The total mass of Brownian loop measure of Riemann surfaces for large genus

Abstract

Let $\mathcal{M}_{g,n}(\mathbf{L})$ be the moduli space of hyperbolic surfaces of genus $g$ with $n \geq 0$ hyperbolic ends of widths $\mathbf{L} \in \mathbb{R}_{\geq 0}^n$. We regard the total mass $|μ_X^κ|$ of the Brownian loop measure with the killing rate $κ$ as a random variable on $\mathcal{M}_{g,n}(\mathbf{L})$. Under the condition $|\mathbf{L}|^2 =o(g)$ as $g \to \infty$, we obtain the following two main results: $(1)$ For any $κ> 0$, the expected value of $|μ_X^κ|$ on all non-peripheral homotopy classes over $\mathcal{M}_{g,n}(\mathbf{L})$ converges to an explicit function of $κ$, which blows up at the rate $ \log \left(\frac{1}κ\right)$ as $κ\to 0^+$. $(2)$ For $κ=0$, over $\mathcal{M}_{g,n}(\mathbf{L})$ the expected value of $|μ_X|$ on homotopy classes of (iterates of) all non-peripheral simple closed geodesics is asymptotically $\frac{1}{2} \log g$.