论文标题
一维硬核模型中的最大差距
Maximum gaps in one-dimensional hard-core models
论文作者
论文摘要
我们研究一维硬核模型中最大间隙大小的分布。首先，我们将长度2 $的长度杆随机打包到长度$ l $的间隔上，但要受杆不重叠的硬核约束。我们发现，在饱和的包装中，具有较高尺寸的差距为$ 2 -o（1/l）$之间，但是所有$ε> 0 $的尺寸的差距至少有$ 2-1/l^{1-ε} $。 随后，我们研究了硬核过程的变体，这是Torquato和Stillinger引入的一维幽灵硬核模型。在此型号中，我们将长度2 $的长度随机包装到长度$ l $的间隔上，以使杆既没有与先前放置的杆重叠，也不是先前被视为候选杆。我们发现，在无限的时间限制中，具有很高的概率，相邻杆之间的最大差距小于$ \ log l $，但至少$（\ log l）^{1-ε} $对于所有$ε> 0。
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 $2 - 1/L^{1-ε}$ for all $ε> 0$. 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 $(\log L)^{1-ε}$ for all $ε> 0.$
