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Critical Properties of an Ising Model with Dilute Long-range Interactions

Statistical mechanical models with local interactions in $d>1$ dimension can be regarded as $d=1$ dimensional models with regular long range interactions. In this paper we study the critical properties of Ising models having $V$ sites, each having $z$ randomly chosen neighbors. For $z=2$ the model reduces to the $d=1$ Ising model. For $z= \infty$ we get a mean field model. We find that for finite $z > 2$ the system has a second order phase transition characterized by a length scale $L={\rm ln}V$ and mean field critical exponents that are independent of $z$.