Some remarks on the wigner distribution

For a generalized Wigner ensemble of random $N$-th order matrices $A_N(\omega)$ the formula
$$ \lim_{N\to\infty}\frac1N\ln\int\det A_N(\omega)d\omega= \lim_{N\to\infty}\int\ln\det A_N(\omega)d\omega. $$
is obtained. This formula implies that $[\det A_N(\omega)]^{1/N}$ is strongly selfaveraging. In addition, it enables one to apply an integral with respect to anticommuting variables to the calculation of the limiting distribution. It can be shown that for this integral the Hartree–Fokapproximationgives the exact answer in the limit $N\to\infty$.