Scaling of the conductance and resistance of square lattices with an exponentially wide spectrum of the resistances of links

The conductance $\overline{G}$ and $\overline{G^{-1}}$ resistance average over realizations of disorder have been calculated for various sizes of square lattices $L$. In contrast with different direction of change in the two quantities at percolation in lattices with the binary spread of conductances of links ($g_i=0$ or $1$), it has been found that the mean conductance and resistance of lattices decrease simultaneously with an increase in $L$ in the case of an exponential distribution of local conductances $g_i=\exp(-k x_i)$, where $x_i\in [0,1]$ are random numbers. When $L$ is smaller than the disorder length $L_0=bk^\nu$, $\overline{G}(L)$ and $\overline{G^{-1}}(L)$ are proportional to $L^{-n}$ with $n=k/5$ and $n=k/6$, respectively. A similar behavior is characteristic of the distributions of conductances of links, which simulate a transition between the open and tunneling regimes in semiconducting lattices of antidots created in a two-dimensional electron gas.