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

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