The scaling window of the 5D Ising model with free boundary conditions

Abstract

The five-dimensional Ising model with free boundary conditions has recently received a renewed interest in a debate concerning the finite-size scaling of the susceptibility near the critical temperature. We provide evidence in favour of the conventional scaling picture, where the susceptibility scales as $O(L^2)$ inside a critical scaling window of width $O(1/L^2)$. Our results are based on Monte Carlo data gathered on system sizes up to $L=79$ (ca. three billion spins) for a wide range of temperatures near the critical point. We analyse the magnetisation distribution, the susceptibility and also the scaling and distribution of the size of the Fortuin-Kasteleyn cluster containing the origin. The probability of this cluster reaching the boundary determines the correlation length, and its behaviour agrees with the mean field critical exponent $\delta=3$, that the scaling window has width $O(1/L^2)$.