arXiv subjects

Vadim Gorin

Publications and source records attributed to Vadim Gorin.

At least 55 records · Page 3Linked to original sources

arXiv2016

Representations of classical Lie groups and quantized free convolution

We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the notions of the free convolution and the free projection. We further prove that if one replaces counting measures with others coming from the work of Perelomov and Popov on the higher order Casimir operators for classical groups, then the operations on the measures turn into the free convolution and projection themselves. We also explain the relation between our results and limit shape theorems for uniformly random lozenge tilings with and without axial symmetry.

math.RT↗

arXiv2016

Spherically Symmetric Random Permutations

We consider random permutations which are spherically symmetric with respect to a metric on the symmetric group $S_n$ and are consistent as $n$ varies. The extreme infinitely spherically symmetric permutation-valued processes are identified for the Hamming, Kendall-tau and Caley metrics. The proofs in all three cases are based on a unified approach through stochastic monotonicity.

math.PR↗

arXiv2016

Moments Match between the KPZ Equation and the Airy Point Process

The results of Amir-Corwin-Quastel, Calabrese-Le Doussal-Rosso, Dotsenko, and Sasamoto-Spohn imply that the one-point distribution of the solution of the KPZ equation with the narrow wedge initial condition coincides with that for a multiplicative statistics of the Airy determinantal random point process. Taking Taylor coefficients of the two sides yields moment identities. We provide a simple direct proof of those via a combinatorial match of their multivariate integral representations.

math-ph↗

arXiv2016

Stochastic Airy semigroup through tridiagonal matrices

We determine the operator limit for large powers of random tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy$_β$ process, which describes the largest eigenvalues in the $β$ ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Ramírez, Rider, and Virág. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable.

math.PR↗

arXiv2015

Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite-dimensional unitary group and their $q$-deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE-eigenvalues distribution in the limit. We also investigate similar behavior for alternating sign matrices (equivalently, six-vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in $O(n=1)$ dense loop model.

math.RT↗

arXiv2015

A quantization of the harmonic analysis on the infinite-dimensional unitary group

The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(\infty). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which replace the nonexisting two-sided regular representation (Olshanski, J. Funct. Anal., 2003, arXiv:0109193). The required decomposition is governed by certain probability measures on an infinite-dimensional space Ω, which is a dual object to U(\infty). A way to describe those measures is to convert them into determinantal point processes on the real line, it turned out that their correlation kernels are computable in explicit form --- they admit a closed expression in terms of the Gauss hypergeometric function 2-F-1 (Borodin and Olshanski, Ann. Math., 2005, arXiv:0109194). In the present work we describe a (nonevident) q-discretization of the whole construction. This leads us to a new family of determinantal point processes. We reveal its connection with an exotic finite system of q-discrete orthogonal polynomials --- the so-called pseudo big q-Jacobi polynomials. The new point processes live on a double q-lattice and we show that their correlation kernels are expressed through the basic hypergeometric function 2-ϕ-1. A crucial novel ingredient of our approach is an extended version G of the Gelfand-Tsetlin graph (the conventional graph describes the Gelfand-Tsetlin branching rule for irreducible representations of unitary groups). We find the q-boundary of G, thus extending previously known results (Gorin, Adv. Math., 2012, arXiv:1011.1769).

math.RT↗

arXiv2015

Lectures on integrable probability

These are lecture notes for a mini-course given at the St. Petersburg School in Probability and Statistical Physics in June 2012. Topics include integrable models of random growth, determinantal point processes, Schur processes and Markov dynamics on them, Macdonald processes and their application to asymptotics of directed polymers in random media.

math.PR↗

arXiv2014

Stochastic monotonicity in Young graph and Thoma theorem

We show that the order on probability measures, inherited from the dominance order on the Young diagrams, is preserved under natural maps reducing the number of boxes in a diagram by $1$. As a corollary we give a new proof of the Thoma theorem on the structure of characters of the infinite symmetric group. We present several conjectures generalizing our result. One of them (if it is true) would imply the Kerov's conjecture on the classification of all homomorphisms from the algebra of symmetric functions into $\mathbb R$ which are non-negative on Hall--Littlewood polynomials.

math.CO↗

arXiv2014

Interacting particle systems at the edge of multilevel Dyson Brownian motions

We study the joint asymptotic behavior of spacings between particles at the edge of multilevel Dyson Brownian motions, when the number of levels tends to infinity. Despite the global interactions between particles in multilevel Dyson Brownian motions, we observe a decoupling phenomenon in the limit: the global interactions become negligible and only the local interactions remain. The resulting limiting objects are interacting particle systems which can be described as Brownian versions of certain totally asymmetric exclusion processes. This is the first appearance of a particle system with local interactions in the context of general $β$ random matrix models.

math.PR↗

arXiv2014

Stochastic six-vertex model

We study the asymmetric six-vertex model in the quadrant with parameters on the stochastic line. We show that the random height function of the model converges to an explicit deterministic limit shape as the mesh size tends to 0. We further prove that the one-point fluctuations around the limit shape are asymptotically governed by the GUE Tracy-Widom distribution. We also explain an equivalent formulation of our model as an interacting particle system, which can be viewed as a discrete time generalization of ASEP started from the step initial condition. Our results confirm an earlier prediction of Gwa and Spohn (1992) that this system belongs to the KPZ universality class.

