RANDOM MATRICES WITH DISCRETE SPECTRUM AND FINITE TODA CHAINS

1991 ◽  
Vol 06 (39) ◽  
pp. 3627-3633 ◽  
Author(s):  
AL. R. KAVALOV ◽  
R. L. MKRTCHYAN ◽  
L. A. ZURABYAN
Keyword(s):  
Matrix Models ◽  
Discrete Spectrum ◽  
Random Matrices ◽  
Real Axis ◽  
Random Matrix ◽  
Toda Chain ◽  
String Equation ◽  
Finite Set ◽  
Random Matrix Models ◽  
Set Of Points

Restricting the eigenvalues of matrices in random matrix models produces different models (Hermitian, unitary, (anti)symmetric, Penner's, etc.). We consider the model in which the eigenvalues receive values from some discrete finite set of points, establish the connection of such a model with a finite Toda chain and study the details of this connection. We derive also the string equation, which in the limit, when eigenvalues become dense on a real axis, tends to the usual string equation.

scholarly journals Identifiability of parametric random matrix models

2019 ◽  
Vol 22 (03) ◽  
pp. 1950018
Author(s):  
Tomohiro Hayase
Keyword(s):  
Probability Theory ◽  
Matrix Models ◽  
Random Matrices ◽  
Random Matrix ◽  
Free Probability ◽  
Parameter Identifiability ◽  
Free Probability Theory ◽  
Noise Type ◽  
Random Matrix Models ◽  
Spectral Distributions

We investigate parameter identifiability of spectral distributions of random matrices. In particular, we treat compound Wishart type and signal-plus-noise type. We show that each model is identifiable up to some kind of rotation of parameter space. Our method is based on free probability theory.

Large N-limit for random matrices with external source with three distinct eigenvalues

2016 ◽  
Vol 05 (02) ◽  
pp. 1650005
Author(s):  
Jian Xu ◽  
Engui Fan ◽  
Yang Chen
Keyword(s):  
Matrix Models ◽  
Random Matrices ◽  
Random Matrix ◽  
Hilbert Problem ◽  
External Source ◽  
Universal Behavior ◽  
Large N ◽  
Random Matrix Models ◽  
Riemann Hilbert Problem ◽  
Sine Kernel

In this paper, we analyze the large N-limit for random matrix with external source with three distinct eigenvalues. And we confine ourselves in the Hermite case and the three distinct eigenvalues are [Formula: see text]. For the case [Formula: see text], we establish the universal behavior of local eigenvalue correlations in the limit [Formula: see text], which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. The result can be obtained by analyzing [Formula: see text] Riemann–Hilbert problem via nonlinear steepest decent method.

scholarly journals Hidden noise structure and random matrix models of stock correlations

2012 ◽  
Vol 12 (4) ◽  
pp. 567-572 ◽  
Author(s):  
Ivailo I. Dimov ◽  
Petter N. Kolm ◽  
Lee Maclin ◽  
Dan Y. C. Shiber
Keyword(s):  
Matrix Models ◽  
Random Matrix ◽  
Noise Structure ◽  
Random Matrix Models

scholarly journals Random matrix models for phase diagrams

2011 ◽  
Vol 74 (10) ◽  
pp. 102001 ◽  
Author(s):  
B Vanderheyden ◽  
A D Jackson
Keyword(s):  
Matrix Models ◽  
Phase Diagrams ◽  
Random Matrix ◽  
Random Matrix Models

scholarly journals Non-Hermitian random matrix models: Free random variable approach

1997 ◽  
Vol 55 (4) ◽  
pp. 4100-4106 ◽  
Author(s):  
Romuald A. Janik ◽  
Maciej A. Nowak ◽  
Gábor Papp ◽  
Jochen Wambach ◽  
Ismail Zahed
Keyword(s):  
Matrix Models ◽  
Random Matrix ◽  
Random Variable ◽  
Variable Approach ◽  
Random Matrix Models

scholarly journals FREE DETERMINISTIC EQUIVALENTS, RECTANGULAR RANDOM MATRIX MODELS, AND OPERATOR-VALUED FREE PROBABILITY THEORY

2012 ◽  
Vol 01 (02) ◽  
pp. 1150008 ◽  
Author(s):  
ROLAND SPEICHER ◽  
CARLOS VARGAS
Keyword(s):  
Matrix Models ◽  
Random Matrix ◽  
Collective Behavior ◽  
Free Probability ◽  
Matrix Problem ◽  
Free Probability Theory ◽  
Random Matrix Model ◽  
Deterministic Equivalent ◽  
Cauchy Transforms ◽  
Random Matrix Models

Motivated by the asymptotic collective behavior of random and deterministic matrices, we propose an approximation (called "free deterministic equivalent") to quite general random matrix models, by replacing the matrices with operators satisfying certain freeness relations. We comment on the relation between our free deterministic equivalent and deterministic equivalents considered in the engineering literature. We do not only consider the case of square matrices, but also show how rectangular matrices can be treated. Furthermore, we emphasize how operator-valued free probability techniques can be used to solve our free deterministic equivalents. As an illustration of our methods we show how the free deterministic equivalent of a random matrix model from [6] can be treated and we thus recover in a conceptual way the results from [6]. On a technical level, we generalize a result from scalar-valued free probability, by showing that randomly rotated deterministic matrices of different sizes are asymptotically free from deterministic rectangular matrices, with amalgamation over a certain algebra of projections. In Appendix A, we show how estimates for differences between Cauchy transforms can be extended from a neighborhood of infinity to a region close to the real axis. This is of some relevance if one wants to compare the original random matrix problem with its free deterministic equivalent.

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