Finite size scaling functions of the phase transition in the ferromagnetic Ising model on random regular graphs

We discuss the finite-size scaling of ferromagnetic Ising model on random regular graphs. These graphs are locally tree-like, and in limit large graphs, Bethe approximation gives exact free energy per site. In thermodynamic limit, these show a phase transition. This transition is rounded off for finite verify theory prediction that this rounding described terms variable $[T/T_c -1] S^{1/2}$ (where $T$ $T_c$ temperature critical respectively, $S$ number sites graph), $\textit{not}$ power diameter graph, which varies as $\log S$. determine theoretical functions specific heat capacity magnetic susceptibility absolute value magnetization closed form compare them to Monte Carlo simulations.