Statistical Mechanics on the Random Locally Tree-like Layered Lattice

Abstract

This thesis defines a variation of a regular random graph called the Random Locally Tree-like Layered Lattice (RLTL). Such lattices are of interest because the Bethe approximation becomes asymptotically exact on them. Unlike the usual Bethe lattice, the RLTL is finite and can be studied numerically. This makes studying problems in statistical physics on the RLTL more tractable. We study the geometrical structure of this graph and the behavior of the Ising model on it. Under the geometrical structure, we explore the properties of the diameter and the radius of gyration of this graph, study the $r$ dependence on the average number of distinct sites present at distance $r$ from a site and also look at the average length shortest loop passing through a site. We find that the average diameter of an r-regular RLTL is of order $[log_{r-1}(S)]$ to leading order in $S$, where $S$ is the total number of sites in the graph. The radius of gyration of the RLTL was also found to be linear in $log(S)$. We then look at the Ising model on the RLTL and look at the finite-size scaling on the lattice. The finite-size effects of the Ising model on random graphs is not well studied in previous literature. On the RLTL, we study the finite-size scaling for the magnetic susceptibility ($\chi$) and deviations in the specific heat capacity from the theoretical value ($\Delta C_v$). We find that the magnetic susceptibility per site obeys a scaling of the form $\chi = S^{1/2} \; f(\varepsilon S^{1/2})$ and the deviations in the specific heat has a scaling of the form $\Delta C_v = g(\varepsilon S^{1/2})$, where $\epsilon = (T - T_c)/T_c$. We propose a theory which explains the observed scaling and explicitly calculate the exact scaling functions.