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Occupation Probabilities and Fluctuations in the Asymmetric Simple Inclusion Process

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.