math.PR↗

arXiv2014

Multilevel Dyson Brownian motions via Jack polynomials

We introduce multilevel versions of Dyson Brownian motions of arbitrary parameter $β>0$, generalizing the interlacing reflected Brownian motions of Warren for $β=2$. Such processes unify $β$ corners processes and Dyson Brownian motions in a single object. Our approach is based on the approximation by certain multilevel discrete Markov chains of independent interest, which are defined by means of Jack symmetric polynomials. In particular, this approach allows to show that the levels in a multilevel Dyson Brownian motion are intertwined (at least for $β\ge 1$) and to give the corresponding link explicitly.

math.PR↗

arXiv2014

Record-dependent measures on the symmetric groups

A probability measure $P_n$ on the symmetric group ${\mathfrak S}_n$ is said to be record-dependent if $P_n(σ)$ depends only on the set of records of a permutation $σ\in{\mathfrak S}_n$. A sequence $P=(P_n)_{n\in{\mathbb N}}$ of consistent record-dependent measures determines a random order on $\mathbb N$. In this paper we describe the extreme elements of the convex set of such $P$. This problem turns out to be related to the study of asymptotic behavior of permutation-valued growth processes, to random extensions of partial orders, and to the measures on the Young-Fibonacci lattice.

math.PR↗

arXiv2014

From Alternating Sign Matrices to the Gaussian Unitary Ensemble

The aim of this note is to prove that fluctuations of uniformly random alternating sign matrices (equivalently, configurations of the six-vertex model with domain wall boundary conditions) near the boundary are described by the Gaussian Unitary Ensemble and the GUE-corners process.

math.PR↗

arXiv2013

Finite traces and representations of the group of infinite matrices over a finite field

The article is devoted to the representation theory of locally compact infinite-dimensional group $\mathbb{GLB}$ of almost upper-triangular infinite matrices over the finite field with $q$ elements. This group was defined by S.K., A.V., and Andrei Zelevinsky in 1982 as an adequate $n=\infty$ analogue of general linear groups $\mathbb{GL}(n,q)$. It serves as an alternative to $\mathbb{GL}(\infty,q)$, whose representation theory is poor. Our most important results are the description of semi-finite unipotent traces (characters) of the group $\mathbb{ GLB}$ via certain probability measures on the Borel subgroup $\mathbb{B}$ and the construction of the corresponding von Neumann factor representations of type $II_\infty$. As a main tool we use the subalgebra $\mathcal A(\mathbb{ GLB})$ of smooth functions in the group algebra $L_1(\mathbb{GLB})$. This subalgebra is an inductive limit of the finite--dimensional group algebras ${\mathbb C}(\mathbb{GL}(n,q))$ under parabolic embeddings. As in other examples of the asymptotic representation theory we discover remarkable properties of the infinite case which does not take place for finite groups, like multiplicativity of indecomposable characters or connections to probabilistic concepts. The infinite dimensional Iwahori-Hecke algebra $\mathcal H_q(\infty)$ plays a special role in our considerations and allows to understand the deep analogy of the developed theory with the representation theory of infinite symmetric group $S(\infty)$ which had been intensively studied in numerous previous papers.

math.RT↗

arXiv2013

Observables of Macdonald processes

We present a framework for computing averages of various observables of Macdonald processes. This leads to new contour--integral formulas for averages of a large class of multilevel observables, as well as Fredholm determinants for averages of two different single level observables.

math.PR↗

arXiv2012

A pattern theorem for random sorting networks

A sorting network is a shortest path from 12..n to n..21 in the Cayley graph of the symmetric group S(n) generated by nearest-neighbor swaps. A pattern is a sequence of swaps that forms an initial segment of some sorting network. We prove that in a uniformly random n-element sorting network, any fixed pattern occurs in at least cn^2 disjoint space-time locations, with probability tending to 1 exponentially fast as n tends to infinity. Here c is a positive constant which depends on the choice of pattern. As a consequence, the probability that the uniformly random sorting network is geometrically realizable tends to 0.

math.PR↗

arXiv2012

The q-Gelfand-Tsetlin graph, Gibbs measures and q-Toeplitz matrices

The problem of the description of finite factor representations of the infinite-dimensional unitary group, investigated by Voiculescu in 1976, is equivalent to the description of all totally positive Toeplitz matrices. Vershik-Kerov showed that this problem is also equivalent to the description of the simplex of central (i.e. possessing a certain Gibbs property) measures on paths in the Gelfand-Tsetlin graph. We study a quantum version of the latter problem. We introduce a notion of a q-centrality and describe the simplex of all q-central measures on paths in the Gelfand-Tsetlin graph. Conjecturally, q-central measurets are related to representations of the quantized universal enveloping algebra U_ε(gl_\infty). We also define a class of q-Toeplitz matrices and show that every extreme q-central measure corresponds to a q-Toeplitz matrix with non-negative minors. Finally, our results can be viewed as a classification theorem for certain Gibbs measures on rhombus tilings of the halfplane. We use a class of q-interpolation polynomials related to Schur functions. One of the key ingredients of our proofs is the binomial formula for these polynomials proved by Okounkov.

math.RT↗

arXiv2012

Markov processes of infinitely many nonintersecting random walks

Consider an N-dimensional Markov chain obtained from N one-dimensional random walks by Doob h-transform with the q-Vandermonde determinant. We prove that as N becomes large, these Markov chains converge to an infinite-dimensional Feller Markov process. The dynamical correlation functions of the limit process are determinantal with an explicit correlation kernel. The key idea is to identify random point processes on Z with q-Gibbs measures on Gelfand-Tsetlin schemes and construct Markov processes on the latter space. Independently, we analyze the large time behavior of PushASEP with finitely many particles and particle-dependent jump rates (it arises as a marginal of our dynamics on Gelfand-Tsetlin schemes). The asymptotics is given by a product of a marginal of the GUE-minor process and geometric distributions.

math.PR↗