Nature of the "Griffiths" singularity in dilute magnets

Abstract

The nature of the singular behavior pointed out by Griffiths for $H=0\mathrm{}$ in dilute magnets is investigated. It is argued that for concentration $p$ less than that for formation of an infinite cluster, all derivatives of $M\left(H\right)\mathrm{}$ are finite. The nonanalyticity in $M\left(H\right)\mathrm{}$ is due to a branch cut along the imaginary $H$ axis having weight $\exp \left[\frac{-\left(\mathrm{const}\right)}{\mathrm{}\left|H\right|\mathrm{}}\right]$ for $\mathrm{}\left|H\right|\mathrm{}\to 0$, and is thus too weak to be experimentally observable. Some numerical and exact analytic results for the dilute magnet on a Bethe lattice are presented.