Two main directions of my research are:

- Asymptotic analysis of
*integrable*stochastic systems through exact formulas and algebraic techniques. - Development of robust methods for proving the
*universality*of the known asymptotic behaviors.

The list of my publications is available at Google Scholar Citations, arXiv, and below.

*Keywords:* 2d statistical mechanics, random matrices,
interacting particle systems, asymptotic representation theory.

Articles in chronological order (point at a title to see the abstract):

- Non-intersecting paths and Hahn orthogonal
polynomial ensemble, We compute the bulk
limit of the correlation functions for the uniform measure on
lozenge tilings of a hexagon. The limiting determinantal process
is a translation invariant extension of the discrete sine
process, which also describes the ergodic Gibbs measure of an
appropriate slope.
**Functional Analysis and its Applications**, 42 (3) (2008), 180-197, arXiv:0708.2349 - Shuffling algorithm for boxed plane
partitions (joint paper with A.Borodin),
We introduce discrete time Markov chains that preserve uniform
measures on boxed plane partitions. Elementary Markov steps
change the size of the box from (a x b x c) to ((a-1) x (b+1) x
c) or ((a+1) x (b-1) x c). Algorithmic realization of each step
involves O((a+b)c) operations. One application is an efficient
perfect random sampling algorithm for uniformly distributed
boxed plane partitions.
Trajectories of our Markov chains can be viewed as random point configurations in the three-dimensional lattice. We compute the bulk limits of the correlation functions of the resulting random point process on suitable two-dimensional sections. The limiting correlation functions define a two-dimensional determinantal point processes with certain Gibbs properties.

**Advances in Mathematics**, 220 (6) (2009). 1739-1770, arXiv: 0804.3071 (See also Random Tilings Generator and its configuration file) - Disjointness of representations arising in
harmonic analysis on the infinite-dimensional unitary group, We prove pairwise disjointness of
representations T_{z,w} of the infinite-dimensional unitary
group. These representations provide a natural generalization of
the regular representation for the case of "big" group
U(\infty). They were introduced and studied by G.Olshanski and
A.Borodin.
Disjointness of the representations can be reduced to disjointness of certain probability measures on the space of paths in the Gelfand-Tsetlin graph. We prove the latter disjointness using probabilistic and combinatorial methods.

**Functional Analysis and its Applications**, 44:2, (2010) 92-105, arXiv: 0805.2660 - Non-colliding Jacobi processes as limits of
Markov chains on Gelfand-Tsetlin graph,
We introduce a stochastic dynamics related to the measures that
arise in harmonic analysis on the infinite-dimensional unitary
group. Our dynamics is obtained as a limit of a sequence of
natural Markov chains on Gelfand-Tsetlin graph. We compute
finite-dimensional distributions of the limit Markov process,
the generator and eigenfunctions of the semigroup related to
this process. The limit process can be identified with Doob
h-transform of a family of independent diffusions. Space-time
correlation functions of the limit process have a determinantal
form.
**Journal of Mathematical Sciences**, Vol. 158, No. 6, 2009, 819-837 (translated from**Zapiski Nauchnykh Seminarov POMI**, Vol. 360 (2008), pp. 91-123), arXiv: 0812.3146 - q-Distributions on plane partitions (joint
paper with A.Borodin and E.M.Rains), We
introduce elliptic weights of boxed plane partitions and prove
that they give rise to a generalization of MacMahon's product
formula for the number of plane partitions in a box. We then
focus on the most general positive degenerations of these
weights that are related to orthogonal polynomials; they form
three two-dimensional families. For distributions from these
families we prove two types of results.
First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm.

Second, we consider a limit when all dimensions of the box grow and plane partitions become large, and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also be viewed as slopes of tangent planes to the hypothetical limit shape) are encoded by a single quadratic polynomial.

