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Observables of Macdonald processes


Authors: Alexei Borodin, Ivan Corwin, Vadim Gorin and Shamil Shakirov
Journal: Trans. Amer. Math. Soc. 368 (2016), 1517-1558
MSC (2010): Primary 05E05
DOI: https://doi.org/10.1090/tran/6359
Published electronically: June 18, 2015
MathSciNet review: 3449217
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Abstract: We present a framework for computing averages of various observables of Macdonald processes. This leads to new contour-integral formulas for averages of a large class of multilevel observables, as well as Fredholm determinants for averages of two different single level observables.


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  • [A] A. Aggrawal, Correlation functions of the Schur process through Macdonald difference operators.
    In preparation.
  • [ACQ] Gideon Amir, Ivan Corwin, and Jeremy Quastel, Probability distribution of the free energy of the continuum directed random polymer in $ 1+1$ dimensions, Comm. Pure Appl. Math. 64 (2011), no. 4, 466-537. MR 2796514 (2012b:60304), https://doi.org/10.1002/cpa.20347
  • [B] A. M. Borodin, Limit Jordan normal form of large triangular matrices over a finite field, Funktsional. Anal. i Prilozhen. 29 (1995), no. 4, 72-75 (Russian); English transl., Funct. Anal. Appl. 29 (1995), no. 4, 279-281 (1996). MR 1375543 (97c:11036), https://doi.org/10.1007/BF01077476
  • [BC] Alexei Borodin and Ivan Corwin, Macdonald processes, Probab. Theory Related Fields 158 (2014), no. 1-2, 225-400. MR 3152785, https://doi.org/10.1007/s00440-013-0482-3
  • [BC2] A. Borodin and I. Corwin,
    Discrete time $ q$-TASEPs,
    Int. Math Rev. Not., to appear, arXiv:1305.2972.
  • [BCF] Alexei Borodin, Ivan Corwin, and Patrik Ferrari, Free energy fluctuations for directed polymers in random media in $ 1+1$ dimension, Comm. Pure Appl. Math. 67 (2014), no. 7, 1129-1214. MR 3207195
  • [BCFV] A. Borodin, I. Corwin, P. L. Ferrari, and B. Vető, Stationary solution of 1d KPZ equation. In preparation.
  • [BCR] Alexei Borodin, Ivan Corwin, and Daniel Remenik, Log-gamma polymer free energy fluctuations via a Fredholm determinant identity, Comm. Math. Phys. 324 (2013), no. 1, 215-232. MR 3116323, https://doi.org/10.1007/s00220-013-1750-x
  • [BCS] Alexei Borodin, Ivan Corwin, and Tomohiro Sasamoto, From duality to determinants for $ q$-TASEP and ASEP, Ann. Probab. 42 (2014), no. 6, 2314-2382. MR 3265169, https://doi.org/10.1214/13-AOP868
  • [BG] A. Borodin and V. Gorin, Lectures on integrable probability,
    arXiv:1212.3351.
  • [BG2] A. Borodin and V. Gorin, General $ \beta $ Jacobi corners process and the Gaussian free field,
    arXiv:1305.3627.
  • [BO] Alexei Borodin and Grigori Olshanski, $ Z$-measures on partitions and their scaling limits, European J. Combin. 26 (2005), no. 6, 795-834. MR 2143199 (2006d:60018), https://doi.org/10.1016/j.ejc.2004.06.003
  • [BP] A. Borodin and L. Petrov, Nearest neighbor Markov dynamics on Macdonald processes,
    arXiv:1305.5501.
  • [COSZ] Ivan Corwin, Neil O'Connell, Timo Seppäläinen, and Nikolaos Zygouras, Tropical combinatorics and Whittaker functions, Duke Math. J. 163 (2014), no. 3, 513-563. MR 3165422, https://doi.org/10.1215/00127094-2410289
  • [CP] I. Corwin and L. Petrov, The q-PushASEP: A new integrable model for traffic in $ 1+1$ dimension, arXiv:1308.3124.
