Painlevé VI and the Unitary Jacobi Ensembles

The six Painlevé transcendants which originally appeared in the studies of ordinary differential equations have been found numerous applications in physical problems. The well‐known examples among which include symmetry reduction of the Ernst equation which arises from stationary axial symmetric Einstein manifold and the spin‐spin correlation functions of the two‐dimensional Ising model in the work of McCoy, Tracy, and Wu. The problem we study in this paper originates from random matrix theory, namely, the smallest eigenvalues distribution of the finite n Jacobi unitary ensembles which was first investigated by Tracy and Widom. This is equivalent to the computation of the probability that the spectrum is free of eigenvalues on the interval . Such ensembles also appears in multivariate statistics known as the double‐Wishart distribution. We consider a more general model where the Jacobi weight is perturbed by a discontinuous factor and study the associated finite Hankel determinant. It is shown that the logarithmic derivative of Hankel determinant satisfies a particular σ‐form of Painlevé VI, which holds for the gap probability as well. We also compute exactly the leading term of the gap probability as .     

A Universality Property of Gaussian Analytic Functions

We consider random analytic functions defined on the unit disk of the complex plane f(z) = ?_n=0^￥ a_n X_n zⁿf(z) = \sum_{n=0}^{\infty} a_{n} X_{n} z^{n}, where the X _n ’s are i.i.d., complex-valued random variables with mean zero and unit variance. The coefficients a _n are chosen so that f(z) is defined on a domain of ℂ carrying a planar or hyperbolic geometry, and Ef(z)[`(f(w))]\mathbf{E}f(z)\overline{f(w)} is covariant with respect to the isometry group. The corresponding Gaussian analytic functions have been much studied, and their zero sets have been considered in detail in a monograph by Hough, Krishnapur, Peres, and Virág. We show that for non-Gaussian coefficients, the zero set converges in distribution to that of the Gaussian analytic functions as one transports isometrically to the boundary of the domain. The proof is elementary and general.     

Matrix Measures, Random Moments, and Gaussian Ensembles

We consider the moment space M_n\mathcal{M}_{n} corresponding to p×p real or complex matrix measures defined on the interval [0,1]. The asymptotic properties of the first k components of a uniformly distributed vector (S_1,n, ... , S_n,n)^* ~ U (M_n)(S_{1,n}, \dots , S_{n,n})^{*} \sim\mathcal{U} (\mathcal{M}_{n}) are studied as n→∞. In particular, it is shown that an appropriately centered and standardized version of the vector (S _1,n ,…,S _k,n )^∗ converges weakly to a vector of k independent p×p Gaussian ensembles. For the proof of our results, we use some new relations between ordinary moments and canonical moments of matrix measures which are of their own interest. In particular, it is shown that the first k canonical moments corresponding to the uniform distribution on the real or complex moment space M_n\mathcal{M}_{n} are independent multivariate Beta-distributed random variables and that each of these random variables converges in distribution (as the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively.     

A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices

We present simple proofs of several basic facts of the global regime (the existence and the form of the nonrandom limiting Normalized Counting Measure of eigenvalues, and the central limit theorem for the trace of the resolvent) for ensembles of random matrices whose probability law involves the Gaussian distribution. The main difference with previous proofs is the systematic use of the Poincare-Nash inequality, allowing us to obtain the O(n ⁻²) bounds for the variance of the normalized trace of the resolvent that are valid up to the real axis in the spectral parameter. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 790–817, June, 2005.     

On the Unitary Binary Group

One consider the variety of the unitary binary group in n variables and it is shown that this algebraic variety is rational and it has n² parameters. Then it is given a parametrical rational representation of this variety.AMS Subject Classification (1991) 20G20     

Local properties of some Gaussian processes

The purpose of this paper is twofold. First we obtain exact formulae for the Hausdorff dimensions of the level sets and graph, and of the image of a fixed time set, for a Gaussian process with stationary increments and monotone incremental variance. Inequalities for these dimensions have been obtained by Kahane in the case where the process is the sum of a trigonometric series with random coefficients. Secondly we obtain some precise results for the brownian motion process. We consider when an image set has positive Lebesgue measure and when the zero set has positive capacity with respect to a given kernel. We show that certain conditions, which Kahane has shown to be sufficient to ensure these properties, are also necessary.     

