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Hole event for random holomorphic sections on compact Riemann surfaces
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Hole event for random holomorphic sections on compact Riemann surfaces
abstract
Let $X$ be a compact Riemann surface and $\mathcal L$ be a positive line bundle on it. We study the conditional zero expectation of all the holomorphic sections of $\mathcal L^n$ 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 $\nu$. This gives rise to a surprising forbidden set.
Forward citations
Cited by 1 Pith paper
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Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights
For every β>0, conditioning a power-exponential Gaussian analytic function to have no zeros in D(0,r) makes the scaled zero measure converge to a limit supported on {|z|=1}∪{|z|≥e^{1/β}}, avoiding {1<|z|<e^{1/β}}.
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