**Selecta Mathematica, New Series**, 16:4, (2010) 731-789, arXiv:0905.0679 (See also Random Tilings Generator and its configuration file) - 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_\epsilon(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.
**Advances in Mathematics**, 229 (2012), no. 1, 201-266, arXiv:1011.1769
(Note that the journal version of this paper contains a misprint in
the part related to q-Toeplitz matrices, this misprint is corrected in
the latest arXiv version.)
- Estimation of multivariate observation-error
statistics for AMSU-A data (joint paper with M.Tsyrulnikov), Advanced Microwave Sounding Unit A (AMSU-A)
observation-error covariances are objectively estimated by
comparing satellite radiances with radiosonde data. Channels 6–8
are examined as being weakly dependent on the surface and on the
stratosphere above the radiosonde top level. Significant
horizontal, interchannel, temporal, and intersatellite
correlations are found. Besides, cross correlations between
satellite and forecast (background) errors (largely disregarded
in practical data assimilation) proved to be far from zero. The
directional isotropy hypothesis is found to be valid for
satellite error correlations. Dependencies on the scan position,
the season, and the satellite are also checked. Bootstrap
simulations demonstrate that the estimated covariances are
statistically significant. The estimated correlations are shown
to be caused by the satellite errors in question and not by
other (nonsatellite) factors.
**Monthly Weather Review**, 139 (2011) no. 12, 3765-3780. - Markov processes of infinitely many
nonintersecting random walks (joint paper with A.Borodin), 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.
**Probability Theory and Related Fields**155 (2013), no. 3-4, 935-997, arXiv:1106.1299 - Block characters of the symmetric groups
(joint paper with A.Gnedin and S.Kerov),
Block character of a finite symmetric group S(n) is a positive
definite function which depends only on the number of cycles in
permutation. We describe the cone of block characters by
identifying its extreme rays, and find relations of the
characters to descent representations and the coinvariant
algebra of S(n). The decomposition of extreme block characters
into the sum of characters of irreducible representations gives
rise to certain limit shape theorems for random Young diagrams.
We also study counterparts of the block characters for the
infinite symmetric group S(\infty) along with their connection
to the Thoma characters of the infinite linear group
GL(\infty,q) over a Galois field.
**Journal of Algebraic Combinatorics**, 38 (2013), no. 1, 79-101, arXiv:1108.5044 - A pattern theorem for random sorting
networks (joint paper with O.Angel and A.Holroyd),
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.
**Electronic Journal of Probability**, 17 (2012), no. 99, 1-16, arXiv:1110.0160 - Record-dependent measures on the symmetric
group (joint paper with A.Gnedin), A
probability measure P_n on the symmetric group S(n) is said to
be record-dependent if P_n(s) depends only on the set of records
of a permutation s from S(n). A sequence P_n, n=1,2,.., of
consistent record-dependent measures determines a random order
on positive integers. 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.
**Random Structures and Algorithms**, 46, no. 4 (2015), 688-706. arXiv:1202.3680 - What can be made out of cubes?
This is an article in Russian addressed mostly to advanced
high-school students. I review classical results on enumeration
of 2d and 3d Young diagrams and show some recent developments in
the study of random diagrams.
**Quantum**, 2012, no. 3,. - Limits of Multilevel TASEP and similar
processes (joint paper with M.Shkolnikov),
We study the asymptotic behavior of a class of stochastic
dynamics on interlacing particle configurations (also known as
Gelfand-Tsetlin patterns). Examples of such dynamics include, in
particular, a multi-layer extension of TASEP and particle
dynamics related to the shuffling algorithm for domino tilings
of the Aztec diamond. We prove that the process of reflected
interlacing Brownian motions introduced by Warren in \cite{W}
serves as a universal scaling limit for such dynamics.
**Annales de l'Institut Henri Poincare: Probabilites et Statistiques**, 51, no. 1 (2015), 18-27. arXiv:1206.3817 - Finite traces and representations of the
group of infinite matrices over a finite field(joint paper with
S.Kerov, A.Vershik), The article is
devoted to the representation theory of locally compact
infinite-dimensional group 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=infinite analogue of general linear groups
GL(n,q). It serves as an alternative to GL(infinity,q), whose
representation theory is poor.
Our most important results are the description of semi-finite unipotent traces (characters) of the group GLB via certain probability measures on the Borel subgroup B and the construction of the corresponding von Neumann factor representations of type II_infinity. As a main tool we use the subalgebra A(GLB) of smooth functions in the group algebra L_1(GLB). This subalgebra is an inductive limit of the finite-dimensional group algebras C(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 H_q(infinity) 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(infinity) which had been intensively studied in numerous previous papers.

