Asymptotics of Plancherel measures for the infinite-dimensional unitary group

We study a two-dimensional family of probability measures on infinite Gelfand-Tsetlin schemes induced by a distinguished family of extreme characters of the infinite-dimensional unitary group. These measures are unitary group analogs of the well-known Plancherel measures for symmetric groups.We show that any measure from our family defines a determinantal point process on Z₊×Z, and we prove that in appropriate scaling limits, such processes converge to two different extensions of the discrete sine process as well as to the extended Airy and Pearcey processes.     

The Ground State Energy of the Massless Spin-Boson Model

We provide an explicit combinatorial expansion for the ground state energy of the massless spin-Boson model as a power series in the coupling parameter. Our method uses the technique of cluster expansion in constructive quantum field theory and takes as a starting point the functional integral representation and its reduction to an Ising model on the real line with long range interactions. We prove the analyticity of our expansion and provide an explicit lower bound on the radius of convergence. We do not need multiscale nor renormalization group analysis. A connection to the loop-erased random walk is indicated.     

Second order freeness and fluctuations of random matrices: II. Unitary random matrices

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We show how the concept of “second order freeness”, which was introduced in Part I, allows one to understand global fluctuations of Haar distributed unitary random matrices. In particular, independence between the unitary ensemble and another ensemble goes in the large N limit over into asymptotic second order freeness. Two important consequences of our general theory are: (i) we obtain a natural generalization of a theorem of Diaconis and Shahshahani to the case of several independent unitary matrices; (ii) we can show that global fluctuations in unitarily invariant multi-matrix models are not universal.     

Differential equations for septic theta functions

We demonstrate that quotients of septic theta functions appearing in Ramanujan’s Notebooks and in Klein’s work satisfy a new coupled system of nonlinear differential equations with symmetric form. This differential system bears a close resemblance to an analogous system for quintic theta functions. The proof extends an elementary technique used by Ramanujan to prove the classical differential system for normalized Eisenstein series on the full modular group. In the course of our work, we show that Klein’s quartic relation induces symmetric representations for low-weight Eisenstein series in terms of weight one modular forms of level seven.     

Harmonic analysis on the infinite-dimensional unitary group

The goal of harmonic analysis on the infinite-dimensional unitary group is to decompose a certain family of unitary representations of this group, which is a substitute for the nonexisting regular representations and depends on two complex parameters (Olshanski, 2003). In the case of noninteger parameters, the decomposing measure is described in terms of determinantal point processes (Borondin and Olshanski, 2005). The aim of the present paper is to describe the decomposition for integer parameters; in this case, the spectrum of the decompositions changes drastically. A similar result was earlier obtained for the infinite symmetric group (Kerov, Olshanski, and Vershik, 2004), but the case of the unitary group turned out to be much more complicated. In the proof we use Gustafson’s multilateral summation formula for hypergeometric series. Bibliography: 6 titles.     

Optimal comparison of the <Emphasis Type="Italic">p</Emphasis>-norms of Dirichlet polynomials

In this sequel to Part I of this series [8], we present a different approach to bounding the expected number of real zeroes of random polynomials with real independent identically distributed coefficients or more generally, exchangeable coefficients. We show that the mean number of real zeroes does not grow faster than the logarithm of the degree. The main ingredients of our approach are Descartes’ rule of signs and a new anti-concentration inequality for the symmetric group. This paper can be read independently of part I in this series.     

Stationary self-similar random fields on the integer lattice

We establish several methods for constructing stationary self-similar random fields (ssf's) on the integer lattice by “random wavelet expansion”, which stands for representation of random fields by sums of randomly scaled and translated functions, or more generally, by composites of random functionals and deterministic wavelet expansion. To construct ssf's on the integer lattice, random wavelet expansion is applied to the indicator functions of unit cubes at integer sites. We demonstrate how to construct Gaussian, symmetric stable, and Poisson ssf's by random wavelet expansion with mother wavelets having compact support or non-compact support. We also generalize ssf's to stationary random fields which are invariant under independent scaling along different coordinate axes. Finally, we investigate the construction of ssf's by combining wavelet expansion and multiple stochastic integrals.     

复完备随机内积模上的随机酉算子群的Stone表示定理

设[a,b]是有限实区间,(S,∥·∥)是完备的随机赋范模并赋予(ε,λ)-拓扑.在本文中,我们首先引进了从[a,b]到S的抽象值函数的Riemann积分并给出值域几乎处处有界的连续函数Riemann可积的一个充分条件.然后我们研究了随机谱测度和随机测度之间的关系.最后,在上述两个准备工作的基础之上,我们建立了复完备随机内积模上随机酉算子群的Stone表示定理.     

A method for determining the partial indices of symmetric matrix functions

We propose a method for determining the partial indices of matrix functions with some symmetries. It rests on the canonical factorization criteria of the author’s previous articles. We show that the method is efficient for the symmetric classes of matrix functions: unitary, hermitian, orthogonal, circular, symmetric, and others. We apply one of our results on the partial indices of Hermitian matrix functions and find effective well-posedness conditions for a generalized scalar Riemann problem (the Markushevich problem).     

