For random hyperbolic surfaces of large genus, the variance of the weighted count of closed geodesics in short intervals [X, X+H] with H=o(X) is asymptotically 2H log X as genus tends to infinity.
arXiv:2507.20653 [math.NT] 18, 20
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Closed geodesics in short intervals for random hyperbolic surfaces
For random hyperbolic surfaces of large genus, the variance of the weighted count of closed geodesics in short intervals [X, X+H] with H=o(X) is asymptotically 2H log X as genus tends to infinity.