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Within an Miller-Abrahams random resistor model, finding the critical resistance when there is spatial disorder is simple as there is the bonding criterion

$\int_0^{r_c} 4 \pi N r^2 dr = B_c \approx 2.7$

in three dimensions where $N$ is the density of sites. Solve for $r_c$ and then put into the resistance as $R_c \sim e^{2 r_c/a}$.

When there is $\textbf{both}$ spatial and energy disorder, a similar procedure is used albeit in four dimensions but the bonding criterion is the same in essence. This gives rise to the so-called variable range hopping.

My question is whether there is a similar simple procedure if there exists $\textbf{only}$ energy disorder. It seems that the normal bonding number $B_c$ would be invalid since that is coming from the geometry. I suppose instead of a critical radius, there would be a critical energy scale and that the problem could be much more difficult if the density of states is non-constant.

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