On integrable directed polymer models on the square lattice

Abstract

In a recent work Povolotsky 24 provided a three-parameter family of stochastic particle systems with zero-range interactions in one dimension which are integrable by coordinate Bethe ansatz. Using these results we obtain the corresponding condition for integrability of a class of directed polymer models with random weights on the square lattice. Analyzing the solutions we find, besides known cases, a new two-parameter family of integrable DP model, which we call the Inverse-Beta polymer, and provide its Bethe ansatz solution.PACS numbers: I. INTRODUCTION AND MAIN RESULTS A. overviewThere is considerable recent interest in exact solutions for models in the universality class of the 1D stochastic growth Kardar-Parisi-Zhang equation (KPZ) 1 . Models in the KPZ class share the same large time statistics, also found to be related to the universal statistics of large random matrices 2 . Methods developped in the context of quantum integrability are exploited and broadly extended to solve a variety of 1D stochastic models. The Bethe ansatz solution of the attractive delta Bose gas (the Lieb-Liniger model 3,4 ) was combined with the replica method 5 , to obtain exact solutions for the KPZ equation directly in the continuum and at arbitrary time, for the main classes of initial conditions (droplet, flat, stationary, half-space) 6-15 . The Cole-Hopf mapping h ∼ ln Z is used, where h is the height of the KPZ interface and Z the partition sum of a directed polymer in a random potential (DP). Hence in the continuum, studying KPZ growth is equivalent to studying the DP model, an equilibrium statistical mechanics problem with quenched disorder. The time in KPZ growth becomes the length of the polymer t. The replica Bethe ansatz (RBA) method then allows to calculate the integer moments Z n and, from them, to retrieve the probability distribution function (PDF) of Z. Since the last step is non-rigorous because of the fast growth of these moments, the mathematical community has concentrated on the exact solution of discrete models, which in favorable cases, do not suffer from the moment growth problem. Discrete models, such as the PNG growth model [16][17][18] , the TASEP and ASEP particle transport model 19,20 and discrete DP models 17,21,22 played a pionneering role in unveiling the universal statistics of the KPZ class at large time (the Airy processes). Recently they have been considerably generalized, unveiling a very rich underlying "stochastic integrability" structure 23-33 . Since in suitable limits (e.g. ASEP with weak asymmetry, q-TASEP with q → 1, semi-discrete DP) they converge to the continuum KPZ equation, they also led to some recent rigorous results for KPZ at arbitrary time [35][36][37][38][39] .Besides their interest in relation to KPZ growth, directed polymers are also important in a variety of fields. This includes optimization and glasses 40,41 , vortex lines in superconductors 42 , domain walls in magnets 43 , disordered conductors 44 , Burgers equation in fluid mechanics 45 , exploration-exploitation tradeof...

Tracy-Widom Asymptotics for a River Delta Model

“…History of the model and related results. Bernoulli-exponential FPP was first introduced in [13], which introduced an exactly solvable model called the beta random walk in random environment (RWRE) and studied Bernoulli-exponential FPP as a low temperature limit of this model (see also the physics works [49,50] further studying the Beta RWRE and some variants). The beta RWRE was shown to be exactly solvable in [13] by viewing it as a limit of q-Hahn TASEP, a Bethe ansatz solvable particle system introduced in [44].…”

The q-Hahn PushTASEP

“…The generalized negative binomial beta distributions reduce to their standard counterparts. In the caseν = 1 2 , the resulting recursion is quite similar, though different from the one satisfied by the inverse beta polymer partition function [TLD15]. In particular, the choice of parameters for the beta random variable depends on whether Z(i − 1, t) or Z(i, t − 1) is greater.…”

“…The main new feature is that in our model, the distribution of the weights depends on the ratio of the partition functions immediately to the left and below (see Definition 4.1). We had initially expected that the inverse beta polymer of [TLD15] would arise from our pushing system, but presently this does not seem to be the case (though perhaps the inverse beta polymer could be included into a 4-parameter family of pushing systems whose existence we speculate on below). However, in the q → 1 limit we do arrive (see Lemma 4.3 and Theorem 4.4) at a solution to the following recursion relation, which bears great similarity to that satisfied by the inverse beta polymer: WhenZ(i, t − 1) >Z(i − 1, t),…”

Non-compact Quantum Spin Chains as Integrable Stochastic Particle Processes

“…Also different rational limits of the q-Hahn process including the ones beyond the stochastic sector where discussed in context of directed polymers in random environment, see e.g. [25].…”