Cumulants of the q-semicircular law, Tutte polynomials, and heaps

Matthieu Josuat-Vergès

doi:10.46298/dmtcs.3074

Abstract

The q-semicircular law as introduced by Bożejko and Speicher interpolates between the Gaussian law and the semicircular law, and its moments have a combinatorial interpretation in terms of matchings and crossings. We prove that the cumulants of this law are, up to some factor, polynomials in q with nonnegative coefficients. This is done by showing that they are obtained by an enumeration of connected matchings, weighted by the evaluation at (1,q) of a Tutte polynomial. The two particular cases q=0 and q=2 have also alternative proofs, related with the fact that these particular evaluation of the Tutte polynomials count some orientations on graphs. Our methods also give a combinatorial model for the cumulants of the free Poisson law. La loi q-semicirculaire introduite par Bożejko et Speicher interpole entre la loi gaussienne et la loi semi-circulaire, et ses moments ont une interprétation combinatoire en termes de couplages et croisements. Nous prouvons que les cumulants de cette loi sont, à un facteur près, des polynômes en q à coefficients positifs. La méthode consiste à obtenir ces cumulants par une énumération de couplages connexes, pondérés par l’évaluation en (1,q) d'un polynôme de Tutte. Les cas particuliers q=0 et q=2 ont une preuve alternative, reliè au fait que des évaluations particulières du polynôme de Tutte comptent certaines orientations de graphes. Nos méthodes donnent aussi un modèle combinatoire aux cumulants de la loi de Poisson libre.

Highlights

Let us consider the sequence mn(q) defined by the generating function mn(q)zn n≥0 = −1 [1]q z 2 − [2]qz2 1 − ... where [i]q 1−qi 1−qFor example, m0(q) m2(q) 1, m4(q)2 + q, and the odd values are 0.The generating function being a Stieltjes continued fraction, mn(q) is the nth moment of a symmetric
We show in Theorem 1 that this k2n(q) can be given a meaning as a generating function of connected matchings
Besides the moments {mn(q)}n≥0, the qsemicircular law can be characterized by its cumulants {kn(q)}n≥1 formally defined by kn (q) log mn (q) n≥1 n≥0 or by its free cumulants {cn(q)}n≥1 ([Nica and Speicher(2006)]) formally defined by 1 + C(zM (z)) = M (z) where M (z) = mn(q)zn, C(z) = cn(q)zn

Summary

Introduction

Let us consider the sequence mn(q) defined by the generating function mn(q)zn n≥0. 1 −. It would be interesting to explain why the same combinatorial objects appear both for c2n(q) and k2n(q) This suggests that there exists some quantity that interpolates between the classical and free cumulants of the q-semicircular law, building a noncommutative probability theory that encompasses the classical and free ones appear to be elusive. It means that building such an interpolation would rely on the q-semicircular law and its moments, but on its realization as a noncommutative random variable

Preliminaries

Cumulants of the free Poisson law