Critical properties of a dilute Ising model in the percolation limit

The critical behavior of the dilute Ising model on a Bethe lattice of coordination $z$ is studied in the percolation limit ($T\to 0$) by means of a new moment-expansion procedure. Exact numerical results for the magnetization as a function of magnetic field at the critical concentration, $p=p_c$, are presented for $z=3\mathrm{}$ and $z=4\mathrm{}$. In particular, the critical exponent ${\delta }_P$ is expected to agree well for very low fields with the approximate value ${\delta }_P=2\mathrm{}$ obtained by Essam et al. The nature of deviations from the value ${\delta }_P=2\mathrm{}$ is discussed and their consequences are noted. The case $p\ne p_c$ is also discussed.