Directed polymers and the quantum Toda lattice

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.

Published in at http://dx.doi.org/10.1214/10-AOP632 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (16)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• The one-dimensional KPZ equation: an exact solution and its universality
• Free-energy distribution of the directed polymer at high temperature
• Asymptotics in ASEP with Step Initial Condition
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• Exact height distributions for the KPZ equation with narrow wedge initial condition
• A Fredholm Determinant Representation in ASEP
• The crossover regime for the weakly asymmetric simple exclusion process
• The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class
• Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers
• Bounds for scaling exponents for a 1+1 dimensional directed polymer in a Brownian environment
• Reflected planar Brownian motions, intertwining relations and crossing probabilities
• Baxter operator and Archimedean Hecke algebra
• Attempted Bethe ansatz solution for one-dimensional directed polymers in random media
• The propagator of the attractive delta-Bose gas in one dimension
• Formulas for Joint Probabilities for the Asymmetric Simple Exclusion Process

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• Tropical Combinatorics and Whittaker functions
• Stochastic higher spin vertex models on the line
• Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
• Airy processes and variational problems
• Spectral theory for the q-Boson particle system
• The strict-weak lattice polymer
• The ASEP and determinantal point processes
• Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz
• A phase transition for $q$-TASEP with a few slower particles
• A multi-layer extension of the stochastic heat equation
• Directed random polymers via nested contour integrals
• Large deviation rate functions for the partition function in a log-gamma distributed random potential
• A q-weighted version of the Robinson-Schensted algorithm
• Two time distribution in Brownian directed percolation
• Uniform convergence to the Airy line ensemble
• Scaling limit for Brownian motions with one-sided collisions
• Tracy-Widom asymptotics for q-TASEP
• Half-space Macdonald processes
• Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles
• Moments and Lyapunov exponents for the parabolic Anderson model
• Intermediate disorder limits for multi-layer semi-discrete directed polymers
• Determinantal structures in the O'Connell-Yor directed random polymer model
• On integrable directed polymer models on the square lattice
• Large Deviations For Sticky Brownian Motions
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• Dynamics of the box-ball system with random initial conditions via Pitman's transformation
• Large deviations of the free energy in the O'Connell-Yor polymer
• Log-Gamma directed polymer with fixed endpoints via the replica Bethe Ansatz
• Intermediate disorder directed polymers and the multi-layer extension of the stochastic heat equation
• Interlacing Diffusions
• Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process
• Free energy distribution of the stationary O'Connell-Yor directed random polymer model
• Finite-temperature and finite-time scaling of the directed polymer free-energy with respect to its geometrical fluctuations
• Geometric RSK and the Toda lattice
• Survival probability of mutually killing Brownian motions and the O'Connell process
• Dualities in quantum integrable many-body systems and integrable probabilities -- I
• Stochasticization of Solutions to the Yang-Baxter Equation
• Interacting diffusions on positive definite matrices
• Scaling limits for non-intersecting polymers and Whittaker measures
• Upper tail bounds for stationary KPZ models
• Fluctuation Exponents of the KPZ equation on a large torus
• Stationary measures for two dual families of finite and zero temperature models of directed polymers on the square lattice
• KPZ fluctuations in finite volume
• Determinantal structures in space inhomogeneous dynamics on interlacing arrays
• Free field theory and observables of periodic Macdonald processes
• Random ballistic growth and diffusion in symmetric spaces
• Reciprocal Time Relation of Noncolliding Brownian Motion with Drift
• Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings
• Borodin-Péché fluctuations of the free energy in directed random polymer models
• Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices
• Matrix Whittaker processes
• From elongated spanning trees to vicious random walks
• System of Complex Brownian Motions Associated with the O'Connell Process
• Tracy-Widom asymptotics for a random polymer model with gamma-distributed weights
• A stationary model of non-intersecting directed polymers
• Stochastic Bäcklund transformations
• Characterization of $H$-Brownian Gibbsian line ensembles
• Exponential ergodicity and convergence for generalized reflected Brownian motion
• Discrete Whittaker processes
• On two-dimensional extensions of Bougerol's identity in law