  • [GLO] Anton Gerasimov, Dimitri Lebedev, and Sergey Oblezin, On $ q$-deformed $ {\mathfrak{gl}}_{l+1}$-Whittaker function. I, II, III, Comm. Math. Phys. 294 (2010), no. 1, 97-119., https://doi.org/10.1007/s00220-009-0917-y, MR2575477 (2011d:17023); ibid, 121-143. MR2575478 (2011d:17024); Lett. Math. Phys. 97 (2011), no. 1, 1-24. MR2802312 (2012f:17030)
  • [FHHSY] B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, and S. Yanagida, A commutative algebra on degenerate $ \mathbb{CP}^1$ and Macdonald polynomials, J. Math. Phys. 50 (2009), no. 9, 095215, 42. MR 2566895 (2011c:33041), https://doi.org/10.1063/1.3192773
  • [FR] Peter J. Forrester and Eric M. Rains, Interpretations of some parameter dependent generalizations of classical matrix ensembles, Probab. Theory Related Fields 131 (2005), no. 1, 1-61. MR 2105043 (2006g:05222), https://doi.org/10.1007/s00440-004-0375-6
  • [F1] J. Fulman, Probabilistic measures and algorithms arising from the Macdonald symmetric functions,
    arXiv:math/971223.
  • [F2] Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51-85. MR 1864086 (2002i:60012), https://doi.org/10.1090/S0273-0979-01-00920-X
  • [GKV] Vadim Gorin, Sergei Kerov, and Anatoly Vershik, Finite traces and representations of the group of infinite matrices over a finite field, Adv. Math. 254 (2014), 331-395. MR 3161102, https://doi.org/10.1016/j.aim.2013.12.028
  • [GS] V. Gorin and M. Shkolnikov, Multilevel Dyson Brownian Motions via Jack polynomials. In preparation.
  • [Ke1] S. Kerov, The boundary of Young lattice and random Young tableaux, Formal power series and algebraic combinatorics (New Brunswick, NJ, 1994), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 24, Amer. Math. Soc., Providence, RI, 1996, pp. 133-158. MR 1363510 (96i:05177)
  • [Ke2] S. V. Kerov, Asymptotic representation theory of the symmetric group and its applications in analysis, translated from the Russian manuscript by N. V. Tsilevich, with a foreword by A. Vershik and comments by G. Olshanski, Translations of Mathematical Monographs, vol. 219, American Mathematical Society, Providence, RI, 2003. MR 1984868 (2005b:20021)
  • [KOO] Sergei Kerov, Andrei Okounkov, and Grigori Olshanski, The boundary of the Young graph with Jack edge multiplicities, Internat. Math. Res. Notices 4 (1998), 173-199. MR 1609628 (99f:05120), https://doi.org/10.1155/S1073792898000154
  • [Ki] J. F. C. Kingman, Random partitions in population genetics, Proc. Roy. Soc. London Ser. A 361 (1978), no. 1704, 1-20. MR 0526801 (58 #26167)
  • [K] C. Krattenthaler, Advanced determinant calculus, The Andrews Festschrift (Maratea, 1998). Sém. Lothar. Combin. 42 (1999), Art. B42q, 67 pp. (electronic). MR 1701596 (2002i:05013)
  • [M] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., with contributions by A. Zelevinsky, Oxford Science Publications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. MR 1354144 (96h:05207)
  • [NS] M. Noumi and A. Sano, An infinite family of higher-order difference operators that commute with Ruijsenaars operators of type A.
    In preparation.
  • [OC] Neil O'Connell, Directed polymers and the quantum Toda lattice, Ann. Probab. 40 (2012), no. 2, 437-458. MR 2952082, https://doi.org/10.1214/10-AOP632
  • [OCY] Neil O'Connell and Marc Yor, Brownian analogues of Burke's theorem, Stochastic Process. Appl. 96 (2001), no. 2, 285-304. MR 1865759 (2002h:60175), https://doi.org/10.1016/S0304-4149(01)00119-3
  • [O1] Andrei Okounkov, Infinite wedge and random partitions, Selecta Math. (N.S.) 7 (2001), no. 1, 57-81. MR 1856553 (2002f:60019), https://doi.org/10.1007/PL00001398
