The Circular Law for random regular digraphs

Let x be a complex random variable with mean zero and bounded variance σ2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ1, …, λn be the eigenvalues of [Formula: see text]. Define the empirical spectral distributionμn of Nn by the formula [Formula: see text] The following well-known conjecture has been open since the 1950's: Circular Law Conjecture: μn converges to the uniform distribution μ∞ over the unit disk as n tends to infinity. We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 0. Our method builds and improves upon earlier work of Girko, Bai, Götze–Tikhomirov, and Pan–Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.

Download Full-text

Circular law, extreme singular values and potential theory

Journal of Multivariate Analysis ◽

10.1016/j.jmva.2009.08.005 ◽

2010 ◽

Vol 101 (3) ◽

pp. 645-656 ◽

Cited By ~ 31

Author(s):

Guangming Pan ◽

Wang Zhou

Keyword(s):

Potential Theory ◽

Singular Values ◽

Circular Law

Download Full-text

The feasibility and stability of large complex biological networks: a random matrix approach

10.1101/223651 ◽

2017 ◽

Author(s):

Lewi Stone

Keyword(s):

Random Matrix ◽

Biological Networks ◽

Matrix Theory ◽

Biological Systems ◽

Theoretical Work ◽

Circular Law ◽

Large Complex ◽

Degree Of Stability ◽

The Stability ◽

Almost All

AbstractIn his theoretical work of the 70’s, Robert May introduced a Random Matrix Theory (RMT) approach for studying the stability of large complex biological systems. Unlike the established paradigm, May demonstrated that complexity leads to instability in generic models of biological networks. The RMT approach has since similarly been applied in many disciplines. Central to the approach is the famous “circular law” that describes the eigenvalue distribution of an important class of random matrices. However the “circular law” generally does not apply for ecological and biological systems in which density-dependence (DD) operates. Here we directly determine the far more complicated eigenvalue distributions of complex DD systems. A simple mathematical solution falls out, that allows us to explore the connection between feasible systems (i.e., having all equilibrium populations positive) and stability. In particular, for these RMT systems, almost all feasible systems are stable. The degree of stability, or resilience, is shown to depend on the minimum equilibrium population, and not directly on factors such as network topology.

Download Full-text