## The Annals of Probability

### A probabilistic interpretation of the Macdonald polynomials

#### Abstract

The two-parameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of $k$ with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary distribution a new two-parameter family of measures on partitions, the inverse of the Macdonald weight (rescaled). The uniform distribution on cycles of permutations and the Ewens sampling formula are special cases. The Markov chain is a version of the auxiliary variables algorithm of statistical physics. Properties of the Macdonald polynomials allow a sharp analysis of the running time. In natural cases, a bounded number of steps suffice for arbitrarily large $k$.

#### Article information

Source
Ann. Probab., Volume 40, Number 5 (2012), 1861-1896.

Dates
First available in Project Euclid: 8 October 2012

https://projecteuclid.org/euclid.aop/1349703310

Digital Object Identifier
doi:10.1214/11-AOP674

Mathematical Reviews number (MathSciNet)
MR3025704

Zentralblatt MATH identifier
1255.05194

#### Citation

Diaconis, Persi; Ram, Arun. A probabilistic interpretation of the Macdonald polynomials. Ann. Probab. 40 (2012), no. 5, 1861--1896. doi:10.1214/11-AOP674. https://projecteuclid.org/euclid.aop/1349703310

#### References

• [1] Aldous, D. and Diaconis, P. (1999). Longest increasing subsequences: From patience sorting to the Baik–Deift–Johansson theorem. Bull. Amer. Math. Soc. (N.S.) 36 413–432.
• [2] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3–48.
• [3] Andersen, H. C. and Diaconis, P. (2007). Hit and run as a unifying device. J. Soc. Fr. Stat. & Rev. Stat. Appl. 148 5–28.
• [4] Andrews, G. E. (1998). The Theory of Partitions. Cambridge Univ. Press, Cambridge.
• [5] Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. Eur. Math. Soc., Zürich.
• [6] Assaf, S. H. (2007). Dual equivalence graphs, ribbon tableaux and Macdonald polynomials. Ph.D. thesis, Dept. Mathematics, Univ. California, Berkeley.
• [7] Awata, H., Kubo, H., Odake, S. and Shiraishi, J. (1996). Quantum ${\mathscr{W}}_{N}$ algebras and Macdonald polynomials. Comm. Math. Phys. 179 401–416.
• [8] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
• [9] Betz, V., Ueltschi, D. and Velenik, Y. (2011). Random permutations with cycle weights. Ann. Appl. Probab. 21 312–331.
• [10] Billingsley, P. (1972). On the distribution of large prime divisors. Period. Math. Hungar. 2 283–289.
• [11] Borgs, C., Chayes, J. T., Frieze, A., Kim, J. H., Tetali, P., Vigoda, E. and Vu, V. H. (1999). Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics. In 40th Annual Symposium on Foundations of Computer Science (New York, 1999) 218–229. IEEE Comput. Soc., Los Alamitos, CA.
• [12] Borodin, A., Okounkov, A. and Olshanski, G. (2000). Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 481–515 (electronic).
• [13] Brémaud, P. (1999). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Texts in Applied Mathematics 31. Springer, New York.
• [14] Ceccherini-Silberstein, T., Scarabotti, F. and Tolli, F. (2008). Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains. Cambridge Studies in Advanced Mathematics 108. Cambridge Univ. Press, Cambridge.
• [15] Cherednik, I. (1992). Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and Macdonald’s operators. Int. Math. Res. Not. IMRN 9 171–180.
• [16] Diaconis, P. and Hanlon, P. (1992). Eigen-analysis for some examples of the Metropolis algorithm. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991). Contemporary Mathematics 138 99–117. Amer. Math. Soc., Providence, RI.
• [17] Diaconis, P. and Holmes, S. P. (2002). Random walks on trees and matchings. Electron. J. Probab. 7 17 pp. (electronic).
• [18] Diaconis, P., Mayer-Wolf, E., Zeitouni, O. and Zerner, M. P. W. (2004). The Poisson–Dirichlet law is the unique invariant distribution for uniform split-merge transformations. Ann. Probab. 32 915–938.
• [19] Diaconis, P. and Ram, A. (2000). Analysis of systematic scan Metropolis algorithms using Iwahori–Hecke algebra techniques. Michigan Math. J. 48 157–190. Dedicated to William Fulton on the occasion of his 60th birthday.
• [20] Diaconis, P. and Ram, A. (2010). A probabilistic interpretation of the Macdonald polynomials. Available at arXiv:1007.4779.
• [21] Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159–179.
• [22] Edwards, R. G. and Sokal, A. D. (1988). Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm. Phys. Rev. D (3) 38 2009–2012.
• [23] Ercolani, N. M. and Ueltschi, D. (2011). Cycle structure of random permutations with cycle weights. Available at arXiv:1102.4796.
• [24] Fristedt, B. (1993). The structure of random partitions of large integers. Trans. Amer. Math. Soc. 337 703–735.
