# Random sequential adsorption of k-mers on the fully-connected lattice: probability distributions of the covering time and extreme value statistics

## TL;DR

This paper analyzes the probability distributions and extreme value statistics of the time needed to fully cover a lattice with k-mers through random sequential adsorption, revealing different scaling behaviors and fluctuation regimes.

## Contribution

It provides explicit probability distributions and scaling laws for the covering time in various regimes, including new extreme value distributions for large coverage gaps.

## Key findings

- Poisson distribution of covering time in low coverage limit
- Gaussian fluctuations in intermediate coverage regime
- Divergence of scaling functions near full coverage

## Abstract

We study the random sequential adsorption of $k$-mers on the fully-connected lattice with $N=kn$ sites. The probability distribution $T_n(s,t)$ of the time $t$ needed to cover the lattice with $s$ $k$-mers is obtained using a generating function approach. In the low coverage scaling limit where $s,n,t\to\infty$ with $y=s/n^{1/2}={\mathrm O}(1)$ the random variable $t-s$ follows a Poisson distribution with mean $ky^2/2$. In the intermediate coverage scaling limit, when both $s$ and $n-s$ are ${\mathrm O}(n)$, the mean value and the variance of the covering time are growing as $n$ and the fluctuations are Gaussian. When full coverage is approached the scaling functions diverge, which is the signal of a new scaling behaviour. Indeed, when $u=n-s={\mathrm O}(1)$, the mean value of the covering time grows as $n^k$ and the variance as $n^{2k}$, thus $t$ is strongly fluctuating and no longer self-averaging. In this scaling regime the fluctuations are governed, for each value of $k$, by a different extreme value distribution, indexed by $u$. Explicit results are obtained for monomers (generalized Gumbel distribution) and dimers.