Decay of two-point functions for (d+1)-dimensional percolation, ising and Potts models withd-dimensional disorder

Abstract

Let$\left\{ {J_{\left\langle {x,y} \right\rangle } } \right\}_{\left\langle {x,y} \right\rangle \subset Z^d }$ and$\left\{ {K_x } \right\}_{x \in Z^d }$ be independent sets of nonnegative i.i.d.r.v.'s, <x,y> denoting a pair of nearest neighbors inZ ^d; let β, γ>0. We consider the random systems: 1. A bond Bernoulli percolation model onZ ^d+1 with random occupation probabilities

2. Ferromagnetic random Ising-Potts models onZ ^d+1; in the Ising case the Hamiltonian is

$$H = - \beta \sum\limits_t {\sum\limits_{\left\langle {x,y} \right\rangle } {J_{\left\langle {x,y} \right\rangle } \sigma (x,t)\sigma (y,t) - } \gamma \sum\limits_x {\sum\limits_t {K_x \sigma (x,t)} \sigma (x,t + 1)} } $$

For such (d+1)-dimensional systems withd-dimensional disorder we prove: (i) for anyd≧1, if β and γ are small, then, with probability one, the two-point functions decay exponentially in thed-dimensional distance and faster than polynomially in the remaining dimension, (ii) ifd≧2, then, with probability one, we have long-range order for either and β with γ sufficiently large of β sufficiently large and any γ.