# The moduli space of twisted Laplacians
and random matrix theory

Jens Marklof  Jens Marklof, School of Mathematics, University of Bristol,
Bristol BS8 1UG, U.K.
<EMAIL_ADDRESS>and  Laura Monk  Laura Monk, School of Mathematics,
University of Bristol, Bristol BS8 1UG, U.K.
<EMAIL_ADDRESS>
(Date: 15 July 2024)

###### Abstract.

Rudnick recently proved that the spectral number variance for the Laplacian of
a large compact hyperbolic surface converges, in a certain scaling limit and
when averaged with respect to the Weil-Petersson measure on moduli space, to
the number variance of the Gaussian Orthogonal Ensemble of random matrix
theory. In this note we extend Rudnick’s approach to show convergence to the
Gaussian Unitary Ensemble for twisted Laplacians which break time-reversal
symmetry, and to the Gaussian Symplectic Ensemble for Dirac operators. This
addresses a question of Naud, who obtained analogous results for twisted
Laplacians on high genus random covers of a fixed compact surface.

Research supported by EPSRC grant EP/W007010/1. Data supporting this study are
included within the article. MSC (2020): 81Q50 (Primary); 11F72, 30F60, 58J50
(Secondary)

###### Contents

  1. 1 Introduction
  2. 2 The affine mapping class group
  3. 3 Invariant measures and Fourier transforms
  4. 4 Length-holonomy functions
  5. 5 Twisted Laplacians
  6. 6 Main Lemmas
  7. 7 Proof of Theorem 1
  8. 8 Dirac operators and GSE statistics

##  1\. Introduction

One of the long-standing conjectures in quantum chaos is that the spectral
statistics of quantum systems with chaotic classical limit are governed by
random matrix theory [4]. Despite convincing heuristics [18, 8], there is
currently not a single example where this phenomenon can be established
rigorously. What is more, counter examples are known [5, 6, 9], so the
conjecture can only be assumed to hold generically. Rudnick recently pointed
out that random matrix statistics emerge in a certain scaling regime for the
spectrum of the Laplacian on hyperbolic surfaces of large genus [15], if one
averages over the moduli space of such surfaces with respect to the Weil-
Petersson measure. (See also [2] for an early numerical study of spectral
statistics on average over moduli space.) These statistics are given by the
Gaussian Orthogonal Ensemble (GOE), which is consistent with Bohigas, Giannoni
and Schmit’s conjecture [4], since the underlying classical dynamics (the
geodesic flow on the hyperbolic surfaces) is time-reversal invariant. Twisted
Laplacians, which model the Aharonov-Bohm effect on hyperbolic surfaces, break
time-reversal symmetry, and their statistics are expected to follow the
statistics from the Gaussian Unitary Ensemble (GUE) [4]. The purpose of this
paper is to construct the moduli space of such twisted Laplacians, which
allows us to extend Rudnick’s result to these operators, and prove
compatibility with GUE statistics in the same scaling regime for twisted
Laplacians on large genus hyperbolic surface with random flux. Analogous
results, both for GUE and for GOE statistics, were established by Naud [13] in
the case of random covers of a fixed compact hyperbolic surface with and
without Aharonov-Bohm flux. The mechanism that produces the difference between
GUE and GOE through the application of semiclassical trace formulas was first
pointed out by Berry [3] in his diagonal approximation argument.

The framework developed in this paper also produces analogous results for
twisted Dirac operators, where we establish convergence to the Gaussian
Symplectic Ensemble (GSE) in the presence of time-reversal symmetry, and
otherwise to GUE.

Rudnick and Wigman have refined the results of [15] by proving a central limit
theorem for the smoothed spectral counting function [16] and asymptotic almost
sure convergence of the spectral variance to GOE [17], in the same scaling
limit used in [15] and the present paper. We will leave these and other
possible extensions to the GUE and GSE setting to future study.

The set-up for twisted Laplacians is as follows; cf. [14] for more details and
background. We represent a compact oriented hyperbolic surface of genus $g$ as
$X=\Gamma\backslash{\mathbb{H}}$, where
$\Gamma\subset\operatorname{PSL}(2,{\mathbb{R}})$ is a co-compact Fuchsian
group acting on the complex upper half plane
${\mathbb{H}}={\mathbb{R}}+\mathrm{i}{\mathbb{R}}_{>0}$ by Möbius
transformations. The Laplacian
$\Delta=y^{2}(\partial_{x}^{2}+\partial_{y}^{2})$ acts on twice differentiable
functions $\varphi:{\mathbb{H}}\to{\mathbb{C}}$ parametrised by
$z=x+\mathrm{i}y$. Given a unitary character $\chi:\Gamma\to{\mathbb{C}}$, we
restrict to the Hilbert space ${\mathcal{H}}_{\chi}$ of functions satisfying
the relation

(1.1) $\varphi(\eta\cdot z)=\chi(\eta)\,\varphi(z)\quad\text{for all
$\eta\in\Gamma$,}$

and with inner product

(1.2)
$\langle\varphi_{1},\varphi_{2}\rangle=\int_{\Gamma\backslash{\mathbb{H}}}\overline{\varphi_{1}(z)}\;\varphi_{2}(z)\;\frac{dx\,dy}{y^{2}}.$

The restriction of $\Delta$ to ${\mathcal{H}}_{\chi}$ is essentially self-
adjoint; we denote its self-adjoint extension by $\Delta_{\chi}$. The operator
$-\Delta_{\chi}$ has non-negative discrete spectrum

(1.3) $0\leq\lambda_{0}\leq\lambda_{1}\leq\ldots\to\infty$

with only accumulation point at $\infty$. If $\chi$ is not a real-valued
function, we will refer to $\Delta_{\chi}$ as a twisted Laplacian. This
operator models the Aharonov-Bohm effect of a free quantum particle on the
surface subject to $2g$ magnetic flux lines $\theta_{1},\ldots,\theta_{2g}$.
Explicitly,
$\chi(\eta)=\chi_{\text{\boldmath$\theta$}}(\eta)=\exp(2\pi\mathrm{i}(\theta_{1}[\eta]_{1}+\ldots+\theta_{2g}[\eta]_{2g}))$
with $[\eta]_{j}$ the topological winding number around the $j$th flux line of
the closed geodesic corresponding to $\eta$. Since $\chi\neq\overline{\chi}$,
the twisted Laplacian violates time-reversal symmetry, and one expects the
spectral statistics to be described by the unitary (GUE) rather than the
orthogonal (GOE) random matrix ensemble.

It is convient to define $r_{j}=\sqrt{\lambda_{j}-\frac{1}{4}}$ with the
branch of the square root chosen so that $\arg r_{j}\in[0,\frac{\pi}{2}]$.
That is, $r_{j}\in{\mathbb{R}}_{\geq 0}$ for $\lambda_{j}\geq\frac{1}{4}$ and
$r_{j}\in\mathrm{i}(0,\tfrac{1}{2}]$ for $\lambda_{j}<\frac{1}{4}$. The main
tool to analyse the distribution of $r_{j}$ is the Selberg trace formula,
which relates spectral to geometric data. For any even test function

(1.4) $h(r)=\int_{\mathbb{R}}\hat{h}(u)\,\mathrm{e}^{\mathrm{i}ur}du,$

with $\hat{h}\in\operatorname{C{}}^{\infty}({\mathbb{R}})$ compactly
supported, we have

(1.5) $\sum_{j=0}^{\infty}h(r_{j})=(g-1)\int_{\mathbb{R}}h(r)r\tanh(\pi
r)\,dr+\sum_{n=1}^{\infty}\sum_{\\{\eta\\}}\chi(\eta)^{n}\,\frac{\ell_{\eta}\hat{h}(n\ell_{\eta})}{2\sinh(n\ell_{\eta}/2)}.$

Here the second sum is over all conjugacy classes
$\\{\eta\\}=\\{\delta\eta\delta^{-1}:\delta\in\Gamma\\}$ of primitive elements
$\eta\in\Gamma$, $\eta\neq 1$ ($\eta$ is called primitive in $\Gamma$, if
$\eta=\delta^{n}$ for some $\delta\in\Gamma$ implies $n=\pm 1$). The set of
primitive conjugacy classes $\\{\eta\\}$ is in one-to-one correspondence with
primitive closed oriented geodesics on $X$, and their lengths are denoted by
$\ell_{\eta}$.

Following Rudnick [15], we consider the distribution of the smoothed counting
function of the $r_{j}$,

(1.6) $N_{f,L,\tau}=\sum_{j=0}^{\infty}h_{L,\tau}(r_{j}),\qquad
h_{L,\tau}(r)=f(L(r-\tau))+f(L(r+\tau)),$

with $L,\tau>0$ and assuming the Fourier transform of the window function

(1.7)
$\hat{f}(u)=\frac{1}{2\pi}\int_{\mathbb{R}}f(u)\,\mathrm{e}^{-\mathrm{i}ur}du$

is smooth and compactly supported. This ensures our assumptions on
$h=h_{L,\tau}$ are satisfied. More precisely, we are interested in the
statistical distribution of $N_{f,L,\tau}$ for random choices of
$(X,{\text{\boldmath$\theta$}})$.

The main new construction presented in this paper is a family of natural
probability measures $\mu_{g,q}$ on the moduli space
$\widetilde{\mathcal{M}}_{g}$ of the pair $(X,{\text{\boldmath$\theta$}})$ for
given genus $g$. It may be viewed as an extension of the classical Weil-
Petersson probability measure $\mu_{g}^{\mathrm{WP}}$ on the moduli space
${\mathcal{M}}_{g}$ of hyperbolic surfaces $X$ of genus $g$ considered in
[15]. The index $q$ indicates whether the measure on $\theta$ is Lebesgue
($q=\infty$) or discrete supported on rationals with denominator
$q\in{\mathbb{N}}$. The cases $q=1$ and $q=2$ correspond to the trivial resp.
general real characters $\chi(\eta)\in\\{\pm 1\\}$ (where we expect GOE), and
$3\leq q\leq\infty$ to complex characters (where we expect GUE).

