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## Macdonald polynomials and extended Gelfand–Tsetlin graph

Selecta Mathematica, New Series. 2021. Vol. 27. Article 41.

Using Okounkov’s *q*-integral representation of Macdonald polynomials we construct an infinite sequence Ω1,Ω2,Ω3,… of countable sets linked by transition probabilities from Ω*𝑁* to Ω*𝑁*−1 for each *𝑁*=2,3,…. The elements of the sets Ω*𝑁* are the vertices of the extended Gelfand–Tsetlin graph, and the transition probabilities depend on the two Macdonald parameters, *q* and *t*. These data determine a family of Markov chains, and the main result is the description of their entrance boundaries. This work has its origin in asymptotic representation theory. In the subsequent paper, the main result is applied to large-*N* limit transition in (*q*, *t*)-deformed *N*-particle beta-ensembles.

Olshanski G., Communications in Mathematical Physics 2021 Vol. 385 P. 595-631

We introduce and study a family of (q, t)-deformed discrete N-particle beta ensembles, where q and t are the parameters of Macdonald polynomials. The main result is the existence of a large-N limit transition leading to random point processes with infinitely many particles. ...

Added: June 22, 2021

Bezrukavnikov R., Finkelberg M. V., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163-190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 20, 2014

Olshanski G., Горин В. Е., Функциональный анализ и его приложения 2015 Т. 49 № 3 С. 70-74

Мы определяем новый комбинаторный объект — расширенный граф Гельфанда–Цетлина с копереходными вероятностями, зависящими от параметра q. Граница этого графа допускает явное описание. Мы вводим семейство вероятностных мер на границе и описываем их корреляционные функции. Эти меры являются q-аналогом спектральных мер, ранее исследованных в контексте задачи гармонического анализа на бесконечномерной унитарной группе. ...

Added: October 22, 2015

Olshanski G., Working papers by Cornell University. Series math "arxiv.org" 2020

Using Okounkov's q-integral representation of Macdonald polynomials we construct an infinite sequence Ω1,Ω2,Ω3,… of countable sets linked by transition probabilities from ΩN to ΩN−1 for each N=2,3,…. The elements of the sets ΩN are the vertices of the extended Gelfand-Tsetlin graph, and the transition probabilities depend on the two Macdonald parameters, q and t. These ...

Added: January 19, 2021

Finkelberg M., Braverman A., Shiraishi J., Providence : American Mathematical Society, 2014

Let G be an almost simple simply connected complex Lie group, and let G/U be its base affine space. In this paper we formulate a conjecture which provides a new geometric interpretation of the Macdonald polynomials associated to G via perverse coherent sheaves on the scheme of formal arcs in the affinizationof G/U. We prove ...

Added: March 5, 2015

Gorin V., Olshanski G., Journal of Functional Analysis 2016 Vol. 270 No. 1 P. 375-418

The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(∞). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which replace the nonexisting two-sided regular representation (Olshanski [31]). The required decomposition is governed by certain probability measures on an infinite-dimensional space ...

Added: September 3, 2015

Khoroshkin A., / arXiv.org. Series 1312.7053 "1312". 2013. No. 1312.7053.

The aim of this paper is to introduce the categorical setup which helps us to relate the theory of Macdonald polynomials and the theory of Weyl modules for current Lie algebras discovered by V.\,Chari and collaborators. We identify Macdonald pairing with the homological pairing on the ring of characters of the Lie algebra of currents. ...

Added: February 14, 2014

Belomestny D., Moulines E., Samsonov S., Statistics and Computing 2022 Vol. 32 No. 1 Article 16

In this paper, we propose an efficient variance reduction approach for additive functionals of Markov chains relying on a novel discrete-time martingale representation. Our approach is fully non-asymptotic and does not require the knowledge of the stationary distribution (and even any type of ergodicity) or specific structure of the underlying density. By rigorously analyzing the ...

Added: August 31, 2020

Konakov V., Mammen E., Probability Theory and Related Fields 2000 Vol. 117 No. 4 P. 551-587

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition
densities are proved. ...

Added: October 15, 2012

Cherednik I., Feigin E., Advances in Mathematics 2015 Vol. 282 P. 220-264

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t→∞ and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the ...

