Lattices with variable and constant occupation density and q-exponential distribution

Abstract

In this paper we test the hypothesis that $q$-exponential distribution fits better on distributions arising from lattices with a heterogeneous topology than a homogeneous topology. We compare two lattices: the first is the typical square lattice with a constant occupation density $p$ (the lattice used in standard percolation theory), and the second is a lattice constructed with a gradient of $p$. In the homogeneous lattice the occupied number of neighbors of each cell is the same (on average) for the full lattice, otherwise in the $p$-gradient lattice this number changes along the lattice. In this sense the $p$-gradient lattice shows a more complex topology than the homogeneous lattice. We fit the $q$-exponential and the stretched exponential distribution on the cluster size distribution that arises in the lattices. We observe that the $q$-exponential fits better on the $p$-gradient lattice than on a constant $p$ lattice. On the other hand, the stretched exponential distribution fits equally well on both lattices.