We will now state our principal result and postpone the precise definition of
the extended Weil-Petersson measure $\mu_{g,q}$ and a discussion of its
properties to the remainder of the paper. We denote by
${\mathbb{E}}_{g}^{\mathrm{WP}}$ and ${\mathbb{E}}_{g,q}$ the corresponding
expected values defined by

${\mathbb{E}}_{g}^{\mathrm{WP}}[\ldots]=\int_{{\mathcal{M}}_{g}}[\ldots]\,d\mu_{g}^{\mathrm{WP}},\qquad{\mathbb{E}}_{g,q}[\ldots]=\int_{\widetilde{\mathcal{M}}_{g}}[\ldots]\,d\mu_{g,q}.$

Let $\Sigma_{g,q}^{2}(L,\tau;f)$ denote the variance of the counting function
$N_{f,L,\tau}$, defined as

(1.8)
$\Sigma_{g,q}^{2}(L,\tau;f)={\mathbb{E}}_{g,q}\bigg{[}\big{|}N_{f,L,\tau}-{\mathbb{E}}_{g,q}[N_{f,L,\tau}]\big{|}^{2}\bigg{]}.$

######  Theorem 1.

Fix $\tau>0$. For $f:{\mathbb{R}}\to{\mathbb{C}}$ with smooth, compactly
supported Fourier tansform, we have, for $L\geq 2$,

(1.9)
$\lim_{g\to\infty}\Sigma_{g,q}^{2}(L,\tau;f)=\begin{cases}\Sigma_{\mathrm{GOE}}^{2}(f)+O(L^{-2}\log
L)&(q=1,2)\\\ \Sigma_{\mathrm{GUE}}^{2}(f)+O(L^{-2}\log L)&(3\leq
q\leq\infty),\end{cases}$

with the random matrix densities

(1.10)
$\Sigma_{\mathrm{GOE}}^{2}(f)=\int_{\mathbb{R}}2|x|\,|\hat{f}(x)|^{2}dx,\qquad\Sigma_{\mathrm{GUE}}^{2}(f)=\int_{\mathbb{R}}|x|\,|\hat{f}(x)|^{2}dx.$

The analogous result for Dirac operators is given in Section 8.

The plan for this paper is as follows. In Section 2 we define a fiber bundle
over the Teichmüller space of hyperbolic surfaces $X$ of genus $g$, and a
natural action of the mapping class group on this extended Teichmüller space.
The fiber at each point $X$ is a torus which parametrises the twist parameter
of the Laplacian. By quotienting the extended Teichmüller space by the mapping
class group, we obtain a natural moduli space of twisted Laplacians. We then
define invariant probability measures on this space in Section 3, which model
continuous and discrete random twists. The projection of these measures to the
base moduli space is the classical Weil-Petersson measure. Although not
essential for subsequent discussion, we also introduce the Fourier transform
on the fiber, and describe the action of the mapping class group on Fourier
space. In Section 4 we focus on a particular class of functions on the
extended moduli space. These are length-holonomy functions which generalise
Mirzakhani’s geometric functions and appear on the geometric side of the
Selberg trace formula for twisted Laplacians. The precise parametrisation of
twisted Laplacians in terms of the extended Teichmüller space is explained in
Section 5. Section 6 states and proves the main lemmas that are required for
the proof of Theorem 1 in Section 7. We extend our approach to Dirac operators
with and without time-reversal symmetry in Section 8, and prove convergence to
GSE resp. GUE statistics.

##  2\. The affine mapping class group

Let $S_{g}$ be a given compact, connected, oriented, smooth surface of genus
$g\geq 2$, which we shall refer to as the base surface. The Teichmüller space
$\mathcal{T}_{g}$ is the set of equivalence classes of marked hyperbolic
surfaces $(X,\phi)$, where (i) $X$ is a hyperbolic surface, (ii) $\phi$ is a
diffeomorphism $S_{g}\rightarrow X$ (the marking), and (iii)
$(X_{1},\phi_{1})$, $(X_{2},\phi_{2})$ are called equivalent if there exists
an isometry $m:X_{1}\rightarrow X_{2}$ such that $m\circ\phi_{1}$ is isotopic
to $\phi_{2}$.

The mapping class group $\mathrm{MCG_{g}}$ of the base surface $S_{g}$ is
defined as the group of positive (i.e., orientation-preserving)
diffeomorphisms $\psi:S_{g}\rightarrow S_{g}$ modulo isotopies. The mapping
class group acts naturally on the Teichmüller space $\mathcal{T}_{g}$ by

(2.1) $\psi\cdot(X,\phi)=(X,\phi\circ\psi^{-1}).$

This action allows us to “forget” the marking, and so the quotient
$\mathcal{M}_{g}:=\mathrm{MCG_{g}}\backslash\mathcal{T}_{g}$ yields the moduli
space of hyperbolic surfaces $X$ of genus $g$ modulo isometries.

The first homology $H_{1}(S_{g};{\mathbb{Z}})$ of the base surface $S_{g}$ is
the abelianisation of its fundamental group $\pi_{1}(S_{g})$. Let us fix a
basis for $H_{1}(S_{g};{\mathbb{Z}})$ by choosing the family
$a_{1},\ldots,a_{g}$, $b_{1},\ldots,b_{g}$ of fundamental oriented loops, so
that

(2.2) $\pi_{1}(S_{g})=\langle
a_{i},b_{i}\,|\,[a_{1},b_{1}]\cdots[a_{g},b_{g}]\rangle,$

where $[a,b]=aba^{-1}b^{-1}$ is the commutator. $H_{1}(S_{g};{\mathbb{Z}})$
can be equipped with a symplectic form $\hat{\iota}$ for which
$a_{1},\ldots,a_{g},b_{1},\ldots,b_{g}$ form the standard symplectic basis, so
that $\hat{\iota}$ is represented by the matrix

(2.3) $J_{g}=\begin{pmatrix}0&I_{g}\\\ -I_{g}&0\end{pmatrix}.$

Any positive diffeomorphism $\psi:S_{g}\rightarrow S_{g}$ induces an
automorphism $\psi_{\ast}:H_{1}(S_{g};{\mathbb{Z}})\rightarrow
H_{1}(S_{g};{\mathbb{Z}})$, which preserves $\hat{\iota}$, and this
automorphism does not depend on the isotopy class of $\psi$. This allows us to
define the symplectic representation of the mapping class group,

(2.4)
$\mathrm{MCG_{g}}\rightarrow\operatorname{Sp}(2g,{\mathbb{Z}}),\qquad\psi\mapsto
M_{\psi},$

which associates with each $\psi\in\mathrm{MCG_{g}}$ a symplectic matrix
$M_{\psi}$ representing $\psi_{\ast}$ in the basis $(a_{i},b_{i})_{1\leq i\leq
g}$. Here $\operatorname{Sp}(2g,{\mathbb{Z}})$ denotes the set of $2g\times
2g$ matrices $M$ with integer coefficients satisfying
$\,{}^{\mathrm{t}}\\!MJ_{g}M=J_{g}$.

In order to define the affine mapping class group $\Lambda_{g}({\mathbb{R}})$,
we need to attach additional data to the surface $X$, in the form of a row
vector ${\text{\boldmath$\theta$}}\in{\mathbb{R}}^{2g}$ (which will later have
the physical interpretation of an Aharonov-Bohm flux). We denote by
$\widetilde{\mathcal{T}}_{g}={\mathcal{T}}_{g}\times{\mathbb{R}}^{2g}$ the
extended Teichmüller space.

A natural action of $\mathrm{MCG_{g}}$ on the extended Teichmüller space
$\widetilde{\mathcal{T}}_{g}$ is given by

(2.5)
$\psi\cdot(X,\phi,{\text{\boldmath$\theta$}})=(X,\phi\circ\psi^{-1},{\text{\boldmath$\theta$}}M_{\psi}^{-1}).$

We furthermore define the ${\mathbb{R}}^{2g}$ action on
$\widetilde{\mathcal{T}}_{g}$ by translation of the vector component by

(2.6)
${\text{\boldmath$\eta$}}\cdot(X,\phi,{\text{\boldmath$\theta$}})=(X,\phi,{\text{\boldmath$\theta$}}+{\text{\boldmath$\eta$}}).$

This suggests the definition of the semidirect product, the affine mapping
class group

(2.7) $\Lambda_{g}({\mathbb{R}})=\mathrm{MCG_{g}}\ltimes{\mathbb{R}}^{2g}$

with multiplication law

(2.8)
$(\psi,{\text{\boldmath$\eta$}})(\psi^{\prime},{\text{\boldmath$\eta$}}^{\prime}):=(\psi\circ\psi^{\prime},{\text{\boldmath$\eta$}}+{\text{\boldmath$\eta$}}^{\prime}M_{\psi}^{-1}).$

Note that
$(\psi,{\text{\boldmath$\eta$}})=(\operatorname{id},{\text{\boldmath$\eta$}})(\psi,{\text{\boldmath$0$}})$
and that $A_{g}({\mathbb{R}})=\\{\operatorname{id}\\}\ltimes{\mathbb{R}}^{2g}$
is a normal subgroup of $\Lambda_{g}({\mathbb{R}})$. The action of
$\Lambda_{g}({\mathbb{R}})$ on $\widetilde{\mathcal{T}}_{g}$ is thus

(2.9)
$(\psi,{\text{\boldmath$\eta$}})\cdot(X,\phi,{\text{\boldmath$\theta$}}):=(X,\phi\circ\psi^{-1},{\text{\boldmath$\eta$}}+{\text{\boldmath$\theta$}}M_{\psi}^{-1}).$

Define the subgroup
$\Lambda_{g}({\mathbb{Z}})=\mathrm{MCG_{g}}\ltimes{\mathbb{Z}}^{2g}$ and its
normal subgroup
$A_{g}({\mathbb{Z}})=\\{\operatorname{id}\\}\ltimes{\mathbb{Z}}^{2g}$.
Furthermore, set ${\mathbb{T}}^{2g}={\mathbb{R}}^{2g}/{\mathbb{Z}}^{2g}$. We
then have the following.

######  Proposition 1.

The extended moduli space
$\widetilde{\mathcal{M}}_{g}:=\Lambda_{g}({\mathbb{Z}})\backslash\widetilde{\mathcal{T}}_{g}$
is a non-trivial fiber bundle over ${\mathcal{M}}_{g}$ with fiber
${\mathbb{T}}^{2g}$.

