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

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 -> infinity with y = s/n(1/2) = 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 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 = 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.