Theory of a Two-Dimensional Ising Model with Random Impurities. II. Spin Correlation Functions

Abstract

We continue our investigation of an Ising model with immobile random impurities by studying the spin-spin correlation functions. These correlations are not probability-1 objects and have a probability distribution. When the random bonds have the particular distribution function studied in the first paper of this series, we demonstrate that the average value and the second moment of the temperature derivatives of these correlations are infinitely differentiable but fail to be analytic at $T_c$, the temperature at which the observable specific heat fails to be analytic. When $T<\mathrm{}{T}_c$, we consider $S_{\infty }\mathrm{}\left(l\right)=limit\; of〈{\sigma }_{0,0}{\sigma }_{l,m}〉\text{as}m\to \infty$. This limit is not independent of $l$. In the special case that the random bonds are symmetrically distributed about the $l$th row, the geometric mean of $S_{\infty }\mathrm{}\left(l\right)\mathrm{}$ is computed and shown to vanish exponentially rapidly when $T\to T_c-$. We contrast this with a lower bound that shows that the spontaneous magnetization can vanish no more rapidly than $T_c-T$, and present a description of how the local magnetization $S_{\infty }{\mathrm{}\left(l\right)\mathrm{}}^{\frac{1}{2}}$ behaves as $T\to T_c-$.