Markov Duality Research Articles

A generalized dynamic asymmetric exclusion process: orthogonal dualities and degenerations

Abstract In this paper, a generalized version of dynamic asymmetric simple exclusion process (ASEP) is introduced, and it is shown that the process has a Markov duality property with the same process on the reversed lattice. The duality functions are multivariate q-Racah polynomials, and the corresponding orthogonality measure is the reversible measure of the process. By taking limits in the generator of dynamic ASEP, its reversible measure, and the duality functions, we obtain orthogonal and triangular dualities for several other interacting particle systems. In this sense, the duality of dynamic ASEP sits on top of a hierarchy of many dualities. For the construction of the process, we rely on representation theory of the quantum algebra U q ( sl 2 ) . In the standard representation, the generator of generalized ASEP can be constructed from the coproduct of the Casimir. After a suitable change of representation, we obtain the generator of dynamic ASEP. The corresponding intertwiner is constructed from q-Krawtchouk polynomials, which arise as eigenfunctions of twisted primitive elements. This gives a duality between dynamic ASEP and generalized ASEP with q-Krawtchouk polynomials as duality functions. Using this duality, we show the (almost) self-duality of dynamic ASEP.

Parameter Permutation Symmetry in Particle Systems and Random Polymers

Many integrable stochastic particle systems in one space dimension (such as TASEP - totally asymmetric simple exclusion process - and its various deformations, with a notable exception of ASEP) remain integrable when we equip each particle $x_i$ with its own jump rate parameter $\nu_i$. It is a consequence of integrability that the distribution of each particle $x_n(t)$ in a system started from the step initial configuration depends on the parameters $\nu_j$, $j\le n$, in a symmetric way. A transposition $\nu_n \leftrightarrow \nu_{n+1}$ of the parameters thus affects only the distribution of $x_n(t)$. For $q$-Hahn TASEP and its degenerations ($q$-TASEP and directed beta polymer) we realize the transposition $\nu_n \leftrightarrow \nu_{n+1}$ as an explicit Markov swap operator acting on the single particle $x_n(t)$. For beta polymer, the swap operator can be interpreted as a simple modification of the lattice on which the polymer is considered. Our main tools are Markov duality and contour integral formulas for joint moments. In particular, our constructions lead to a continuous time Markov process $\mathsf{Q}^{(\mathsf{t})}$ preserving the time $\mathsf{t}$ distribution of the $q$-TASEP (with step initial configuration, where $\mathsf{t}\in \mathbb{R}_{>0}$ is fixed). The dual system is a certain transient modification of the stochastic $q$-Boson system. We identify asymptotic survival probabilities of this transient process with $q$-moments of the $q$-TASEP, and use this to show the convergence of the process $\mathsf{Q}^{(\mathsf{t})}$ with arbitrary initial data to its stationary distribution. Setting $q=0$, we recover the results about the usual TASEP established recently in [arXiv:1907.09155] by a different approach based on Gibbs ensembles of interlacing particles in two dimensions.

Two Dualities: Markov and Schur–Weyl

Abstract We show that quantum Schur–Weyl duality leads to Markov duality for a variety of asymmetric interacting particle systems. In particular, we consider the following three cases: (1) Using a Schur–Weyl duality between a two-parameter quantum group and a two-parameter Hecke algebra from [6], we recover the Markov self-duality of multi-species ASEP previously discovered in [23] and [3]. (2) From a Schur–Weyl duality between a co-ideal subalgebra of a quantum group and a Hecke algebra of type B [2], we find a Markov duality for a multi-species open ASEP on the semi-infinite line. The duality functional has not previously appeared in the literature. (3) A “fused” Hecke algebra from [15] leads to a new process, which we call braided ASEP. In braided ASEP, up to $m$ particles may occupy a site and up to $m$ particles may jump at a time. The Schur–Weyl duality between this Hecke algebra and a quantum group lead to a Markov duality. The duality function had previously appeared as the duality function of the multi-species ASEP$(q,m/2)$ [23] and the stochastic multi-species higher spin vertex model [24].

