Power Law Decay for Systems of Randomly Coupled Differential Equations

Abstract

We consider large random matrices X with centered, independent entries but possibly different variances. We compute the normalized trace of f (X)g(X * ) for f, g functions analytic on the spectrum of X. We use these results to compute the long time asymptotics for systems of coupled differential equations with random coefficients. We show that when the coupling is critical the norm squared of the solution decays like t −1/2 .

Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach

“…where r out is the spectral radius. This result has been already obtained for the Ginibre ensemble [10] and, recently, for matrices with independent identically distributed (iid) entries [41]. Applications of the developed formalism are presented in Sec.…”

Universal transient behavior in large dynamical systems on networks

“…To grasp generic properties of large systems, it is natural to take A as a random matrix of pairwise interactions [5,19,41] and further average the squared norm (2) over the disorder S N (t) = |y(t)| 2 . This setting has been considered in the literature before for systems with fully connected topology [30,[42][43][44][45]. There, S(t) = lim N →∞ S N (t) does not depend on the fine details of the underlying ensemble but only on its spectral radius and thus enjoys a high degree of universality.…”

“…In the large N limit, and for A = −µ1 + X/ √ N , with X having independent identically distributed (i.i.d.) entries with arXiv:1906.10634v2 [nlin.AO] 29 Aug 2019 zero mean and finite moments [30,43], S(t) reads…”

Optimal lower bound on the least singular value of the shifted Ginibre ensemble

“…Girko's circular law [Girko 1984;Bai 1997;Tao and Vu 2008;Geman 1986;Bai and Yin 1986;Bordenave et al 2018]). Also the joint probability density function of all Ginibre eigenvalues, as well as their local correlation functions are explicitly known; see [Ginibre 1965] and[Mehta 1967] for the relatively simple complex case, and [Lehmann and Sommers 1991;Edelman 1997;Borodin and Sinclair 2009;Forrester and Nagao 2007] for the more involved real case, where the appearance of N 1=2 real eigenvalues causes a singularity in the local density. However, eigenvalues of X give no direct information on the singular values of X z and the extensive literature on the Ginibre spectrum is not applicable.…”

“…Supersymmetry is a compelling method originated in physics [Efetov 1997;Guhr 2011;Verbaarschot et al 1985] to produce surprising identities related to random matrices, whose potential has not yet been fully exploited in mathematics. It has been especially successful in deriving rigorous results on Gaussian random band matrices [Bao and Erdős 2017;Disertori et al 2002;2018;Shcherbina 2014;Shcherbina and Shcherbina 2016;, sometimes even beyond the Gaussian case [Shcherbina 2011;2013;Spencer 2011], as well as on overlaps of non-Hermitian Ginibre eigenvectors [Fyodorov 2018].…”