Overcrowding for zeros of Hyperbolic Gaussian analytic functions

Abstract We consider the family { f L } L &gt;0 of Gaussian analytic functions in the unit disk, distinguished by the invariance of their zero set with respect to hyperbolic isometries. Let n L ( r ) be the number of zeros of f L in a disk of radius r . We study the asymptotic probability of the rare event where there is an overcrowding of the zeros as r ↑ 1, i.e., for every L &gt; 0, we are looking for the asymptotics of the probability ℙ[ n L ( r ) ≥ V ( r )] with V ( r ) large compared to the $${\mathbb E}[n_{L}(r)]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>]</mml:mo> </mml:math> . Peres and Virág showed that for L = 1 (and only then) the zero set forms a determinantal point process, making many explicit computations possible. Curiously, contrary to the much better understood planar model, it appears that for L &lt; 1 the exponential order of decay of the probability of overcrowding when V is close to $${\mathbb E}[n_{L}(r)]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>]</mml:mo> </mml:math> is much less than the probability of a deficit of zeros.