  We evaluate the mean escape time of overdamped particles over potential
barriers in short-correlated quenched Gaussian disorder potentials in one
dimension at low temperature. The thermally activated escape is very sensitive
to the form of the \emph{tail} of the potential barrier probability
distribution. We evaluate this tail by using the optimal fluctuation method.
For monotone decreasing autocovariances we reproduce the tail obtained by
Lopatin and Vinokur (2001). However, for nonmonotonic autocovariances of the
disorder potential we show that the tail changes. It is much higher when the
disorder potential exhibits negative autocorrelations, and it is much lower
when the autocovariance is nonmonotonic but everywhere positive. This leads to
an \emph{exponential} increase or decrease, respectively, of the mean escape
time.