A short note on Markov duality in multi–species higher spin stochastic vertex models

We show that the multi–species higher spin stochastic vertex model, also called the Uq(An(1)) vertex model, satisfies a duality where the indicator function has the form {η[i,n]x≥ξ[i,n]x}. In other words, for every particle in the ξ configuration of species i at vertex x, there must be a particle of species j≥i at vertex x in the η configuration. For these duality functions, the dual process has fewer particles than the original process, making it suitable for applications. The proof follows by applying charge reversal to previously discovered duality functions, which also results in open boundary conditions. As a special case, we recover the duality for the stochastic six vertex model recently found by Y. Lin.

Stochastic PDE Limit of the Six Vertex Model

We study the stochastic six vertex model and prove that under weak asymmetry scaling (i.e., when the parameter $$\Delta \rightarrow 1^+$$ so as to zoom into the ferroelectric/disordered phase critical point) its height function fluctuations converge to the solution to the Kardar–Parisi–Zhang (KPZ) equation. We also prove that the one-dimensional family of stochastic Gibbs states for the symmetric six vertex model converge under the same scaling to the stationary solution to the stochastic Burgers equation. Our proofs rely upon the Markov (self) duality of our model. The starting point is an exact microscopic Hopf–Cole transform for the stochastic six vertex model which follows from the model’s known one-particle Markov self-duality. Given this transform, the crucial step is to establish self-averaging for specific quadratic function of the transformed height function. We use the model’s two-particle self-duality to produce explicit expressions (as Bethe ansatz contour integrals) for conditional expectations from which we extract time-decorrelation and hence self-averaging in time. The crux of our Markov duality method is that the entire convergence result reduces to precise estimates on the one-particle and two-particle transition probabilities. Previous to our work, Markov dualities had only been used to prove convergence of particle systems to linear Gaussian SPDEs (e.g. the stochastic heat equation with additive noise).

The q-Hahn PushTASEP

Abstract We introduce the $q$-Hahn PushTASEP—an integrable stochastic interacting particle system that is a three-parameter generalization of the PushTASEP, a well-known close relative of the TASEP (totally asymmetric simple exclusion process). The transition probabilities in the $q$-Hahn PushTASEP are expressed through the $_4\phi _3$ basic hypergeometric function. Under suitable limits, the $q$-Hahn PushTASEP degenerates to all known integrable (1+1)-dimensional stochastic systems with a pushing mechanism. One can thus view our new system as a pushing counterpart of the $q$-Hahn TASEP introduced by Povolotsky [37]. We establish Markov duality relations and contour integral formulas for the $q$-Hahn PushTASEP. In a $q\to 1$ limit of our process we arrive at a random recursion, which, in a special case, appears to be similar to the inverse-Beta polymer model. However, unlike in recursions for Beta polymer models, the weights (i.e., the coefficients of the recursion) in our model depend on the previous values of the partition function in a nontrivial manner.

Markov duality for stochastic six vertex model

We prove that Schutz’s ASEP Markov duality functional is also a Markov duality functional for the stochastic six vertex model. We introduce a new method that uses induction on the number of particles to prove the Markov duality.

An Algebraic Construction of Duality Functions for the Stochastic $${\mathcal{U}_q( A_n^{(1)})}$$ Vertex Model and Its Degenerations

