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Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2020

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

Loïc Turban

Résumé

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.

Dates et versions

hal-02431015 , version 1 (07-01-2020)

Identifiants

Citer

Loïc Turban. Random sequential adsorption of k -mers on the fully-connected lattice: probability distributions of the covering time and extreme value statistics. Journal of Physics A: Mathematical and Theoretical, 2020, 53 (3), pp.035001. ⟨10.1088/1751-8121/ab59f5⟩. ⟨hal-02431015⟩
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