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. 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 the following: (i) an inverse square-root law of occupation, (ii) a square-root law of fluctuation, and (iii) a Rayleigh law for the distribution of interexit times. The universality of these results is discussed.

Figure 1

The nonempty interval $\{k,\cdots ,k+m-1\}$ becomes empty if, and only if, all interval sites other than site $k+m-1$ are empty and the particles that occupy site $k+m-1$ hop to site $k+m$.

Figure 2

The empty interval $\{k,...,k+m-1\}$ becomes nonempty if, and only if, site $k-1$ is occupied and the particles that occupy it hop to site $k$.

Figure 3

During the time interval $(t,t+▵)$, the incremental load $L(t;1,m)$ can change either due to the arrival of a particle to the first site or due to the opening of gate $m$.

Figure 4

During the time interval $(t,t+▵)$, the incremental load $L(t;k,m)$ can change either due to the opening of gate $k-1$ or due to the opening of gate $k+m-1$.

Figure 5

Top panel: Equation (55) defines a boundary value problem for $G(z;k,m)$. The PGF $G(z;k,m)$ is determined by a weighted average of its southern and northwestern neighbors in the positive quadrant of the $(k,m)$ plane. The boundary PGFs $G(z;k,0)$ and $G(z;1,m)$ are given by Eqs. (45) and (49), respectively. Bottom panel: A three-step algorithm can be used in order to solve the boundary value problem for the PGF $G(z;k,m)$: (i) start at the left boundary and solve for the column that stands to its right; (ii) treat the newly solved column as the new left boundary and iterate; and (iii) stop at the $k^{\mathrm{th}}$ column and obtain the desired solution.

Figure 6

Expressing $G(z;k,m)$ as a weighted sum over known boundary functions. All boundary functions must be properly weighted and taken into account. The weight of each boundary function is given by the number of legitimate paths leading to it, multiplied by $1/2$ raised to the power of the path length. Paths are made out of south ($\downarrow$) and northwest ($\nwarrow$) steps only and must not pass through another boundary function except the one lying at the end of the path. Some boundary functions can be reached via several different paths while others cannot be reached at all (the latter are discarded in the computation of the sum).