Random covers of finite homogeneous lattices

Abstract

We develop and extend some results for the scheme of independent random elements distributed on a finite lattice. In particular, we introduce the concept of the cover of a homogeneous lattice Ln of rank n and derive the exact equations and estimations for the number of covers with a given number of blocks and for the total covers number of the lattice Ln. A theorem about the asymptotic normality of the blocks number in a random equiprobable cover of the lattice Ln is proved. The concept of the cover index of the lattice Ln, that extend the notion of the cover index of a finite set by its independent random subsets, is introduced. Applying the lattice moments method, the limit distribution as n→∞ for the cover index of a subspace lattice of the n-dimensional vector space over a finite field is determined.