Directed polymers and the quantum Toda lattice

Annals of Probability

Volumn 40, Issue 2, 2012, Pages 437-458

  O'Connell, Neil ^a  

^a UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Random matrices; Whittaker functions

Indexed keywords

EID: 84860379569     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AOP632     Document Type: Article

References (62)

1. AMIR, G., CORWIN, I. and QUASTEL, J. (2011). Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Comm. Pure Appl. Math. 64 466-537.
2. BARYSHNIKOV, Y. (2001). GUEs and queues. Probab. Theory Related Fields 119 256-274. MR1818248
3. BAUDOIN, F. and O'CONNELL, N. (2011). Exponential functionals of Brownian motion and class one Whittaker functions. Ann. Inst. H. Poincaré Probab. Statist. 47 1096-1120.
4. BERENSTEIN, A. and KAZHDAN, D. (2000). Geometric and unipotent crystals. GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. Special Volume, Part I 188-236. MR1826254
5. BIANE, P., BOUGEROL, P. andO'CONNELL, N. (2005). Littelmann paths and Brownian paths. Duke Math. J. 130 127-167. MR2176549
6. BIANE, P., BOUGEROL, P. andO'CONNELL, N. (2009). Continuous crystal and Duistermaat-Heckman measure for Coxeter groups. Adv. Math. 221 1522-1583. MR2522427
7. BOUGEROL, P. and JEULIN, T. (2002). Paths in Weyl chambers and random matrices. Probab. Theory Related Fields 124 517-543. MR1942321
8. BUMP, D. (1984). Automorphic Forms on GL(3, R). Lecture Notes in Math. 1083. Springer, Berlin. MR0765698
9. BUMP, D. (1989). The Rankin Selberg method: A survey. In Number Theory, Trace Formulas and Discrete Groups (K. E. Aubert, E. Bombieri and D. Goldfeld, eds.) 49-109. Academic Press, New York. MR0993311
10. BUMP, D. and FRIEDBERG, S. (1990). The exterior square automorphic L-functions on GL(n). In Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part II (Ramat Aviv, 1989). Israel Math. Conf. Proc. 3 47-65. Weizmann, Jerusalem. MR1159108
11. BUMP, D. and HUNTLEY, J. (1995). Unramified Whittaker functions for GL(3, R). J. Anal. Math. 65 19-44. MR1335367
12. CALABRESE, P., LE DOUSSAL, P. and ROSSO, A. (2010). Free-energy distribution of the directed polymer at high temperature. EPL 90 20002.
13. DOTSENKO, V. (2010). Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers. J. Stat. Mech. P07010.
14. DOTSENKO, V. (2010). Bethe ansatz derivation of the Tracy-Widom distribution for onedimensional directed polymers. EPL 90 20003.
15. DOTSENKO, V. and KLUMOV, B. (2010). Bethe ansatz solution for one-dimensional directed polymers in random media. J. Stat. Mech. Theory Exp. 3 P03022, 42. MR2629606
16. DUBÉDAT, J. (2004). Reflected planar Brownian motions, intertwining relations and crossing probabilities. Ann. Inst. H. Poincaré Probab. Statist. 40 539-552. MR2086013
17. EVANS, M. and SWARTZ, T. (2000). Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford Univ. Press, Oxford. MR1859163
18. GERASIMOV, A., KHARCHEV, S. and LEBEDEV, D. (2004). Representation theory and quantum inverse scattering method: The open Toda chain and the hyperbolic Sutherland model. Int. Math. Res. Not. 17 823-854. MR2040074
19. GERASIMOV, A., KHARCHEV, S., LEBEDEV, D. OBLEZIN, S. (2006). On a Gauss-Givental representation of quantum Toda chain wave function. Int. Math. Res. Not. Art. ID 96489, 23. MR2219213
