· 2013· Preprint
Characteristic polynomials of random matrices
- 175citations
- 2013year
Short summary
Moments of random matrix characteristic polynomial determinants are universal functions, transitioning to Airy or Bessel problems at spectral edges and exhibiting new phenomena with external source matrices.
AI-generated from the title and abstract; the full text is not read.
Key points
- Moments of random variable determinants det(λ−X) are universal functions under appropriate scaling.
- Spectral edges exhibit a crossover to Airy or Bessel problems.
- External source matrices introduce new phenomena and critical points.
- Simple formulas exist for computing moments with arbitrary source matrices.
AI-generated from the title and abstract; the full text is not read.
Abstract
We have discussed earlier the correlation functions of the random variables det(λ−X) in which X is a random matrix. In particular the moments of the distribution of these random variables are universal functions, when measured in the appropriate units of the level spacing. When the λ’s, instead of belonging to the bulk of the spectrum, approach the edge, a cross-over takes place to an Airy or to a Bessel problem, and we consider here these modified classes of universality. Furthermore, when an external matrix source is added to the probability distribution of X, various new phenomenons may occur and one can tune the spectrum of this source matrix to new critical points. Again there are remarkably simple formulae for arbitrary source matrices, which allow us to compute the moments of the characteristic polynomials in these cases as well.
The authors' abstract, as published at the source., 2013 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics