Averages of determinants of Laplacians over moduli spaces for large genus

Yuxin He and Yunhui Wu Yau Mathematical Sciences Center and Department of Mathematical Sciences, Tsinghua University, Beijing, China [email protected] [email protected]

Abstract.

Let $\mathcal{M}_{g}$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. We view the regularized determinant $\log\det(\Delta_{X})$ of Laplacian as a function on $\mathcal{M}_{g}$ and show that there exists a universal constant $E>0$ such that as $g\to\infty$ ,

(1)

the expected value of $\left|\frac{\log\det(\Delta_{X})}{4\pi(g-1)}-E\right|$ over $\mathcal{M}_{g}$ has rate of decay $g^{-\delta}$ for some uniform constant $\delta\in(0,1)$ ;
(2)

the expected value of $\left|\frac{\log\det(\Delta_{X})}{4\pi(g-1)}\right|^{\beta}$ over $\mathcal{M}_{g}$ approaches to $E^{\beta}$ whenever $\beta\in[1,2)$ .

1. Introduction

Let $X$ be a closed hyperbolic surface of genus $g$ $(g>1)$ . The spectrum of the Laplacian $\Delta_{X}$ of $X$ on $L^{2}(X)$ is a discrete closed subset in $\mathbb{R}^{\geq 0}$ and consists of eigenvalues. We enumerate them, counted with multiplicity, in the following increasing order:

0=\lambda_{0}(X)<\lambda_{1}(X)\leq\lambda_{2}(X)\leq\cdots\to+\infty.

For $z\in\mathbb{C}$ , the spectral zeta function of $X$ is given as

\zeta_{X}(z)=\sum_{i=1}^{\infty}\frac{1}{\lambda_{i}^{z}(X)}.

From Weyl’s Law, the function $\zeta_{X}(z)$ is well-defined and holomorphic when $\Re{z}>1$ . The regularized determinant is usually defined by

\log\det(\Delta_{X})\overset{\text{def}}{=}-\zeta_{X}^{\prime}(0),

provided that $\zeta_{X}(\cdot)$ has an analytic extension to $z=0$ . It is known from e.g. Hoker-Phong [5] and Sarnak [25] that

(1)

\det(\Delta_{X})=Z_{0}^{\prime}(1)e^{E\cdot 4\pi(g-1)},

where

E=\frac{-1+2\log 2\pi+8\xi^{\prime}(-1)}{8\pi}\approx 0.0538,

and $Z_{0}(s)$ is the Selberg zeta function

Z_{0}(s)=\Pi_{\gamma}\Pi_{k=0}^{\infty}\left(1-e^{(k+s)\ell_{\gamma}(X)}\right)

where $\gamma$ runs over all primitive closed geodesics on $X$ and $\ell_{\gamma}(X)$ is the length of $\gamma$ on $X$ . It is known by Wolpert [29] that the magnitudes $\left|\log\det(\Delta_{X})\right|$ and $\left|\log Z_{0}^{\prime}(1)\right|$ are proper functions on moduli space $\mathcal{M}_{g}$ of Riemann surfaces of genus $g$ , that is, they will be divergent when the surface $X$ goes to the boundary of $\mathcal{M}_{g}$ .

In this work, we view $\log\det(\Delta_{X})$ and $\log Z_{0}^{\prime}(1)$ as random variables with respect to the probability measure $\mathop{\rm Prob}\nolimits_{\rm WP}^{g}$ on $\mathcal{M}_{g}$ given by the Weil-Petersson metric. Naud is the first one to study their asymptotic behaviors for large genus and shows in [21] that

Theorem (Naud).

For any $\epsilon>0$ ,

(2)

\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ \frac{\left|\log Z_{0}^{\prime}(1)\right|}{4\pi(g-1)}<\epsilon\right)=1.

Actually it is shown in [21] that the property in (2) also holds for the other two models of random hyperbolic surfaces: random cover [14, 15] and Brooks-Makover [3]. In the proof, Naud essentially only requires uniform spectral gaps and suitable countings of closed geodesics for large genus. In this paper, we focus on the Weil-Petersson model and use the techniques developed in [22, 32] to show that

Theorem 1.