20. GERASIMOV, A., LEBEDEV, D. and OBLEZIN, S. (2008). Baxter operator and Archimedean Hecke algebra. Comm. Math. Phys. 284 867-896. MR2452597
21. GERASIMOV, A., LEBEDEV, D. and OBLEZIN, S. (2011). New integral representations of Whittaker functions for classical Lie groups. Unpublished manuscript. Available at arXiv:0705.2886.
22. GIVENTAL, A. (1997). Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture. In Topics in Singularity Theory. Amer. Math. Soc. Transl. Ser. 2 180 103-115. Amer. Math. Soc., Providence, RI. MR1767115
23. GRAVNER, J., TRACY, C. A. WIDOM, H. (2001). Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Statist. Phys. 102 1085-1132. MR1830441
24. HARTMAN, P. WATSON, G. S. (1974). "Normal" distribution functions on spheres and the modified Bessel functions. Ann. Probab. 2 593-607. MR0370687
25. HASHIZUME, M. (1982). Whittaker functions on semisimple Lie groups. Hiroshima Math. J. 12 259-293. MR0665496
26. ISHII, T. STADE, E. (2007). New formulas for Whittaker functions on GL(n, R). J. Funct. Anal. 244 289-314. MR2294485
27. JACQUET, H. (2004). Integral representation of Whittaker functions. In Contributions to Automorphic Forms, Geometry, and Number Theory (H. Hida, D. Ramakrishnan and F. Shahidi, eds.) 373-419. Johns Hopkins Univ. Press, Baltimore, MD. MR2058615
28. JOE, D. KIM, B. (2003). Equivariant mirrors and the Virasoro conjecture for flag manifolds. Int. Math. Res. Not. 15 859-882. MR1952524
29. JOHANSSON, K. (2004). Determinantal processes with number variance saturation. Comm. Math. Phys. 252 111-148. MR2103906
30. JONES, L. O'CONNELL, N. (2006). Weyl chambers, symmetric spaces and number variance saturation. ALEA Lat. Am. J. Probab. Math. Stat. 2 91-118. MR2249664
31. KHARCHEV, S. LEBEDEV, D. (1999). Integral representation for the eigenfunctions of a quantum periodic Toda chain. Lett. Math. Phys. 50 53-77. MR1751619
32. KHARCHEV, S. LEBEDEV, D. (2000). Eigenfunctions of GL(N, R) Toda chain: The Mellin-Barnes representation. JETP Lett. 71 235-238.
33. KHARCHEV, S. LEBEDEV, D. (2001). Integral representations for the eigenfunctions of quantum open and periodic Toda chains from the QISM formalism. J. Phys. A 34 2247-2258. MR1831292
34. KIRILLOV, A. N. (2001). Introduction to tropical combinatorics. In Physics and Combinatorics, 2000 (Nagoya) (A. N. Kirillov and N. Liskova, eds.) 82-150. World Sci. Publ., River Edge, NJ. MR1872253
35. KOSTANT, B. (1977). Quantisation and representation theory. In Representation Theory of Lie Groups, Proc. SRC/LMS Research Symposium, Oxford 1977. LMS Lecture Notes 34 287-316. Cambridge Univ. Press, Cambridge.
36. MATSUMOTO, H. YOR, M. (1999). A version of Pitman's 2M-X theorem for geometric Brownian motions. C. R. Acad. Sci. Paris Sér. I Math. 328 1067-1074. MR1696208
37. MATSUMOTO, H. YOR, M. (2000). An analogue of Pitman's 2M - X theorem for exponential Wiener functionals. I. A time-inversion approach. Nagoya Math. J. 159 125-166. MR1783567
38. MORIARTY, J. O'CONNELL, N. (2007). On the free energy of a directed polymer in a Brownian environment. Markov Process. Related Fields 13 251-266. MR2343849
39. NOUMI, M. YAMADA, Y. (2004). Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions. In Representation Theory of Algebraic Groups and Quantum Groups. Adv. Stud. Pure Math. 40 371-442. Math. Soc. Japan, Tokyo. MR2074600
