×

Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. (English) Zbl 1390.05240

The authors develop a method for studying the limit, as the number of variables tends to infinity, of symmetric polynomials arising in the context of representation theory as well as in its application to probability and statistical mechanics. Applications of the computational technique presented are also given; in particular, in the field of representation theory, to the approximation of characters of \(U(\infty)\) as well as of its \(q\)-deformed characters. Other applications concern the behaviour of uniformly random lozenge tilings of large polygonal domains and the GUE-eigenvalues distribution in the limit; the asymptotic behavior, as the size tends to infinity, of a functional of the alternating sign matrices of size \(n\), which describe the configurations of the six-vertex model with domain wall b.c.’s; the asymptotic expansions of the boundary-to-boundary current for the completely packed \(O(n=1)\) dense loop model.

MSC:

05E05 Symmetric functions and generalizations
22E99 Lie groups
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F99 Limit theorems in probability theory

Software:

ROBBINS
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Aldous, D. J. (1985). Exchangeability and related topics. In École D’été de Probabilités de Saint-Flour , XIII- 1983. Lecture Notes in Math. 1117 1-198. Springer, Berlin. · Zbl 0562.60042
[2] Ayyer, A. and Romik, D. (2013). New enumeration formulas for alternating sign matrices and square ice partition functions. Adv. Math. 235 161-186. · Zbl 1259.05010
[3] Baryshnikov, Y. (2001). GUEs and queues. Probab. Theory Related Fields 119 256-274. · Zbl 0980.60042
[4] Baxter, R. J. (2007). Exactly Solved Models in Statistical Mechanics . Dover, New York. · Zbl 1201.60091
[5] Behrend, R. E. (2013). Multiply-refined enumeration of alternating sign matrices. Adv. Math. 245 439-499. · Zbl 1283.05051
[6] Behrend, R. E., Di Francesco, P. and Zinn-Justin, P. (2012). On the weighted enumeration of alternating sign matrices and descending plane partitions. J. Combin. Theory Ser. A 119 331-363. · Zbl 1232.05037
[7] Billingsley, P. (1995). Probability and Measure , 3rd ed. Wiley, New York. · Zbl 0822.60002
[8] Borodin, A. and Kuan, J. (2010). Random surface growth with a wall and Plancherel measures for \(\mathrm{O}(\infty)\). Comm. Pure Appl. Math. 63 831-894. · Zbl 1193.82013
[9] Borodin, A. and Olshanski, G. (2012). The boundary of the Gelfand-Tsetlin graph: A new approach. Adv. Math. 230 1738-1779. · Zbl 1245.05131
[10] Boyer, R. P. (1983). Infinite traces of AF-algebras and characters of \({\mathrm{U}}(\infty)\). J. Operator Theory 9 205-236. · Zbl 0552.46029
[11] Boyer, R. P. (1992). Characters and factor representations of the infinite-dimensional classical groups. J. Operator Theory 28 281-307. · Zbl 0813.22001
[12] Bressoud, D. M. (1999). Proofs and Confirmations : The Story of the Alternating Sign Matrix Conjecture . Cambridge Univ. Press, Cambridge. · Zbl 0944.05001
[13] Bufetov, A. and Gorin, V. (2013). Representations of classical Lie groups and quantized free convolution. Available at . arXiv:1311.5780 · Zbl 1326.22012
[14] Cantini, L. and Sportiello, A. (2011). Proof of the Razumov-Stroganov conjecture. J. Combin. Theory Ser. A 118 1549-1574. · Zbl 1232.05038
[15] Cohn, H., Kenyon, R. and Propp, J. (2001). A variational principle for domino tilings. J. Amer. Math. Soc. 14 297-346 (electronic). · Zbl 1037.82016
[16] Cohn, H., Larsen, M. and Propp, J. (1998). The shape of a typical boxed plane partition. New York J. Math. 4 137-165 (electronic). · Zbl 0908.60083
[17] Colomo, F. and Pronko, A. G. (2005). On two-point boundary correlations in the six-vertex model with domain wall boundary conditions. J. Stat. Mech. Theory Exp. 5 05010, 21 pp. (electronic). · Zbl 1049.82006
[18] Colomo, F. and Pronko, A. G. (2008). Emptiness formation probability in the domain-wall six-vertex model. Nuclear Phys. B 798 340-362. · Zbl 1234.82007
[19] Colomo, F. and Pronko, A. G. (2010). The limit shape of large alternating sign matrices. SIAM J. Discrete Math. 24 1558-1571. · Zbl 1223.82011
