Spectral theory for the q-Boson particle system

Abstract

We develop spectral theory for the generator of the q-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the q-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with q-TASEP, this leads to moment formulas which characterize the fixed time distribution of q-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our q-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O'Connell-Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation / Kardar-Parisi-Zhang equation.

63 pages, 5 figures

References in corpus (21)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Asymptotics in ASEP with Step Initial Condition
• An exact solution for the KPZ equation with flat initial conditions
• Interaction quenches in the 1D Bose gas
• The KPZ equation with flat initial condition and the directed polymer with one free end
• Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity
• Stationary correlations for the 1D KPZ equation
• On integrability of zero-range chipping models with factorized steady state
• The One-dimensional KPZ Equation and the Airy Process
• Bethe ansatz solution of zero-range process with nonuniform stationary state
• The Asymmetric Avalanche Process
• A q-weighted version of the Robinson-Schensted algorithm
• Shocks in the asymmetric exclusion process with internal degree of freedom
• Two-time free energy distribution function in (1+1) directed polymers
• Two-point generating function of the free energy for a directed polymer in a random medium
• The Bose Gas and Asymmetric Simple Exclusion Process on the Half-Line
• Diagonalization of an Integrable Discretization of the Repulsive Delta Bose Gas on the Circle
• Replica Bethe ansatz derivation of the GOE Tracy-Widom distribution in one-dimensional directed polymers with free boundary conditions
• Diagonalization of the infinite q-boson system
• The semi-infinite q-boson system with boundary interaction
• Distribution function of the endpoint fluctuations of one-dimensional directed polymers in a random potential

Cited by in corpus (27)

• Random-walk in Beta-distributed random environment
• The strict-weak lattice polymer
• Directed random polymers via nested contour integrals
• Lectures on Integrable probability: Stochastic vertex models and symmetric functions
• An algebraic construction of duality functions for the stochastic U_q(A_n^{(1)}) vertex model and its degenerations
• Tracy-Widom asymptotics for q-TASEP
• Half-space Macdonald processes
• Half-space stationary Kardar-Parisi-Zhang equation
• Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz
• Exact formulas for random growth with half-flat initial data
• Large Deviations For Sticky Brownian Motions
• A deformation of affine Hecke algebra and integrable stochastic particle system
• Height fluctuations for the stationary KPZ equation
• Orthogonality of Bethe Ansatz eigenfunctions for the Laplacian on a hyperoctahedral Weyl alcove
• The crossing probability for directed polymers in random media
• The Transition Probability of the $q$-TAZRP ($q$-Bosons) with Inhomogeneous Jump Rates
• Log-Gamma directed polymer with fixed endpoints via the replica Bethe Ansatz
• The q-PushASEP: A New Integrable Model for Traffic in 1+1 Dimension
• Completeness of the Bethe Ansatz for an open $q$-boson system with integrable boundary interactions
• Tracy-Widom limit of q-Hahn TASEP
• Observables of coloured stochastic vertex models and their polymer limits
• Diagonalization of the infinite q-boson system
• Observables of stochastic colored vertex models and local relation
• Hidden diagonal integrability of $q$-Hahn vertex model and Beta polymer model
• Current statistics in the q-boson zero range process
• On the eigenfunctions for the multi-species q-Boson system
• Boundary Interactions for the Semi-Infinite q-Boson System and Hyperoctahedral Hall-Littlewood Polynomials