Critical behavior of random spin systems at the percolation threshold

Ising and Heisenberg ferromagnets and spin-glasses are studied at the percolation threshold for power-law bond distributions, P(J)∼‖J${\Vert }^{\alpha \mathrm{-}1}$ for J→0. Nonuniversal, α-dependent, critical behavior is found. A hierarchical model for the ‘‘backbone’’ yields, via finite-size scaling, expressions for the exponent t in the relation Υ∝(p-$p_c$ ${)}^t$ for the generalized stiffness coefficient, and for the percolation-to-thermal crossover exponent φ.