**Advances in Mathematics**, 254 (2014), 331-395, arXiv:1209.4945 - Lectures on integrable probability (joint
paper with A.Borodin), 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. In:
**Probability and Statistical Physics in St. Petersburg, Proceedings of Symposia in Pure Mathematics,**Vol.\ 91, 155--214. AMS 2016. arXiv:1212.3351 - Asymptotics of symmetric polynomials with
applications to statistical mechanics and representation theory
(joint paper with G.Panova), 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.
**Annals of Probability**, 43, no. 6, (2015), 3052-3132, arXiv:1301.0634 - Are atmospheric-model tendency errors
perceivable from routine observations? (joint paper with
M.Tsyrulnikov), In predictability
experiments with simulated model errors (ME) and the COSMO
model, reproducibility of ME from finite-time
model-minus-observed tendencies is studied. It is found that in
1-h to 6-h tendencies, ME appear to be too heavily contaminated
by noises due to, first, initial errors and, second, trajectory
drift as a result of ME themselves. The resulting
reproducibility error is far above the acceptable level. The
conclusion is drawn that the accuracy and coverage of current
routine observations are far from being sufficient to reliably
estimate ME.
**COSMO Newsletter**No. 13: April 2013, 3-18, www.cosmo-model.org. - General beta Jacobi corners process and the
Gaussian Free Field (joint paper with A.Borodin),
We prove that the two-dimensional Gaussian Free Field describes
the asymptotics of global fluctuations of a multilevel extension
of the general beta Jacobi random matrix ensembles. Our approach
is based on the connection of the Jacobi ensembles to a
degeneration of the Macdonald processes that parallels the
degeneration of the Macdonald polynomials to to the
Heckman-Opdam hypergeometric functions (of type A). We also
discuss the beta goes to infinity limit.
**Communications on Pure and Applied Mathematics**, 68, no. 10 (2015), 1774-1844. arXiv:1305.3627 - Observables of Macdonald processes (joint
paper with A.Borodin, I.Corwin and S.Shakirov),
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.
**Transactions of American Mathematical Society**, 368 (2016), 1517-1558. arxiv:1306.0659 - 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.
**Communications in Mathematical Physics**,332, no. 1 (2014), 437-447. arXiv:1306.6347 - Representations of classical Lie groups and
quantized free convolution (joint paper with A.Bufetov),
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.

**Geometric and Functional Analysis (GAFA)**, 25, no. 3 (2015), 763-814, arXiv:1311.5780 - Multilevel Dyson Brownian motions via Jack
polynomials (joint paper with M.Shkolnikov),
We introduce multilevel versions of Dyson Brownian motions of
arbitrary parameter beta>0, generalizing the interlacing
reflected Brownian motions of Warren for beta=2. Such processes
unify beta 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 beta>=1) and to give the corresponding link
explicitly.
**Probability Theory and Related Fields**, 163, no. 3, 413-463. (2015) arXiv:1401.5595 - Stochastic six-vertex model (joint paper
with A.Borodin and I.Corwin), 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.
**Duke Mathematical Journal**, 165, no. 3 (2016), 563-624. arXiv:1407.6729 - Interacting particle systems at the edge of
multilevel Dyson Brownian motions (joint paper with M.Shkolnikov),
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 beta random
matrix models.
**Advances in Mathematics**304 (2017), 90-130, arXiv:1409.2016 - Stochastic monotonicity in Young graph and
Thoma theorem (joint paper with A.Bufetov),
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

**R**which are non-negative on Hall-Littlewood polynomials.**International Mathematics Research Notices**, 2015 (23): 12920-12940. arXiv:1411.3307 - Determinantal measures related to big
q-Jacobi polynomials (joint paper with G.Olshanski),
We define a novel combinatorial object—the extended
Gelfand—Tsetlin graph with cotransition probabilities depending
on a parameter q. The boundary of this graph admits an explicit
description. We introduce a family of probability measures on
the boundary and describe their correlation functions. These
measures are a q-analogue of the spectral measures studied
earlier in the context of the problem of harmonic analysis on
the infinite-dimensional unitary group.
**Functional Analysis and Its Applications**, 49, no. 3 (2015), 214-217. - A quantization of the harmonic analysis on
the infinite-dimensional unitary group (joint paper with
G.Olshanski), 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
\Omega, 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-\phi-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).

**Journal of Functional Analysis**, 270, 375-418 (2016). arXiv:1504.06832 - Gaussian asymptotics of discrete
beta-ensembles (joint paper with A.Borodin and A.Guionnet), We introduce and study stochastic N-particle
ensembles which are discretizations for general-beta log-gases
of random matrix theory. The examples include random tilings,
families of non-intersecting paths, (z,w)-measures, etc. We
prove that under technical assumptions on general analytic
potential, the global fluctuations for such ensembles are
asymptotically Gaussian as N tends to infinity. The covariance
is universal and coincides with its counterpart in random matrix
theory.
Our main tool is an appropriate discrete version of the Schwinger-Dyson (or loop) equations, which originates in the work of Nekrasov and his collaborators.

**Publications mathématiques de l'IHÉS**125, no. 1 (2017). arXiv:1505.03760 - Stochastic Airy semigroup through tridiagonal
matrices (joint paper with M.Shkolnikov),
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_beta process,
which describes the largest eigenvalues in the beta ensembles of
random matrix theory. Another consequence is a Feynman-Kac
formula for the stochastic Airy operator of Ramirez, Rider, and
Virag.
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.

