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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…

Probability · Mathematics 2014-10-28 Alexei Borodin , Ivan Corwin , Tomohiro Sasamoto

The asymmetric simple exclusion process (ASEP) is a paradigm for non-equilibrium physics that appears as a building block to model various low-dimensional transport phenomena, ranging from intracellular traffic to quantum dots. We review…

Statistical Mechanics · Physics 2015-05-27 Kirone Mallick

We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process.…

Mathematical Physics · Physics 2008-01-20 Alexei Borodin , Patrik L. Ferrari , Michael Prähofer , Tomohiro Sasamoto

The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more…

Probability · Mathematics 2018-10-30 Jinho Baik , Zhipeng Liu

We consider the q-TASEP that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on Z for q in [0,1) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the…

Probability · Mathematics 2019-05-20 Patrik L. Ferrari , Balint Veto

We consider the asymmetric simple exclusion process (ASEP) with forward hopping rate 1, backward hopping rate q and periodic boundary conditions. We show that the Bethe equations of ASEP can be decoupled, at all order in perturbation in the…

Statistical Mechanics · Physics 2021-09-08 Sylvain Prolhac

We consider TASEP in continuous time with non-random initial conditions and arbitrary fixed density of particles rho. We show GOE Tracy-Widom universality of the one-point fluctuations of the associated height function. The result phrased…

Probability · Mathematics 2018-05-01 Patrik L. Ferrari , Alessandra Occelli

We study the spatial correlations of the one-dimensional KPZ surface for the flat initial condition. It is shown that the multi-point joint distribution for the height is given by a Fredholm determinant, with its kernel in the scaling limit…

Statistical Mechanics · Physics 2009-11-11 T. Sasamoto

The relaxation time limit of the one-point distribution of the spatially periodic totally asymmetric simple exclusion process is expected to be the universal one point distribution for the models in the KPZ universality class in a periodic…

Probability · Mathematics 2020-08-18 Jinho Baik , Zhipeng Liu , Guilherme L. F. Silva

We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for…

Mathematical Physics · Physics 2022-02-10 Stéphane Ouvry , Alexios P. Polychronakos

We derive the moderate deviation principles for the fluctuation fields of the facilitated exclusion process (FEP) in one dimension when the process starts from its stationary measure, both in the symmetric and asymmetric cases. The main…

Probability · Mathematics 2025-05-05 Linjie Zhao

Domains of attraction are identified for the universality classes of one-point asymptotic fluctuations for the Kardar-Parisi-Zhang (KPZ) equation with general initial data. The criterion is based on a large deviation rate function for the…

Probability · Mathematics 2020-10-15 Jeremy Quastel , Daniel Remenik

In this paper, we study an exact solution of the asymmetric simple exclusion process on a periodic lattice of finite sites with two typical updates, i.e., random and parallel. Then, we find that the explicit formulas for the partition…

Statistical Mechanics · Physics 2009-05-19 Masahiro Kanai , Katsuhiro Nishinari , Tetsuji Tokihiro

We study the Facilitated TASEP, an interacting particle system on the one dimensional integer lattice. We prove that starting from step initial condition, the position of the rightmost particle has Tracy Widom GSE statistics on a cube root…

Probability · Mathematics 2024-12-13 Jinho Baik , Guillaume Barraquand , Ivan Corwin , Toufic Suidan

The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in TASEP with the step-type initial condition. Calculated is the multi-time joint distribution…

Mathematical Physics · Physics 2007-08-18 T. Imamura , T. Sasamoto

We note that a tridiagonal matrix representation of the algebra of the partially asymmetric exclusion process (PASEP) lends itself to interpretation as the transfer matrix for weighted Motzkin lattice paths. A continued fraction…

Statistical Mechanics · Physics 2009-07-27 R. A. Blythe , W. Janke , D. A. Johnston , R. Kenna

We study the one-dimensional discrete time totally asymmetric simple exclusion process with parallel update rules on a spatially periodic domain. A multi-point space-time joint distribution formula is obtained for general initial…

Probability · Mathematics 2022-01-10 Yuchen Liao

We show that the stationary measure for the totally asymmetric simple exclusion process on a segment with open boundaries is given by a marginal of a two-line measure with a simple and explicit description. We use this representation to…

Probability · Mathematics 2025-02-11 Wlodek Bryc , Pavel Zatitskii

In this paper we consider the totally asymmetric simple exclusion process, with non-random initial condition having three regions of constant densities of particles. From left to right, the densities of the three regions are increasing.…

Mathematical Physics · Physics 2020-01-08 Patrik L. Ferrari , Peter Nejjar

The one-dimensional polynuclear growth model with external sources at edges is studied. The height fluctuation at the origin is known to be given by either the Gaussian, the GUE Tracy-Widom distribution, or certain distributions called…

Mathematical Physics · Physics 2007-05-23 T. Imamura , T. Sasamoto