very well with the numerical calculation of $\varkappa_c(n)$ for the full non-truncated $\det\textbf{T}$ (see insert in Fig. \[kappa-power-law\]).

For the particular case of the power law correlation function $c(x)\propto 1/x^{1/2}$ where for small $\varkappa$ the term $\Delta D$ is finite and nonzero, thus providing the violation of the mER, the diffusivity can be presented in a more appropriate form. It was shown that the approximation (\[long-c\]) is valid for $x\rightarrow\infty$. Hence, if $c(x)\rightarrow \pm(a/x)^{1/2}$ for large $x$, then $$G(y,z)= 4\varkappa^4 a Z^2_2(y)Z^2_2(z)\left[1-c(y)\right] \left[1-c(z)\right]
\label{n=1/2}$$ and $$\Delta D = \frac{4\varkappa^4 a v^3}{D_0^2}\left\{\int\limits_0^\infty dy e^{-\gamma y}Z^2_2(y)\left[1-c(y)\right]\right\}^2.
\label{DDn=1/2}$$ For the power law $c(x)$ with $n=1/2$ $\varkappa_c=0.8425...$ and is defined by the mechanism II. Nontrivial approximate value of $\Delta D$ could be calculated by the saddle point method for $\varkappa\rightarrow 1$ in close analogy with the calculation of $v$ in Ref. but, unfortunately, this region is unphysical due to the divergence of $Z_4(x,y,z)$. It is interesting to note that does not indicate any singularity at $\varkappa\rightarrow\varkappa_c$ and $\Delta D$ remains finite at that point.

Correlation functions $c(x)$ having local extrema
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Such correlation functions demonstrate very remarkable results for the critical $\varkappa_c$ if that value is defined by the mechanism II. Details my be 