Benjamini-Schramm limits of Masur-Smillie-Veech random translation surfaces of area g and genus g are Poisson translation planes of intensity 4.
Large Genus Asymptotics for Siegel – Veech Constants
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Benjamini-Schramm limits of high genus translation surfaces: research announcement
Benjamini-Schramm limits of Masur-Smillie-Veech random translation surfaces of area g and genus g are Poisson translation planes of intensity 4.