The ASEP and determinantal point processes

Abstract

We introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line, with correlation kernels defined via spectral projections for the associated Jacobi matrices. For classical weights, we show how such ensembles arise as limits of various hypergeometric orthogonal polynomials ensembles. We then prove that the q-Laplace transform of the height function of the ASEP with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete Laguerre ensemble --- a member of the new family. This allows us to obtain the large time asymptotics of the ASEP in three limit regimes: (a) for finitely many rightmost particles; (b) GUE Tracy-Widom asymptotics of the height function; (c) KPZ asymptotics of the height function for the ASEP with weak asymmetry. We also give similar results for two instances of the stochastic six vertex model in a quadrant. The proofs are based on limit transitions for the corresponding determinantal point processes.

47 pages

References in corpus (9)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Asymptotics in ASEP with Step Initial Condition
• From Random Matrices to Stochastic Operators
• A Fredholm Determinant Representation in ASEP
• Orthogonal polynomial ensembles in probability theory
• Asymptotics of Plancherel-type random partitions
• Lectures on Integrable probability: Stochastic vertex models and symmetric functions
• Moments Match between the KPZ Equation and the Airy Point Process
• Stochastic higher spin six vertex model and Madconald measures

Cited by in corpus (27)

• Stochastic six-vertex model in a half-quadrant and half-line open ASEP
• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
• Coulomb-gas electrostatics controls large fluctuations of the KPZ equation
• Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
• Lower tail of the KPZ equation
• Algebraic Heun operator and band-time limiting
• Shift-invariance for vertex models and polymers
• The free boundary Schur process and applications I
• Moments Match between the KPZ Equation and the Airy Point Process
• Half-space Macdonald processes
• Generalizations of TASEP in discrete and continuous inhomogeneous space
• Stochastic higher spin six vertex model and Madconald measures
• Observables of coloured stochastic vertex models and their polymer limits
• Some recent progress in singular stochastic PDEs
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• Markov duality and Bethe ansatz formula for half-line open ASEP
• Determinantal point processes and fermion quasifree states
• Tracy-Widom asymptotics for a river delta model
• Stochastic Telegraph Equation Limit for the Stochastic Six Vertex Model
• Partial Isometries, Duality, and Determinantal Point Processes
• Moments of the SHE under delta initial measure
• GUE $\times$ GUE limit law at hard shocks in ASEP
• On the $q$-TASEP with a random initial condition
• The stochastic six-vertex model speed process
• Fluctuations of Young diagrams for symplectic groups and semiclassical orthogonal polynomials
• Cutoff and discrete Product Structure in ASEP
• Determinantal Point Processes, Stochastic Log-Gases, and Beyond