Dilute random magnets in one dimension

We study, in this paper, a system of localized spins distributed randomly over a one-dimensional array of sites. Expressions for the low-temperature behavior of the magnetic susceptibility, the heat capacity, the correlation length, and the order-parameter susceptibility of the system are derived. The probability distribution of the coupling constant changes with the concentration of localized spins from one that is singular in the dilute range to a nonsingular one. Associated with this change are changes in the critical exponents from nonuniversal to universal behavior, as well as specific crossover phenomena. The scaling relation $d\nu >2-\alpha$ is violated.