Maximum gaps in one-dimensional hard-core models

We study the distribution of the maximum gap size in one-dimensional hard-core models. First, we randomly sequentially pack rods of length $2$ onto an interval of length $L$, subject to the hard-core constraint that rods do not overlap. We find that in a saturated packing, with high probability there is no gap of size $2 - o(1/L)$ between adjacent rods, but there are gaps of size at least $2 - 1/L^{1-\epsilon}$ for all $\epsilon > 0$. We subsequently study a variant of the hard-core process, the one-dimensional ghost hard-core model introduced by Torquato and Stillinger. In this model, we randomly sequentially pack rods of length $2$ onto an interval of length $L$, such that placed rods neither overlap with previously placed rods nor previously considered candidate rods. We find that in the infinite time limit, with high probability the maximum gap between adjacent rods is smaller than $\log L$ but at least $(\log L)^{1-\epsilon}$ for all $\epsilon > 0.$