Concentration of Random Determinants and Permanent Estimators

We show that the absolute value of the determinant of a matrix with random independent (but not necessarily i.i.d.) entries is strongly concentrated around its mean. As an application, we show that Godsil–Gutman and Barvinok estimators for the permanent of a strictly positive matrix give subexponential approximation ratios with high probability. A positive answer to the main conjecture of the paper would lead to polynomial approximation ratios in the above problem.