###### Proof.

We have that
$A_{g}({\mathbb{Z}})\backslash\widetilde{\mathcal{T}}_{g}={\mathcal{T}}_{g}\times{\mathbb{T}}^{2g}$.
That is, we can view
$A_{g}({\mathbb{Z}})\backslash\widetilde{\mathcal{T}}_{g}$ has a trivial fiber
bundle over ${\mathcal{T}}_{g}$ with fiber ${\mathbb{T}}^{2g}$. This proves
that $\widetilde{\mathcal{M}}_{g}$ is a torus bundle over ${\mathcal{M}}_{g}$.
By the normality of $A_{g}({\mathbb{Z}})$, the action of the mapping class
group defined in (2.5) induces an action on
${\mathcal{T}}_{g}\times{\mathbb{T}}^{2g}$ given by

(2.10)
$\psi\cdot(X,\phi,{\text{\boldmath$\theta$}}+{\mathbb{Z}}^{2g})=(X,\phi\circ\psi^{-1},{\text{\boldmath$\theta$}}M_{\psi}^{-1}+{\mathbb{Z}}^{2g}).$

Note that $\psi\mapsto M_{\psi}\in\operatorname{Sp}(2g,{\mathbb{Z}})$ acts
non-trivially on ${\mathbb{T}}^{2g}$, hence the bundle is non-trivial. ∎

The proof of Proposition 1 shows that if ${\mathcal{F}}_{g}$ is a fundamental
domain for the action of $\mathrm{MCG_{g}}$ in ${\mathcal{T}}_{g}$, then
$\widetilde{\mathcal{F}}_{g}={\mathcal{F}}_{g}\times(-\tfrac{1}{2},\tfrac{1}{2}]^{2g}$
is a fundamental domain for the action of $\Lambda_{g}({\mathbb{Z}})$ in
$\widetilde{\mathcal{T}}_{g}$.

We furthermore note that ${\mathcal{M}}_{g}$ can be represented as
$\Lambda_{g}({\mathbb{R}})\backslash\widetilde{\mathcal{T}}_{g}$, and the base
projection $\Pi:\widetilde{\mathcal{M}}_{g}\to{\mathcal{M}}_{g}$ is then given
by
$\Lambda_{g}({\mathbb{Z}})\cdot(X,\phi,{\text{\boldmath$\theta$}})\mapsto\Lambda_{g}({\mathbb{R}})\cdot(X,\phi,{\mathbb{R}}^{2g})$.

##  3\. Invariant measures and Fourier transforms

We denote by $\operatorname{Vol}_{g}^{\mathrm{WP}}$ the Weil-Petersson measure
on ${\mathcal{T}}_{g}$. Since it is $\mathrm{MCG_{g}}$-invariant, it descends
to a measure $\mu_{g}^{\mathrm{WP}}$ on ${\mathcal{M}}_{g}$. This measure is
finite and we normalise it to be a probability measure. That is, for any Borel
set ${\mathcal{B}}\subset{\mathcal{M}}_{g}$,

(3.1)
$\mu_{g}^{\mathrm{WP}}({\mathcal{B}})=\frac{\operatorname{Vol}_{g}^{\mathrm{WP}}({\mathcal{F}}_{g}\cap\pi^{-1}{\mathcal{B}})}{\operatorname{Vol}_{g}^{\mathrm{WP}}({\mathcal{F}}_{g})},$

where $\pi$ is the canonical projection
${\mathcal{T}}_{g}\to{\mathcal{M}}_{g}$ and ${\mathcal{F}}_{g}$ is a
measurable fundamental domain of $\mathrm{MCG_{g}}$ in ${\mathcal{T}}_{g}$.

Let us now lift this measure to natural measures on the extended Teichmüller
and moduli spaces. For $q\in{\mathbb{N}}\cup\\{\infty\\}$, we define the
measure $\operatorname{Vol}_{g,q}$ on $\widetilde{\mathcal{T}}_{g}$ by

(3.2)
$\operatorname{Vol}_{g,q}=\operatorname{Vol}_{g}^{\mathrm{WP}}\times\nu_{q}^{2g}$

where $\nu_{\infty}^{k}=\operatorname{Vol}_{{\mathbb{R}}^{k}}$ is the standard
Lebesgue measure and otherwise

(3.3)
$\nu_{q}^{k}=\frac{1}{q^{k}}\sum_{{\text{\boldmath$m$}}\in{\mathbb{Z}}^{k}}\delta_{q^{-1}{\text{\boldmath$m$}}}.$

Here $\delta_{\text{\boldmath$\eta$}}$ denotes the Dirac mass at $\eta$. The
point of these definitions is that each $\nu_{q}^{2g}$ is invariant under the
action of $\operatorname{Sp}(2g,{\mathbb{Z}})$ (by linear transformations) and
translation invariant by a subgroup containing ${\mathbb{Z}}^{2g}$. Therefore
they descend to natural $\operatorname{Sp}(2g,{\mathbb{Z}})$-invariant
probability measures on ${\mathbb{T}}^{2g}$, which for simplicity we will also
denote by $\nu_{q}^{2g}$. Explicitly,
$\nu_{\infty}^{k}=\operatorname{Vol}_{{\mathbb{T}}^{k}}$ the Haar-Lebesgue
probability measure and, for $q\in{\mathbb{N}}$,

(3.4)
$\nu_{q}^{k}=\frac{1}{q^{k}}\sum_{{\text{\boldmath$m$}}\in{\mathbb{Z}}_{q}^{k}}\delta_{q^{-1}{\text{\boldmath$m$}}}$

with the shorthand ${\mathbb{Z}}_{q}={\mathbb{Z}}/q{\mathbb{Z}}$.

With these assumptions, the measure $\operatorname{Vol}_{g,q}$ is invariant
under the extended mapping class group $\Lambda_{g}({\mathbb{Z}})$, hence
descends, after the appropriate normalisation, to a probability measure
$\mu_{g,q}$ defined on Borel sets
${\mathcal{B}}\subset\widetilde{\mathcal{M}}_{g}$ by

(3.5)
$\mu_{g,q}({\mathcal{B}})=\frac{\operatorname{Vol}_{g,q}(\widetilde{\mathcal{F}}_{g}\cap\pi^{-1}{\mathcal{B}})}{\operatorname{Vol}_{g,q}(\widetilde{\mathcal{F}}_{g})}.$

Note that
$\operatorname{Vol}_{g,q}(\widetilde{\mathcal{F}}_{g})=\operatorname{Vol}_{g}({\mathcal{F}}_{g})$,
since $\nu_{q}^{2g}({\mathbb{T}}^{2g})=1$.

In the following we denote by $\,{}^{\mathrm{t}}\\!{\mathbb{Z}}_{q}^{2g}$ the
set of integer column vectors $\,{}^{\mathrm{t}}\\!{\text{\boldmath$m$}}$. For
functions $F:{\mathcal{T}}_{g}\times{\mathbb{T}}^{2g}\to{\mathbb{C}}$ so that
$F(X,\phi,\,\cdot\,)\in\operatorname{L{}}^{1}({\mathbb{T}}^{2g},\nu_{q}^{2g})$,
we define the fiber Fourier transform
$\widehat{F}_{q}:{\mathcal{T}}_{g}\times\,^{\mathrm{t}}\\!{\mathbb{Z}}_{q}^{2g}\to{\mathbb{C}}$
by

(3.6)
$\widehat{F}_{q}(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})=\int_{{\mathbb{T}}^{2g}}F(X,\phi,{\text{\boldmath$\theta$}})\,e(-{\text{\boldmath$\theta$}}\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})\,d\nu_{q}^{2g}({\text{\boldmath$\theta$}}),$

with $e(x)=\mathrm{e}^{2i\pi x}$. This is the standard Fourier transform for
$q=\infty$ (put ${\mathbb{Z}}_{q}={\mathbb{Z}}$), and the discrete Fourier
transform if $q\in{\mathbb{N}}$ (here
${\mathbb{Z}}_{q}={\mathbb{Z}}/q{\mathbb{Z}}$ as above). Of particular
interest is the zeroth Fourier coefficient, which we denote by

(3.7)
$\breve{F}_{q}(X,\phi)=\widehat{F}_{q}(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$0$}}).$

We define the dual action of $\mathrm{MCG_{g}}$ on
${\mathcal{T}}_{g}\times\,^{\mathrm{t}}\\!{\mathbb{Z}}^{2g}$ by

(3.8)
$\psi\star(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})=(X,\phi\circ\psi^{-1},M_{\psi}\,^{\mathrm{t}}\\!{\text{\boldmath$m$}}).$

Then

(3.9)
$\widehat{F}_{q}(\psi\star(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}}))=\int_{{\mathbb{T}}^{2g}}F(\psi\cdot(X,\phi,{\text{\boldmath$\theta$}}))\,e(-{\text{\boldmath$\theta$}}\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})\,d\nu_{q}^{2g}({\text{\boldmath$\theta$}}).$

Hence, if $F$ is invariant under $\mathrm{MCG_{g}}$, and thus can be viewed as
a function on $\widetilde{\mathcal{M}}_{g}$, then

(3.10)
$\widehat{F}_{q}(\psi\star(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}}))=\widehat{F}_{q}(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})$

and

(3.11) $\breve{F}_{q}(\psi\cdot(X,\phi))=\breve{F}_{q}(X,\phi)$

for all $\psi\in\mathrm{MCG_{g}}$.

In the following, we identify $\mathrm{MCG_{g}}$-invariant functions of
${\mathcal{T}}_{g}$ with functions of ${\mathcal{M}}_{g}$, and
$\Lambda_{g}({\mathbb{Z}})$-invariant functions of
$\widetilde{\mathcal{T}}_{g}$ with functions of $\widetilde{\mathcal{M}}_{g}$,
using the same notation.

With this convention, we can define the fiber Fourier transform for
$F\in\operatorname{L{}}^{1}(\widetilde{\mathcal{M}}_{g},\mu_{g,q})$ by

(3.12)
$\widehat{F}_{q}(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})=\int_{\Pi^{-1}(X,\phi)}F(X,\phi,{\text{\boldmath$\theta$}})\,e(-{\text{\boldmath$\theta$}}\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})\,d\nu_{q}^{2g}({\text{\boldmath$\theta$}})$

where $\widehat{F}_{q}$ is now viewed as a function of the quotient
$\widetilde{\mathcal{M}}_{g,q}^{\star}:=\mathrm{MCG_{g}}\backslash({\mathcal{T}}_{g}\times\,^{\mathrm{t}}\\!{\mathbb{Z}}_{q}^{2g})$
with respect to the $\star$-action. The zeroth Fourier coefficient
$\breve{F}_{q}$ is identified with a function of ${\mathcal{M}}_{g}$. The
above discussion in particular implies the following.

