On a Limiting Distribution of Singular Values of Random Band Matrices
Anna Lytova, Leonid Pastur
TL;DR
This paper derives an equation for the limiting distribution of singular values of large random band matrices and identifies conditions for convergence to the quarter-circle law, including special cases like lower triangular matrices.
Contribution
It provides a new equation for the Stieltjes transform of singular value distributions and explores conditions for convergence to known laws, extending understanding of large random band matrices.
Findings
Derived an equation for the Stieltjes transform of the distribution
Identified conditions for convergence to the quarter-circle law
Analyzed properties of lower triangular random matrices
Abstract
An equation is obtained for the Stieltjes transform of the normalized distribution of singular values of non-symmetric band random matrices in the limit when the band width and rank of the matrix simultaneously tend to infinity. Conditions under which this limit agrees with the quarter-circle law are found. An interesting particular case of lower triangular random matrices is also considered and certain properties of the corresponding limiting singular value distribution are given.
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Taxonomy
TopicsRandom Matrices and Applications · Stochastic processes and statistical mechanics · Spectral Theory in Mathematical Physics