High Temperature Expansion Study of the Nishimori multicritical Point in Two and Four Dimensions

We study the two and four dimensional Nishimori multicritical point via high temperature expansions for the $\pm J$ distribution, random-bond, Ising model. In $2d$ we estimate the the critical exponents along the Nishimori line to be $\gamma=2.37\pm 0.05$, $\nu=1.32\pm 0.08$. These, and earlier $3d$ estimates $\gamma =1.80\pm 0.15$, $\nu=0.85\pm 0.08$ are remarkably close to the critical exponents for percolation, which are known to be $\gamma=43/18$, $\nu=4/3$ in $d=2$ and $\gamma=1.805\pm0.02$ and $\nu=0.875\pm 0.008$ in $d=3$. However, the estimated $4d$ Nishimori exponents $\gamma=1.80\pm 0.15$, $\nu=1.0\pm 0.1$, are quite distinct from the $4d$ percolation results $\gamma=1.435\pm 0.015$, $\nu=0.678\pm 0.05$.