  • [O2] A. Okounkov, $ {\rm BC}$-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials, Transform. Groups 3 (1998), no. 2, 181-207. MR 1628453 (99h:33061), https://doi.org/10.1007/BF01236432
  • [OO] Andrei Okounkov and Grigori Olshanski, Asymptotics of Jack polynomials as the number of variables goes to infinity, Internat. Math. Res. Notices 13 (1998), 641-682. MR 1636541 (2001i:05156), https://doi.org/10.1155/S1073792898000403
  • [OR] Andrei Okounkov and Nikolai Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003), no. 3, 581-603 (electronic). MR 1969205 (2004b:60033), https://doi.org/10.1090/S0894-0347-03-00425-9
  • [P] L. A. Petrov, A two-parameter family of infinite-dimensional diffusions on the Kingman simplex, Funktsional. Anal. i Prilozhen. 43 (2009), no. 4, 45-66 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 43 (2009), no. 4, 279-296. MR 2596654 (2011c:60259), https://doi.org/10.1007/s10688-009-0036-8
  • [R1] Eric M. Rains, Transformations of elliptic hypergeometric integrals, Ann. of Math. (2) 171 (2010), no. 1, 169-243. MR 2630038 (2011i:33046), https://doi.org/10.4007/annals.2010.171.169
  • [R2] Eric M. Rains, Limits of elliptic hypergeometric integrals, Ramanujan J. 18 (2009), no. 3, 257-306. MR 2495549 (2010b:33021), https://doi.org/10.1007/s11139-007-9055-3
  • [SS] T. Sasamoto and H. Spohn,
    One-dimensional KPZ equation: an exact solution and its universality,
    Phys. Rev. Lett. 104 (2010), 23.
  • [Se] Timo Seppäläinen, Scaling for a one-dimensional directed polymer with boundary conditions, Ann. Probab. 40 (2012), no. 1, 19-73. MR 2917766, https://doi.org/10.1214/10-AOP617
  • [Sh] Jun'ichi Shiraishi, A family of integral transformations and basic hypergeometric series, Comm. Math. Phys. 263 (2006), no. 2, 439-460. MR 2207651 (2007c:33019), https://doi.org/10.1007/s00220-005-1504-5
  • [V] Mirjana Vuletić, A generalization of MacMahon's formula, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2789-2804. MR 2471939 (2009m:05192), https://doi.org/10.1090/S0002-9947-08-04753-3
  • [W] S. Ole Warnaar, Bisymmetric functions, Macdonald polynomials and $ \mathfrak{sl}_3$ basic hypergeometric series, Compos. Math. 144 (2008), no. 2, 271-303. MR 2406113 (2009d:33051), https://doi.org/10.1112/S0010437X07003211
  • [Z] Andrey V. Zelevinsky, Representations of finite classical groups, A Hopf algebra approach, Lecture Notes in Mathematics, vol. 869, Springer-Verlag, Berlin-New York, 1981. MR 643482 (83k:20017)

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Additional Information

Alexei Borodin
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 – and – Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, Russia
Email: borodin@math.mit.edu

Ivan Corwin
Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027 – and – Clay Mathematics Institute, 10 Memorial Boulevard, Suite 902, Providence, Rhode Island 02903 – and – Institut Henri Poincaré, 11 Rue Pierre et Marie Curie, 75005 Paris, France
Email: ivan.corwin@gmail.com

Vadim Gorin
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 – and – Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, Russia
Email: vadicgor@gmail.com

Shamil Shakirov
Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720
Email: shakirov@itep.ru

DOI: https://doi.org/10.1090/tran/6359
Received by editor(s): June 10, 2013
Received by editor(s) in revised form: December 6, 2013, and December 16, 2013
Published electronically: June 18, 2015
Dedicated: To the memory of A. Zelevinsky
Article copyright: © Copyright 2015 American Mathematical Society

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