Tropical Combinatorics and Whittaker functions

Abstract

We establish a fundamental connection between the geometric RSK correspondence and GL(N,R)-Whittaker functions, analogous to the well known relationship between the RSK correspondence and Schur functions. This gives rise to a natural family of measures associated with GL(N,R)-Whittaker functions which are the analogues in this setting of the Schur measures on integer partitions. The corresponding analogue of the Cauchy-Littlewood identity can be seen as a generalisation of an integral identity for GL(N,R)-Whittaker functions due to Bump and Stade. As an application, we obtain an explicit integral formula for the Laplace transform of the law of the partition function associated with a one-dimensional directed polymer model with log-gamma weights recently introduced by one of the authors (TS).

31 pages,6 figures, updated introduction

References in corpus (4)

• Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity
• Airy kernel with two sets of parameters in directed percolation and random matrix theory
• Second class particles and cube root asymptotics for Hammersley's process
• Baxter operator and Archimedean Hecke algebra

Cited by in corpus (42)

• Spectral theory for the q-Boson particle system
• The strict-weak lattice polymer
• Ratios of partition functions for the log-gamma polymer
• Universal Kardar-Parisi-Zhang dynamics in integrable quantum systems
• Directed random polymers via nested contour integrals
• A multi-layer extension of the stochastic heat equation
• A q-weighted version of the Robinson-Schensted algorithm
• Half-space Macdonald processes
• Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles
• Determinantal structures in the O'Connell-Yor directed random polymer model
• Revisiting Directed Polymers with heavy-tailed disorder
• Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer
• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• The crossing probability for directed polymers in random media
• 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
• The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall
• Geometric RSK and the Toda lattice
• RSK in last passage percolation: a unified approach
• Crossing probability for directed polymers in random media: exact tail of the distribution
• Interacting diffusions on positive definite matrices
• Scaling limits for non-intersecting polymers and Whittaker measures
• Midpoint distribution of directed polymers in the stationary regime: exact result through linear response
• Stochastic growth in time dependent environments
• Hidden diagonal integrability of $q$-Hahn vertex model and Beta polymer model
• Mutually avoiding paths in random media and largests eigenvalues of random matrices
• Free field theory and observables of periodic Macdonald processes
• Log-gamma directed polymer with one free end via coordinate Bethe Ansatz
• The geometric Burge correspondence and the partition function of polymer replicas
• The Ghost-Box-Ball System: A Unified Perspective on Soliton Cellular Automata, the RSK Algorithm and Phase Shifts
• Borodin-Péché fluctuations of the free energy in directed random polymer models
• Matrix Whittaker processes
• GOE and ${\mathrm{Airy}}_{2\to 1}$ marginal distribution via symplectic Schur functions
• Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices
• A stationary model of non-intersecting directed polymers
• Characterization of $H$-Brownian Gibbsian line ensembles
• Bispectral dual difference equations for the quantum Toda chain with boundary perturbations
• Tracy-Widom fluctuations in 2D random Schrodinger operators
• Intertwining the Busemann process of the directed polymer model
• Interlacing networks: birational RSK, the octahedron recurrence, and Schur function identities
• Tropical integrable systems and Young tableaux: Shape equivalence and Littlewood-Richardson correspondence
• Discrete Whittaker processes