Point-to-line polymers and orthogonal Whittaker functions

Abstract

We study a one dimensional directed polymer model in an inverse-gamma random environment, known as the log-gamma polymer, in three different geometries: point-to-line, point-to-half line and when the polymer is restricted to a half space with end point lying free on the corresponding half line.Via the use of A.N.Kirillov's geometric Robinson-Schensted-Knuth correspondence, we compute the Laplace transform of the partition functions in the above geometries in terms of orthogonal Whittaker functions, thus obtaining new connections between the ubiquitous class of Whittaker functions and exactly solvable probabilistic models. In the case of the first two geometries we also provide multiple contour integral formulae for the corresponding Laplace transforms. Passing to the zero-temperature limit, we obtain new formulae for the corresponding last passage percolation problems with exponential weights.

39 pages, 5 figures. Typos corrected

References in corpus (11)

• Asymptotics in ASEP with Step Initial Condition
• Fluctuation properties of the TASEP with periodic initial configuration
• An exact solution for the KPZ equation with flat initial conditions
• The KPZ equation with flat initial condition and the directed polymer with one free end
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• Littelmann path model for geometric crystals, Whittaker functions on Lie groups and Brownian motion
• Log-gamma directed polymer with one free end via coordinate Bethe Ansatz
• New Integral Representations of Whittaker Functions for Classical Lie Groups
• Whittaker functions on orthogonal groups of odd degree
• Whittaker processes and Landau-Ginzburg potentials for flag manifolds
• Positive temperature dynamics on Gelfand-Tsetlin patterns restricted by wall

Cited by in corpus (14)

• Half-space Macdonald processes
• Stationary half-space last passage percolation
• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation
• Log-gamma directed polymer with one free end via coordinate Bethe Ansatz
• Correlations for symplectic and orthogonal Schur measures
• The geometric Burge correspondence and the partition function of polymer replicas
• Matrix Whittaker processes
• Positive temperature dynamics on Gelfand-Tsetlin patterns restricted by wall
• On the combinatorics of last passage percolation in a quarter square and ${\mathrm{GOE}}^2$ fluctuations
• Wave functions for quantum integrable particle systems via partial confluences of multivariate hypergeometric functions
• Determinantal point processes from symplectic and orthogonal characters and applications
• Crossover in the log-gamma polymer from the replica coordinate Bethe Ansatz
• The half-space Airy stat process