Finite free cumulants: Multiplicative convolutions, genus expansion and infinitesimal distributions
Octavio Arizmendi, Jorge Garza-Vargas, Daniel Perales
Abstract
Given two polynomials <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis x right-parenthesis comma q left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p(x), q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we give a combinatorial formula for the finite free cumulants of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis x right-parenthesis squared-times Subscript d Baseline q left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo> ⊠ </mml:mo> <mml:mi>d</mml:mi> </mml:msub> <mml:mi>q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p(x)\boxtimes _d q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="squared-times Subscript d Baseline"> <mml:semantics> <mml:msub> <mml:mo> ⊠ </mml:mo> <mml:mi>d</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\boxtimes _d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> converges to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="squared-times"> <mml:semantics> <mml:mo> ⊠ </mml:mo> <mml:annotation encoding="application/x-tex">\boxtimes</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building on our results we give a new short and conceptual proof of a recent result (see J. Hoskins and Z. Kabluchko [Exp. Math. (2021), pp. 1–27]; S. Steinerberger [Exp. Math. (2021), pp. 1–6]) that connects root distributions of polynomial derivatives with free fractional convolution powers.