A recent paper \cite{KMMO} introduced the stochastic U_q(A_n^{(1)}) vertex model. The stochastic S-matrix is related to the R-matrix of the quantum group U_q(A_n^{(1)}) by a gauge transformation. We will show that a certain function D^+_{\mu} intertwines with the transfer matrix and its space reversal. When interpreting the transfer matrix as the transition matrix of a discrete-time totally asymmetric particle system on the one-dimensional lattice Z, the function D^+_{\mu} becomes a Markov duality function D_{\mu} which only depends on q and the vertical spin parameters \mu_x. By considering degenerations in the spectral parameter, the duality results also hold on a finite lattice with closed boundary conditions, and for a continuous-time degeneration. This duality function had previously appeared in a multi-species ASEP(q,j) process. The proof here uses that the R-matrix intertwines with the co-product, but does not explicitly use the Yang-Baxter equation. It will also be shown that the stochastic U_q(A_n^{(1)}) is a multi-species version of a stochastic vertex model studied in \cite{BP,CP}. This will be done by generalizing the fusion process of \cite{CP} and showing that it matches the fusion of \cite{KRL} up to the gauge transformation. We also show, by direct computation, that the multi-species q-Hahn Boson process (which arises at a special value of the spectral parameter) also satisfies duality with respect to D_0, generalizing the single-species result of \cite{C}.

Dynamic ASEP, Duality, and Continuous q−1-Hermite Polynomials

Abstract We demonstrate a Markov duality between the dynamic asymmetric simple exclusion process (ASEP) and the standard ASEP. We then apply this to step initial data, as well as a half-stationary initial data (which we introduce). While investigating the duality for half-stationary initial data, we uncover and utilize connections to the continuous q−1-Hermite polynomials. Finally, we introduce a family of stationary initial data which are related to the indeterminate moment problem associated with these q−1-Hermite polynomials.

The $q$-Hahn asymmetric exclusion process

We introduce new integrable exclusion and zero-range processes on the one-dimensional lattice that generalize the $q$-Hahn TASEP and the $q$-Hahn Boson (zero-range) process introduced in [J. Phys. A 46 (2013) 465205, 25] and further studied in [Int. Math. Res. Not. IMRN 14 (2015) 5577–5603], by allowing jumps in both directions. Owing to a Markov duality, we prove moment formulas for the locations of particles in the exclusion process. This leads to a Fredholm determinant formula that characterizes the distribution of the location of any particle. We show that the model-dependent constants that arise in the limit theorems predicted by the KPZ scaling theory are recovered by a steepest descent analysis of the Fredholm determinant. For some choice of the parameters, our model specializes to the multi-particle-asymmetric diffusion model introduced in [Phys. Rev. E 58 (1998) 4181]. In that case, we make a precise asymptotic analysis that confirms KPZ universality predictions. Surprisingly, we also prove that in the partially asymmetric case, the location of the first particle also enjoys cube-root fluctuations which follow Tracy–Widom GUE statistics.

Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz

We develop spectral theory for the q-Hahn stochastic particle system introduced recently by Povolotsky. That is, we establish a Plancherel type isomorphism result which implies completeness and biorthogonality statements for the Bethe ansatz eigenfunctions of the system. Owing to a Markov duality with the q-Hahn TASEP (a discrete-time generalization of TASEP with particles' jump distribution being the orthogonality weight for the classical q-Hahn orthogonal polynomials), we write down moment formulas which characterize the fixed time distribution of the q-Hahn TASEP with general initial data. The Bethe ansatz eigenfunctions of the q-Hahn system degenerate into eigenfunctions of other (not necessarily stochastic) interacting particle systems solvable by the coordinate Bethe ansatz. This includes the ASEP, the (asymmetric) six-vertex model, and the Heisenberg XXZ spin chain (all models are on the infinite lattice). In this way, each of the latter systems possesses a spectral theory, too. In particular, biorthogonality of the ASEP eigenfunctions which follows from the corresponding q-Hahn statement implies symmetrization identities of Tracy and Widom (for ASEP with either step or step Bernoulli initial configuration) as corollaries. Another degeneration takes the q-Hahn system to the q-Boson particle system (dual to q-TASEP) studied in detail in our previous paper (2013). Thus, at the spectral theory level we unify two discrete-space regularizations of the Kardar-Parisi-Zhang equation / stochastic heat equation, namely, q-TASEP and ASEP.

Spectral theory for the -Boson particle system

AbstractWe 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 ($q$-deformed totally asymmetric simple exclusion process), 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.