Spatial persistence of fluctuating interfaces

We show that the probability, P-0(l). that the height of a fluctuating (d 1)-dimensional interface in it steady state stays above its initial value up to a distance l, along any linear cut in the d-dimensional space, decays as P-0(l) similar to l(-theta). Here theta is a ''spatial" persistence exp onent, and takes different values, theta (s) or theta (0), depending on how the point from which l is measured is specified. These exponents are shown to map onto corresponding temporal persistence exponents for a generalized d = 1 random-walk equation. The exponent theta (0) is nontrivial even for Gaussian interfaces.