• [25] Fulman, J. (2002). Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 51–85.
• [26] Garsia, A. and Remmel, J. B. (2005). Breakthroughs in the theory of Macdonald polynomials. Proc. Natl. Acad. Sci. USA 102 3891–3894 (electronic).
• [27] Ghosh, J. K. and Ramamoorthi, R. V. (2003). Bayesian Nonparametrics. Springer, New York.
• [28] Gontcharoff, V. (1944). Du domaine de l’analyse combinatoire. Bull. Acad. Sci. USSR Sér. Math. [Izvestia Akad. Nauk SSSR] 8 3–48.
• [29] Gordon, I. (2003). On the quotient ring by diagonal invariants. Invent. Math. 153 503–518.
• [30] Haglund, J., Haiman, M. and Loehr, N. (2005). A combinatorial formula for Macdonald polynomials. J. Amer. Math. Soc. 18 735–761 (electronic).
• [31] Haglund, J., Haiman, M. and Loehr, N. (2005). Combinatorial theory of Macdonald polynomials. I. Proof of Haglund’s formula. Proc. Natl. Acad. Sci. USA 102 2690–2696 (electronic).
• [32] Haglund, J., Haiman, M. and Loehr, N. (2008). A combinatorial formula for nonsymmetric Macdonald polynomials. Amer. J. Math. 130 359–383.
• [33] Haiman, M. (2006). Cherednik algebras, Macdonald polynomials and combinatorics. In International Congress of Mathematicians, Vol. III 843–872. Eur. Math. Soc., Zürich.
• [34] Hanlon, P. (1992). A Markov chain on the symmetric group and Jack symmetric functions. Discrete Math. 99 123–140.
• [35] Hoppe, F. M. (1987). The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25 123–159.
• [36] Hora, A. and Obata, N. (2007). Quantum Probability and Spectral Analysis of Graphs. Springer, Berlin. With a foreword by Luigi Accardi.
• [37] Jiang, J. (2011). Multiplicative measures on partitions, asymptotic theory. Preprint, Dept. Mathematics, Stanford Univ.
• [38] Kerov, S. V. (2003). Asymptotic Representation Theory of the Symmetric Group and Its Applications in Analysis. Translations of Mathematical Monographs 219. Amer. Math. Soc., Providence, RI. Translated from the Russian manuscript by N. V. Tsilevich, With a foreword by A. Vershik and comments by G. Olshanski.
• [39] Knop, F. and Sahi, S. (1997). A recursion and a combinatorial formula for Jack polynomials. Invent. Math. 128 9–22.
• [40] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI. With a chapter by James G. Propp and David B. Wilson.
• [41] Logan, B. F. and Shepp, L. A. (1977). A variational problem for random Young tableaux. Adv. Math. 26 206–222.
• [42] Macdonald, I. G. (1995). Symmetric Functions and Hall Polynomials, 2nd ed. The Clarendon Press Oxford Univ. Press, New York. With contributions by A. Zelevinsky, Oxford Science Publications.
• [43] Macdonald, I. G. (2000/01). Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45 Art. B45a, 40 pp. (electronic).
• [44] Macdonald, I. G. (2003). Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics 157. Cambridge Univ. Press, Cambridge.
• [45] Newman, M. E. J. and Barkema, G. T. (1999). Monte Carlo Methods in Statistical Physics. The Clarendon Press Oxford Univ. Press, New York.
• [46] Okounkov, A. (2001). Infinite wedge and random partitions. Selecta Math. (N.S.) 7 57–81.
• [47] Okounkov, A. (2002). Symmetric functions and random partitions. In Symmetric Functions 2001: Surveys of Developments and Perspectives. NATO Sci. Ser. II Math. Phys. Chem. 74 223–252. Kluwer Academic, Dordrecht.
• [48] Okounkov, A. (2005). The uses of random partitions. In XIVth International Congress on Mathematical Physics 379–403. World Sci. Publ., Hackensack, NJ.
• [49] Olshanski, G. (2011). Random permutations and related topics. In The Oxford Handbook on Random Matrix Theory (G. Akermann, J. Baik and P. Di Francesco, eds.). Oxford Univ. Press. To appear. Available at http://www.bookdepository.co.uk/Oxford-Handbook-Random-Matrix-Theory-Gernot-Akemann/9780199574001?b=-3&t=-26#Bibliographicdata-26.
• [50] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard.
• [51] Ram, A. and Yip, M. (2011). A combinatorial formula for Macdonald polynomials. Adv. Math. 226 309–331.
• [52] Saloff-Coste, L. (1997). Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (Saint-Flour, 1996). Lecture Notes in Math. 1665 301–413. Springer, Berlin.
• [53] Stanley, R. P. (1989). Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 76–115.
• [54] Vershik, A. M. (1996). Statistical mechanics of combinatorial partitions, and their limit configurations. Funktsional. Anal. i Prilozhen. 30 19–39, 96.
• [55] Veršik, A. M. and Kerov, S. V. (1977). Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Dokl. Akad. Nauk SSSR 233 1024–1027.
• [56] Yakubovich, Y. (2009). Ergodicity of multiplicative statistics. Available at arXiv:0901.4655.
• [57] Zhao, J. T. (2011). Universality results for measures on partitions. Preprint, Dept. Mathematics, Stanford Univ.