Occupation Probabilities and Fluctuations in the Asymmetric Simple Inclusion Process

TL;DR

This paper derives exact formulas for occupation probabilities in the ASIP model, revealing universal asymptotic laws for occupation, fluctuations, and inter-exit times, and connects these to Catalan's trapezoids.

Contribution

It provides the first exact closed-form expression for occupation probabilities in ASIP and links these to combinatorial structures, advancing understanding of its asymptotic behavior.

Findings

Exact probability formulas in terms of Catalan's trapezoids

Inverse square root law of occupation

Square root law of fluctuations

Abstract

The Asymmetric Simple Inclusion Process (ASIP), a lattice-gas model of unidirectional transport and aggregation, was recently proposed as an `inclusion' counterpart of the Asymmetric Simple Exclusion Process (ASEP). In this paper we present an exact closed-form expression for the probability that a given number of particles occupies a given set of consecutive lattice sites. Our results are expressed in terms of the entries of Catalan's trapezoids --- number arrays which generalize Catalan's numbers and Catalan's triangle. We further prove that the ASIP is asymptotically governed by: (i) an inverse square root law of occupation; (ii) a square root law of fluctuation; and (iii) a Rayleigh law for the distribution of inter-exit times. The universality of these results is discussed.