Observables of Macdonald processes

Borodin, Alexei; Corwin, Ivan; Gorin, Vadim; Shakirov, Shamil

doi:10.1090/tran/6359

Observables of Macdonald processes
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by Alexei Borodin, Ivan Corwin, Vadim Gorin and Shamil Shakirov PDF

Trans. Amer. Math. Soc. 368 (2016), 1517-1558 Request permission

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.

References

A. Aggrawal, Correlation functions of the Schur process through Macdonald difference operators. In preparation.
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, DOI 10.1002/cpa.20347
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, DOI 10.1007/BF01077476
Alexei Borodin and Ivan Corwin, Macdonald processes, Probab. Theory Related Fields 158 (2014), no. 1-2, 225–400. MR 3152785, DOI 10.1007/s00440-013-0482-3
A. Borodin and I. Corwin, Discrete time $q$–TASEPs, Int. Math Rev. Not., to appear, arXiv:1305.2972.
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, DOI 10.1002/cpa.21520
A. Borodin, I. Corwin, P. L. Ferrari, and B. Vető, Stationary solution of 1d KPZ equation. In preparation.
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, DOI 10.1007/s00220-013-1750-x
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, DOI 10.1214/13-AOP868
A. Borodin and V. Gorin, Lectures on integrable probability, arXiv:1212.3351.
A. Borodin and V. Gorin, General $\beta$ Jacobi corners process and the Gaussian free field, arXiv:1305.3627.
Alexei Borodin and Grigori Olshanski, $Z$-measures on partitions and their scaling limits, European J. Combin. 26 (2005), no. 6, 795–834. MR 2143199, DOI 10.1016/j.ejc.2004.06.003
A. Borodin and L. Petrov, Nearest neighbor Markov dynamics on Macdonald processes, arXiv:1305.5501.
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, DOI 10.1215/00127094-2410289
I. Corwin and L. Petrov, The q-PushASEP: A new integrable model for traffic in $1+1$ dimension, arXiv:1308.3124.
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., DOI 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)
B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, and S. Yanagida, A commutative algebra on degenerate $\Bbb {CP}^1$ and Macdonald polynomials, J. Math. Phys. 50 (2009), no. 9, 095215, 42. MR 2566895, DOI 10.1063/1.3192773
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, DOI 10.1007/s00440-004-0375-6
J. Fulman, Probabilistic measures and algorithms arising from the Macdonald symmetric functions, arXiv:math/971223.
Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51–85. MR 1864086, DOI 10.1090/S0273-0979-01-00920-X
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, DOI 10.1016/j.aim.2013.12.028
V. Gorin and M. Shkolnikov, Multilevel Dyson Brownian Motions via Jack polynomials. In preparation.
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, DOI 10.1007/bf02362775
S. V. Kerov, Asymptotic representation theory of the symmetric group and its applications in analysis, Translations of Mathematical Monographs, vol. 219, American Mathematical Society, Providence, RI, 2003. Translated from the Russian manuscript by N. V. Tsilevich; With a foreword by A. Vershik and comments by G. Olshanski. MR 1984868, DOI 10.1090/mmono/219
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, DOI 10.1155/S1073792898000154
J. F. C. Kingman, Random partitions in population genetics, Proc. Roy. Soc. London Ser. A 361 (1978), no. 1704, 1–20. MR 526801, DOI 10.1098/rspa.1978.0089
C. Krattenthaler, Advanced determinant calculus, Sém. Lothar. Combin. 42 (1999), Art. B42q, 67. The Andrews Festschrift (Maratea, 1998). MR 1701596
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
M. Noumi and A. Sano, An infinite family of higher-order difference operators that commute with Ruijsenaars operators of type A. In preparation.
Neil O’Connell, Directed polymers and the quantum Toda lattice, Ann. Probab. 40 (2012), no. 2, 437–458. MR 2952082, DOI 10.1214/10-AOP632
Neil O’Connell and Marc Yor, Brownian analogues of Burke’s theorem, Stochastic Process. Appl. 96 (2001), no. 2, 285–304. MR 1865759, DOI 10.1016/S0304-4149(01)00119-3
Andrei Okounkov, Infinite wedge and random partitions, Selecta Math. (N.S.) 7 (2001), no. 1, 57–81. MR 1856553, DOI 10.1007/PL00001398
A. Okounkov, $\textrm {BC}$-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials, Transform. Groups 3 (1998), no. 2, 181–207. MR 1628453, DOI 10.1007/BF01236432
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, DOI 10.1155/S1073792898000403
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. MR 1969205, DOI 10.1090/S0894-0347-03-00425-9
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, DOI 10.1007/s10688-009-0036-8
Eric M. Rains, Transformations of elliptic hypergeometric integrals, Ann. of Math. (2) 171 (2010), no. 1, 169–243. MR 2630038, DOI 10.4007/annals.2010.171.169
Eric M. Rains, Limits of elliptic hypergeometric integrals, Ramanujan J. 18 (2009), no. 3, 257–306. MR 2495549, DOI 10.1007/s11139-007-9055-3
T. Sasamoto and H. Spohn, One-dimensional KPZ equation: an exact solution and its universality, Phys. Rev. Lett. 104 (2010), 23.
Timo Seppäläinen, Scaling for a one-dimensional directed polymer with boundary conditions, Ann. Probab. 40 (2012), no. 1, 19–73. MR 2917766, DOI 10.1214/10-AOP617
Jun’ichi Shiraishi, A family of integral transformations and basic hypergeometric series, Comm. Math. Phys. 263 (2006), no. 2, 439–460. MR 2207651, DOI 10.1007/s00220-005-1504-5
Mirjana Vuletić, A generalization of MacMahon’s formula, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2789–2804. MR 2471939, DOI 10.1090/S0002-9947-08-04753-3
S. Ole Warnaar, Bisymmetric functions, Macdonald polynomials and $\mathfrak {sl}_3$ basic hypergeometric series, Compos. Math. 144 (2008), no. 2, 271–303. MR 2406113, DOI 10.1112/S0010437X07003211
Andrey V. Zelevinsky, Representations of finite classical groups, Lecture Notes in Mathematics, vol. 869, Springer-Verlag, Berlin-New York, 1981. A Hopf algebra approach. MR 643482