**Annals of Probability**46, no. 4 (2018), 2287-2344. arXiv:1601.06800 - Bulk universality for random lozenge tilings
near straight boundaries and for tensor products.
We prove that the asymptotic of the bulk local statistics in
models of random lozenge tilings is universal in the vicinity of
straight boundaries of the tiled domains. The result applies to
uniformly random lozenge tilings of large polygonal domains on
triangular lattice and to the probability measures describing
the decomposition in Gelfand-Tsetlin bases of tensor products of
representations of unitary groups. In a weaker form our theorem
also applies to random domino tilings.
**Communications in Mathematical Physics**354, no. 1 (2017), 317–344. arXiv:1603.02707 - Fluctuations of particle systems determined
by Schur generating functions (joint paper with A.Bufetov), We develop a new toolbox for the analysis of
the global behavior of stochastic discrete particle systems. We
introduce and study the notion of the Schur generating function
of a random discrete configuration. Our main result provides a
Central Limit Theorem (CLT) for such a configuration given
certain conditions on the Schur generating function. As
applications of this approach, we prove CLT's for several
probabilistic models coming from asymptotic representation
theory and statistical physics, including random lozenge and
domino tilings, non-intersecting random walks, decompositions of
tensor products of representations of unitary groups.
**Advances in Mathematics**, 338, no. 7 (2018), 702-781. arXiv:1604.01110 - Moments match between the KPZ equation and
the Airy point process (joint paper with A.Borodin)
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.
**SIGMA**12 (Special issue in honor of P.Deift and C.Tracy; 2016), 102. arXiv:1608.01557 - Universality of local statistics for
noncolliding random walks (joint paper with L.Petrov), to appear
in
**Annals of Probability**. We consider the N-particle noncolliding Bernoulli random walk - a discrete time Markov process in Z^N obtained from a collection of N independent simple random walks with unit steps by conditioning that they never collide. We study the asymptotic behavior of local statistics of this process started from an arbitrary initial configuration on short times T much smaller than N as N tends to infinity. We show that if the particle density of the initial configuration is bounded away from 0 and 1 down to scales D much smaller than T in a neighborhood of size Q much larger than T of some location x (i.e., x is in the "bulk"), and the initial configuration is balanced in a certain sense, then the space-time local statistics at x are asymptotically governed by the extended discrete sine process (which can be identified with a translation invariant ergodic Gibbs measure on lozenge tilings of the plane). We also establish similar results for certain types of random initial data. Our proofs are based on a detailed analysis of the determinantal correlation kernel for the noncolliding Bernoulli random walk.The noncolliding Bernoulli random walk is a discrete analogue of the beta=2 Dyson Brownian Motion whose local statistics are universality governed by the continuous sine process. Our results parallel the ones in the continuous case. In addition, we naturally include situations with inhomogeneous local particle density on scale T, which nontrivially affects parameters of the limiting extended sine process, and in a particular case leads to a new behavior.

arXiv:1608.03243 - Spherically Symmetric Random Permutations
(joint paper with A.Gnedin), We consider
random permutations which are spherically symmetric with respect
to a metric on the symmetric group Sn 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.
to appear in
**Random Structures and Algorithms.**arXiv:1611.01860 - Interlacing adjacent levels of beta-Jacobi
corners processes(joint paper with L.Zhang),
We study the asymptotic of the global fluctuations for the
difference between two adjacent levels in the beta-Jacobi
corners process (multilevel and general beta extension of the
classical Jacobi ensemble of random matrices). The limit is
identified with the derivative of the 2d Gaussian Free Field.
Our main tool is integral forms for the (Macdonald-type)
difference operators originating from the shuffle algebra.
to appear in
**Probability Theory and Related Fields**. arXiv:1612.02321 - Random sorting networks: local statistics
via random matrix laws (joint paper with M.Rahman),
This paper derives the bulk local limit of the swap process of
uniformly random sorting networks. The limit object is defined
through a deterministic procedure, a local version of the
Edelman-Greene algorithm, applied to a two dimensional
determinantal point process with explicit kernel. The latter
describes the asymptotic joint law near 0 of the eigenvalues of
the corners in the antisymmetric Gaussian Unitary Ensemble. In
particular, the limiting distribution of the first swap time at
a given position is identified with the limiting distribution
for the closest to 0 eigenvalue in the antisymmetric GUE. The
proofs rely on the determinantal structure and a double contour
integral representation for the kernel of random Poissonized
Young tableaux. to appear in
**Probability Theory and Related Fields.**arXiv:1702.07895 - Crystallization of random matrix orbits
(joint paper with A.Marcus), Three
operations on eigenvalues of real/complex/quaternion
(corresponding to β=1,2,4) matrices, obtained from cutting out
principal corners, adding, and multiplying matrices can be
extrapolated to general values of β>0 through associated
special functions. We show that β→∞ limit for these operations
leads to the finite free projection, additive convolution, and
multiplicative convolution, respectively. The limit is the most
transparent for cutting out the corners, where the joint
distribution of the eigenvalues of principal corners of a
uniformly-random general β self-adjoint matrix with fixed
eigenvalues is known as β-corners process. We show that as β→∞
these eigenvalues crystallize on the irregular lattice of all
the roots of derivatives of a single polynomial. In the second
order, we observe a version of the discrete Gaussian Free Field
(dGFF) put on top of this lattice, which provides a new
explanation of why the (continuous) Gaussian Free Field governs
the global asymptotics of random matrix ensembles.
to appear in
**International Mathematics Research Notices**. arXiv:1706.07393 - Fourier transform on high-dimensional
unitary groups with applications to random tilings (joint paper
with A.Bufetov) A combination of direct
and inverse Fourier transforms on the unitary group U(N)
identifies normalized characters with probability measures on
N-tuples of integers. We develop the N→∞ version of this
correspondence by matching the asymptotics of partial
derivatives at the identity of logarithm of characters with Law
of Large Numbers and Central Limit Theorem for global behavior
of corresponding random N-tuples.

