A unified fluctuation formula for one-cut β\beta-ensembles of random matrices

Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut $\beta$-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut $\beta$-ensemble. As particular cases of the main result we consider the classical $\beta$-Gaussian, $\beta$-Wishart and $\beta$-Jacobi ensembles, for which we derive previously available results as well as new ones within a unified simple framework. We also discuss the connections between the problem of trace fluctuations for the Gaussian Unitary Ensemble and the enumeration of planar maps.Comment: 16 pages, 4 figures, 3 tables. Revised version where references have been added and typos correcte

The normal distribution is ⊞\boxplus-infinitely divisible

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a subfamily Askey-Wimp-Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution.Comment: AMS LaTeX, 29 pages, using tikz and 3 eps figures; new proof including infinite divisibility of certain Askey-Wilson-Kerov distibution

Relations between cumulants in noncommutative probability

We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed.Comment: 27 pages, 7 figures, AMS LaTe

Cumulants, lattice paths, and orthogonal polynomials

A formula expressing free cumulants in terms of the Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and Lagrange inversion. For the converse we discuss Gessel-Viennot theory to express Hankel determinants in terms of various cumulants.Comment: 11 pages, AMS LaTeX, uses pstricks; revised according to referee's suggestions, in particular cut down last section and corrected some wrong attribution

Simple maps, Hurwitz numbers, and Topological Recursion

We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder. We show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks. We also obtain an elegant formula for cylinders. These relations reproduce the relation between moments and free cumulants established by Collins et al. math.OA/0606431, and implement the symplectic transformation $x \leftrightarrow y$ on the spectral curve in the context of topological recursion. We conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of $x$ and $y$. We propose an argument to prove this statement conditionally to a mild version of symplectic invariance for the $1$-hermitian matrix model, which is believed to be true but has not been proved yet. Our argument relies on an (unconditional) matrix model interpretation of fully simple maps, via the formal hermitian matrix model with external field. We also deduce a universal relation between generating series of fully simple maps and of ordinary maps, which involves double monotone Hurwitz numbers. In particular, (ordinary) maps without internal faces -- which are generated by the Gaussian Unitary Ensemble -- and with boundary perimeters $(\lambda_1,\ldots,\lambda_n)$ are strictly monotone double Hurwitz numbers with ramifications $\lambda$ above $\infty$ and $(2,\ldots,2)$ above $0$. Combining with a recent result of Dubrovin et al. math-ph/1612.02333, this implies an ELSV-like formula for these Hurwitz numbers.Comment: 66 pages, 7 figure

The splitting process in free probability theory

Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Moebius calculus over the lattice of non-crossing partitions. In this work we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf algebra structure on the linear span of non-crossing partitions. The generating series of free moments is seen as a character on this Hopf algebra. It is characterized by solving a linear fixed point equation that relates it to the generating series of free cumulants. These phenomena are explained through a process similar to (though different from) the arborification process familiar in the theory of dynamical systems, and originating in Cayley's work

Monotone, free, and boolean cumulants: a shuffle algebra approach

The theory of cumulants is revisited in the "Rota way", that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each other using the pre-Lie Magnus expansion together with shuffle and half-shuffle logarithms.Comment: final versio