Current Distribution and Random Matrix Ensembles for an Integrable Asymmetric Fragmentation Process

Abstract

We calculate the time-evolution of a discrete-time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size-dependent rates. In the continuous-time limit the process turns into the totally asymmetric simple exclusion process (only pieces of size 1 break off a given cluster). We express the exact solution of master equation for the process in terms of a determinant which can be derived using the Bethe ansatz. From this determinant we compute the distribution of the current across an arbitrary bond which after appropriate scaling is given by the distribution of the largest eigenvalue of the Gaussian unitary ensemble of random matrices. This result confirms universality of the scaling form of the current distribution in the KPZ universality class and suggests that there is a link between integrable particle systems and random matrix ensembles.

Distribution of a Particle’s Position in the ASEP with the Alternating Initial Condition

“…By using the technique used in [12] Rákos and Schütz [11] obtained the same result as Johansson [7] for the TASEP with step initial condition. Rákos and Schütz's method originates from the Bethe Ansatz while Johansson used a combinatorial argument.…”

“…That the asymptotics on the fluctuation of x m (t) or T (x, t) are related to the GUE TracyWidom distribution in random matrix theory and that fluctuations are in the regime of the t 1/3 scale are well known for various situations [2,7,11,18]. Besides the step initial condition, the TASEP with the alternating initial condition also has been investigated.…”

“…This initial condition assumes that all even sites are occupied and all odd sites are empty at t = 0 or vice-versa. While the current fluctuation of the TASEP (more generally ASEP) with step initial condition in the long time limit is governed by the GUE Tracy-Widom distribution [2,7,11,14,18], on the other hand, the current fluctuation of the TASEP with the alternating initial condition is related to the GOE Tracy-Widom distribution [3-6, 10, 15]. The appearance of the GOE Tracy-Widom distribution in growth models goes back to [1] by Baik and Rains and a breakthrough in the TASEP was led by Sasamoto [13].…”

Transition Probabilities of the Bethe Ansatz Solvable Interacting Particle Systems

“…Hence, if u(x 1 , x 2 ; t) satisfies (17), it also satisfies (16). This is the boundary condition suggested in [1,2] as a combination of the boundary conditions of the TASEP and the droppush model.…”

A multi-dimensional Markov chain and the Meixner ensemble

“…The Fredholm determinant formula has the advantage that it is much better suited for computation of asymptotics. The argument in this paper leading to (15) gives an alternative approach, starting from the definitions, to this formula.…”