×

zbMATH — the first resource for mathematics

Probability distribution of the free energy of the continuum directed random polymer in \(1 + 1\) dimensions. (English) Zbl 1222.82070
The fully stochastic version of the Kardar-Parisi-Zhang equation used to model a stochastic growth of one-dimensional interface, while driven by the Gaussian space-time white noise, is still far from being a satisfactory theory, specifically since it is restricted to an equilibrium regime. The main goal of the present article is to address the far from equilibrium regime, which can be most conveniently stated in terms of a stochastic heat equation with delta function initial condition. The main result is an exact formula for the probability distribution for the free energy of the continuum directed random polymer in \(1+1\) dimensions. Told otherwise one obtains the one-point distribution for the pertinent stochastic heat equation, or for the KPZ equation with narrow wedge initial conditions. Explicit formulas are obtained for the one-dimensional crossover (marginal) distributions which interpolate between standard Gaussian distribution for small times and the GUE Tracy-Widom one for large times. The proof heavily relies on a rigorous steepest descent analysis of the Tracy-Widom formula for the antisymmetric simple exclusion process with antishock initial data.

MSC:
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82D60 Statistical mechanical studies of polymers
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
15B52 Random matrices (algebraic aspects)
35Q53 KdV equations (Korteweg-de Vries equations)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Balázs , M. Quastel , J. Seppäläinen , T. Scaling exponent for the Hopf-Cole solution of KPZ/stochastic Burgers 2009
[2] Bertini, The stochastic heat equation: Feynman-Kac formula and intermittence, J. Statist. Phys. 78 pp 1377– (1995) · Zbl 1080.60508 · doi:10.1007/BF02180136
[3] Bertini, Stochastic Burgers and KPZ equations from particle systems, Comm. Math. Phys. 183 pp 571– (1997) · Zbl 0874.60059 · doi:10.1007/s002200050044
[4] Billingsley, Wiley Series in Probability and Statistics: Probability and Statistics, in: Convergence of probability measures (1999) · Zbl 0944.60003 · doi:10.1002/9780470316962
[5] Borodin, Fredholm determinants, Jimbo-Miwa-Ueno \(\tau\)-functions, and representation theory, Comm. Pure Appl. Math. 55 pp 1160– (2002) · Zbl 1033.34089 · doi:10.1002/cpa.10042
[6] Borodin, Large time asymptotics of growth models on space-like paths. I. PushASEP, Electron. J. Probab. 13 pp 1380– (2008) · Zbl 1187.82084 · doi:10.1214/EJP.v13-541
[7] Borodin, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 pp 481– (2000) · Zbl 0938.05061 · doi:10.1090/S0894-0347-00-00337-4
[8] Calabrese, Free-energy distribution of the directed polymer at high temperature, Euro. Phys. Lett. 90 pp 20002– (2010) · doi:10.1209/0295-5075/90/20002
[9] Chan, Scaling limits of Wick ordered KPZ equation, Comm. Math. Phys. 209 pp 671– (2000) · Zbl 0956.60077 · doi:10.1007/PL00020963
[10] Corwin, Limit processes for TASEP with shocks and rarefaction fans, J. Stat. Phys. 140 pp 232– (2010) · Zbl 1197.82078 · doi:10.1007/s10955-010-9995-7
[11] Deift, Courant Lecture Notes in Mathematics, 3, in: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach (1999) · Zbl 0997.47033
[12] Dotsenko, Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers, Euro. Phys. Lett. 90 pp 20003– (2010) · doi:10.1209/0295-5075/90/20003
[13] Forster, Large-distance and long-time properties of a randomly stirred fluid, Phys. Rev. A (3) 16 pp 732– (1977) · doi:10.1103/PhysRevA.16.732
[14] Gärtner, Convergence towards Burgers’ equation and propagation of chaos for weakly asymmetric exclusion processes, Stochastic Process. Appl. 27 pp 233– (1988) · Zbl 0643.60094 · doi:10.1016/0304-4149(87)90040-8