######  Proposition 2.

If $F\in\operatorname{L{}}^{1}(\widetilde{\mathcal{M}}_{g},\mu_{g,q})$ then
$\breve{F}_{q}\in\operatorname{L{}}^{1}({\mathcal{M}}_{g},\mu_{g}^{\mathrm{WP}})$
with

(3.13)
$\int_{\widetilde{\mathcal{M}}_{g}}F\,d\mu_{g,q}=\int_{{\mathcal{M}}_{g}}\breve{F}_{g}\,d\mu_{g}^{\mathrm{WP}}.$

We will now apply the above formula to a class of functions that arise in the
geometric analysis of twisted Laplacians. The reduction to a standard integral
over moduli space will then allow us to apply Mirzakhani’s volume formulas.

##  4\. Length-holonomy functions

Given a closed non-contractible oriented loop $\gamma$ on $S_{g}$, let
$\ell_{\gamma}(X,\phi)$ denote the length of the closed geodesic
$\phi(\gamma)$ on $X$, and $\operatorname{Stab}_{\gamma}$ the stabiliser of
$\gamma$ in $\mathrm{MCG_{g}}$. We denote by $[\gamma]\in
H_{1}(S_{g};\mathbb{Z})$ the holonomy of $\gamma$ represented as a column
vector written in the basis $(a_{i},b_{i})_{1\leq i\leq g}$, as defined above.

Let $f:{\mathbb{R}}_{>0}\times{\mathbb{T}}\to{\mathbb{C}}$. We naturally
associate to $f$ and $\gamma$ a function $f_{\gamma}$ defined on
$\widetilde{\mathcal{T}}_{g}$ by

(4.1)
$f_{\gamma}(X,\phi,{\text{\boldmath$\theta$}})=f(\ell_{\gamma}(X,\phi),{\text{\boldmath$\theta$}}\,[\gamma]).$

For $\psi\in\operatorname{Stab}_{\gamma}$, we have
$\ell_{\gamma}(X,\phi\circ\psi^{-1})=\ell_{\gamma}(X,\phi)$ and
${\text{\boldmath$\theta$}}M_{\psi}^{-1}\,[\gamma]={\text{\boldmath$\theta$}}\,[\psi^{-1}(\gamma)]={\text{\boldmath$\theta$}}\,[\gamma]$,
and therefore

(4.2)
$f_{\gamma}(\psi\cdot(X,\phi,{\text{\boldmath$\theta$}}))=f_{\gamma}(X,\phi,{\text{\boldmath$\theta$}}).$

We now define, for $f$ compactly supported, the function

(4.3)
$F_{\gamma}(X,\phi,{\text{\boldmath$\theta$}})=\sum_{\psi\in\mathrm{MCG_{g}}/\operatorname{Stab}_{\gamma}}f_{\gamma}(\psi\cdot(X,\phi,{\text{\boldmath$\theta$}})).$

This function is, by construction, $\mathrm{MCG_{g}}$-invariant, i.e.,
$F_{\gamma}(\psi\cdot(X,\phi,{\text{\boldmath$\theta$}}))=F_{\gamma}(X,\phi,{\text{\boldmath$\theta$}})$
for all $\psi\in\mathrm{MCG_{g}}$, and may therefore be viewed as a function
on $\widetilde{\mathcal{M}}_{g}$. Its fiber Fourier transform is as follows.

######  Proposition 3.

For $\gamma$ as above, $q\leq\infty$,
${\text{\boldmath$m$}}\in{\mathbb{Z}}_{q}^{2g}$ and $f$ with compact support,
we have

(4.4)
$\widehat{F}_{\gamma,q}(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})=\sum_{\psi\in\mathrm{MCG_{g}}/\operatorname{Stab}_{\gamma}}\widehat{f}_{\gamma,q}(\psi\star(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})),$

where

(4.5)
$\widehat{f}_{\gamma,q}(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})=\sum_{n\in{\mathbb{Z}}_{q}}\widehat{f}_{q}(\ell_{\gamma}(X,\phi),n)\;{\mathbbm{1}}(\,^{\mathrm{t}}\\!{\text{\boldmath$m$}}=n[\gamma]\bmod
q\,^{\mathrm{t}}\\!{\mathbb{Z}}^{2g})$

and, for $(x,n)\in{\mathbb{R}}_{>0}\times{\mathbb{Z}}_{q}$,

(4.6)
$\widehat{f}_{q}(x,n)=\int_{\mathbb{T}}f(x,y)\,e(-ny)\,d\nu_{q}^{1}(dy).$

###### Proof.

Relation (4.4) follows from (3.9). Formula (4.5) follows from the identity

(4.7)
$\int_{{\mathbb{T}}^{2g}}e({\text{\boldmath$\theta$}}\,^{\mathrm{t}}\\!{\text{\boldmath$s$}})\,d\nu_{q}^{2g}({\text{\boldmath$\theta$}})={\mathbbm{1}}({\text{\boldmath$s$}}={\text{\boldmath$0$}}\bmod
q{\mathbb{Z}}^{2g})$

applied to
$\,{}^{\mathrm{t}}\\!{\text{\boldmath$s$}}=n[\gamma]-\,^{\mathrm{t}}\\!{\text{\boldmath$m$}}$.
∎

In the special case of the zeroth Fourier coefficient, the formula reads

(4.8)
$\breve{F}_{\gamma,q}(X,\phi)=\sum_{\psi\in\mathrm{MCG_{g}}/\operatorname{Stab}_{\gamma}}\breve{f}_{\gamma,q}(\ell_{\gamma}(\psi\cdot(X,\phi))$

where

(4.9)
$\breve{f}_{\gamma,q}(x)=\sum_{n\in\ker_{q}[\gamma]}\widehat{f}_{q}(x,n),\qquad\ker_{q}[\gamma]=\\{n\in{\mathbb{Z}}_{q}:n[\gamma]=\,^{\mathrm{t}}\\!{\text{\boldmath$0$}}\bmod
q\,^{\mathrm{t}}\\!{\mathbb{Z}}^{2g}\\}.$

Note that although $[\psi(\gamma)]$ depends on the choice of
$\psi\in\mathrm{MCG_{g}}$, the condition
$n[\psi(\gamma)]=\,^{\mathrm{t}}\\!{\text{\boldmath$0$}}\bmod
q{\mathbb{Z}}^{2g}$ is independent of that choice, since
$\,{}^{\mathrm{t}}\\!{\text{\boldmath$0$}}$ is a fixed point of
$\operatorname{Sp}(2g,{\mathbb{Z}})$. Hence
$\ker_{q}[\gamma]=\ker_{q}[\psi(\gamma)]$ for $\psi\in\mathrm{MCG_{g}}$. We
conclude that $\breve{F}_{\gamma,q}(X,\phi)$ is $\mathrm{MCG_{g}}$-invariant
and therefore a well-defined function on the moduli space ${\mathcal{M}}_{g}$.
In fact, $\breve{F}_{\gamma,q}(X,\phi)$ belongs to the family of “geometric
functions” studied by Mirzakhani when $\gamma$ is simple [10, 11], and by
Anantharaman and the second author for general loops [1].

We extend the above discussion to multi-loops
${\text{\boldmath$\gamma$}}=(\gamma_{1},\ldots,\gamma_{k})$, with each
$\gamma_{j}$ a closed non-contractible oriented loop $S_{g}$. We set
${\text{\boldmath$\ell$}}_{\text{\boldmath$\gamma$}}=(\ell_{\gamma_{1}},\ldots,\ell_{\gamma_{k}})$
and

(4.10)
$\operatorname{Stab}_{\text{\boldmath$\gamma$}}=\\{\psi\in\mathrm{MCG_{g}}:\psi(\gamma_{j})=\gamma_{j}\;\forall
j=1,\ldots,k\\}.$

In the following,
$[{\text{\boldmath$\gamma$}}]=([\gamma_{1}],\ldots,[\gamma_{k}])\in
H_{1}(S_{g};\mathbb{Z})^{k}$ is represented as a $(2g\times k)$ matrix. For
$f:{\mathbb{R}}_{>0}^{k}\times{\mathbb{T}}^{k}\to{\mathbb{C}}$ with compact
support, define

(4.11)
$f_{\text{\boldmath$\gamma$}}(X,\phi,{\text{\boldmath$\theta$}})=f({\text{\boldmath$\ell$}}_{\text{\boldmath$\gamma$}}(X,\phi),{\text{\boldmath$\theta$}}\,[{\text{\boldmath$\gamma$}}])$

and

(4.12)
$F_{\text{\boldmath$\gamma$}}(X,\phi,{\text{\boldmath$\theta$}})=\sum_{\psi\in\mathrm{MCG_{g}}/\operatorname{Stab}_{\text{\boldmath$\gamma$}}}f_{\text{\boldmath$\gamma$}}(\psi\cdot(X,\phi,{\text{\boldmath$\theta$}})).$

We have the same $\mathrm{MCG_{g}}$-invariance as above, i.e.,
$F_{\text{\boldmath$\gamma$}}(\psi\cdot(X,\phi,{\text{\boldmath$\theta$}}))=F_{\text{\boldmath$\gamma$}}(X,\phi,{\text{\boldmath$\theta$}})$
for all $\psi\in\mathrm{MCG_{g}}$, and can therefore view it as a function on
$\widetilde{\mathcal{M}}_{g}$. Its fiber Fourier transform is given in the
following proposition.

