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

Abstract

In this paper we give the distribution of the position of a particle in the asymmetric simple exclusion process (ASEP) with the alternating initial condition. That is, we find P(X m (t) ≤ x) where X m (t) is the position of the particle at time t which was at m = 2k − 1, k ∈ Z at t = 0. As in the ASEP with step initial condition, there arises a new combinatorial identity for the alternating initial condition, and this identity relates the integrand of the integral formula for P(X m (t) ≤ x) to a determinantal form together with an extra product.

Exact formulas for random growth with half-flat initial data

“…This identity was discovered and checked for small values of N on Mathematica by Le Doussal and Calabrese. The formula can, in fact, be derived as a suitable limit of an analogous symmetrization identity proved in [22] in the context of ASEP with flat initial condition (see Lemma 2 in that paper).…”

Current Moments of 1D ASEP by Duality

“…In [25], Tracy and Widom succeeded in computing the distribution of the particle position for the ASEP with general parameter values using the transition probability derived from the Bethe ansatz. For recent developments see [8,23,24,[26][27][28][29][30][31].…”

The Current Distribution of the Multiparticle Hopping Asymmetric Diffusion Model

“…where the contour C is a sufficiently large circle so that all singularities are included in the circle. Comparing (8) with the kernel of the ASEP in [16], we notice that there is an extra term vξ−u 1−τ and τ in (9) corresponds to τ −1 in [16].…”

“…An initial configuration such that all positive integers are occupied and all other sites are empty at t = 0 in the MADM corresponds to the half flat initial condition in the PushASEP such that all positive even integer sites are occupied and all others are empty. There is an explicit probability formula for the ASEP with the half flat initial condition [8] but we report that any trial to find such an explicit formula in a closed form for the PushASEP with the half flat initial condition was unsuccessful.…”