Hole event for random holomorphic sections on compact Riemann surfaces

Let X be a compact Riemann surface and L be a positive line bundle on it. We study the conditional zero expectation of all the holomorphic sections of Lⁿ which do not vanish on D for some fixed open subset D of X. We prove that as n tends to infinity, the zeros of these sections are equidistributed outside D with respect to a probability measure ν. This gives rise to a surprising forbidden set.