AI 中文总结
本文研究一维随机序列吸附（RSA）的离散键合模型，分析键合间隙的渐近分布及多尺度键合的渐近行为，拓展了Rényi停车常数相关的理论结果。
AI 中文摘要
固定正整数k≥2，对于n≥k，考虑一排n个分子。从n−k+1个相邻k元组分子中均匀随机选取一个并将这k个分子键合。接着，从所有剩余的相邻k元组中再次均匀随机选取一个并键合，重复此过程直至不存在相邻k元组为止。记M_k^(n)为键合分子数的期望值。已知m_k := lim(n→∞) M_k^(n)/n的显式积分公式，也已知m_∞ := lim(k→∞) m_k的显式公式。常数m_∞被称为Rényi停车常数，是上述离散堆积问题的连续模拟的极限堆积密度，这些均属于随机序列吸附（RSA）模型。本文第一部分研究键合k元组之间出现的大小为0,1,…,k−1的间隙，表明在对k网格进行缩放后，当k→∞时，晶格上离散问题的期望间隙的经验分布弱收敛于上述连续模拟中已知的适当间隙分布。第二部分考虑离散键合问题的两种不同模型，其中同时发生k₁键合和k₂键合，满足2≤k₁<k₂，得到了m_k类似量的显式公式，并分别在k₂→∞（k₁固定）以及k₁、k₂以特定比值同时→∞的情况下研究这些类似量的渐近行为。
英文摘要
Fix a positive integer $k\ge2$, and for $n\ge k$, consider a row of $n$ molecules. From among the $n-k+1$ nearest-neighbor $k$-tuples of molecules, select one uniformly at random and bond the $k$ molecules. Now, from all the remaining nearest-neighbor $k$-tuples, again select one uniformly at random and bond the $k$ molecules. Continue like this until there are no nearest-neighbor $k$-tuples left. Let $M^{(n)}_k$ denote the expected value of the number of bonded molecules. An explicit integral formula for $m_k:=\lim_{n\to\infty}\frac{M^{(n)}_k}n$ is known, and an explicit formula for $m_\infty:=\lim_{k\to\infty}m_k$ is known. The constant $m_\infty$, known as the Rényi parking constant, arises as the limiting packing density for a continuous analog of the above discrete packing problems. These are all models of what is called random sequential adsorption (RSA). The first part of this paper studies the gaps of sizes $0,1,\cdots, k-1$ that arise between bonded $k$-tuples and shows that after scaling the $k$-grid, when $k\to\infty$ the empirical distribution of expected gaps in the discrete problem on the lattice converges weakly to an appropriate gap distribution that is known to hold for the above noted continuous analog. The second part of the this paper considers two different models of the discrete bonding problem when both $k_1$-bonding and $k_2$-bonding occur, with $2\le k_1<k_2$. Explicit formulas are obtained for the analogs of $m_k$, and the asymptotic behavior of these analogs is studied both when $k_2\to\infty$ with $k_1$ fixed, and when $k_1,k_2\to\infty$ at certain ratios.