######  Proposition 4.

For $\gamma$ a multi-loop as above, $q\leq\infty$,
${\text{\boldmath$m$}}\in{\mathbb{Z}}_{q}^{2g}$ and $f$ with compact support,
we have

(4.13)
$\widehat{F}_{{\text{\boldmath$\gamma$}},q}(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})=\sum_{\psi\in\mathrm{MCG_{g}}/\operatorname{Stab}_{\text{\boldmath$\gamma$}}}\widehat{f}_{{\text{\boldmath$\gamma$}},q}(\psi\star(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}}))$

where

(4.14)
$\widehat{f}_{{\text{\boldmath$\gamma$}},q}(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$m$}})=\sum_{{\text{\boldmath$n$}}\in{\mathbb{Z}}_{q}^{k}}\widehat{f}_{q}({\text{\boldmath$\ell$}}_{\text{\boldmath$\gamma$}}(X,\phi),{\text{\boldmath$n$}})\;{\mathbbm{1}}(\,^{\mathrm{t}}\\!{\text{\boldmath$m$}}=[{\text{\boldmath$\gamma$}}]\,^{\mathrm{t}}\\!{\text{\boldmath$n$}}\bmod
q\,^{\mathrm{t}}\\!{\mathbb{Z}}^{2g})$

and

(4.15)
$\widehat{f}_{q}({\text{\boldmath$x$}},{\text{\boldmath$n$}})=\int_{{\mathbb{T}}^{k}}f({\text{\boldmath$x$}},{\text{\boldmath$y$}})\,e(-{\text{\boldmath$n$}}\,^{\mathrm{t}}\\!{\text{\boldmath$y$}})\,d\nu_{q}^{k}({\text{\boldmath$y$}}).$

###### Proof.

This is analogous to the proof of Proposition 3. ∎

The zeroth Fourier coefficient
$\breve{F}_{{\text{\boldmath$\gamma$}},q}(X,\phi)=\widehat{F}_{{\text{\boldmath$\gamma$}},q}(X,\phi,\,^{\mathrm{t}}\\!{\text{\boldmath$0$}})$
is again $\mathrm{MCG_{g}}$-invariant and thus well-defined on
${\mathcal{M}}_{g}$. Specifying ${\text{\boldmath$m$}}={\text{\boldmath$0$}}$
in (4.13), (4.14) yields formulas similar to (4.8) and (4.9) in the case of
multi-loops.

##  5\. Twisted Laplacians

We will now explain how the data $(X,\phi,{\text{\boldmath$\theta$}})$ is
related to the Fuchsian group $\Gamma$ and its character $\chi_{\gamma}$ used
in the introduction to define the twisted Laplacian.

We start with $\Gamma=\Gamma(X)$, which we realise as the fundamental group
$\pi_{1}(X)$ of $X$, so that $X=\Gamma\backslash{\mathbb{H}}$. We define a
basis of fundamental loops on $X$ by

$\tilde{a}_{1}=\phi(a_{1}),\ldots,\tilde{a}_{g}=\phi(a_{g}),\tilde{b}_{1}=\phi(b_{1}),\ldots,\tilde{b}_{g}=\phi(b_{g}).$

By cutting $X$ along these loops we obtain a fundamental domain of $\Gamma(X)$
in ${\mathbb{H}}$, which is a $4g$-sided polygon with geodesic boundaries,
labeled by

$\tilde{a}_{1},\tilde{b}_{1},\tilde{a}_{1}^{-1},\tilde{b}_{1}^{-1},\tilde{a}_{2},\tilde{b}_{2},\ldots,\tilde{a}_{g}^{-1},\tilde{b}_{g}^{-1}.$

Let $\delta_{j}$ be the Möbius transformation that maps the side labeled
$\tilde{a}_{j}$ to $\tilde{a}_{j}^{-1}$, and let $\eta_{j}$ be the Möbius
transformation that maps the side labeled $\tilde{b}_{j}^{-1}$ to
$\tilde{b}_{j}$. The elements $\delta_{j},\eta_{j}$ now generate $\Gamma(X)$
and we have a group isomorphism $\iota_{X}:\pi_{1}(X)\to\Gamma(X)$ defined by
$\iota_{X}(\tilde{a}_{j})=\eta_{j}$, $\iota_{X}(\tilde{b}_{j})=\delta_{j}$.
Thus $\iota_{(X,\phi)}=\iota_{X}\circ\phi$ yields an isomorphism
$\iota_{(X,\phi)}:\pi_{1}(S_{g})\to\Gamma(X)$.

Given ${\text{\boldmath$\theta$}}\in{\mathbb{R}}^{2g}$, we define the unitary
character

(5.1)
$\chi_{\text{\boldmath$\theta$}}:\pi_{1}(S_{g})\to{\mathbb{C}},\qquad\gamma\mapsto
e({\text{\boldmath$\theta$}}\,[\gamma]),$

where $[\gamma]\in H_{1}(S_{g};\mathbb{Z})$ is represented as a column vector
written in the basis $a_{1},\ldots,a_{g}$, $b_{1},\ldots,b_{g}$.

For every $(X,\phi,{\text{\boldmath$\theta$}})\in\widetilde{\mathcal{T}}_{g}$
we can now define the unitary character
$\chi_{(X,\phi,{\text{\boldmath$\theta$}})}:\Gamma(X)\to{\mathbb{C}}$ by

(5.2)
$\chi_{(X,\phi,{\text{\boldmath$\theta$}})}=\chi_{\text{\boldmath$\theta$}}\circ\iota_{(X,\phi)}^{-1}=\chi_{\text{\boldmath$\theta$}}\circ\phi^{-1}\circ\iota_{X}^{-1}.$

Note that, for $\psi\in\mathrm{MCG_{g}}$, we have

(5.3)
$\chi_{\psi\cdot(X,\phi,{\text{\boldmath$\theta$}})}(\eta)=e({\text{\boldmath$\theta$}}M_{\psi}^{-1}\,[(\phi\circ\psi^{-1})^{-1}\circ\iota_{X}^{-1}(\eta)])=e({\text{\boldmath$\theta$}}\,[\phi^{-1}\circ\iota_{X}^{-1}(\eta)])=\chi_{(X,\phi,{\text{\boldmath$\theta$}})}(\eta).$

Furthermore, for ${\text{\boldmath$m$}}\in{\mathbb{Z}}^{2g}$, we have

(5.4)
$\chi_{(X,\phi,{\text{\boldmath$\theta$}}+{\text{\boldmath$m$}})}(\eta)=\chi_{(X,\phi,{\text{\boldmath$\theta$}})}(\eta).$

Therefore the character family
$\chi=\chi_{(X,\phi,{\text{\boldmath$\theta$}})}$ is invariant under
$\Lambda_{g}({\mathbb{Z}})$ and hence parametrised by the extended moduli
space $\widetilde{\mathcal{M}}_{g}$. This in turn yields a
$\widetilde{\mathcal{M}}_{g}$-family of twisted Laplacians
$\Delta_{(X,\phi,{\text{\boldmath$\theta$}})}=\Delta_{\chi}$.

We will consider $\Delta_{(X,\phi,{\text{\boldmath$\theta$}})}$ as random
operators by drawing $(X,\phi,{\text{\boldmath$\theta$}})$ with respect to the
probability measures $\mu_{g,q}$ defined in (3.5). For $q=1$, the character is
trivial, and for $q=2$ real-valued. The following proposition states that for
$q\geq 3$ the characters are real-valued with vanishing probability in the
limit of large genus.

######  Proposition 5.

For every $(X,\phi)\in{\mathcal{T}}_{g}$, we have

(5.5)
$\nu_{q}^{2g}\big{\\{}{\text{\boldmath$\theta$}}\in{\mathbb{T}}^{2g}:\chi_{(X,\phi,{\text{\boldmath$\theta$}})}^{2}=\operatorname{id}\big{\\}}=\begin{cases}0&(q=\infty)\\\
q^{-2g}&(q\in 2{\mathbb{N}}+1)\\\ (q/2)^{-2g}&(q\in
2{\mathbb{N}}).\end{cases}$

###### Proof.

This follows from the observation that
$\chi_{(X,\phi,{\text{\boldmath$\theta$}})}^{2}=\operatorname{id}$ if and only
if
${\text{\boldmath$\theta$}}\in\\{0,\frac{1}{2}\\}^{2g}\bmod{\mathbb{Z}}^{2g}$.
∎

This ensures that real-valued characters with GOE statistics are exceptional
and will not spoil the GUE statistics of the full ensemble.

##  6\. Main Lemmas

We now state the main lemmas required for the proof of Theorem 1. Recall that
non-contractible oriented loops on $S_{g}$ are identified with oriented closed
geodesics on $X$, which in turn correspond to conjugacy classes of non-trivial
group elements; primitivity is preserved under these identifications.

######  Lemma 1.

For $f:{\mathbb{R}}_{>0}\to{\mathbb{C}}$ with compact support,
$\ell_{\gamma}=\ell_{\gamma}(X,\phi)$,

(6.1)
${\mathbb{E}}_{g,q}\Bigg{[}\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\\\
\mathrm{prim.}\end{subarray}}\frac{\ell_{\gamma}\,f(n\ell_{\gamma})\,e({\text{\boldmath$\theta$}}[\gamma]n)}{2\sinh(n\ell_{\gamma}/2)}\Bigg{]}=\begin{cases}0&(q=\infty)\\\
\displaystyle{\mathbb{E}}_{g}^{\mathrm{WP}}\bigg{[}\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\\\
\mathrm{prim.}\end{subarray}}\frac{\ell_{\gamma}f(nq^{*}_{\gamma}\ell_{\gamma})}{2\sinh(nq^{*}_{\gamma}\ell_{\gamma}/2)}\bigg{]}&(q\in{\mathbb{N}})\end{cases}$

where each sum is over non-contractible oriented primitive loops and
$q^{*}_{\gamma}=q/\gcd([\gamma],q)$.

In analogy with the comment after equation (4.8), we note that
$q^{*}_{\psi(\gamma)}=q^{*}_{\gamma}$ for $\psi\in\mathrm{MCG_{g}}$. This
follows directly from the fact that
$M_{\psi}\in\operatorname{Sp}(2g,{\mathbb{Z}})$ and so
$\gcd(M_{\psi}[\gamma])=\gcd([\gamma])$.

