scholarly journals Integrable probability: stochastic vertex models and symmetric functions

2018 ◽  
Author(s):  
Alexei Borodin ◽  
Leonid Petrov
Keyword(s):  
Symmetric Functions ◽  
Height Function ◽  
Rational Functions ◽  
Universality Class ◽  
Integral Representations ◽  
Baxter Equation ◽  
Partition Functions ◽  
Vertex Model ◽  
Schur Polynomials ◽  
Higher Spin

This chapter presents the study of a homogeneous stochastic higher spin six-vertex model in a quadrant. For this model concise integral representations for multipoint q-moments of the height function and for the q-correlation functions are derived. At least in the case of the step initial condition, these formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six-vertex model, ASEP, various q-TASEPs, and associated zero-range processes. The arguments are largely based on properties of a family of symmetric rational functions that can be defined as partition functions of the higher spin six-vertex model for suitable domains; they generalize classical Hall–Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang–Baxter equation for the higher spin six-vertex model.

scholarly journals Littlewood–Richardson coefficients for Grothendieck polynomials from integrability

2019 ◽  
Vol 2019 (757) ◽  
pp. 159-195 ◽  
Author(s):  
Michael Wheeler ◽  
Paul Zinn-Justin
Keyword(s):  
Baxter Equation ◽  
Partition Functions ◽  
Vertex Model ◽  
Structure Constants ◽  
Product Rules ◽  
Grothendieck Polynomials

AbstractWe study the Littlewood–Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant K-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang–Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood–Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson–Tao and Vakil.

scholarly journals YANG–BAXTER FIELD FOR SPIN HALL–LITTLEWOOD SYMMETRIC FUNCTIONS

2019 ◽  
Vol 7 ◽  
Author(s):  
ALEXEY BUFETOV ◽  
LEONID PETROV
Keyword(s):  
Probability Distribution ◽  
Symmetric Functions ◽  
Exclusion Process ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Vertex Model ◽  
Joint Distributions ◽  
Simple Exclusion Process ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion

Employing bijectivization of summation identities, we introduce local stochastic moves based on the Yang–Baxter equation for $U_{q}(\widehat{\mathfrak{sl}_{2}})$ . Combining these moves leads to a new object which we call the spin Hall–Littlewood Yang–Baxter field—a probability distribution on two-dimensional arrays of particle configurations on the discrete line. We identify joint distributions along down-right paths in the Yang–Baxter field with spin Hall–Littlewood processes, a generalization of Schur processes. We consider various degenerations of the Yang–Baxter field leading to new dynamic versions of the stochastic six-vertex model and of the Asymmetric Simple Exclusion Process.

scholarly journals Higher spin six vertex model and symmetric rational functions

2016 ◽  
Vol 24 (2) ◽  
pp. 751-874 ◽  
Author(s):  
Alexei Borodin ◽  
Leonid Petrov
Keyword(s):  
Rational Functions ◽  
Vertex Model ◽  
Higher Spin

scholarly journals On the Yang–Baxter equation for the six-vertex model

2014 ◽  
Vol 882 ◽  
pp. 70-96 ◽  
Author(s):  
Vladimir V. Mangazeev
Keyword(s):  
Baxter Equation ◽  
Vertex Model

scholarly journals Symmetric Functions for the Generating Matrix of the Yangian of $\mathfrak{gl}_n({\Bbb C})$

10.37236/217 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Natasha Rozhkovskaya
Keyword(s):  
Symmetric Functions ◽  
Combinatorial Identities ◽  
Schur Polynomials ◽  
Generating Matrix

Analogues of classical combinatorial identities for elementary and homogeneous symmetric functions with coefficients in the Yangian are proved. As a corollary, similar relations are deduced for shifted Schur polynomials.

Statement B and the Yang-Baxter Equation

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Statistical Mechanics ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Transfer Matrix ◽  
Baxter Equation ◽  
Partition Functions ◽  
Alternate Proof ◽  
Crystal Basis ◽  
Exactly Solved Models ◽  
Multiple Dirichlet Series

This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.

scholarly journals Boundary matrices for the higher spin six vertex model

2019 ◽  
Vol 945 ◽  
pp. 114665 ◽  
Author(s):  
Vladimir V. Mangazeev ◽  
Xilin Lu
Keyword(s):  
Vertex Model ◽  
Higher Spin

scholarly journals Six-vertex Models and the GUE-corners Process

2018 ◽  
Vol 2020 (6) ◽  
pp. 1794-1881
Author(s):  
Evgeni Dimitrov
Keyword(s):  
Correlation Functions ◽  
Probability Distributions ◽  
Gibbs Measures ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Difference Operators ◽  
Vertex Model ◽  
Gaussian Unitary Ensemble ◽  
Vertex Models ◽  
Higher Spin

Abstract We consider a class of probability distributions on the six-vertex model, which originates from the higher spin vertex models of [13]. We define operators, inspired by the Macdonald difference operators, which extract various correlation functions, measuring the probability of observing different arrow configurations. For the class of models we consider, the correlation functions can be expressed in terms of multiple contour integrals, which are suitable for asymptotic analysis. For a particular choice of parameters we analyze the limit of the correlation functions through the steepest descent method. Combining this asymptotic statement with some new results about Gibbs measures on Gelfand–Tsetlin cones and patterns, we show that the asymptotic behavior of our six-vertex model near the boundary is described by the Gaussian Unitary Ensemble-corners process.

Series representing transcendental numbers that are not U-numbers

2015 ◽  
Vol 11 (03) ◽  
pp. 869-892
Author(s):  
Emre Alkan
Keyword(s):  
Rational Functions ◽  
Rational Numbers ◽  
Upper Bounds ◽  
Integral Representations ◽  
Binomial Coefficients ◽  
Elementary Functions ◽  
Linear Combinations ◽  
Type Series ◽  
Transcendental Numbers ◽  
Mathematical Constants

Using integral representations with carefully chosen rational functions as integrands, we find new families of transcendental numbers that are not U-numbers, according to Mahler's classification, represented by a series whose terms involve rising factorials and reciprocals of binomial coefficients analogous to Apéry type series. Explicit descriptions of these numbers are given as linear combinations with coefficients lying in a suitable real algebraic extension of rational numbers using elementary functions evaluated at arguments belonging to the same field. In this way, concrete examples of transcendental numbers which can be expressed as combinations of classical mathematical constants such as π and Baker periods are given together with upper bounds on their wn measures.

Sign in / Sign up

Export Citation Format

Share Document