Some conditional probabilities in the TASEP with second class particles

Abstract

In this paper we consider the TASEP with second class particles which consists of k first class particles and N − k second class particles. We assume that all first class particles are initially located to the left of the leftmost second class particle. Under this assumption, we find the probability that the first class particles are at x, x + 1, · · · , x + k − 1 and these positions are still to the left of the leftmost second class particle at time t. If we additionally assume that the initial positions of the particles are 1, · · · , N , that is, step initial condition, then the formula of the probability does not depend on k and is very similar to a formula for the TASEP (without second class particles) with step initial condition.

Exact Formulas of the Transition Probabilities of the Multi-Species Asymmetric Simple Exclusion Process

“…We first review the basic concepts used in [1,8,9] to extend the results for the TASEP with second class particles there to the multi-species ASEP. The state space of the multi-species ASEP with N particles is countable, so we may view P (Y,ν) (X, π; t) as a matrix element of an infinite matrix, denoted P(t), which is a member of a probability semigroup {P(t) : t ≥ 0}.…”

“…The approach we take in this paper is similar to that in [4,11,12] because the model in this paper is a generalization of the two-species ASEP. We first introduce some notations and generalize the section 2.1 in [12] to the asymmetric case.…”

“…where A ν,π σ was not given explicitly except a special case in [17]. Recently, for the two-species TASEP, the author gave a formula of A ν,π σ when ν = π [11,12].…”

Simplified Forms of the Transition Probabilities of the Two-Species ASEP with Some Initial Orders of Particles

“…The transition probability from the initial state (Y, ν) to state (X, π) at time t is denoted by P (Y,ν) (X, π; t). Earlier works on the transition probabilities and some distributions in the multi-species ASEP are found in [3,6,7,10,11,12,17,18]. Also, the multi-species ASEP can be considered as a special case of the coloured stochastic vertex model (see [1,2]).…”

Integrability of the multi-species TASEP with species-dependent rates

“…If a particle belonging to l tries to jump to the site occupied by a particle belonging to l ′ ≥ l, the jump is prohibited but if a particle belonging to l ′ tries to jump to the site occupied by a particle belonging to l < l ′ , then the jump occurs by interchanging positions. The transition probabilities and some determinantal formulas for the multi-species ASEP or its special cases were found in [1,[3][4][5][6]11]. Also, for some special initial conditions with a single second class particle, some distributions and their asymptotics were studied in [2,8].…”