Spectral theory for the -Boson particle system

Abstract

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.

The Strict-Weak Lattice Polymer

“…Then, using an identity from [10] and a variant of an argument from [9], they arrive at the result of Theorem 1. 7.…”

“…Using the below reduction of the true evolution to the free evolution, it is possible to diagonalize the true evolution equation via coordinate Bethe ansatz. We do not pursue this further here, but reference, for example [7,8,16].…”

Emergence of jams in the generalized totally asymmetric simple exclusion process

“…where ξ = 2c(1 − c)/θ is the effective correlation length in the system size scale, which depends on the transition parameter θ defined in (28).…”

“…It was later rediscovered in [25], within a totally different context as an integrable generalization of the TASEP. Finally, it was shown to be a particular q = 0 limiting case of the general three parametric Bethe ansatz-solvable stochastic chipping model [26], also referred to as a q-Hahn or (q,μ,ν)−boson process [27,28]. In turn, it contains already known TASEPs with parallel and sequential update as particular cases.…”

Tracy–Widom asymptotics for $q$-TASEP

“…The duality of q-TASEP and q-TAZRP was proved in [6] and as a consequence, joint moment formulas for multiple particle positions were obtained for q-TASEP however they are not of Fredholm determinant form. An explicit formula for the transition probabilities of q-TAZRP with a finite number of particles was recently found in two different ways: in [5] by using the spectral theory for the q-Boson particle system and in [14] by using the Bethe ansatz. In these two papers, the distribution of the left-most particle's position after a fixed time for general initial condition was also characterized.…”