Finite-size corrections to disordered Ising models on random regular graphs

We derive the analytical expression for the first finite-size correction to the average free energy of disordered Ising models on random regular graphs. The formula can be physically interpreted as a weighted sum over all non-self-intersecting loops in the graph, the weight being the free-energy shift due to the addition of the loop to an infinite tree.

Figure 1

Pictorial representation of the argument, given in Sec. 4, to compute the free-energy shift due to the addition of a loop to a large tree graph. An open chain embedded in a tree graph (left), its removal from the tree (center), and the addition of a loop (right) are shown.

Figure 2

(Main panel) Average energy density $e\left(\beta \right)$ of a spin glass on a RRG with connectivity $c=4$, bimodal random couplings $J=\pm 1$, and uniform external magnetic field $H_{\mathrm{ext}}=0.7$. The profiles of $e\left(\beta \right)$ are drawn for different system sizes: From top to bottom $N=64,256,1024$. (Inset) The coefficient $e_1\left(\beta \right)$ of the $1/N$ correction to the thermodynamic energy density, measured experimentally by means of Eq. (25). The vertical dashed lines, both in the main panel and in the inset, mark the position of the critical point. Notice that the coefficient $e_1$ does not depend on the size $N$ only up to $\beta \sim 0.8$. Above this value there is an evident $N$ dependence, which is more and more pronounced as one gets closer to the critical point. Exactly at the critical temperature, the coefficient $e_1$ diverges when the system size $N$ goes to infinity. This divergence is the signal of the onset of a different type of scaling of the finite-size corrections at the critical point and, indeed, in all the critical domain.

Figure 3

Finite-size corrections to the energy density of a spin glass model on a RRG with connectivity $c=4$ and bimodal random couplings $J=\pm 1$. The various panels refer to different values of the external uniform magnetic field. The results obtained from Monte Carlo simulations are compared with the analytical values predicted from Eqs. (14) and (31). The vertical dashed lines mark the positions of the critical temperatures.