  The propagator for the activity in a broad class of self-organized critical
models obeys an imaginary-time Schr\"odinger equation with a nonlocal,
history-dependent potential representing memory. Consequently, the probability
for an avalanche to spread beyond a distance $r$ in time $t$ has an anomalous
tail $\exp{[-C\,x^{1/(D-1)}]}$ for $x=r^D/t \gg 1$ and $D>2$, indicative of
glassy dynamics. The theory is verified for an exactly solvable model, where
$D=4$ and $C=3/4$, and for the Bak-Sneppen model where it is tested
numerically.
