Under |L|^2 = o(g) as g → ∞, the expected total mass of the Brownian loop measure on non-peripheral homotopy classes converges to an explicit κ-dependent function diverging as log(1/κ), and for κ=0 it is asymptotically (1/2) log g on simple closed geodesics.
Random hyperbolic surfaces of large genus have first eigenvalues greater than
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The total mass of Brownian loop measure of Riemann surfaces for large genus
Under |L|^2 = o(g) as g → ∞, the expected total mass of the Brownian loop measure on non-peripheral homotopy classes converges to an explicit κ-dependent function diverging as log(1/κ), and for κ=0 it is asymptotically (1/2) log g on simple closed geodesics.