The Weil-Petersson volume of the moduli space 
of genus-
hyperbolic surfaces with 
labeled boundary components of lengths 
is
 |
where 
is the Weil-Petersson symplectic form and 
. In Fenchel-Nielsen
coordinates with length and twist variables 
, its normalization is 
. The integral uses
orbifold volume, and a boundary length 0 represents a cusp (Mirzakhani 2007). Write
.
For fixed 
,
Li and Wang (2026) reported
 |
as 
,
identifying the normalization constant as 
. Li and Wang (2026) also reported a moving-saddle
expansion uniform over all 
, with relative error 
at leading order and 
after optimal truncation, for some 
. The fixed-
formula displayed above is not the uniform leading term. The
full uniformity and normalization arguments require independent assessment. Li and
Wang (2026) report AI assistance with mathematical derivation and verification.
Li and Wang (2026) further reported a limiting primitive unoriented length spectrum for a random surface drawn from the probability
measure given by normalized Weil-Petersson volume on 
as 
and 
. The limit is a Poisson background
of intensity 
together with independent clusters given by the complete primitive spectra of pairs of pants having two cusps. The waist lengths of
those two-cusp pairs of pants form a Poisson process
with intensity 
.
The clusters retain correlations, so the full limit is not Poisson. This extension
is subject to the same independent-assessment qualification as the volume asymptotics.
