Universal Distribution Functions in Two-Dimensional Localized Systems

We find the conductance distribution function of the two-dimensional Anderson model in the strongly localized limit. The fluctuations of $\ln g$ grow with lateral size as $L^{1/3}$ and follow a universal distribution that depends on the type of leads. For narrow leads, it is the Tracy-Widom distribution, which appears in the problem of the largest eigenvalue of random matrices from the Gaussian unitary ensemble and in many other problems like the longest increasing subsequence of a permutation, directed polymers, or polynuclear growth. We also show that for wide leads the conductance follows a related, but different, distribution.

Figure 1

Histograms of $\ln g$ versus the scaled variable $\chi$ for several sizes and disorders of the Anderson model with narrow (solid symbols) and wide (empty symbols) leads. The continuous lines correspond to $F_2^{\prime }(\chi )$ and $F_0^{\prime }(\chi )$.