scholarly journals Local circular law for the product of a deterministic matrix with a random matrix

2017 ◽  
Vol 22 (0) ◽  
Author(s):  
Haokai Xi ◽  
Fan Yang ◽  
Jun Yin
Keyword(s):  
Random Matrix ◽  
Circular Law ◽  
Deterministic Matrix

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

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.

scholarly journals The strong circular law: A combinatorial view

2020 ◽  
pp. 2150031 ◽  
Author(s):  
Vishesh Jain
Keyword(s):  
Error Rate ◽  
Random Matrix ◽  
Random Variable ◽  
Singular Value ◽  
Operator Norm ◽  
Additive Combinatorics ◽  
Counting Problem ◽  
Circular Law ◽  
Matrix Formula ◽  
Least Singular Value

Let [Formula: see text] be an [Formula: see text] complex random matrix, each of whose entries is an independent copy of a centered complex random variable [Formula: see text] with finite nonzero variance [Formula: see text]. The strong circular law, proved by Tao and Vu, states that almost surely, as [Formula: see text], the empirical spectral distribution of [Formula: see text] converges to the uniform distribution on the unit disc in [Formula: see text]. A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the random matrix [Formula: see text] (where [Formula: see text] is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using machinery from additive combinatorics or any net arguments), we show that for any fixed complex matrix [Formula: see text] with operator norm at most [Formula: see text] and for all [Formula: see text], [Formula: see text] where [Formula: see text] is the least singular value of [Formula: see text] and [Formula: see text] are positive absolute constants. Our result is optimal up to the constants [Formula: see text] and the inverse exponential-type error rate improves upon the inverse polynomial error rate due to Tao and Vu. Our proof relies on the solution to the so-called counting problem in inverse Littlewood–Offord theory, developed by Ferber, Luh, Samotij, and the author, a novel complex anti-concentration inequality, and a “rounding trick” based on controlling the [Formula: see text] operator norm of heavy-tailed random matrices.

SSA, Random Matrix Theory, and Noise-Reduced Correlations

2016 ◽  
Author(s):  
Jan W Dash ◽  
Xipei Yang ◽  
Mario Bondioli ◽  
Harvey J. Stein
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory

Modeling of Integration Processes of Ukraine to EU Using Random Matrix Theory

2017 ◽  
Author(s):  
Hanna Danylchuk ◽  
Liubov Kibalnyk ◽  
Olexandr Serdiuk
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory

History—an overview

2018 ◽  
Author(s):  
Oriol Bohigas ◽  
Hans A. Weidenmüller
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Rapid Development ◽  
The Past ◽  
History Of ◽  
And Mathematics

An overview of the history of random matrix theory (RMT) is provided in this chapter. Starting from its inception, the authors sketch the history of RMT until about 1990, focusing their attention on the first four decades of RMT. Later developments are partially covered. In the past 20 years RMT has experienced rapid development and has expanded into a number of areas of physics and mathematics.

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