Asymptotics of Symmetric Polynomials with Applications to Statistical Mechanics and Representation Theory
نویسندگان
چکیده
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.
منابع مشابه
Synopsis: Dual Equivalence Graphs, Ribbon Tableaux and Macdonald Polynomials
The primary focus of this dissertation is symmetric function theory. The main objectives are to present a new combinatorial construction which may be used to establish the symmetry and Schur positivity of a function expressed in terms of monomials, and to use this method to find a combinatorial description of the Schur expansion for two important classes of symmetric functions, namely LLT and M...
متن کاملAsymptotics of characters of symmetric groups: Structure of Kerov character polynomials
We study asymptotics of characters of the symmetric groups on a fixed conjugacy class. It was proved by Kerov that such a character can be expressed as a polynomial in free cumulants of the Young diagram (certain functionals describing the shape of the Young diagram). We show that for each genus there exists a universal symmetric polynomial which gives the coefficients of the part of Kerov char...
متن کاملGeneralized trace formula and asymptotics of the averaged Turan determinant for polynomials orthogonal with a discrete Sobolev inner product
Let be a finite positive Borel measure supported on [−1, 1] and introduce the discrete Sobolev-type inner product 〈f, g〉 = ∫ 1 −1 f (x)g(x) d (x)+ K ∑ k=1 Nk ∑ i=0 Mk,if (ak)g (ak), where the mass points ak belong to [−1, 1], and Mk,i > 0(i = 0, 1, . . . , Nk). In this paper, we obtain generalized trace formula and asymptotics of the averagedTuran determinant for the Sobolev-type orthogonal pol...
متن کاملComputer Generation of Characteristic Polynomials of Edge-Weighted Graphs, Heterographs, and Directed Graphs
The computer code developed previously (K. Balasubramanian, J . Computational Chern., 5,387 (1984)) for the characteristic polynomials of ordinary (nonweighted) graphs is extended in this investigation to edge-weighted graphs, heterographs (vertex-weighted), graphs with loops, directed graphs, and signed graphs. This extension leads to a number of important applications of this code to several ...
متن کاملA review on symmetric games: theory, comparison and applications
Game theory models decision makers' behaviors in strategic situations. Since the structures of games are different, behavior and preferences of the players are different in various types of game. This paper reviews various situations of games. Here, characteristics of some common games are discussed and compared. Specifically, we focus on a group of games called symmetric games including Prison...
متن کامل