  We show experimentally and theoretically that the persistence of large
deviations in equilibrium step fluctuations is characterized by an infinite
family of independent exponents. These exponents are obtained by carefully
analyzing dynamical experimental images of Al/Si(111) and Ag(111) equilibrium
steps fluctuating at high (970K) and low (320K) temperatures respectively, and
by quantitatively interpreting our observations on the basis of the
corresponding coarse-grained discrete and continuum theoretical models for
thermal surface step fluctuations under attachment/detachment
(``high-temperature'') and edge-diffusion limited kinetics
(``low-temperature'') respectively.