######  Lemma 2.

For $f:{\mathbb{R}}_{>0}\to{\mathbb{C}}$ with compact support,
$\ell_{\gamma}=\ell_{\gamma}(X,\phi)$,

(6.2)
${\mathbb{E}}_{g,q}\Bigg{[}\bigg{|}\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\\\
\mathrm{prim.}\end{subarray}}\frac{\ell_{\gamma}\,f(n\ell_{\gamma})\,e({\text{\boldmath$\theta$}}[\gamma]n)}{2\sinh(n\ell_{\gamma}/2)}\bigg{|}^{2}\Bigg{]}\\\
=\displaystyle{\mathbb{E}}_{g}^{\mathrm{WP}}\Bigg{[}\sum_{n_{1},n_{2}=1}^{\infty}\sum_{\begin{subarray}{c}\gamma_{1},\gamma_{2}\;\mathrm{prim.}\\\
n_{1}[\gamma_{1}]=n_{2}[\gamma_{2}]\\\ \bmod
q{\mathbb{Z}}^{2g}\end{subarray}}\frac{\ell_{\gamma_{1}}f(n_{1}\ell_{\gamma_{1}})}{2\sinh(n_{1}\ell_{\gamma_{1}}/2)}\;\frac{\ell_{\gamma_{2}}\overline{f}(n_{2}\ell_{\gamma_{2}})}{2\sinh(n_{2}\ell_{\gamma_{2}}/2)}\Bigg{]},$

where each sum is over non-contractible oriented primitive loops; remove
“$\bmod q{\mathbb{Z}}^{2g}$” if $q=\infty$.

###### Proof.

The sums appearing in Lemmas 1 and 2 are a special case of expressions of the
form

(6.3) $\sum_{\begin{subarray}{c}{\text{\boldmath$\gamma$}}\\\
\text{prim.}\end{subarray}}\rho({\text{\boldmath$\ell$}}_{{\text{\boldmath$\gamma$}}}(X,\phi)\,^{\mathrm{t}}\\!{\text{\boldmath$n$}})\;e({\text{\boldmath$\theta$}}\,[{\text{\boldmath$\gamma$}}]\,^{\mathrm{t}}\\!{\text{\boldmath$n$}}),$

where the sum is over oriented multi-loops, where each loop is primitive.

Specifically, for Lemma 1 take $k=1$ and

(6.4) $\rho(x)=\frac{xf(nx)}{2\sinh(nx/2)}$

with $f$ as in the lemma; for Lemma 2 take $k=2$ with
${\text{\boldmath$n$}}=(n_{1},-n_{2})$ (rather than
${\text{\boldmath$n$}}=(n_{1},n_{2})$ as before), and

(6.5)
$\rho(x_{1},x_{2})=\frac{x_{1}f(n_{1}x_{1})}{2\sinh(n_{1}x_{1}/2)}\;\frac{x_{2}\overline{f}(n_{2}x_{2})}{2\sinh(n_{2}x_{2}/2)}.$

We divide the summation in (6.3) into orbits under the mapping class group,
i.e., write it as

(6.6)
$\sum_{\begin{subarray}{c}{\text{\boldmath$\gamma$}}\bmod\mathrm{MCG_{g}}\\\
\text{prim.}\end{subarray}}F_{\text{\boldmath$\gamma$}}(X,\phi,{\text{\boldmath$\theta$}})$

where the sum runs over representatives $\gamma$ of $\mathrm{MCG_{g}}$-orbits
of primitive multi-loops, and

(6.7)
$F_{\text{\boldmath$\gamma$}}(X,\phi,{\text{\boldmath$\theta$}})=\sum_{\psi\in\mathrm{MCG_{g}}/\operatorname{Stab}_{\text{\boldmath$\gamma$}}}f_{\text{\boldmath$\gamma$}}(\psi\cdot(X,\phi,{\text{\boldmath$\theta$}})).$

with

(6.8)
$f_{\text{\boldmath$\gamma$}}(X,\phi,{\text{\boldmath$\theta$}})=\rho({\text{\boldmath$\ell$}}_{{\text{\boldmath$\gamma$}}}(X,\phi)\,^{\mathrm{t}}\\!{\text{\boldmath$n$}})\;e({\text{\boldmath$\theta$}}\,[{\text{\boldmath$\gamma$}}]\,^{\mathrm{t}}\\!{\text{\boldmath$n$}}).$

Both lemmas now follow from Proposition 4. ∎

##  7\. Proof of Theorem 1

For $q\in{\mathbb{N}}$ set

(7.1)
$I_{f,q}(L,\tau)=\frac{4}{L}\int_{0}^{\infty}\sum_{n=1}^{\infty}\hat{f}\left(\frac{nqx}{L}\right)\frac{\sinh^{2}(x/2)}{\sinh(nqx)}\cos(nq\tau
x)\,dx,$

and for $q=\infty$, set $I_{f,q}(L,\tau)=0$.

######  Proposition 6.

Under the assumptions for Theorem 1, we have

(7.2)
$\lim_{g\to\infty}{\mathbb{E}}_{g,q}\Bigg{[}\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\\\
\mathrm{prim.}\end{subarray}}\frac{\ell_{\gamma}\,\hat{h}_{L,\tau}(n\ell_{\gamma})\,e({\text{\boldmath$\theta$}}[\gamma]n)}{2\sinh(n\ell_{\gamma}/2)}\Bigg{]}=I_{f,q}(L,\tau).$

###### Proof.

The case $q=\infty$ follows trivially from Lemma 1. Assume therefore
$q<\infty$. Lemma 1 reduces the problem to the $g\to\infty$ asymptotics of

(7.3)
${\mathbb{E}}_{g}^{\mathrm{WP}}\Bigg{[}\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\\\
\mathrm{prim.}\end{subarray}}\frac{\ell_{\gamma}\hat{h}_{L,\tau}(nq^{*}_{\gamma}\ell_{\gamma})}{2\sinh(nq^{*}_{\gamma}\ell_{\gamma}/2)}\Bigg{]}.$

Rudnick’s case corresponds to the case $q=1$ (where $q^{*}_{\gamma}=1$), and
the proof of the above statement follows from the identical argument. The
terms that are shown to vanish as $g\to\infty$ for $q=1$ [15, §4.3] dominate
the same terms for $q\geq 2$, and therefore also these vanish. The only
surviving contribution comes from non-separating simple oriented loops, which
form a double orbit under the mapping class group, one of $\gamma$ such that
$[\gamma]=\,^{\mathrm{t}}\\!(1,0,\ldots,0)$, and the other of $\gamma^{-1}$.
This means that $\gcd([\gamma],q)=1$ and therefore $q_{\gamma}^{*}=q$. The
evaluation of

(7.4)
$\lim_{g\to\infty}{\mathbb{E}}_{g}^{\mathrm{WP}}\Bigg{[}\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\\\
\mathrm{non-sep.}\\\
\mathrm{simple}\end{subarray}}\frac{\ell_{\gamma}\hat{h}_{L,\tau}(nq\ell_{\gamma})}{2\sinh(nq\ell_{\gamma}/2)}\Bigg{]}$

follows directly from Mirzakhani’s volume formula [10] and asymptotics [11];
cf. [15, §4.3]. ∎

######  Proposition 7.

Under the assumptions for Theorem 1, we have

(7.5)
$\lim_{g\to\infty}{\mathbb{E}}_{g,q}\Bigg{[}\bigg{|}\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\\\
\mathrm{prim.}\end{subarray}}\frac{\ell_{\gamma}\,\hat{h}_{L,\tau}(n\ell_{\gamma})\,e({\text{\boldmath$\theta$}}[\gamma]n)}{2\sinh(n\ell_{\gamma}/2)}\bigg{|}^{2}\Bigg{]}\\\
=\begin{cases}\Sigma_{\mathrm{GOE}}^{2}(f)+|I_{f,q}(L,\tau)|^{2}+O(L^{-2}\log
L)&(q=1,2)\\\ \Sigma_{\mathrm{GUE}}^{2}(f)+|I_{f,q}(L,\tau)|^{2}+O(L^{-2}\log
L)&(3\leq q\leq\infty).\end{cases}$

###### Proof.

Let us first consider $q<\infty$. By Lemma 2, we need to find the large $g$
asymptotics of

(7.6)
${\mathbb{E}}_{g}^{\mathrm{WP}}\Bigg{[}\sum_{n_{1},n_{2}=1}^{\infty}\sum_{\begin{subarray}{c}\gamma_{1},\gamma_{2}\;\mathrm{prim.}\\\
n_{1}[\gamma_{1}]=n_{2}[\gamma_{2}]\\\ \bmod
q{\mathbb{Z}}^{2g}\end{subarray}}\frac{\ell_{\gamma_{1}}\hat{h}_{L,\tau}(n_{1}\ell_{\gamma_{1}})}{2\sinh(n_{1}\ell_{\gamma_{1}}/2)}\;\frac{\ell_{\gamma_{2}}\overline{\hat{h}}_{L,\tau}(n_{2}\ell_{\gamma_{2}})}{2\sinh(n_{2}\ell_{\gamma_{2}}/2)}\Bigg{]}.$

As above, the lower order terms for $q=1$ dominate those for $q\geq 2$, and
hence we can refer to the estimates of [15, §5.2, §6]. The only leading order
terms come from pairs $(\gamma_{1},\gamma_{2})$ of simple non-separating
oriented loops in one of the following three constellations: (i)
$\gamma_{1}=\gamma_{2}$, (ii) $\gamma_{1}=\gamma_{2}^{-1}$, or (iii)
$\gamma_{1}\cap\gamma_{2}=\emptyset$.

