for $c(x)=1/\left[(x/a)^3+1\right]$ (right side of the plot) and $c(x)=1/\left[(x/b)^5+1\right]$ (left side).[]{data-label="mix"}](figure5.eps){width="3in"}

Power law correlation function $c(x)\propto 1/x^n$
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Use of $\det\textbf{T}_{\rm tr}$ gives us the possibility to carry out an analytical calculation of the critical line $\varkappa_c(n)$ for the power law correlation function (\[phase-power\]) (more exactly, we assume only that for large $x$ $c(x)\approx (a/x)^n$). Power law correlation functions are typical for amorphous organic semiconductors, though the corresponding models usually provide Gaussian or approximately Gaussian DOS [@Novikov:2584; @Novikov:2532]. For the approximately exponential DOS considered in Ref. we should expect $c_U(x)\propto 1/x^5$ and $c(x)\propto 1/x^{5/2}$ [@Novikov:24504], though parameter $U_0$ is typically very small and the resulting total DOS in amorphous organic semiconductors is mostly formed by other contributions.

If we assume, in close resemblance to the discussion in Appendix \[appB\], that the minimal value of $\det\textbf{T}_{\rm tr}$ is achieving at $x=z$ and for relevant values of $x$ $(x/a)^n\gg 1$ (this is indeed the case for $n\gg 1$, in practice $n\approx 2-3$ is large enough), then $c(x+z)=c(2x)\approx c/2^n$, here $c=c(x)$. Hence, $$\det \textbf{T}_{\rm tr}\approx \left(1-\varkappa^2\right)^2 +
2\varkappa^2(1-\varkappa^2)c^2-\varkappa^2(1+\varkappa)^2\frac{c^2}{2^{2n}}
+2\varkappa^3(1+\varkappa)\frac{c^3}{2^n}.
\label{det-simple}$$ Minimization of this expression with respect to $c$ and subsequent solution of the equation $\det\textbf{T}_{\rm tr}(c_{\rm min})=0$ gives $$\frac{1-\varkappa_c}{1+\varkappa_c}=\frac{1}{2^{2n+3}}.
\label{pl-min}$$ This solution agrees 