There exists a uniform constant $0<\delta<1$ such that as $g\to\infty$ ,

\frac{1}{\mathop{\rm Vol}_{\mathrm{WP}}(\mathcal{M}_{g})}\int_{\mathcal{M}_{g}}\frac{\left|\log Z_{0}^{\prime}(1)\right|}{4\pi(g-1)}dX=O\left(\frac{1}{g^{\delta}}\right),

where the implied constant is universal and $\mathop{\rm Vol}_{\mathrm{WP}}(\mathcal{M}_{g})$ is the Weil-Petersson volume of $\mathcal{M}_{g}$ .

It would be interesting to study the optimal choice of $\delta$ in Theorem 1. By using Markov’s inequality, Theorem 1 clearly implies the result of Naud above. Moreover, combining with (1) it also implies that

\lim\limits_{g\to\infty}\frac{1}{\mathop{\rm Vol}_{\mathrm{WP}}(\mathcal{M}_{g})}\int_{\mathcal{M}_{g}}\frac{\left|\log\det(\Delta_{X})\right|}{4\pi(g-1)}dX=E.

Indeed, our second result is as follows.

Theorem 2.

For any $\beta\in(0,2)$ ,

\lim\limits_{g\to\infty}\frac{1}{\mathop{\rm Vol}_{\mathrm{WP}}(\mathcal{M}_{g})}\int_{\mathcal{M}_{g}}\left|\frac{\log\det(\Delta_{X})}{4\pi(g-1)}\right|^{\beta}dX=E^{\beta}.

For any $\beta\geq 2$ ,

\int_{\mathcal{M}_{g}}\left|\log\det(\Delta_{X})\right|^{\beta}dX=\infty.

For any $\beta\in(0,1)$ , this is due to Naud [21].

Notations.

We say two positive functions

f_{1}(g)\prec f_{2}(g)\quad\emph{or}\quad f_{2}(g)\succ f_{1}(g)\quad\emph{or}\quad f_{1}(g)=O(f_{2}(g))

if there exists a universal constant $C>0$ , independent of $g$ , such that $f_{1}(g)\leq C\cdot f_{2}(g)$ ; and we say

f_{1}(g)\asymp f_{2}(g)

if $f_{1}(g)\prec f_{2}(g)$ and $f_{2}(g)\prec f_{1}(g)$ .

Plan of the paper.

In Section 2 we will provide a review of relevant background materials and show that for $\beta>0$ , $\mathbb{E}_{\rm WP}^{g}\left[\left(\frac{1}{\mathop{\rm sys}(X)}\right)^{\beta}\right]\asymp 1$ if and only if $\beta\in(0,2)$ which will be applied in the proof of Theorem 2. Then by assuming (2) and closely following [21], in Section 3 we will complete the proof of Theorem 2. In the last Section, we will use the techniques developed in [22, 32] to finish the proof of Theorem 1.

Acknowledgement

We would like to thank all the participants in our seminar on Teichmüller theory for helpful discussions and comments on this project. We also would like to thank Frédéric Naud for his interests on this work. The second named author is partially supported by the NSFC grant No. $12171263$ , $12361141813$ and $12425107$ .

2. Preliminary

In this section, we set up notations and basic propositions on the regularized determinant of the Laplacian and the Weil-Petersson model of random surfaces.

2.1. Determinants of Laplacians

Let $X_{g,n}$ with $2g-2+n\geq 1$ be a complete hyperbolic surface of genus $g$ with $n$ geodesic boundaries. By the Gauss-Bonnet formula, its hyperbolic area satisfies $\mathrm{Area}(X_{g,n})=2\pi(2g-2+n)$ . For a closed geodesic $\gamma$ on $X_{g,n}$ , we use $\ell(\gamma)$ or $\ell_{\gamma}(X_{g,n})$ to denote its length.

