Random-Length Random Walks and Finite-Size Scaling in High Dimensions

We address a long-standing debate regarding the finite-size scaling (FSS) of the Ising model in high dimensions, by introducing a random-length random walk model, which we then study rigorously. We prove that this model exhibits the same universal FSS behavior previously conjectured for the self-avoiding walk and Ising model on finite boxes in high-dimensional lattices. Our results show that the mean walk length of the random walk model controls the scaling behavior of the corresponding Green’s function. We numerically demonstrate the universality of our rigorous findings by extensive Monte Carlo simulations of the Ising model and self-avoiding walk on five-dimensional hypercubic lattices with free and periodic boundaries.

Figure 1

Appropriate scaled two-point functions of the Ising model and SAW on five-dimensional hypercubic lattices with periodic boundaries onto the scaling variable $y=\Vert \mathbf{x}\Vert /L^{(d-\mu )/(d-2)}$. (a) Anomalous FSS scaling at $z_c$ onto the ansatz in Eq. (1) with $\mu =d/2$. When $\Vert \mathbf{x}\Vert \approx L/2$, the two-point functions display the anomalous FSS behavior $g(\mathbf{x})\asymp L^{-d/2}$, in contrast to the standard mean field prediction $g(\mathbf{x})\asymp L^{2-d}$. (b) Standard mean-field scaling at the pseudocritical point $z_L=z_c-a{L}^{-2}$ onto the ansatz in Eq. (1) with $\mu =2$. In contrast to the critical PBC case, the two-point functions display the standard mean-field scaling behavior $g(\mathbf{x})\asymp L^{2-d}$ when $\Vert \mathbf{x}\Vert \approx L/2$.

Figure 2

Two-point functions of the Ising model and SAW on five-dimensional hypercubic lattices with free boundaries. (a) Standard mean-field scaling $g(\mathbf{x})\asymp \Vert \mathbf{x}{\Vert }^{2-d}$ at $z_c$. (b) Anomalous FSS at the pseudocritical point ${\tilde{z}}_L=z_c+a_L{L}^{-2}$ onto the scaling variable $y=\Vert \mathbf{x}\Vert /L^{(d-\mu )/(d-2)}$ with $\mu =d/2$. The two-point functions collapse except at distances close to the boundary. This shows that $g(\mathbf{x})$ displays the same FSS behavior as on periodic boundaries at criticality.

Figure 3

FSS behavior of the mean walk length $N_{\mathrm{SAW}}$ (a) and the susceptibility ${\chi }_{\mathrm{Ising},\mathrm{SAW}}$ (b) with periodic boundary conditions in five dimensions. The diamonds (squares) display SAW (Ising) data. To emphasize universality, the Ising and SAW data were translated onto the same curve. In each figure, the line on the top corresponds to the critical scaling behavior $N$, $\chi \asymp L^{d/2}$. The remaining lines (top to bottom) correspond to the scaling behavior $N$, $\chi \asymp L^{\lambda }$ at pseudocritical points $z_L(\lambda )=z_c-a{L}^{\lambda }$ ($a>0$) with $\lambda =2.5$, 2, 1.5, 1.

Figure 4

FSS behavior of the mean walk length $N_{\mathrm{SAW}}$ with free boundary conditions in five dimensions. The inset shows the convergence of $a_L$ in the SAW (circles) and Ising model (triangles).