In case (i), we have two orbits of $\mathrm{MCG_{g}}$ represented by $\gamma$
with $[\gamma]=(1,0,\ldots,0)$ and its inverse $\gamma^{-1}$. The contribution
of these terms to (7.6) is thus

(7.7) $\sum_{\begin{subarray}{c}n_{1},n_{2}=1\\\ n_{1}=n_{2}\bmod
q\end{subarray}}^{\infty}{\mathbb{E}}_{g}^{\mathrm{WP}}\Bigg{[}\sum_{\psi\in\mathrm{MCG_{g}}/\operatorname{Stab}_{\gamma}}\frac{\ell_{\psi(\gamma)}^{2}\hat{h}_{L,\tau}(n_{1}\ell_{\psi(\gamma)})\overline{\hat{h}}_{L,\tau}(n_{2}\ell_{\psi(\gamma)})}{4\sinh(n_{1}\ell_{\psi(\gamma)}/2)\sinh(n_{2}\ell_{\psi(\gamma)}/2)}\Bigg{]}.$

Rudnick shows in [15, Lemma 5.2] that for $q=1$ terms with $n_{1}+n_{2}\geq 3$
contribute at most $O(L^{-2}\log L)$, and hence the same is true for the
contribution of those terms to (7.7). The remaining terms are $n_{1}=n_{2}=1$,
and is worked out in [15, (5.1)],

(7.8)
${\mathbb{E}}_{g}^{\mathrm{WP}}\Bigg{[}\sum_{\psi\in\mathrm{MCG_{g}}/\operatorname{Stab}_{\gamma}}\frac{\ell_{\psi(\gamma)}^{2}\hat{h}_{L,\tau}(\ell_{\psi(\gamma)})\overline{\hat{h}}_{L,\tau}(\ell_{\psi(\gamma)})}{4\sinh(\ell_{\psi(\gamma)}/2)\sinh(\ell_{\psi(\gamma)}/2)}\Bigg{]}=\int_{\mathbb{R}}|x|\,|\hat{f}(x)|^{2}dx+O(L^{-2}).$

Case (ii) is analogous, again we have two orbits of $\mathrm{MCG_{g}}$
represented by $\gamma$ with $[\gamma]=(1,0,\ldots,0)$ and its inverse
$\gamma^{-1}$. Since $\ell_{\psi(\gamma)}=\ell_{\psi(\gamma^{-1})}$, the
contribution of these terms is now

(7.9) $\sum_{\begin{subarray}{c}n_{1},n_{2}=1\\\ n_{1}=-n_{2}\bmod
q\end{subarray}}^{\infty}{\mathbb{E}}_{g}^{\mathrm{WP}}\Bigg{[}\sum_{\psi\in\mathrm{MCG_{g}}/\operatorname{Stab}_{\gamma}}\frac{\ell_{\psi(\gamma)}^{2}\hat{h}_{L,\tau}(n_{1}\ell_{\psi(\gamma)})\overline{\hat{h}}_{L,\tau}(n_{2}\ell_{\psi(\gamma)})}{4\sinh(n_{1}\ell_{\psi(\gamma)}/2)\sinh(n_{2}\ell_{\psi(\gamma)}/2)}\Bigg{]}.$

As before, the $q=1$ case bounds above terms with $n_{1}+n_{2}\geq 3$ by
$O(L^{-2}\log L)$. This leaves the case $n_{1}=n_{2}=1$, which is compatible
with the condition $n_{1}=-n_{2}\bmod q$ only for $q=1$ or $q=2$. In the
latter case (7.9) becomes (7.8). We have thus explained the GOE resp. GUE term
in (7.5).

In the remaining case (iii), [15, Lemma 5.3], the corresponding pairs of loops
can be generated by four orbits of the mapping class group
$\mathrm{MCG_{g}}({\text{\boldmath$\gamma$}})$, with
${\text{\boldmath$\gamma$}}=(\gamma_{1},\gamma_{2})$ as well as the
orientation-reversed $(\gamma_{1}^{-1},\gamma_{2})$,
$(\gamma_{1},\gamma_{2}^{-1})$, $(\gamma_{1}^{-1},\gamma_{2}^{-1})$. Since
both $\gamma_{1}$ and $\gamma_{2}$ are simple non-separating and disjoint, we
may choose representatives such that $[\gamma_{1}]=(1,0,\ldots,0)$ and
$[\gamma_{2}]=(0,1,0,\ldots,0)$. This means that the condition
$n_{1}[\gamma_{1}]=n_{2}[\gamma_{2}]\bmod q{\mathbb{Z}}^{2g}$ simplifies to
$n_{1},n_{2}=0\bmod q$. The contribution from the four options is the same,
and we therefore have the total contribution given by

(7.10)
$4\sum_{n_{1},n_{2}=1}^{\infty}{\mathbb{E}}_{g}^{\mathrm{WP}}\Bigg{[}\sum_{\psi\in\mathrm{MCG_{g}}/\operatorname{Stab}_{(\gamma_{1},\gamma_{2})}}\frac{\ell_{\psi_{1}(\gamma)}\ell_{\psi_{2}(\gamma)}\hat{h}_{L,\tau}(n_{1}q\ell_{\psi_{1}(\gamma)})\overline{\hat{h}}_{L,\tau}(n_{2}q\ell_{\psi_{2}(\gamma)})}{4\sinh(n_{1}q\ell_{\psi_{1}(\gamma)}/2)\sinh(n_{2}q\ell_{\psi_{2}(\gamma)}/2)}\Bigg{]}.$

The same argument as in [15, Lemma 5.3] can be used to show that the terms for
$\gamma_{1}$ and $\gamma_{2}$ decouple for large genus $g$, i.e., the
$g\to\infty$ limit of (7.10) is given by $|I_{f,q}(L,\tau)|^{2}$.

The above proof remains the same for $q=\infty$ by dropping “mod $q$”; the
contribution from case (iii) is trivially zero. ∎

In view of Selberg’s trace formula (1.5), Propositions 6 and 7 now imply
Theorem 1.

##  8\. Dirac operators and GSE statistics

The objective of this final section is to explain how the above construction
can be extended to study random Dirac operators, and in particular to prove
GSE-statistics in the case of spin-systems with time-reversal invariance. We
refer the reader to [7, 12] for a more detailed introduction to Dirac
operators on hyperbolic surfaces.

The Dirac operator on the hyperbolic plane $\mathbb{H}$ is given as the
matrix-valued operator

(8.1)
$\mathrm{D}=i\begin{pmatrix}0&iy\partial_{x}+y\partial_{y}-\frac{1}{2}\\\
-iy\partial_{x}+y\partial_{y}-\frac{1}{2}&0\end{pmatrix}$

acting on differentiable functions ${\mathbb{H}}\rightarrow{\mathbb{C}}^{2}$.
We now consider a discrete co-compact subgroup $\Gamma$ of
$\operatorname{SL}(2,{\mathbb{R}})$ rather than of
$\operatorname{PSL}(2,{\mathbb{R}})=\operatorname{SL}(2,{\mathbb{R}})/\\{\pm
1\\}$ as in the setting of Laplacians. We assume $\Gamma$ is strictly
hyperbolic, i.e., the only elements with $|\operatorname{tr}\eta|\leq 2$ are
$\eta=\pm 1$. The action of $\eta\in\operatorname{SL}(2,{\mathbb{R}})$ on
${\mathbb{H}}\times\operatorname{S{}}^{1}$ (with $\operatorname{S{}}^{1}$ the
unit circle in the complex plane) is defined as

(8.2)
$(z,\mathrm{e}^{\mathrm{i}t})\mapsto\left(\frac{az+b}{cz+d},\frac{cz+d}{|cz+d|}\mathrm{e}^{\mathrm{i}t}\right),\qquad\eta=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{SL}(2,{\mathbb{R}}).$

In particular, $-1$ acts by
$(z,\mathrm{e}^{\mathrm{i}t})\mapsto(z,-\mathrm{e}^{\mathrm{i}t})$. For
$\eta\in\Gamma$ as above, set

(8.3) $j_{\eta}(z)=\frac{cz+d}{|cz+d|},\qquad
J_{\eta}(z)=\begin{pmatrix}j_{\eta}(z)&0\\\ 0&j_{\eta}(\bar{z})\end{pmatrix}.$

We assume in the following that $-1\in\Gamma$, and fix a unitary character
$\chi:\Gamma\rightarrow{\mathbb{C}}$ such that $\chi(-1)=-1$. We consider the
action of $\mathrm{D}$ on the space of functions
${\mathbb{H}}\rightarrow{\mathbb{C}}^{2}$ satisfying

(8.4) $\varphi(\eta\cdot z)=\chi(\eta)\,J_{\eta}(z)\,\varphi(z)\qquad\text{for
all }\eta\in\Gamma,$

and define the Dirac operator $\mathrm{D}_{\chi}$ for
$X=\Gamma\backslash{\mathbb{H}}$ to be the self-adjoint extension of the
restriction of $\mathrm{D}$ to this space of automorphic forms.

Due to the chiral symmetry, the spectrum of the operator $\mathrm{D}_{\chi}$
is always symmetric around $0$. We denote by
$0\leq\lambda_{0}\leq\lambda_{1}\leq\ldots\to\infty$ its non-negative
eigenvalues with multiplicity, including half of the eigenvalues equal to $0$.
As for the Laplacian, this counting function can be studied by means of the
trace formula [7]

(8.5)
$\sum_{j=0}^{\infty}h(\lambda_{j})=(g-1)\int_{\mathbb{R}}h(r)\,r\coth(\pi
r)\,dr+\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\\{\eta\\}\\\
\operatorname{tr}\eta>2\end{subarray}}\chi(\eta)^{n}\,\frac{\ell_{\eta}\hat{h}(n\ell_{\eta})}{2\sinh(n\ell_{\eta}/2)}$

where the sum is, as in the classical Selberg trace formula, over
$\Gamma$-conjugacy classes of primitive elements $\eta$, but now with $\Gamma$
considered as a subgroup of $\operatorname{SL}(2,{\mathbb{R}})$, and the
summation restricted to elements with positive trace (which is necessarily
greater than 2 as the corresponding group elements are hyperbolic).

