Macdonald processes

Probability Theory and Related Fields

Volumn 158, Issue 1-2, 2014, Pages 225-400

  Borodin, Alexei ^a   Corwin, Ivan ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84892877156     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-013-0482-3     Document Type: Article

References (113)

1. Alberts, T., Khanin, K., Quastel, J.: The intermediate disorder regime for directed polymers in dimension 1+1. Phys. Rev. Lett. 105, 090603 (2010).
2. Alberts, T., Khanin, K., Quastel, J.: Intermediate disorder regime for 1+1 dimensional directed polymers. arXiv: 1202. 4398.
3. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable models in quantum mechanics, 2nd edn. AMS, Providence (2005).
4. Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions. Commun. Pure Appl. Math. 64, 466-537 (2011).
5. Andjel, E. D.: Invariant measures for the zero range process. Ann. Probab. 10, 525-547 (1982).
6. Andrews, G., Askey, R., Roy, R.: Special functions. Cambridge University Press, Cambridge (2000).
7. Awata, H., Odake, S., Shiraishi, J.: Integral representations of the Macdonald symmetric functions. Commun. Math. Phys. 179, 666-747 (1996).
8. Baik, J., Ben Arous, G., Péché, S.: Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Ann. Probab. 33, 1643-1697 (2005).
9. Balázs, M., Quastel, J., Seppäläinen, T.: Scaling exponent for the Hopf-Cole solution of KPZ/stochastic Burgers. J. Am. Math. Soc. 24, 683-708 (2011).
10. Balazs, M., Seppalainen, T.: Order of current variance and diffusivity in the asymmetric simple exclusion process. Ann. Math. 171, 1237-1265 (2010).
11. Balázs, M., Komjáthy, J., Seppäläinen, T.: Microscopic concavity and fluctuation bounds in a class of deposition processes. Ann. Inst. H. Poincaré B 48, 151-187 (2012).
12. Bangerezako, G.: An introduction to q-difference equations. http://perso. uclouvain. be/alphonse. magnus/gbang/qbook712. pdf. Accessed Oct 2011.
13. Betea, D.: Elliptically distributed lozenge tilings of a hexagon. arXiv: 1110. 4176.
14. Bertini, L., Cancrini, N.: The stochastic heat equation: Feynman-Kac formula and intermittence. J. Stat. Phys. 78, 1377-1401 (1995).
15. Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571-607 (1997).
16. Bolthausen, E.: A note on diffusion of directed polymers in a random environment. Commun. Math. Phys. 123, 529-534 (1989).
17. Bornemann, F.: On the numerical evaluation of Fredholm determinants. Math. Comp. 79, 871-915 (2010).
18. Borodin, A.: Schur dynamics of the Schur processes. Adv. Math. 228, 2268-2291 (2011).
19. Borodin, A., Corwin, I.: On moments of the parabolic Anderson model. arXiv: 1211. 7125.
20. Borodin, A., Corwin, I., Ferrari, P. L.: Free energy fluctuations for directed polymers in random media in 1+1 dimension. arXiv: 1204. 1024.
21. Borodin, A., Corwin, I., Gorin, V., Shakirov, S.: (2013).
22. Borodin, A., Corwin, I., Ferrari, P. L., Veto, B.: (2013).
23. Borodin, A., Corwin, I., Remenik, D.: Log-Gamma polymer free energy fluctuations via a Fredholm determinant, identity. arXiv: 1206. 4573.
24. Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for q-TASEP and ASEP. arXiv: 1207. 5035.
25. Borodin, A., Duits, M.: Limits of determinantal processes near a tacnode. Ann. Inst. H. Poincare B 47, 243-258 (2011).
26. Borodin, A., Ferrari, P. L.: Anisotropic growth of random surfaces in 2+1 dimensions. arXiv: 0804. 3035.
27. Borodin, A., Gorin, V.: Markov processes of infinitely many nonintersecting random walks. Probab. Theory Relat. Fields. doi: 10. 1007/s00440-012-0417-4 (2013).