Added: September 3, 2015

Igor Kheifets, Saikkonen P., Econometric Reviews 2020 Vol. 39 No. 39 P. 407-414

Smooth transition autoregressive models are widely used to capture nonlinearities in univariate and multivariate time series. Existence of stationary solution is typically assumed, implicitly or explicitly. In this paper we describe conditions for stationarity and ergodicity of vector STAR models. The key condition is that the joint spectral radius of certain matrices is below 1, ...

Added: February 23, 2021

Feigin E., Makedonskyi I., Orr D., Advances in Mathematics 2018 Vol. 330 P. 997-1033

We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric q-Whittaker functions. In particular, we show that the series part of the nonsymmetric q-Whittaker function is a generating function for the graded characters of generalized global Weyl modules. ...

Added: September 13, 2018

Konakov V., Mammen E., / Cornell University. Series arXiv "math". 2023. No. 2304.10673.

The Robbins-Monro algorithm is a recursive, simulation-based stochastic procedure to approximate the zeros of a function that can be written as an expectation. It is known that under some technical assumptions, Gaussian limit distributions approximate the stochastic performance of the algorithm. Here, we are interested in strong approximations for Robbins-Monro procedures. The main tool for ...

Added: April 24, 2023

Kelbert M., Sazonov I., Gravenor M., Mathematical Biosciences 2016 Vol. 274 P. 45-57

We consider the epidemic dynamics in stochastically interacting population centers coupled by a random migration. ...

Added: February 15, 2016

Окубо Ю. undefined., Journal of Physics: Conference Series 2017 Vol. 804 No. 012036 P. 1-8

We investigate the existence and the orthogonality of the generalized Jack symmetric functions which play an important role in the AGT relations. We show their orthogonality by deforming them to the generalized Macdonald symmetric functions. ...

Added: October 26, 2017

Blank M., Discrete and Continuous Dynamical Systems 2021 Vol. 41 No. 4 P. 1649-1665

We study qualitative properties of the set of recurrent points of
finitely generated free semigroups of measurable maps. In the case of a single
generator the classical Poincare recurrence theorem shows that these properties are closely related to the presence of an invariant measure. Curious, but
otherwise it turns out to be possible that almost all points are ...

Added: October 21, 2020

Konakov V., Mammen E., Probability Theory and Related Fields 2009 Vol. 143 No. 1 P. 137-176

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by ...

Added: December 4, 2012

Feigin E., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2014. No. 1407.6316.

The Cherednik-Orr conjecture expresses the t\to\infty limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove this conjecture in several special cases. ...

Added: August 10, 2014

Konakov V., Mammen E., Bernoulli: a journal of mathematical statistics and probability 2005 Vol. 11 No. 4 P. 591-641

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. We prove Edgeworth-type expansions of order o(n-1-δ),δ>0, for transition densities. For this purpose we apply the parametrix method to represent the transition density as a functional of densities of sums of independent and identically distributed variables. Then we apply Edgeworth expansions ...

Added: December 4, 2012

Bezhaeva Z., Куликов В. Л., Олехова Е. Ф. et al., Труды Математического института им. В.А. Стеклова РАН 2017 Т. 297 С. 38-45

We define an invariant Erdős measure on the compact abelian group of A-adic integers. We also define an A-invariant Erdős measure on the n-dimensional torus. We show the connection between these invariant measures and functions of countable stationary Markov chains. ...

Added: September 7, 2017

Finkelberg M. V., MATHEMATICAL SCIENCES 2013 Vol. 51 No. 596 P. 46-51

This is a survey of the author's and his collaboratots' recent works on the quasiflags' moduli spaces introduced by Gerard Laumon some 25 years ago. These spaces are used in the study of geometric Eisenstein series, quantum cohomology and K-theory of the flag varieties, Weyl modules, Nekrasov partition function of N=2 supersymmetric gauge quantum field ...

Added: February 14, 2013

Vologodsky V., Finkelberg M. V., Bezrukavnikov R., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163-190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 17, 2015

Feigin E., Kato S., Makedonskyi I., Journal fuer die reine und angewandte Mathematik 2020 Vol. 764 P. 181-216

We study the non-symmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the non-symmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on ...

Added: August 12, 2020

Durmus A., Moulines E., Naumov A. et al., / Cornell University. Series arXiv "math". 2023.

In this paper, we establish novel deviation bounds for additive functionals of geometrically ergodic Markov chains similar to Rosenthal and Bernstein-type inequalities for sums of independent random variables. We pay special attention to the dependence of our bounds on the mixing time of the corresponding chain. Our proof technique is, as far as we know, ...

Added: June 18, 2023