Pants decompositions of random surfaces
It is well-known that a compact hyperbolic Riemann surface \(X\) of genus \(g\) can be decomposed as a connected sum of three-holed spheres (usually called pairs of pants). The total pants length of such a decomposition is the sum of the lengths of the curves used for cutting the surface; and the total pants length of \(X\) is the infimum of the total pants length of its decompositions. In this paper the authors show that for any \(\varepsilon>0\) the measure (with respect to the normalized Weil-Petersson volume on the moduli space of hyperbolic surfaces of genus \(g\)) of the set of hyperbolic surfaces of genus \(g\) with total pants length at least \(g^{7/6-\varepsilon}\) tends to~1 as \(g\) tends to~\(+\infty\). The authors also prove a similar result for combinatorial surfaces built by glueing together equilateral triangles to form a closed surface: for any \(\varepsilon>0\) the measure (with respect to the uniform measure on the finite set of combinatorial surfaces obtained by glueing \(N\) triangles) of the set of combinatorial surfaces built with \(N\) triangles and with total pants length at least \(N^{7/6-\varepsilon}\) tends to~1 as \(N\) tends to~\(+\infty\).
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