40. O'CONNELL, N. (2003). A path-transformation for random walks and the Robinson-Schensted correspondence. Trans. Amer. Math. Soc. 355 3669-3697 (electronic). MR1990168
41. O'CONNELL, N. (2003). Random matrices, non-colliding processes and queues. In Séminaire de Probabilités XXXVI. Lecture Notes in Math. 1801 165-182. Springer, Berlin. MR1971584
42. O'CONNELL, N. YOR, M. (2001). Brownian analogues of Burke's theorem. Stochastic Process. Appl. 96 285-304. MR1865759
43. O'CONNELL, N. YOR, M. (2002). A representation for non-colliding random walks. Electron. Comm. Probab. 7 1-12 (electronic). MR1887169
44. O'CONNELL, N. WARREN, J. A multi-layer extension of the stochastic heat equation. Available at arXiv:1104.3509.
45. PITMAN, J. W. (1975). One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. in Appl. Probab. 7 511-526. MR0375485
46. PROLHAC, S. SPOHN, H. (2011). Two-point generating function of the free energy for a directed polymer in a random medium. J. Stat. Mech. P01031.
47. RIETSCH, K. (2006). A mirror construction for the totally nonnegative part of the Peterson variety. Nagoya Math. J. 183 105-142. MR2253887
48. ROGERS, L. C. G. PITMAN, J. W. (1981). Markov functions. Ann. Probab. 9 573-582. MR0624684
49. SASAMOTO, T. SPOHN, H. (2010). Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B 834 523-542. MR2628936
50. SASAMOTO, T. SPOHN, H. (2010). One-dimensional Kardar-Parisi-Zhang equation: An exact solution and its universality. Phys. Rev. Lett. 104 230602.
51. SASAMOTO, T. SPOHN, H. (2010). The crossover regime for the weakly asymmetric simple exclusion process. J. Stat. Phys. 140 209-231. MR2659278
52. SASAMOTO, T. SPOHN, H. (2010). The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class. J. Stat. Mech. P11013.
53. SEMENOV-TIAN-SHANSKY, M. (1994). Quantisation of open Toda lattices. In Dynamical Systems VII: Integrable Systems, Nonholonomic Dynamical Systems (V. I. Arnol'd and S. P. Novikov, eds.). Encyclopaedia of Mathematical Sciences 16 116-225. Springer, Berlin.
54. SEPPÄLÄINEN, T. VALKÓ, B. (2010). Bounds for scaling exponents for a 1 + 1 dimensional directed polymer in a Brownian environment. ALEA Lat. Am. J. Probab. Math. Stat. 7 451-476.
55. STADE, E. (2001). Mellin transforms of GL(n, R) Whittaker functions. Amer. J. Math. 123 121-161. MR1827280
56. STADE, E. (2002). Archimedean L-factors on GL(n) × GL(n) and generalized Barnes integrals. Israel J. Math. 127 201-219. MR1900699
57. TRACY, C. A. andWIDOM, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151-174. MR1257246
58. TRACY, C. A. andWIDOM, H. (2008). Integral formulas for the asymmetric simple exclusion process. Comm. Math. Phys. 279 815-844. MR2386729
59. TRACY, C. A. and WIDOM, H. (2008). A Fredholm determinant representation in ASEP. J. Stat. Phys. 132 291-300. MR2415104
60. TRACY, C. A. and WIDOM, H. (2009). Asymptotics in ASEP with step initial condition. Comm. Math. Phys. 290 129-154. MR2520510
61. TRACY, C. A. and WIDOM, H. (2010). Formulas for joint probabilities for the asymmetric simple exclusion process. J. Math. Phys. 51 063302, 10. MR2675194
62. WARREN, J. (2007). Dyson's Brownian motions, intertwining and interlacing. Electron. J. Probab. 12 573-590. MR2299928