Jamming densities of random sequential adsorption on d-dimensional cubic lattices.

The rate of convergence of the jamming densities to their asymptotic high-dimensional tree approximation is studied, for two types of random sequential adsorption (RSA) processes on a $d$-dimensional cubic lattice. The first RSA process has an exclusion shell around a particle of nearest neighbors in all $d$ dimensions ($N1$ model). In the second process the exclusion shell consists of a $d$-dimensional hypercube with length $k=2$ around a particle ($N2$ model). For the $N1$ model the deviation of the jamming density ${\ensuremath{\rho}}_{r}(d)$ from its asymptotic high $d$ value ${\ensuremath{\rho}}_{\text{asy}}(d)=\frac{ln(1+2d)}{2d}$ vanishes as ${[\frac{ln(1+2d)}{2d}]}^{3.41}$. In addition, it has been shown that the coefficients ${a}_{n}(d)$ of the short-time expansion of the occupation density of this model (at least up to $n=6$) are given for all $d$ by a finite correction sum of order $(n\ensuremath{-}2)$ in $1/d$ to their asymptotic high $d$ limit. The convergence rate of the jamming densities of the $N2$ model to their high $d$ limits ${\ensuremath{\rho}}_{\text{asy}}(d)=d\frac{ln3}{{3}^{d}}$ is slow. For $2\ensuremath{\le}d\ensuremath{\le}4$ the generalized Palasti approximation provides by far a better approximation. For higher $d$ values the jamming densities converge monotonically to the above asymptotic limits, and their decay with $d$ is clearly faster than the decay as ${(0.432\phantom{\rule{0.16em}{0ex}}332...)}^{d}$ predicted by the generalized Palasti approximation.