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<p>We consider the family {<i>f</i><sub><i>L</i></sub>}<sub><i>L</i>&gt;0</sub> of Gaussian analytic functions in the unit disk, distinguished by the invariance of their zero set with respect to hyperbolic isometries. Let <i>n</i><sub><i>L</i></sub>(<i>r</i>) be the number of zeros of <i>f</i><sub><i>L</i></sub> in a disk of radius <i>r</i>. We study the asymptotic probability of the rare event where there is an overcrowding of the zeros as <i>r</i> ↑ 1, i.e., for every <i>L</i> &gt; 0, we are looking for the asymptotics of the probability ℙ[<i>n</i><sub><i>L</i></sub>(<i>r</i>) ≥ <i>V</i>(<i>r</i>)] with <i>V</i>(<i>r</i>) large compared to the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2759_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb E}[n_{L}(r)]\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msub> <mi>n</mi> <mrow> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </math></EquationSource> </InlineEquation>. Peres and Virág showed that for <i>L</i> = 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 <i>L</i> &lt; 1 the exponential order of decay of the probability of overcrowding when <i>V</i> is close to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2759_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb E}[n_{L}(r)]\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msub> <mi>n</mi> <mrow> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </math></EquationSource> </InlineEquation> is much less than the probability of a deficit of zeros.</p>

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Overcrowding for zeros of Hyperbolic Gaussian analytic functions

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  • 发表日期：

    2025年5月9日

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  • Keren Mor Waknin

摘要

We consider the family {fL}L>0 of Gaussian analytic functions in the unit disk, distinguished by the invariance of their zero set with respect to hyperbolic isometries. Let nL(r) be the number of zeros of fL 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 > 0, we are looking for the asymptotics of the probability ℙ[nL(r) ≥ V(r)] with V(r) large compared to the \({\mathbb E}[n_{L}(r)]\) E [ n L ( r ) ] . 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 < 1 the exponential order of decay of the probability of overcrowding when V is close to \({\mathbb E}[n_{L}(r)]\) E [ n L ( r ) ] is much less than the probability of a deficit of zeros.