28. Borodin, A., Gorin, V.: Shuffling algorithm for boxed plane partitions. Adv. Math. 220, 1739-1770 (2009).
29. Borodin, A., Kuan, J.: Random surface growth with a wall and Plancherel measures for O(infinity). Commun. Pure Appl. Math. 63, 831-894 (2010).
30. Borodin, A., Olshanski, G.: Markov processes on the path space of the Gelfand-Tsetlin graph and on its boundary. arXiv: 1009. 2029.
31. Borodin, A., Rains, E.: Eynard-Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121, 291-317 (2005).
32. Calabrese, P., Caux, J. S.: Dynamics of the attractive 1D Bose gas: analytical treatment from integrability. J. Stat. Mech. P08032 (2007).
33. Calabrese, P., Le Doussal, P., Rosso, A.: Free-energy distribution of the directed polymer at high temperature. Euro. Phys. Lett. 90, 20002 (2010).
34. Carmona, R., Molchanov, S.: Parabolic Anderson problem and intermittency. Memoirs of AMS, vol. 518 (1994).
35. Comets, F., Shiga, T., Yoshida, N.: Probabilistic analysis of directed polymers in a random environment: a review. In: Funaki, T., Osada, H. (eds.) Stochastic Analysis on Large Scale Interacting Systems, pp. 115-142. Math. Soc. Japan, Tokyo (2004).
36. Comets, F., Yoshida, N.: Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 5, 1746-1770 (2006).
37. Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. arXiv: 1106. 1596.
38. Corwin, I., Hammond, A.: The H-Brownian Gibbs property of the KPZ line ensemble (2013).
39. Corwin, I., O'Connell, N., Seppäläinen, T., Zygouras, N.: Tropical combinatorics and Whittaker functions. arXiv: 1110. 3489.
40. Corwin, I., Quastel, J.: Crossover distributions at the edge of the rarefaction fan. Ann. Probab. (to appear).
41. Corwin, I., Quastel, J.: Renormalization fixed point of the KPZ universality class. arXiv: 1103. 3422.
42. De Masi, A., Presutti, E., Spohn, H., Wick, W. D.: Asymptotic equivalence of fluctuation fields for reversible exclusion processes with speed change. Ann. Probab. 14, 409-423 (1986).
43. Diaconis, P., Fill, J. A.: Strong stationary times via a new form of duality. Ann. Probab. 18, 1483-1522 (1990).
44. Dorlas, T. C.: Orthogonality and completeness of the Bethe ansatz eigenstates of the nonlinear Schroedinger model. Comm. Math. Phys. 154, 347-376 (1993).
45. Dotsenko, V.: Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers. Euro. Phys. Lett. 90, 20003 (2010).
46. Etingof, P.: Whittaker functions on quantum groups and q-deformed Toda operators. Amer. Math. Soc. Transl. Ser. 2, vol. 194, pp. 9-25. AMS, Providence (1999).
47. Forrester, P. J.: The spectrum edge of random matrix ensembles. Nucl. Phys. B 402, 709-728 (1993).
48. Forrester, P. J., Rains, E.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theo. Rel. Fields 131, 1-61 (2005).
49. Gärtner, J.: Convergence towards Burgers equation and propagation of chaos for weakly asymmetric exclusion process. Stoch. Proc. Appl. 27, 233-260 (1988).
50. Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a Gauss-Givental representation for quantum Toda chain wave function. Int. Math. Res. Notices 96489 (2006).
51. Gerasimov, A., Lebedev, D., Oblezin, S.: On a classical limit of q-deformed Whittaker functions. arXiv: 1101. 4567.
52. Gerasimov, A., Lebedev, D., Oblezin, S.: Baxter operator and Archimedean Hecke algebra. Commun. Math. Phys. 284, 867-896 (2008).
53. ℓ+1-Whittaker function I. Commun. Math. Phys. 294, 97-119 (2010).
54. ℓ+1-Whittaker function III arXiv: 0805. 3754.
55. Givental, A.: Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture. Topics in Singularity Theory., AMS Transl. Ser. 2, vol. 180, pp. 103-115, AMS, Providence (1997).
