Appendix \[appA\] gives for that case $$\det \textbf{T} = \left[1-\varkappa^2+\varkappa^2 c^2(y)\right]
\left[1-\varkappa^2+\varkappa^2 c^2(z)\right]+\varkappa^2 \left(1-\varkappa^2\right)c^2(x)\left[1-c^2(y)\right]\left[1-c^2(z)\right].
\label{exp-c}$$ We see that in this case $\det\textbf{T}$ becomes equal to 0 (and, hence, $Z_4(x,y,z)$ becomes singular) only at $\varkappa=1$, exactly at the critical point of the transition to the dispersive non-equilibrium transport regime.

![Test of the applicability of the simple approximation (\[long-c\]) for $\det \textbf{T}$. Solid lines show the behavior of $\Delta \det \textbf{T}_{\rm lr}(x,y,y)/\Delta \det \textbf{T}(x,y,y)$ where $\Delta \det \textbf{T}=\det \textbf{T}(x,y,y)-Z_2^{-2}(y)$ for various values of $y/a$ (indicated near the corresponding curve) and for correlation function $c(x)=1/\left[(x/a)^{1/2}+1\right]$ ($\varkappa=0.5$).[]{data-label="test-lr"}](figure1.eps){width="3in"}

The situation becomes different for the the case of very long range correlation, where for $x\rightarrow\infty$ we can neglect $y$ and $z$ in $c(x+y)$ etc., assuming $c(x+y)\approx c(x)$ because shows that the typical values of $y$ and $z$ are $\simeq 1/\gamma$. In this situation $C_{ij}\approx c(x)$ for all $i,j$ and for $x\rightarrow\infty$ $$\det \textbf{T}_{\rm lr} \approx \left[1-\varkappa^2+\varkappa^2 c^2(y)\right]
\left[1-\varkappa^2+\varkappa^2 c^2(z)\right]- 4\varkappa^4 c^2(x)\left[1-c(y)\right]\left[1-c(z)\right].
\label{long-c}$$ Validity of the approximation (\[long-c\]) is demonstrated in Fig. \[test-lr\]. We immediately see that for the long range disorder $\det \textbf{T}$, contrary to the case of the exponential $c(x)$, depends on the sign of the auxiliary correlation function. Hence, for the critical correlation function 