A Family of Complex Irreducible Characters Possessed by Unitary and Special Unitary Groups Defined over Local Rings

Abstract

Let 𝒪 be a discrete valuation ring whose residue field 𝒪/𝔭 is finite and has odd characteristic. Let l be a positive integer. Set R = 𝒪/𝔭 ^l and let R = R[θ] be the ring obtained by adjoining to R a square root of a non-square unit. Consider the involution σ of R that fixes R elementwise and sends θ to ? θ. Let V be a free R-module of rank n > 0 endowed with a non-degenerate hermitian form ( , ) relative to σ. Let U _n (R) be the subgroup of GL(V) that preserves ( , ). Let SU _n (R) be the subgroup of all g ∈ U _n (R) whose determinant is equal to one. Let Ψ be the Weil character of U _n (R).

All irreducible constituents of Ψ are determined. An explicit character formula is given for each of them. In particular, all character degrees are computed. For n > 2 the corresponding results are also obtained for the restriction of Ψ to SU _n (R).     

Rate of convergence to the semi-circle law for the Deformed Gaussian Unitary Ensemble

It is shown that the Kolmogorov distance between the expected spectral distribution function of an n × n matrix from the Deformed Gaussian Ensemble and the distribution function of the semi-circle law is of order O(n ^−2/3+v ). Research supported by the DFG-Forschergruppe FOR 399/1. Partially supported by INTAS grant N 03-51-5018, by RFBF grant N 02-01-00233, by RFBR-DFG grant N 04-01-04000, by RF grant of the leading scientific schools NSh-4222.2006.1.     

On a Universality of the Heat Equation

We prove that there are solutions u(t,x) of the heat equation u_t = u_xx such that every continuous function f : [a, b] → ? can be uniformly approximated by a subsequence of u (n, ·), n? ?.     

On Unitary Equivalence of Analytic Toeplitz Operators

ＯｎＵｎｉｔａｒｙＥｑｕｉｖａｌｅｎｃｅｏｆＡｎａｌｙｔｉｃＴｏｅｐｌｉｔｚＯｐｅｒａｔｏｒｓＤｉｎｇＸｕａｎｈａｏ（丁宣浩）（ＤａｘｉａｎＴｅａｃｈｅｒ＇ｓＣｏｌｅｇｅ，Ｄａｘｉａｎ，Ｓｉｃｈｕａｎ，６３５０００）ＡｂｓｔｒａｃｔＩｎｔｈｉｓｐａｐ...     

On Joint Universality of the Riemann Zeta-Function

Mathematical Notes - A theorem is obtained on the approximation of a collection of analytic functions in short intervals by a collection of shifts of the Riemann zeta-function...     

On the Unitary Similarity of Matrix Families

The classical Specht criterion for the unitary similarity between two complex n × n matrices is extended to the unitary similarity between two normal matrix sets of cardinality m. This property means that the algebra generated by a set is closed with respect to the conjugate transpose operation. Similar to the well-known result of Pearcy that supplements Specht's theorem, the proposed extension can be made a finite criterion. The complexity of this criterion depends on n as well as the length l of the algebras under analysis. For a pair of matrices, this complexity can be significantly lower than that of the Specht--Pearcy criterion.     

Local asymptotic normality of families of Gaussian distributions

In the paper, in terms of spectral language, one finds conditions guaranteeing the LAN property.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 136, pp. 13–26, 1984.     

Local times for functions and Gaussian processes

Some necessary and sufficient conditions that a Gaussian process with continuous time have a local time are discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 104–112, 1979.The author is grateful to the participants of the seminar on local times and to Yu. A. Davydov in particular.     

On Harmonic Polynomials Invariant under Unitary Transformations

Mathematical Notes - Unitary transformations and canonical representatives of a family of real-valued harmonic fourth-degree polynomials in three complex variables are studied. The subject relates...     

On a Product Formula for Unitary Groups

For any nonnegative self-adjoint operators A and B in a separableHilbert space, the Trotter-type formula is shown to converge strongly in the norm closureof dom (A^1/2) dom (B^1/2 for some subsequence and for almostevery t R. This result extends to the degenerate case, andto Kato-functions following the method of T. Kato (see ‘Trotter'sproduct formula for an arbitrary pair of self-adjoint contractionsemigroup’, Topics in functional analysis (ed. M. Kac,Academic Press, New York, 1978) 185–195). Moreover, therestrictions on the convergence can be removed by consideringfunctions other than the exponential. 2000 Mathematics SubjectClassification 47D03 (primary), 47B25 (secondary).