Strong Laws of Large Numbers for Sublinear Expectation under Controlled 1st Moment Condition

This paper deals with strong laws of large numbers for sublinear expectation under controlled 1st moment condition. For a sequence of independent random variables, the author obtains a strong law of large numbers under conditions that there is a control random variable whose 1st moment for sublinear expectation is finite. By discussing the relation between sublinear expectation and Choquet expectation, for a sequence of i.i.d random variables, the author illustrates that only the finiteness of uniform 1st moment for sublinear expectation cannot ensure the validity of the strong law of large numbers which in turn reveals that our result does make sense.     

Moments of the Hermitian Matrix Jacobi Process

In this paper, we compute the expectation of traces of powers of the Hermitian matrix Jacobi process for a large enough but fixed size. To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy–Binet formula in order to determine the partitions having nonzero contributions after integration. It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized beta distribution. For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity.     

Partial sum of matrix entries of representations of the symmetric group and its asymptotics

Many aspects of the asymptotics of Plancherel distributed partitions have been studied in the past fifty years, in particular the limit shape, the distribution of the longest rows, connections with random matrix theory and characters of the representation matrices of the symmetric group. Regarding the latter, we extend a celebrated result of Kerov on the asymptotic of Plancherel distributed characters by studying partial trace and partial sum of a representation matrix. We decompose each of these objects into a main term and a reminder, and for each such a decomposition we prove a central limit theorem for the main term. We apply these results to prove a law of large numbers for the partial sum. Our main tool is the expansion of symmetric functions evaluated on Jucys–Murphy elements.     

Renormalization group approach to function approximation and to improving subsequent approximations

We establish a relation between bijective functions and renormalization group transformations and find their renormalization group invariants. For these functions, taking into account that they are globally one-to-one, we propose several improved approximations (compared with the power series expansion) based on this relation. We propose using the obtained approximations to improve the subsequent approximations of physical quantities obtained, in particular, by one of the main calculation techniques in theoretical physics, i.e., by perturbation theory. We illustrate the effectiveness of the renormalization group approximation with several examples: renormalization group approximations of several analytic functions and calculation of the nonharmonic oscillator ground-state energy. We also generalize our approach to the case of set maps, both continuous and discrete.     

Limit distributions of random matrices

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded almost surely. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line. Our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability. We show that the associated matricially free Gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices, including non-Hermitian Gaussian matrices and matrices of Wishart type.     

Probability representation of spin states and inequalities for unitary matrices

We review the probability representation of spin states described by probability distributions (tomograms). We use the relation of a tomogram to unitary group elements to obtain some inequalities for unitary matrices. We present the Cirel’son bound and the entropic inequalities for entangled spin states in the form of relations for functions on the unitary group.     

On the characteristic map of finite unitary groups

In his classic book on symmetric functions, Macdonald describes a remarkable result by Green relating the character theory of the finite general linear group to transition matrices between bases of symmetric functions. This connection allows one to analyze the character theory of the general linear group via symmetric group combinatorics. Using works of Ennola, Kawanaka, Lusztig and Srinivasan, this paper describes the analogous setting for the finite unitary group. In particular, we explain the connection between Deligne–Lusztig theory and Ennola's efforts to generalize Green's work, and from this we deduce various character theoretic results. Applications include calculating certain sums of character degrees, and giving a model of Deligne–Lusztig type for the finite unitary group, which parallels results of Klyachko and Inglis and Saxl for the finite general linear group.     

Curvature of the space of positive Lagrangians

A Lagrangian submanifold in an almost Calabi–Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We compute the Riemann curvature of this metric and show all sectional curvatures are non-positive. The motivation for our calculation comes from mirror symmetry. Roughly speaking, an exact isotopy class of positive Lagrangians corresponds under mirror symmetry to the space of Hermitian metrics on a holomorphic vector bundle. The latter space is an infinite-dimensional analog of the non-compact symmetric space dual to the unitary group, and thus has non-positive curvature.     

A study on the transitivity of probabilistic and fuzzy relations

Given a set of alternatives we consider a fuzzy relation and a probabilistic relation defined on such a set. We investigate the relation between the T-transitivity of the fuzzy relation and the cycle-transitivity of the associated probabilistic relation. We provide a general result, valid for any t-norm and we later provide explicit expressions for important particular cases. We also apply the results obtained to explore the transitivity satisfied by the probabilistic relation defined on a set of random variables. We focus on uniform continuous random variables.     

Unitary graphs and classification of a family of symmetric graphs with complete quotients

A finite graph Γ is called G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. We study a family of symmetric graphs, called the unitary graphs, whose vertices are flags of the Hermitian unital and whose adjacency relations are determined by certain elements of the underlying finite fields. Such graphs admit the unitary groups as groups of automorphisms, and play a significant role in the classification of a family of symmetric graphs with complete quotients such that an associated incidence structure is a doubly point-transitive linear space. We give this classification in the paper and also investigate combinatorial properties of the unitary graphs.     

Genus Expansion for Real Wishart Matrices

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial properties distinguishing the leading order terms. By considering higher cumulants, we give a central-limit-type theorem for the asymptotic distribution around the expected value.