The Annals of Probability

Directed polymers and the quantum Toda lattice

Neil O’Connell

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Abstract

We characterize the law of the partition function of a Brownian directed polymer model in terms of a diffusion process associated with the quantum Toda lattice. The proof is via a multidimensional generalization of a theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion. It is based on a mapping which can be regarded as a geometric variant of the RSK correspondence.

Article information

Source
Ann. Probab., Volume 40, Number 2 (2012), 437-458.

Dates
First available in Project Euclid: 26 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1332772711

Digital Object Identifier
doi:10.1214/10-AOP632

Mathematical Reviews number (MathSciNet)
MR2952082

Zentralblatt MATH identifier
1245.82091

Subjects
Primary: 15A52 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.) 60J65: Brownian motion [See also 58J65] 82D60: Polymers

Keywords
Random matrices Whittaker functions

Citation

O’Connell, Neil. Directed polymers and the quantum Toda lattice. Ann. Probab. 40 (2012), no. 2, 437--458. doi:10.1214/10-AOP632. https://projecteuclid.org/euclid.aop/1332772711


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References

  • [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.
  • [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.
  • [5] Biane, P., Bougerol, P. and O’Connell, N. (2005). Littelmann paths and Brownian paths. Duke Math. J. 130 127–167.
  • [6] Biane, P., Bougerol, P. and O’Connell, N. (2009). Continuous crystal and Duistermaat–Heckman measure for Coxeter groups. Adv. Math. 221 1522–1583.
  • [7] Bougerol, P. and Jeulin, T. (2002). Paths in Weyl chambers and random matrices. Probab. Theory Related Fields 124 517–543.
  • [8] Bump, D. (1984). Automorphic Forms on GL(3, ℝ). Lecture Notes in Math. 1083. Springer, Berlin.
  • [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.
  • [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.
  • [11] Bump, D. and Huntley, J. (1995). Unramified Whittaker functions for GL(3, ℝ). J. Anal. Math. 65 19–44.
  • [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 one-dimensional 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.
  • [16] Dubédat, J. (2004). Reflected planar Brownian motions, intertwining relations and crossing probabilities. Ann. Inst. H. Poincaré Probab. Statist. 40 539–552.
  • [17] Evans, M. and Swartz, T. (2000). Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford Univ. Press, Oxford.
  • [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.
  • [19] Gerasimov, A., Kharchev, S., Lebedev, D. and Oblezin, S. (2006). On a Gauss–Givental representation of quantum Toda chain wave function. Int. Math. Res. Not. Art. ID 96489, 23.
  • [20] Gerasimov, A., Lebedev, D. and Oblezin, S. (2008). Baxter operator and Archimedean Hecke algebra. Comm. Math. Phys. 284 867–896.
  • [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.
  • [23] Gravner, J., Tracy, C. A. and Widom, H. (2001). Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Statist. Phys. 102 1085–1132.
  • [24] Hartman, P. and Watson, G. S. (1974). “Normal” distribution functions on spheres and the modified Bessel functions. Ann. Probab. 2 593–607.
  • [25] Hashizume, M. (1982). Whittaker functions on semisimple Lie groups. Hiroshima Math. J. 12 259–293.
  • [26] Ishii, T. and Stade, E. (2007). New formulas for Whittaker functions on GL(n, ℝ). J. Funct. Anal. 244 289–314.
  • [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.
  • [28] Joe, D. and Kim, B. (2003). Equivariant mirrors and the Virasoro conjecture for flag manifolds. Int. Math. Res. Not. 15 859–882.
  • [29] Johansson, K. (2004). Determinantal processes with number variance saturation. Comm. Math. Phys. 252 111–148.
  • [30] Jones, L. and O’Connell, N. (2006). Weyl chambers, symmetric spaces and number variance saturation. ALEA Lat. Am. J. Probab. Math. Stat. 2 91–118.
  • [31] Kharchev, S. and Lebedev, D. (1999). Integral representation for the eigenfunctions of a quantum periodic Toda chain. Lett. Math. Phys. 50 53–77.
  • [32] Kharchev, S. and Lebedev, D. (2000). Eigenfunctions of GL(N, ℝ) Toda chain: The Mellin–Barnes representation. JETP Lett. 71 235–238.
  • [33] Kharchev, S. and 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.
  • [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.
  • [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. and Yor, M. (1999). A version of Pitman’s 2MX theorem for geometric Brownian motions. C. R. Acad. Sci. Paris Sér. I Math. 328 1067–1074.
  • [37] Matsumoto, H. and Yor, M. (2000). An analogue of Pitman’s 2MX theorem for exponential Wiener functionals. I. A time-inversion approach. Nagoya Math. J. 159 125–166.
  • [38] Moriarty, J. and O’Connell, N. (2007). On the free energy of a directed polymer in a Brownian environment. Markov Process. Related Fields 13 251–266.
  • [39] Noumi, M. and 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.
  • [40] O’Connell, N. (2003). A path-transformation for random walks and the Robinson–Schensted correspondence. Trans. Amer. Math. Soc. 355 3669–3697 (electronic).
  • [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.
  • [42] O’Connell, N. and Yor, M. (2001). Brownian analogues of Burke’s theorem. Stochastic Process. Appl. 96 285–304.
  • [43] O’Connell, N. and Yor, M. (2002). A representation for non-colliding random walks. Electron. Comm. Probab. 7 1–12 (electronic).
  • [45] Pitman, J. W. (1975). One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. in Appl. Probab. 7 511–526.
  • [46] Prolhac, S. and 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.
  • [48] Rogers, L. C. G. and Pitman, J. W. (1981). Markov functions. Ann. Probab. 9 573–582.
  • [49] Sasamoto, T. and Spohn, H. (2010). Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B 834 523–542.
  • [50] Sasamoto, T. and Spohn, H. (2010). One-dimensional Kardar–Parisi–Zhang equation: An exact solution and its universality. Phys. Rev. Lett. 104 230602.
  • [51] Sasamoto, T. and Spohn, H. (2010). The crossover regime for the weakly asymmetric simple exclusion process. J. Stat. Phys. 140 209–231.
  • [52] Sasamoto, T. and 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. and 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.
  • [56] Stade, E. (2002). Archimedean L-factors on GL(n) × GL(n) and generalized Barnes integrals. Israel J. Math. 127 201–219.
  • [57] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.
  • [58] Tracy, C. A. and Widom, H. (2008). Integral formulas for the asymmetric simple exclusion process. Comm. Math. Phys. 279 815–844.
  • [59] Tracy, C. A. and Widom, H. (2008). A Fredholm determinant representation in ASEP. J. Stat. Phys. 132 291–300.
  • [60] Tracy, C. A. and Widom, H. (2009). Asymptotics in ASEP with step initial condition. Comm. Math. Phys. 290 129–154.
  • [61] Tracy, C. A. and Widom, H. (2010). Formulas for joint probabilities for the asymmetric simple exclusion process. J. Math. Phys. 51 063302, 10.
  • [62] Warren, J. (2007). Dyson’s Brownian motions, intertwining and interlacing. Electron. J. Probab. 12 573–590.