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 $n\ge 0$ hyperbolic ends of widths $\mathbf{L}\in {\mathbb{R}}_{\ge 0}^n$. We regard the total mass $|{\mu }_X^{\kappa }|$ of the Brownian loop measure with the killing rate $\kappa$ 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 $\kappa >0$, the expected value of $|{\mu }_X^{\kappa }|$ on all non-peripheral homotopy classes over ${\mathcal{M}}_{g,n}(\mathbf{L})$ converges to an explicit function of $\kappa$, which blows up at the rate $\log \left(\frac{1}{\kappa }\right)$ as $\kappa \to 0^{+}$. $(2)$ For $\kappa =0$, over ${\mathcal{M}}_{g,n}(\mathbf{L})$ the expected value of $|{\mu }_X|$ on homotopy classes of (iterates of) all non-peripheral simple closed geodesics is asymptotically $\frac{1}{2}\log g$.

32 pages, 3 figures. Comments welcome