𝑞-Deformed Gaussian unitary ensemble: Spectral moments and genus-type expansions

The eigenvalue probability density function of the Gaussian unitary ensemble permits a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -extension related to the discrete <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Hermite weight and corresponding <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash upper N squared"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>N</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">1/N^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.