[20] Colomo, F., Pronko, A. G. and Zinn-Justin, P. (2010). The arctic curve of the domain wall six-vertex model in its antiferroelectric regime. J. Stat. Mech. Theory Exp. 3 L03002, 11. · Zbl 1187.82019
[21] Copson, E. T. (1965). Asymptotic Expansions . Cambridge Univ. Press, New York. · Zbl 0123.26001
[22] de Gier, J. (2012). Asymptotic of symmetric polynomials, Open problem session of the program “Random Spatial Processes.” MSRI, Berkeley, Spring 2012.
[23] de Gier, J., Nienhuis, B. and Ponsaing, A. (2010). Exact spin quantum Hall current between boundaries of a lattice strip. Nuclear Phys. B 838 371-390. · Zbl 1206.81156
[24] de Gier, J. and Ponsaing, A. (2011). Separation of variables for symplectic characters. Lett. Math. Phys. 97 61-83. · Zbl 1219.81152
[25] Diaconis, P. and Freedman, D. (1980). Finite exchangeable sequences. Ann. Probab. 8 745-764. · Zbl 0434.60034
[26] Diaconis, P. and Freedman, D. (1984). Partial exchangeability and sufficiency. In Statistics : Applications and New Directions ( Calcutta , 1981) (J. K. Ghosh and J. Roy, eds.) 205-236. Indian Statist. Inst., Calcutta.
[27] Edrei, A. (1953). On the generation function of a doubly infinite, totally positive sequence. Trans. Amer. Math. Soc. 74 367-383. · Zbl 0050.07901
[28] Elkies, N., Kuperberg, G., Larsen, M. and Propp, J. (1992). Alternating-sign matrices and domino tilings. II. J. Algebraic Combin. 1 219-234. · Zbl 0788.05017
[29] Erdelyi, A. (1953). Higher Transcendental Functions. Bateman Manuscript Project 1 . McGraw-Hill, New York.
[30] Erdélyi, A. (1956). Asymptotic Expansions . Dover, New York. · Zbl 0070.29002
[31] Ferrari, P. L. and Spohn, H. (2006). Domino tilings and the six-vertex model at its free-fermion point. J. Phys. A 39 10297-10306. · Zbl 1114.82006
[32] Fonseca, T. and Zinn-Justin, P. (2008). On the doubly refined enumeration of alternating sign matrices and totally symmetric self-complementary plane partitions. Electron. J. Combin. 15 Research Paper 81, 35. · Zbl 1206.05015
[33] Forrester, P. J. and Nordenstam, E. (2009). The anti-symmetric GUE minor process. Mosc. Math. J. 9 749-774, 934. · Zbl 1191.15032
[34] Gorin, V. (2012). The \(q\)-Gelfand-Tsetlin graph, Gibbs measures and \(q\)-Toeplitz matrices. Adv. Math. 229 201-266. · Zbl 1235.22028
[35] Gorin, V. (2014). From alternating sign matrices to the Gaussian unitary ensemble. Available at . arXiv:1306.6347 · Zbl 1303.15038
[36] Guionnet, A. and Maïda, M. (2005). A Fourier view on the \(R\)-transform and related asymptotics of spherical integrals. J. Funct. Anal. 222 435-490. · Zbl 1065.60023
[37] Guionnet, A. and Zeitouni, O. (2002). Large deviations asymptotics for spherical integrals. J. Funct. Anal. 188 461-515. · Zbl 1002.60021
[38] Harish-Chandra (1957). Differential operators on a semisimple Lie algebra. Amer. J. Math. 79 87-120. · Zbl 0072.01901
[39] Harish-Chandra (1957). Fourier transforms on a semisimple Lie algebra. I. Amer. J. Math. 79 193-257. · Zbl 0077.25205
[40] Heckman, G. and Schlichtkrull, H. (1994). Harmonic Analysis and Special Functions on Symmetric Spaces. Perspectives in Mathematics 16 . Academic Press, San Diego, CA. · Zbl 0836.43001
[41] Hough, B. and Jiang, Y. (2012). Asymptotic mixing time analysis of a random walk on the orthogonal group. Available at . arXiv:1211.2031
[42] Itzykson, C. and Zuber, J. B. (1980). The planar approximation. II. J. Math. Phys. 21 411-421. · Zbl 0997.81549
[43] Izergin, A. G. (1987). Partition function of a six-vertex model in a finite volume. Sov. Phys. Dokl. 32 878-879. · Zbl 0875.82015
[44] Johansson, K. and Nordenstam, E. (2006). Eigenvalues of GUE minors. Electron. J. Probab. 11 1342-1371. · Zbl 1127.60047
[45] Kenyon, R. (2009). Lectures on dimers. In Statistical Mechanics. IAS/Park City Math. Ser. 16 191-230. Amer. Math. Soc., Providence, RI. · Zbl 1180.82001
[46] Kenyon, R. and Okounkov, A. (2007). Limit shapes and the complex Burgers equation. Acta Math. 199 263-302. · Zbl 1156.14029
[47] Koekoek, R. and Swarttouw, R. F. (1998). The Askey-scheme of hypergeometric orthogonal polynomials and its \(q\)-analogue. Delft Univ. Technology, Faculty of Information Technology and Systems, Dept. Technical Mathematics and Informatics, Report no. 98-17. Available at .