56. Gohberg, I. C., Krein, M. G.: Introduction to the theory of linear nonselfadjoint operators in Hilbert space. AMS Transl. (1969).
57. Hahn, W.: Beiträge zur Theorie der Heineschen Reihen. Die 24 Integrale der hypergeometrischen q-Differenzengleichung. Das q-Analogon der Laplace-Transformation. Mathematische Nachrichten 2, 340-379 (1949).
58. Harris, T. E.: Additive set-valued Markov processes and graphical methods. Ann. Probab. 6, 355-378 (1978).
59. Heckman, G. J., Opdam, E. M.: Yang's system of particles and Hecke algebras. Ann. Math. 145, 139-173 (1997).
60. Hille, E.: Analytic function theory, vol. II. Giin and Co., Chelsea New York (1977). Republished.
61. Huse, D. A., Henley, C. L.: Pinning and roughening of domain walls in Ising systems due to random impurities. Phys. Rev. Lett. 54, 2708-2711 (1985).
62. Imbrie, J. Z., Spencer, T.: Diffusion of directed polymers in a random environment. J. Stat. Phys. 52, 609-626 (1988).
63. Ishii, T., Stade, E.: New formulas for Whittaker functions on GL(n, ℝ). J. Funct. Anal. 244, 289-314 (2007).
64. Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437-476 (2000).
65. Johansson, K.: Random growth and random matrices. European Congress of Mathematics, vol. I (Barcelona, 2000), 445-456, Progr. Math., vol. 201, Birkhäuser, Basel (2001).
66. Kardar, M.: Replica-Bethe Ansatz studies of two-dimensional interfaces with quenched random impurities. Nucl. Phys. B 290, 582-602 (1987).
67. Kardar, K., Parisi, G., Zhang, Y. Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889-892 (1986).
68. Kerov, S.: Asymptotic representation theory of the symmetric group and its applications in analysis. Translations of mathematical monographs, vol. 219, AMS, Providence (2003).
69. Kerov, S., Okounkov, A., Olshanski, G.: The boundary of the Young graph with Jack edge multiplicities. Int. Math. Res. Notices 173-199 (1998).
70. Kharchev, S., Lebedev, D.: Integral representations for the eigenfunctions of quantum open and periodic Toda chains from the QISM formalism. J. Phys. A 34, 2247-2258 (2001).
71. Kirillov, A. N.: Introduction to tropical combinatorics. In: Kirillov, A. N., Liskova, N. (eds.) Physics and Combinatorics. Proc. Nagoya 2000 2nd Internat. Workshop. World Scientific, Singapore, 82-150, (2001).
72. Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. Spinger, Berlin (1999).
73. Kostant, B.: Quantisation and representation theory. In: Proc. SRC/LMS Research Symposium on Representation Theory of Lie Groups, Oxford. LMS Lecture Notes 34, pp. 287-316. Cambridge University Press, Cambridge (1977).
74. Lax, P. D.: Functional analysis. Wiley, Newyork (2002).
75. Lieb, E. H., Liniger, W.: Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State. Phys. Rev. Lett. 130, 1605-1616 (1963).
76. Liggett, T.: Continuous time Markov processes: an introduction. In: Graduate Studied in Mathematics, vol. 113, AMS, Providence (2010).
77. Liggett, T.: Interacting particle systems. Spinger, Berlin (2005).
78. Liggett, T.: An infinite particle system with zero range interactions. Ann. Probab. 1, 240-253 (1973).
79. Macdonald, I. G.: Symmetric functions and hall polynomials, 2nd edn. Oxford University Press, New York (1999).
80. McGuire, J. B.: Study of exactly soluble one-dimensional N-body problems. J. Math. Phys. 5, 622 (1964).
81. Moreno Flores, G., Quastel, J.: Intermediate disorder for the O'Connell-Yor model (2013).
82. Moriarty, J., O'Connell, N.: On the free energy of a directed polymer in a Brownian environment. Markov Process Relat. Fields 13, 251-266 (2007).