[15] Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 pp 437– (2000) · Zbl 0969.15008 · doi:10.1007/s002200050027
[16] Johansson, Discrete polynuclear growth and determinantal processes, Comm. Math. Phys. 242 pp 277– (2003) · Zbl 1031.60084 · doi:10.1007/s00220-003-0945-y
[17] Kardar, Replica Bethe Ansatz studies of two-dimensional interfaces with quenched random impurities, Nuclear Phys. B 290 pp 582– (1987) · doi:10.1016/0550-3213(87)90203-3
[18] Kardar, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 pp 889– (1986) · Zbl 1101.82329 · doi:10.1103/PhysRevLett.56.889
[19] Kolokolov, Universal and nonuniversal tails of distribution functions in the directed polymer and Kardar-Parisi-Zhang problems, Phys. Rev. B 78 pp 024206– (2008) · doi:10.1103/PhysRevB.78.024206
[20] Konno, Stochastic partial differential equations for some measure-valued diffusions, Probab. Theory Related Fields 79 pp 201– (1988) · Zbl 0631.60058 · doi:10.1007/BF00320919
[21] Lieb, Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Phys. Rev. 130 pp 1605– (1963) · Zbl 0138.23001 · doi:10.1103/PhysRev.130.1605
[22] Liggett, Reprint of the 1985 original. Classics in Mathematics, in: Interacting particle systems (2005)
[23] Prähofer, Scale invariance of the PNG droplet and the Airy process, J. Statist. Phys. 108 pp 1071– (2002) · Zbl 1025.82010 · doi:10.1023/A:1019791415147
[24] Reed, Methods of modern mathematical physics. IV. Analysis of operators (1978) · Zbl 0401.47001
[25] Sasamoto, The crossover regime for the weakly asymmetric simple exclusion process, J. Stat. Phys. 140 pp 209– (2010) · Zbl 1197.82093 · doi:10.1007/s10955-010-9990-z
[26] Sasamoto, Exact height distributions for the KPZ equation with narrow wedge initial condition, Nuclear Phys. B 834 pp 523– (2010) · Zbl 1204.35137 · doi:10.1016/j.nuclphysb.2010.03.026
[27] Sasamoto, One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality, Phys. Rev. Lett. 104 pp 230602– (2010) · doi:10.1103/PhysRevLett.104.230602
[28] Seppäläinen , T. Scaling for a one-dimensional directed polymer with boundary conditions 2009 · Zbl 1254.60098
[29] Simon, Mathematical Surveys and Monographs, 120, in: Trace ideals and their applications (2005) · Zbl 1074.47001
[30] Srivastava, Series associated with the zeta and related functions (2001) · Zbl 1014.33001 · doi:10.1007/978-94-015-9672-5
[31] Tracy, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 pp 151– (1994) · Zbl 0789.35152 · doi:10.1007/BF02100489
[32] Tracy, CRM Proceedings and Lecture Notes, 31, in: Isomonodromic deformations and applications in physics (Montréal, QC, 2000) pp 85– (2002)
[33] Tracy, A Fredholm determinant representation in ASEP, J. Stat. Phys. 132 pp 291– (2008) · Zbl 1144.82045 · doi:10.1007/s10955-008-9562-7
[34] Tracy, Integral formulas for the asymmetric simple exclusion process, Comm. Math. Phys. 279 pp 815– (2008) · Zbl 1148.60080 · doi:10.1007/s00220-008-0443-3
[35] Tracy, Asymptotics in ASEP with step initial condition, Comm. Math. Phys. 290 pp 129– (2009) · Zbl 1184.60036 · doi:10.1007/s00220-009-0761-0
[36] Tracy, Formulas for joint probabilities for the asymmetric simple exclusion process, J. Math. Phys. 51 pp 063302– (2010) · Zbl 1311.60118 · doi:10.1063/1.3431977
[37] van Beijeren, Excess noise for driven diffusive systems, Phys. Rev. Lett. 54 pp 2026– (1985) · doi:10.1103/PhysRevLett.54.2026
[38] Varlamov, Fractional derivatives of products of Airy functions, J. Math. Anal. Appl. 337 pp 667– (2008) · Zbl 1141.33002 · doi:10.1016/j.jmaa.2007.03.098
[39] Walsh, Lecture Notes in Mathematics, 1180, in: École d’été de probabilités de Saint Flour, XIV-1984 pp 265– (1986) · doi:10.1007/BFb0074920
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.