As one application we study fluctuations of the height function of random domino and lozenge tilings of a rich class of domains. In this direction we prove the Kenyon-Okounkov's conjecture (which predicts asymptotic Gaussianity and exact form of the covariance) for a family of non-simply connected polygons. Another application is a central limit theorem for the U(N) quantum random walk with random initial data. arXiv:1712.09925 - The KPZ equation and moments of random
matrices (joint paper with S.Sodin), The
logarithm of the diagonal matrix element of a high power of a
random matrix converges to the Cole-Hopf solution of the
Kardar-Parisi-Zhang equation in the sense of one-point
distributions. to appear in
**Journal of Mathematical Physics, Analysis, Geometry**(Special issue in honor of V.A. Marchenko). arXiv:1801.02574 - A stochastic telegraph equation from the six-vertex model (joint paper with A.Borodin). A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second order linear hyperbolic) telegraph differential equation. The inhomogeneities we consider are proportional to the two-dimensional white noise, and solutions to our equation are two-dimensional random Gaussian fields. We show that such fields arise naturally as asymptotic fluctuations of the height function in a certain limit regime of the stochastic six vertex model in a quadrant. The corresponding law of large numbers -- the limit shape of the height function - is described by the (deterministic) homogeneous telegraph equation. arXiv:1803.09137
- Product matrix processes as limits of random
plane partitions (joint paper with A.Borodin, E.Strahov), We
consider a random process with discrete time formed by singular
values of products of truncations of Haar distributed unitary
matrices. We show that this process can be understood as a
scaling limit of the Schur process, which gives determinantal
formulas for (dynamical) correlation functions and a contour
integral representation for the correlation kernel. The relation
with the Schur processes implies that the continuous limit of
marginals for q-distributed plane partitions coincides with the
joint law of singular values for products of truncations of
Haar-distributed random unitary matrices. We provide structural
reasons for this coincidence that may also extend to other
classes of random matrices. to appear in
**International Mathematics Research Notices.**arXiv:1806.10855 - q-deformed Character Theory for Infinite-Dimensional Symplectic and Orthogonal Groups (joint paper with C.Cuenca). The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding finite-dimensional groups, as the rank tends to infinity. We solve a q-deformed version of the latter problem for orthogonal and symplectic groups, extending previously known results for the unitary group. The proof is based on novel determinantal and double-contour integral formulas for the q-specialized characters. arXiv:1812.06523
- Gaussian fluctuations for products of random matrices (joint paper with Y.Sun). We study global fluctuations for singular values of M-fold products of several right-unitarily invariant N×N random matrix ensembles. As N→∞, we show the fluctuations of their height functions converge to an explicit Gaussian field, which is log-correlated for M fixed and has a white noise component for M→∞ jointly with N. Our technique centers on the study of the multivariate Bessel generating functions of these spectral measures, for which we prove a central limit theorem for global fluctuations via certain conditions on the generating functions. We apply our approach to a number of ensembles, including rectangular Ginibre matrices, truncated Haar-random unitary matrices, and right-unitarily invariant matrices with fixed singular values, using a detailed asymptotic analysis of multivariate Bessel functions to verify the necessary conditions. arXiv:1812.06532