gives us the possibility to generated only limited subset of all exponentially distributed random energies $U(x)$, but we can produce the random field $U(x)$ having an arbitrary nonnegative binary correlation function $c_U(x)$ because $$\label{2a-corrU}
c_U(x)=\left<U(x_1) U(x_2)\right>-\left<U\right>^2=\frac{U_0^2}{2}\left[c_X^2(x)+c_Y^2(x)\right],
\hskip10pt x=x_1-x_2.$$

We may consider the use of auxiliary variables $X(x)$ and $Y(x)$ as just a technical trick to introduce spatial correlation in the exponential random energy landscape. This trick facilitates the calculations and there is no need to assume any physical meaning for $X$ and $Y$. Yet there is some physical justification for the Gaussian trick as a way to introduce spatial correlation to the exponential distribution, because May et al. demonstrated a particular mechanism to produce the almost exponential distribution in amorphous organic materials based essentially on some modification of [@May:136401].

The bivariate distribution for $X$, needed to calculate $Z_2(x)$, has the form $$\label{2a-PDF_GX}
P_G(X_1,X_2,c_X)=\frac{1}{2\pi\sqrt{1-c^2_X}}\exp\left[
-\frac{X_1^2+X_2^2-2c_X X_1X_2}{2\left(1-c^2_X\right)}\right],$$ here $X_i=X(x_i)$, $c_X=c_X(x_1-x_2)$ (the distribution for $Y$ has a similar form) [@Feller:book]. Quadratic form in the exponent of is positively defined if $|c_X|<1$. For $x=0$ we have $c_{X,Y}(0)=\sigma^2=1$ and for $x\rightarrow\infty$ $c_{X,Y}(x)\rightarrow 0$. Correlation function $Z_2(x)$ has been calculated previously [@Novikov:24504] and is equal to $$Z_2(x)=\frac{1}{1-\varkappa^2\left[1-c^2(x)\right]}
\label{Z}$$ for $\varkappa < 1$, here and later we 