  Spatial persistent large deviations probability of surface growth processes
governed by the Edwards-Wilkinson dynamics, $P_x(x,s)$, with $-1 \leq s \leq 1$
is mapped isomorphically onto the temporal persistent large deviations
probability $P_t(t,s)$ associated with the stochastic Markovian random walk
problem. We show using numerical simulations that the infinite family of
spatial persistent large deviations exponents $\theta_x(s)$ characterizing the
power law decay of $P_x(x,s)$ agrees, as predicted on theoretical grounds by
Majumdar and Bray [Phys. Rev. Lett. {\bf 86}, 3700 (2001)] with the numerical
measurements of $\theta_t(s)$, the continuous family of exponents
characterizing the long time power law behavior of $P_t(t,s)$. We also discuss
the simulations of the spatial persistence probability corresponding to a
discrete model in the Mullins-Herring universality class, where our discrete
simulations do not agree well with the theoretical predictions perhaps because
of the severe finite-size corrections which are known to strongly inhibit the
manifestation of the asymptotic continuum behavior in discrete models involving
large values of the dynamical exponent and the associated extremely slow
convergence to the asymptotic regime.
