Proposition: The two finite disorder laws are different

At $n=2$, both autocorrelation functions are deterministic under Rademacher disorder. Under Gaussian disorder, the stationary function has a non-point law. In fact, for every fixed positive lag $t$, the Gaussian and Rademacher laws of $A_{2,J}(t)$ have total-variation distance one.

Proof. The functions in (eq:two-stat)–(eq:two-quench) are even in $a$, hence even in $J$. Rademacher disorder has $|J|=1$ almost surely, so its function pair is deterministic.

For fixed physical lag $\ell >0$, put

For $z>0$,

Thus $F_{\ell }$ is strictly increasing on $[0,1)$. The map $|J|\mapsto |\tanh (J/\sqrt{2})|$ is also strictly increasing. A Gaussian $|J|$ has a continuous distribution. Its autocorrelation value is therefore non-atomic.

The Rademacher value is the value corresponding to $|J|=1$. The singleton containing that value has Rademacher probability one and Gaussian probability zero. This proves the total-variation assertion.