We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
We investigate the critical behavior of the random-bond $\ifmmode\pm\else\textpm\fi{}J$ Ising model on a square lattice at the multicritical Nishimori point in the $T\text{\ensuremath{-}}p$ phase diagram, where
T is the temperature and
p is the disorder parameter (
p=1 corresponds to the pure Ising model). We perform a finite-size scaling analysis of high-statistics Monte Carlo simulations along the Nishimori line defined by $2p\ensuremath{-}1=\text{tanh}(1/T)$, along which the multicritical point lies. The multicritical Nishimori point is located at ${p}^{\ensuremath{\ast}}=0.890\text{ }81(7)$, ${T}^{\ensuremath{\ast}}=0.9528(4)$, and the renormalization-group dimensions of the operators that control the multicritical behavior are
y1=0.655(15) and
y2=0.250(2); they correspond to the thermal exponent $\ensuremath{\nu}\ensuremath{\equiv}1/{y}_{2}=4.00(3)$ and to the crossover exponent $\ensuremath{\phi}\ensuremath{\equiv}{y}_{1}/{y}_{2}=2.62(6)$.