Now let $X=X_{g}$ be a closed hyperbolic surface of genus $g$ $(g>1)$ and $\Delta_{X}$ be its Laplacian. Recall that as in the introduction, the regularized determinant $\log\det(\Delta_{X})=-\zeta_{X}^{\prime}(0)$ . Indeed, it is known from e.g. [5, 25, 21] that it also satisfies the following identity

(3)

\log\det\Delta_{X}=4\pi(g-1)E+\gamma_{0}-\int_{0}^{1}\frac{S_{X}(t)}{t}dt-\int_{1}^{\infty}\frac{S_{X}(t)-1}{t}dt,

where $\gamma_{0}$ is the Euler constant, $E\approx 0.0538$ is the constant given in (1), and

S_{X}(t)=\frac{e^{-t/4}}{(4\pi t)^{\frac{1}{2}}}\sum_{k\geq 1}\sum_{\gamma\in\mathcal{P}}\frac{\ell(\gamma)}{2\sinh(k\ell(\gamma)/2)}e^{-(k\ell(\gamma))^{2}/4t}.

Here $\mathcal{P}$ is the set of oriented primitive closed geodesics on $X$ . One may see [21] and references therein for more details. Selberg’s trace formula gives the following identity:

(4)

\sum_{j=0}^{\infty}e^{-t\lambda_{j}}=S_{X}(t)+\frac{4\pi(g-1)e^{-\frac{t}{4}}}{4\pi}\int_{-\infty}^{\infty}re^{-tr^{2}}\tanh(\pi r)dr.

See [4, Theorem 9.5.3] or [2, Theorem 5.6] for a general form of Selberg’s trace formula. One may also see e.g. [26] and [8].

2.2. Counting geodesics

We will apply the following two countings on closed geodesics. The first one is for general closed geodesics. One may see e.g. [4, Lemma 6.6.4].

Lemma 3.

Let $S$ be a hyperbolic surface of genus $g\geq 2$ and let $L>0$ . There are at most $(g-1)e^{L+6}$ oriented closed geodesics on $S$ of length $\leq L$ which are not iterates of closed geodesics of length $\leq 2\mathop{\rm arcsinh}1$ .

Recall that a closed geodesic $\gamma$ is called filling in a hyperbolic surface $Y$ with geodesic boundaries if each component of $Y\setminus\gamma$ is homotopic to a single point or some boundary component of $Y$ . The second counting is the following bound for filling closed geodesics that is essential in the proof of Theorem 1. One may see [32, Theorem 4] or [31, Theorem 18] for more details.

Theorem 4 (Wu-Xue).

For any $0<\epsilon<\frac{1}{2}$ and $m=2g-2+n\geq 1$ , there exists a constant $c(\epsilon,m)>0$ only depending on $\epsilon$ and $m$ such that for all $L>0$ and any compact hyperbolic surface $Y$ of genus $g$ with $n$ boundary simple closed geodesics, the following inequality holds:

\#_{f}(Y,L)\leq c(\epsilon,m)\cdot e^{L-\frac{1-\epsilon}{2}\ell(\partial Y)},

where $\#_{f}(Y,L)$ is the number of filling closed geodesics in $Y$ of length $\leq L$ , and $\ell(\partial Y)$ is the total length of the boundary closed geodesics of $Y$ .

2.3. The Weil-Petersson model of random surfaces

For $2g-2+n\geq 1$ , let $\mathcal{T}_{g,n}$ be the Teichmüller space of Riemann surfaces with $g$ genus and $n$ punctures. The mapping class group $\mathop{\rm Mod}_{g,n}$ acts on $\mathcal{T}_{g,n}$ keeping the Weil-Petersson symplectic form $\omega_{\mathrm{WP}}$ invariant. The quotient space $\mathcal{M}_{g,n}=\mathcal{T}_{g,n}/\mathop{\rm Mod}_{g,n}$ is the moduli space of Riemann surfaces. Set $\mathcal{T}_{g}=\mathcal{T}_{g,0}$ and $\mathcal{M}_{g}=\mathcal{M}_{g,0}$ . For $L=(L_{1},\cdots,L_{n})\in\mathbb{R}^{n}_{\geq 0}$ , let $\mathcal{T}_{g,n}(L)$ be the Teichmüller space of bordered hyperbolic surfaces with $n$ geodesic boundaries of length $L_{1},\cdots,L_{n}$ , and $\mathcal{M}_{g,n}(L)=\mathcal{T}_{g,n}(L)/\mathop{\rm Mod}_{g,n}$ . In particularly, $\mathcal{T}_{g,n}(0,\cdots,0)=\mathcal{T}_{g,n}$