From duality to determinants for q-TASEP and ASEP

Abstract

We prove duality relations for two interacting particle systems: the $q$-deformed totally asymmetric simple exclusion process ($q$-TASEP) and the asymmetric simple exclusion process (ASEP). Expectations of the duality functionals correspond to certain joint moments of particle locations or integrated currents, respectively. Duality implies that they solve systems of ODEs. These systems are integrable and for particular step and half-stationary initial data we use a nested contour integral ansatz to provide explicit formulas for the systems' solutions, and hence also the moments. We form Laplace transform-like generating functions of these moments and via residue calculus we compute two different types of Fredholm determinant formulas for such generating functions. For ASEP, the first type of formula is new and readily lends itself to asymptotic analysis (as necessary to reprove GUE Tracy--Widom distribution fluctuations for ASEP), while the second type of formula is recognizable as closely related to Tracy and Widom's ASEP formula [Comm. Math. Phys. 279 (2008) 815--844, J. Stat. Phys. 132 (2008) 291--300, Comm. Math. Phys. 290 (2009) 129--154, J. Stat. Phys. 140 (2010) 619--634]. For $q$-TASEP, both formulas coincide with those computed via Borodin and Corwin's Macdonald processes [Probab. Theory Related Fields (2014) 158 225--400]. Both $q$-TASEP and ASEP have limit transitions to the free energy of the continuum directed polymer, the logarithm of the solution of the stochastic heat equation or the Hopf--Cole solution to the Kardar--Parisi--Zhang equation. Thus, $q$-TASEP and ASEP are integrable discretizations of these continuum objects; the systems of ODEs associated to their dualities are deformed discrete quantum delta Bose gases; and the procedure through which we pass from expectations of their duality functionals to characterizing generating functions is a rigorous version of the replica trick in physics.

Published in at http://dx.doi.org/10.1214/13-AOP868 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Asymptotics in ASEP with Step Initial Condition
• A Fredholm Determinant Representation in ASEP
• Bethe ansatz solution of zero-range process with nonuniform stationary state
• Formulas for ASEP with Two-Sided Bernoulli Initial Condition
• Stochastic integrability and the KPZ equation

Cited by in corpus (61)

• A KPZ Cocktail- Shaken, not stirred: Toasting 30 years of kinetically roughened surfaces
• The one-dimensional KPZ equation and its universality class
• Stochastic six-vertex model
• Random-walk in Beta-distributed random environment
• Duality for stochastic models of transport
• Stochastic higher spin vertex models on the line
• Exact solution of the macroscopic fluctuation theory for the symmetric exclusion process
• KPZ equation limit of higher-spin exclusion processes
• Current Fluctuations of the Stationary ASEP and Six-Vertex Model
• The ASEP and determinantal point processes
• Large deviations of a tracer in the symmetric exclusion process
• Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz
• A phase transition for $q$-TASEP with a few slower particles
• Directed random polymers via nested contour integrals
• A q-weighted version of the Robinson-Schensted algorithm
• A Pfaffian representation for flat ASEP
• An algebraic construction of duality functions for the stochastic U_q(A_n^{(1)}) vertex model and its degenerations
• Asymmetric stochastic transport models with ${\mathcal{U}}_q(\mathfrak{su}(1,1))$ symmetry
• Self-Duality for the Two-Component Asymmetric Simple Exclusion Process
• Half-space Macdonald processes
• Tracy-Widom asymptotics for q-TASEP
• Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles
• The $q$-Hahn asymmetric exclusion process
• Moments and Lyapunov exponents for the parabolic Anderson model
• A Multi-species ASEP(q,j) and q-TAZRP with Stochastic Duality
• Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two
• Determinantal structures in the O'Connell-Yor directed random polymer model
• On integrable directed polymer models on the square lattice
• Exact formulas for random growth with half-flat initial data
• Self-duality of Markov processes and intertwining functions
• Integrable Floquet dynamics, generalized exclusion processes and "fused" matrix ansatz
• Integral formulas of ASEP and $q$-TAZRP on a ring
• Integrable stochastic dualities and the deformed Knizhnik-Zamolodchikov equation
• On dualities for SSEP and ASEP with open boundary conditions
• Multispecies totally asymmetric zero range process: I. Multiline process and combinatorial $R$
• Dualities in quantum integrable many-body systems and integrable probabilities -- I
• Tracy-Widom limit of q-Hahn TASEP
• Observables of coloured stochastic vertex models and their polymer limits
• The transition probability and the probability for the left-most particle's position of the q-TAZRP
• Stochasticization of Solutions to the Yang-Baxter Equation
• Random walk on nonnegative integers in beta distributed random environment
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• Solution of the Kolmogorov equation for TASEP
• Limiting current distribution for a two species asymmetric exclusion process
• Transition probability and total crossing events in the multi-species asymmetric exclusion process
• Duality in quantum transport models
• KPZ fluctuations in finite volume
• Regularity comparison of symbolic powers, integral closure of powers and powers of edge ideals
• Log-gamma directed polymer with one free end via coordinate Bethe Ansatz
• A short proof of a symmetry identity for the $(q,\mu ,\nu )$-deformed Binomial distribution
• Product blocking measures and a particle system proof of the Jacobi triple product
• Tracy-Widom asymptotics for a river delta model
• Exact density profile in a tight-binding chain with dephasing noise
• Duality relations for the ASEP conditioned on a low current
• Current statistics in the q-boson zero range process
• Q-zero range has random walking shocks
• A Generalized Dynamic Asymmetric Exclusion Process: Orthogonal Dualities and Degenerations
• Asymptotic fluctuations of geometric q-TASEP, geometric q-PushTASEP and q-PushASEP
• On the $q$-TASEP with a random initial condition
• Exact steady states and fragmentation-induced relaxation in the no-passing asymmetric simple exclusion process
• Integral formula for the propagator of the one-dimensional Hubbard model