can emerge without external fine-tuning is a puzzle. We consider this problem in discrete Markovian systems, where we can leverage results from random matrix theory. Indeed, generic large transition matrices are governed by universal results, which predict the absence of long timescales unless fine-tuned. We consider an ensemble of transition matrices and motivate a temperature-like variable that controls the dynamic range of matrix elements, which we show plays a crucial role in the applicability of the large matrix limit: as the dynamic range increases, a phase transition occurs whereby the random matrix theory result is avoided, and long relaxation times ensue, in the entire `ordered' phase. We furthermore show that this phase transition is accompanied by a drop in the entropy rate and a peak in complexity, as measured by predictive information (Bialek, Nemenman, Tishby, Neural Computation 13(21) 2001). Extending the Markov model to a Hidden Markov model (HMM), we show that observable sequences inherit properties of the hidden sequences, allowing HMMs to be understood in terms of more accessible Markov models. We then apply our findings to fMRI data from 820 human subjects scanned at wakeful rest. We show that the data can be quantitatively understood in terms of the random model, and that brain activity lies close to the phase transition when engaged in unconstrained, task-free cognition -- supporting the brain criticality hypothesis in this context.

Although the literature on the spectra of random networks is rich, the influence of network topology in the high connectivity limit of network spectra remains poorly understood. By considering the configuration model of networks with four distinct degree distributions, we show that the spectral density of the adjacency matrices of highly connected random networks is determined by their degree distribution. In particular, we obtain analytical results for the spectrum of random networks with exponential degree distribution in the high connectivity limit. We also derive a relation between the fourth moment of the eigenvalue distribution and the variance of the degree distribution, which leads to a sufficient condition for the breakdown of the Wigner semicircle law for highly connected random networks. Based on the same relation, we propose a classification scheme of the distinct universal behaviors of the spectral density in the high connectivity limit.

This paper discusses the iterative methods to find the invariant measures of an Iterated Function System with Probability (IFSP), using as an example the IFSP that generates a Cantor set with probabilities, for which the invariant probability density is calculated, and the invariant cumulative probability function, finding that these invariant measures are the solutions of two different functional equations whose solution is the invariant measures of the Iterated Function System. Additionally, the evolution of the Density Distribution Function (DDF) is given by the Perron-Frobenius Operator (PFO) associated with the IFSP, and the invariant DDF is the fixed point of PFO. For obtaining the evolution of the cumulative distribution function (CDF), we use the IFS-type approach applied to non-negative functions introduced by Forte and Vrscay, where associated to each function of the IFSP is introduced a grey level map, an operator T is constructed, applying T iteratively to the CDF of the initial homogeneous distribution, we obtain the evolution of the CDF, and the invariant CDF is the fixed point of the T operator.