[48] Koornwinder, T. H. (1998). Special functions associated with root systems: A first introduction for non-specialists. In Special Functions and Differential Equations ( Madras , 1997) (K. Srinivasa Rao et al., eds.) 10-24. Allied, New Delhi. · Zbl 0924.33008
[49] Korepin, V. E. (1982). Calculation of norms of Bethe wave functions. Comm. Math. Phys. 86 391-418. · Zbl 0531.60096
[50] Kuperberg, G. (1996). Another proof of the alternating-sign matrix conjecture. Int. Math. Res. Not. IMRN 3 139-150. · Zbl 0859.05027
[51] Kuznetsov, V. B. and Sklyanin, E. K. (2006). Factorization of symmetric polynomials. In Jack , Hall-Littlewood and Macdonald Polynomials. Contemp. Math. 417 239-256. Amer. Math. Soc., Providence, RI. · Zbl 1132.33318
[52] Macdonald, I. G. (1998). Symmetric Functions and Hall Polynomials , 2nd ed. Clarendon Press, New York. · Zbl 0899.05068
[53] Macdonald, I. G. (1998). Symmetric Functions and Orthogonal Polynomials. University Lecture Series 12 . Amer. Math. Soc., Providence, RI. · Zbl 0887.05053
[54] Metcalfe, A. P. (2013). Universality properties of Gelfand-Tsetlin patterns. Probab. Theory Related Fields 155 303-346. · Zbl 1271.60017
[55] Nordenstam, E. (2009). Interlaced particles in tilings and random matrices. Doctoral Thesis, KTH.
[56] Okada, S. (2006). Enumeration of symmetry classes of alternating sign matrices and characters of classical groups. J. Algebraic Combin. 23 43-69. · Zbl 1088.05012
[57] Okounkov, A. and Olshanski, G. (1998). Asymptotics of Jack polynomials as the number of variables goes to infinity. Int. Math. Res. Not. IMRN 1998 641-682. · Zbl 0913.33004
[58] Okounkov, A. and Olshanski, G. (2006). Limits of \(BC\)-type orthogonal polynomials as the number of variables goes to infinity. In Jack , Hall-Littlewood and Macdonald Polynomials. Contemp. Math. 417 281-318. Amer. Math. Soc., Providence, RI. · Zbl 1151.33008
[59] Okounkov, A. and Reshetikhin, N. (2006). The birth of a random matrix. Moscow Math. J. 6 553-566, 588. · Zbl 1130.15014
[60] Olshanski, G. and Vershik, A. (1996). Ergodic unitarily invariant measures on the space of infinite Hermitian matrices. In Contemporary Mathematical Physics (R. L. Dobrushin et al., eds.). Amer. Math. Soc. Transl. Ser. 2 175 137-175. Amer. Math. Soc., Providence, RI. · Zbl 0853.22016
[61] Panova, G. (2014). Lozenge tilings with free boundaries. Available at . arXiv:1408.0417 · Zbl 1323.05032
[62] Petrov, L. (2014). Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes. Probab. Theory Related Fields . To appear. Available at . arXiv:1202.3901 · Zbl 1315.60013
[63] Petrov, L. (2014). The boundary of the Gelfand-Tsetlin graph: New proof of Borodin-Olshanski’s formula, and its q-analogue. Moscow Math. J. 14 121-160. · Zbl 1297.05249
[64] Razumov, A. V. and Stroganov, Y. G. (2004). Combinatorial nature of the ground-state vector of the \(\mathrm{O}(1)\) loop model. Theor. Math. Phys. 138 333-337. · Zbl 1178.82020
[65] Smirnov, S. (2009). Critical percolation in the plane. Available at . arXiv:0909.4499
[66] Stroganov, Y. G. (2006). The Izergin-Korepin determinant at a cube root of unity. Teoret. Mat. Fiz. 146 65-76. · Zbl 1177.82042
[67] Vershik, A. M. (1974). Description of invariant measures for the actions of some infinite-dimensional groups. Sov. Math. Dokl. 15 1396-1400. · Zbl 0324.28014
[68] Vershik, A. M. and Kerov, S. V. (1982). Characters and factor-representations of the infinite unitary group. Sov. Math. Dokl. 26 570-574. · Zbl 0524.22017
[69] Voiculescu, D. (1976). Représentations factorielles de type II1 de \(U(\infty)\). J. Math. Pures Appl. (9) 55 1-20. · Zbl 0352.22014
[70] Weyl, H. (1997). The Classical Groups : Their Invariants and Representations . Princeton Univ. Press, Princeton, NJ. · Zbl 1024.20501
[71] Zeilberger, D. (1996). Proof of the alternating sign matrix conjecture. Electron. J. Combin. 3 Research Paper 13, approx. 84 pp. (electronic). · Zbl 0858.05023
[72] Zhelobenko, D. P. (1970). Kompaktnye Gruppy Li i Ikh Predstavleniya . Izdat. “Nauka”, Moscow. · Zbl 0228.22013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.