We study approach to the large-time jammed state of the deposited particles in the model of random sequential adsorption. The convergence laws are usually derived from the argument of Pomeau which includes the assumption of the dominance, at large enough times, of small landing regions into each of which only a single particle can be deposited without overlapping earlier deposited particles and which, after a certain time are no longer created by depositions in larger gaps. The second assumption has been that the size distribution of gaps open for particle-center landing in this large-time small-gaps regime is finite in the limit of zero gap size. We report numerical Monte Carlo studies of a recently introduced model of random sequential adsorption on patterned one-dimensional substrates that suggest that the second assumption must be generalized. We argue that a region exists in the parameter space of the studied model in which the gap-size distribution in the Pomeau large-time regime actually linearly vanishes at zero gap sizes. In another region, the distribution develops a threshold property, i.e., there are no small gaps below a certain gap size. We discuss the implications of these findings for new asymptotic power-law and exponential-modified-by-a-power-law convergences to jamming in irreversible one-dimensional deposition.
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