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$q$-deformed Gaussian unitary ensemble: spectral moments and genus-type expansions
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abstract
The eigenvalue probability density function of the Gaussian unitary ensemble permits a $q$-extension related to the discrete $q$-Hermite weight and corresponding $q$-orthogonal polynomials. A combinatorial counting method is used to specify a positive sum formula for the spectral moments of this model. The leading two terms of the scaled $1/N^2$ genus-type expansion of the moments are evaluated explicitly in terms of the incomplete beta function. Knowledge of these functional forms allows for the smoothed leading eigenvalue density and its first correction to be determined analytically.
Forward citations
Cited by 2 Pith papers
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Spectral analysis of $q$-deformed unitary ensembles with the Al-Salam--Carlitz weight
For the q-Al-Salam-Carlitz unitary ensemble with a<0, the authors derive explicit spectral moments and a limiting density with two soft-to-hard edge phase transitions.
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Zeros and exponential profiles of polynomials II: Examples
Elaborates a companion method that converts exponential coefficient profiles into limiting zero distributions, covering Touchard, Fubini, Eulerian, Narayana, hypergeometric, q-Laguerre, free-probability, and different...
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