  We present estimates on the small singular values of a class of matrices with
independent Gaussian entries and inhomogeneous variance profile, satisfying a
broad-connectedness condition. Using these estimates and concentration of
measure for the spectrum of Gaussian matrices with independent entries, we
prove that for a large class of graphs satisfying an appropriate expansion
property, the Barvinok--Godsil-Gutman estimator for the permanent achieves
sub-exponential errors with high probability.
