DECAY OF 2-POINT FUNCTIONS FOR (D + 1)-DIMENSIONAL PERCOLATION, ISING AND POTTS MODELS WITH D-DIMENSIONAL DISORDER

Let {J < x,y >} < x,y > subset-of Z(d) and {K(x)} x-epsilon-Z(d) be independent sets of nonnegative i.i.d.r.v.'s. < x,y > denoting a pair of nearest neighbors in Z(d); let beta, gamma > O. We consider the random systems: 1. A bond Bernoulli percolation model on Z(d + 1) with random occupation probabilities GRAPHICS 2. Ferromagnetic random Ising-Potts models on Z(d + 1); in the Ising case the Hamiltonian is GRAPHICS For such (d + 1)-dimensional systems with d-dimensional disorder we prove: (i) for any d greater-than-or-equal-to 1, if beta and gamma are small, then, with probability one, the two-point functions decay exponentially in the d-dimensional distance and faster than polynomially in the remaining dimension, (ii) if d greater-than-or-equal-to 2, then, with probability one, we have long-range order for either any beta with gamma sufficiently large or beta sufficiently large and any gamma. [GRAPHICS] For such (d + 1)-dimensional systems with d-dimensional disorder we prove: (i) for any d greater-than-or-equal-to 1, if beta and gamma are small, then, with probability one, the two-point functions decay exponentially in the d-dimensional distance and faster than polynomially in the remaining dimension, (ii) if d greater-that-or-equal-to 2, then, with probability one, we have long-range order for either any beta with gamma sufficiently large or beta sufficiently large and any gamma.