Geodesic
SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES
Forum of Mathematics, Sigma, Tome 8 (2020)

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Given integers $g,n\geqslant 0$ satisfying $2-2g-n0$, let ${\mathcal{M}}_{g,n}$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $g$ with $n$ cusps. We study the global behavior of the Mirzakhani function $B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$ which assigns to $X\in {\mathcal{M}}_{g,n}$ the Thurston measure of the set of measured geodesic laminations on $X$ of hyperbolic length ${\leqslant}1$. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of ${\mathcal{M}}_{g,n}$ and deduce that $B$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $B$ to statistics of counting problems for simple closed hyperbolic geodesics. 
@article{10_1017_fms_2019_49,
     author = {FRANCISCO ARANA-HERRERA and JAYADEV S. ATHREYA},
     title = {SQUARE-INTEGRABILITY {OF} {THE} {MIRZAKHANI} {FUNCTION} {AND} {STATISTICS} {OF} {SIMPLE} {CLOSED} {GEODESICS} {ON} {HYPERBOLIC} {SURFACES}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {8},
     year = {2020},
     doi = {10.1017/fms.2019.49},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2019.49/}
}
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FRANCISCO ARANA-HERRERA; JAYADEV S. ATHREYA. SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2019.49

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