Wigner chaos and the fourth moment

We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer-Major theorem.Comment: Published in at http://dx.doi.org/10.1214/11-AOP657 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: Horn hypergeometric functions of two variables

HYPERDIRE is a project devoted to the creation of a set of Mathematica-based programs for the differential reduction of hypergeometric functions. The current version allows for manipulations involving the full set of Horn-type hypergeometric functions of two variables, including 30 functions.Comment: minor corrections, publushed versio

LpL_p compression, traveling salesmen, and stable walks

We show that if $H$ is a group of polynomial growth whose growth rate is at least quadratic then the $L_p$ compression of the wreath product \Z\bwr H equals $\max{\frac{1}{p},{1/2}}$. We also show that the $L_p$ compression of \Z\bwr \Z equals $\max{\frac{p}{2p-1},\frac23}$ and the $L_p$ compression of (\Z\bwr\Z)_0 (the zero section of \Z\bwr \Z, equipped with the metric induced from \Z\bwr \Z) equals $\max{\frac{p+1}{2p},\frac34}$. The fact that the Hilbert compression exponent of \Z\bwr\Z equals $\frac23$ while the Hilbert compression exponent of (\Z\bwr\Z)_0 equals $\frac34$ is used to show that there exists a Lipschitz function f:(\Z\bwr\Z)_0\to L_2 which cannot be extended to a Lipschitz function defined on all of \Z\bwr \Z

Symmetry and Integrability of Non-Singlet Sectors in Matrix Quantum Mechanics

We study the non-singlet sectors of matrix quantum mechanics (MQM) through an operator algebra which generates the spectrum. The algebra is a nonlinear extension of the W_\infty algebra where the nonlinearity comes from the angular part of the matrix which can not be neglected in the non-singlet sector. The algebra contains an infinite set of commuting generators which can be regarded as the conserved currents of MQM. We derive the spectrum and the eigenfunctions of these conserved quantities by a group theoretical method. An interesting feature of the spectrum of these charges in the non-singlet sectors is that they are identical to those of the singlet sector except for the multiplicities. We also derive the explicit form of these commuting charges in terms of the eigenvalues of the matrix and show that the interaction terms which are typical in Calogero-Sutherland system appear. Finally we discuss the bosonization and rewrite the commuting charges in terms of a free boson together with a finite number of extra degrees of freedom for the non-singlet sectors.Comment: 34 pages, 4 figure

Complex projective structures with Schottky holonomy

A Schottky group in PSL(2, C) induces an open hyperbolic handlebody and its ideal boundary is a closed orientable surface S whose genus is equal to the rank of the Schottky group. This boundary surface is equipped with a (complex) projective structure and its holonomy representation is an epimorphism from pi_1(S) to the Schottky group. We will show that an arbitrary projective structure with the same holonomy representation is obtained by (2 pi-)grafting the basic structure described above.Comment: 52 pages, 14 figure