cases and for the divergence of $Z_4(x,y,z)$ its most favorable value is $U(x+y)=0$ irrespective of all other $U$ terms. Absence of correlation between $U(x+y)$ and other $U$ terms facilitates the fulfilment of this requirement and the easiest way to keep $U(x+y)=0$ without affecting other terms is to set $y\rightarrow\infty$.

This important observation means that we may set $y\rightarrow\infty$ and $c(y+...)=0$ in from the very beginning and consider instead the truncated determinant $$\begin{aligned}
\det \textbf{T}_{\rm tr}=\left(1-\varkappa^2\right)^2+
\varkappa^2(1-\varkappa^2)\left[c^2(z)+c^2(x)\right]
-\varkappa^2(1+\varkappa)^2 c^2(x+z)+\\
\nonumber
+2\varkappa^3(1+\varkappa)c(x)c(x+z)c(z).
\label{red-detT}\end{aligned}$$ In future for all analytic calculations we will use $\det \textbf{T}_{\rm tr}$. Fig. \[w\] shows a typical example of the increase of $\det\textbf{T}$ for the finite $y$.

![Difference between $\det\textbf{T}(x_0,y,x_0)$ and $\det\textbf{T}(x_0,y\rightarrow\infty,x_0)$ where $x_0$ is the optimal distance such as $c(2x_0)=c_{\rm min}$ (here $c_{\rm min}$ is the minimal value of $c(x)$, see explanation for the particular choice of $x_0$ in Appendix \[appB\]) for the correlation function $c(x)=\left(1-x/a\right)/\left[1+s(x/a)^2\right]$ and $c_{\rm min}$ equal to -1 (solid line), -1/3 (dotted line), and -0.1 (broken line). For every case $\varkappa$ was chosen as the critical $\varkappa_c$ according to . Particular value of $c_{\rm min}$ was set by the proper choice of $s$.[]{data-label="w"}](figure4.eps){width="3in"}

The particular role of the exponential correlation function could be easily 