83. Mueller, C.: On the support of solutions to the heat equation with noise. Stochastics 37, 225-246 (1991).
84. Nagao, T., Wadati, M.: Eigenvalue distribution of random matrices at the spectrum edge. J. Phys. Soc. Jpn. 62, 3845-3856 (1993).
85. Noumi, M., Yamada, Y.: Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions. Representation theory of algebraic groups and quantum groups. In: Adv. Stud. Pure Math., vol. 40, pp. 371-442. Math. Soc. Japan, Tokyo (2004).
86. O'Connell, N.: Directed polymers and the quantum Toda lattice. Ann. Probab. 40, 437-458 (2012).
87. O'Connell, N.: Whittaker functions and related stochastic processes. arXiv: 1201. 4849.
88. O'Connell, N., Warren, J.: A multi-layer extension of the stochastic heat equation arXiv: 1104. 3509.
89. O'Connell, N., Yor, M.: Brownian analogues of Burke's theorem. Stoch. Proc. Appl. 96, 285-304 (2001).
90. Okounkov, A.: (Shifted) Macdonald polynomials: q-Integral representation and combinatorial formula. Compos. Math. 112, 147-182 (1998).
91. Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Am. Math. Soc. 16, 581-603 (2003).
92. Okounkov, A.: Infinite wedge and random partitions. Selecta Math. 7, 57-81 (2001).
93. Oxford, S.: The Hamiltonian of the quantized nonlinear Schrödinger equation. Ph. D. thesis, UCLA, (1979).
94. Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071-1106 (2002).
95. Prolhac, S., Spohn, H.: The propagator of the attractive delta-Bose gas in one dimension. J. Math. Phys. 52, 122106 (2011).
96. Ruijsenaars, S. N. M.: Relativistic Toda systems. Commun. Math. Phys. 133, 217-247 (1990).
97. Sasamoto, T., Spohn, H.: One-dimensional KPZ equation: an exact solution and its universality. Phys. Rev. Lett. 104, 23 (2010).
98. Sasamoto, T., Wadati, M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A 31, 6057-6071 (1998).
99. Semenov-Tian-Shansky, M.: Quantisation of open Toda lattices. Dynamical systems VII: integrable systems, nonholonomic dynamical systems. In: Arnol'd, V. I., Novikov, S. P. (eds.) Encyclopaedia of mathematical sciences, vol. 16. Springer, Berlin (1994).
100. Seppäläinen, T.: Existence of hydrodynamics for the totally asymmetric simple K-exclusion process. Ann. Probab. 27, 361-415 (1999).
101. Seppäläinen, T.: Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40, 19-73 (2012).
102. Seppäläinen, T., Valko, B.: Bounds for scaling exponents for a 1+1 dimensional directed polymer in a Brownian environment. Alea 7, 451-476 (2010).
103. Simon, B.: Trace ideals and their applications. AMS, Providence (2005).
104. Spohn, H.: KPZ scaling theory and the semi-discrete directed polymer model. arXiv: 1201. 0645.
105. Tracy, C., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151-174 (1994).
106. Tracy, C., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279, 815-844 (2008) (Erratum. Commun. Math. Phys. 304, 875-878 (2011)).
107. Tracy, C., Widom, H.: A Fredholm determinant representation in ASEP. J. Stat. Phys. 132, 291-300 (2008).
108. Tracy, C., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129-154 (2009).
109. Tracy, C., Widom, H.: Formulas for ASEP with two-sided Bernoulli initial condition. J. Stat. Phys. 140, 619-634 (2010).
110. Tracy, C., Widom, H.: The dynamics of the one-dimensional delta-function Bose gas. J. Phys. A 41, 485204 (2008).
111. Vuletic, M.: A generalization of MacMahon's formula. Trans. Am. Math. Soc. 361, 2789-2804 (2009).
112. Wallach, N.: Real reductive groups II. Academic Press, San Diego (1992).
113. Warren, J.: Dyson's Brownian motions, intertwining and interlacing. Electr. J. Probab. 12, 573-590 (2007).