A new look at some aspects of one-dimensional random sequential adsorption and its continuum limit

Abstract

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 R\'enyi 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.