Let us now construct random ensembles of Dirac operators
$\mathrm{D}_{(X,\phi,{\text{\boldmath$\theta$}})}=\mathrm{D}_{\chi}$ with
$(X,\phi,{\text{\boldmath$\theta$}})$ sampled from
$\widetilde{\mathcal{M}}_{g}$ with respect to the probability measure
$\mu_{g,q}$ defined in Section 3. To define the required character
$\chi=\chi_{(X,\phi,{\text{\boldmath$\theta$}})}$, note that
$\Gamma=\Gamma(X)$ is a double cover of the fundamental group $\pi_{1}(S_{g})$
of $S_{g}$, with the covering map $p_{(X,\phi)}:\Gamma(X)\to\pi_{1}(S_{g})$
given by $p_{(X,\phi)}(\eta)=\iota_{(X,\phi)}^{-1}(\pm\eta)$; recall the
discussion in Section 5. Thus $\Gamma(X)$ is generated by $-1$ and

(8.6) $\eta_{j}\in p_{(X,\phi)}^{-1}(a_{j}),\qquad\delta_{j}\in
p_{(X,\phi)}^{-1}(b_{j})\qquad(j=1,\ldots,g),$

which we choose to have positive trace. Given the character
$\chi_{\text{\boldmath$\theta$}}:\pi_{1}(S_{g})\to{\mathbb{C}}$ as in (5.1),
we now define the unitary character
$\chi_{(X,\phi,{\text{\boldmath$\theta$}})}:\Gamma(X)\to{\mathbb{C}}$ by
specifying its value on the generating set,

(8.7)
$\chi_{(X,\phi,{\text{\boldmath$\theta$}})}(\eta_{j})=\chi_{\text{\boldmath$\theta$}}(a_{j})=e(\theta_{j}),\qquad\chi_{(X,\phi,{\text{\boldmath$\theta$}})}(\delta_{j})=\chi_{\text{\boldmath$\theta$}}(b_{j})=e(\theta_{j+g}),\qquad\chi_{(X,\phi,{\text{\boldmath$\theta$}})}(-1)=-1.$

Thus for a general $\gamma\in\pi_{1}(S_{g})$ and $\eta\in
p_{(X,\phi)}^{-1}(\gamma)$, we have
$\chi_{(X,\phi,{\text{\boldmath$\theta$}})}(\eta)=\pm\chi_{\text{\boldmath$\theta$}}(\gamma)$.
We define $\sigma_{\gamma}\in\\{\pm 1\\}$ by the relation
$\chi_{(X,\phi,{\text{\boldmath$\theta$}})}(\eta)=\sigma_{\gamma}\chi_{\text{\boldmath$\theta$}}(\gamma)$
for $\operatorname{tr}\eta>0$. Note that $\sigma_{\gamma}$ is not
multiplicative in $\gamma$. We have, however,
$\sigma_{\gamma^{-1}}=\sigma_{\gamma}$ and more generally
$\sigma_{\gamma^{n}}=\sigma_{\gamma}^{n}$ for $n\in{\mathbb{Z}}$. To see this,
note that if $\operatorname{tr}\eta>0$ then we have
$\operatorname{tr}\eta^{n}>0$ (this is trivial for $\eta=1$, and a fact for
all $\eta\in\operatorname{SL}(2,{\mathbb{R}})$ with
$\operatorname{tr}\eta>2$). Hence, by the multiplicativity of characters, we
have
$\chi_{(X,\phi,{\text{\boldmath$\theta$}})}(\eta)^{n}=\chi_{(X,\phi,{\text{\boldmath$\theta$}})}(\eta^{n})=\sigma_{\gamma^{n}}\chi_{\text{\boldmath$\theta$}}(\gamma^{n})=\sigma_{\gamma^{n}}\chi_{\text{\boldmath$\theta$}}(\gamma)^{n}$,
which yields $\sigma_{\gamma^{n}}=\sigma_{\gamma}^{n}$.

We now sample ${\text{\boldmath$\theta$}}\in{\mathbb{T}}^{2g}$ with respect to
the probability measure $\nu_{q}^{2g}$ as defined in Section 3. Since the
range of $\chi_{(X,\phi,{\text{\boldmath$\theta$}})}$ contains $-1$, it is
natural to consider only even values of $q$ as well as $q=\infty$.

The case $q=2$ corresponds to real characters $\overline{\chi}=\chi$, which
means $\mathrm{D}_{\chi}$ is symmetric under time-reversal and we have
Karmer’s degeneracy. That is, every eigenvalue $\lambda_{j}$ has even
multiplicity [7, Lemma 2]. This motivates the definition of the following
reduced counting function taking into account the systematic degeneracy,

(8.8)
$N_{f,L,\tau}^{q}=\begin{cases}\frac{1}{2}\sum_{j=0}^{\infty}h_{L,\tau}(\lambda_{j})&(q=2)\\\\[5.0pt]
\sum_{j=0}^{\infty}h_{L,\tau}(\lambda_{j})&(q\geq 4).\end{cases}$

As in the case of the Laplacian, we are interested in the large $g$
asymptotics of the spectral variance of the Dirac operator $D_{\chi}$ on $X$
with $\chi=\chi_{(X,\phi,{\text{\boldmath$\theta$}})}$,

(8.9)
$\widetilde{\Sigma}_{g,q}^{2}(L,\tau;f)={\mathbb{E}}_{g,q}\bigg{[}\big{|}N_{f,L,\tau}^{q}-{\mathbb{E}}_{g,q}[N_{f,L,\tau}^{q}]\big{|}^{2}\bigg{]}.$

The following theorem shows that the spectral statistics of a Dirac operator
is also consistent with the expected random matrix distributions, i.e., GSE
statistics in the presence of time-reversal symmetry, and GUE otherwise.

######  Theorem 2.

Fix $\tau>0$. For $f:{\mathbb{R}}\to{\mathbb{C}}$ with smooth, compactly
supported Fourier tansform, we have, for $L\geq 2$,

(8.10)
$\lim_{g\to\infty}\widetilde{\Sigma}_{g,q}^{2}(L,\tau;f)=\begin{cases}\Sigma_{\mathrm{GSE}}^{2}(f)+O(L^{-2}\log
L)&(q=2)\\\ \Sigma_{\mathrm{GUE}}^{2}(f)+O(L^{-2}\log L)&(q\in
2{\mathbb{N}}+2,\;q=\infty),\end{cases}$

with the random matrix density $\Sigma_{\mathrm{GUE}}^{2}(f)$ defined above
and

(8.11)
$\Sigma_{\mathrm{GSE}}^{2}(f)=\int_{\mathbb{R}}\frac{|x|}{2}\,|\hat{f}(x)|^{2}dx.$

The proof of Theorem 2 mirrors that of Theorem 1, and the key steps are the
following analogues of Propositions 6 and 7.

######  Proposition 8.

Under the assumptions for Theorem 2, we have

(8.12)
$\lim_{g\to\infty}{\mathbb{E}}_{g,q}\Bigg{[}\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\\\
\mathrm{prim.}\end{subarray}}\frac{\sigma_{\gamma}^{n}\ell_{\gamma}\,\hat{h}_{L,\tau}(n\ell_{\gamma})\,e({\text{\boldmath$\theta$}}[\gamma]n)}{2\sinh(n\ell_{\gamma}/2)}\Bigg{]}=I_{f,q}(L,\tau),$

with $I_{f,q}(L,\tau)$ as in (7.1).

###### Proof.

The same argument as in proof of Proposition 6 tells us that only primitive
loops with $\gcd[\gamma]=1$ contribute to the leading order term. The
analogous calculation as in Lemma 1 shows that, for sums over primitive
$\gamma$ with $\gcd[\gamma]=1$, we have

(8.13)
${\mathbb{E}}_{g,q}\Bigg{[}\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\\\
\mathrm{prim.}\\\
\gcd[\gamma]=1\end{subarray}}\frac{\sigma_{\gamma}^{n}\,\ell_{\gamma}\,\hat{h}_{L,\tau}(n\ell_{\gamma})\,e({\text{\boldmath$\theta$}}[\gamma]n)}{2\sinh(n\ell_{\gamma}/2)}\Bigg{]}=\begin{cases}0&(q=\infty)\\\
\displaystyle{\mathbb{E}}_{g}^{\mathrm{WP}}\bigg{[}\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\\\
\mathrm{prim.}\\\
\gcd[\gamma]=1\end{subarray}}\frac{\ell_{\gamma}\hat{h}_{L,\tau}(nq\ell_{\gamma})}{2\sinh(nq\ell_{\gamma}/2)}\bigg{]}&(q\in
2{\mathbb{N}}).\end{cases}$

The key point here is that the condition $\gcd[\gamma]=1$ restricts the sum
over $n$ to multiples of $q$ (as in Lemma 1), and hence
$\sigma_{\gamma}^{n}=1$ for such $n$, since $q$ is even. We can now proceed as
in the proof of Proposition 6. ∎

######  Proposition 9.

Under the assumptions for Theorem 2, we have

(8.14)
$\lim_{g\to\infty}{\mathbb{E}}_{g,q}\Bigg{[}\bigg{|}\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\\\
\mathrm{prim.}\end{subarray}}\frac{\sigma_{\gamma}^{n}\ell_{\gamma}\,\hat{h}_{L,\tau}(n\ell_{\gamma})\,e({\text{\boldmath$\theta$}}[\gamma]n)}{2\sinh(n\ell_{\gamma}/2)}\bigg{|}^{2}\Bigg{]}\\\
=\begin{cases}\Sigma_{\mathrm{GOE}}^{2}(f)+|I_{f,q}(L,\tau)|^{2}+O(L^{-2}\log
L)&(q=2)\\\ \Sigma_{\mathrm{GUE}}^{2}(f)+|I_{f,q}(L,\tau)|^{2}+O(L^{-2}\log
L)&(q\in 2{\mathbb{N}}+2,\;q=\infty).\end{cases}$

###### Proof.

As in the proof of Proposition 7, the only leading order terms in the high
genus limit come again from pairs $(\gamma_{1},\gamma_{2})$ of simple non-
separating oriented loops, such that one of the following holds: (i)
$\gamma_{1}=\gamma_{2}$, (ii) $\gamma_{1}=\gamma_{2}^{-1}$, or (iii)
$\gamma_{1}\cap\gamma_{2}=\emptyset$. To evaluate these terms, we follow the
argument used to prove Proposition 7, where we showed that in case (i) we have
$n_{1}=n_{2}\bmod q$, in case (ii) $n_{1}=-n_{2}\bmod q$, and in case (iii)
$n_{1}=n_{2}=0\bmod q$. Thus, if $q\in 2{\mathbb{N}}$, we have for (i) and
(ii) that
$\sigma_{\gamma_{1}}^{n_{1}}\sigma_{\gamma_{2}}^{n_{2}}=\sigma_{\gamma_{1}}^{n_{1}+n_{2}}=\sigma_{\gamma_{1}}^{n_{1}-n_{2}}=\sigma_{\gamma_{1}}^{q}=1$,
and for (iii)
$\sigma_{\gamma_{1}}^{n_{1}}\sigma_{\gamma_{2}}^{n_{2}}=\sigma_{\gamma_{1}}^{q}\sigma_{\gamma_{2}}^{q}=1$.
The case $q=\infty$ is analogous (drop the mod $q$). This proves that the
leading order terms in the high genus limit are exactly the same as in
Proposition 7. ∎

###### Proof of Theorem 2.

We apply the Selberg trace formula (8.5) for the Dirac operator with test
function $h=h_{L,\tau}$ in order to convert the spectral variance to the
geometric sums of Propositions 8 and 9. Hence the claim is proved when
$q=4,6,\ldots,\infty$. For the remaining case $q=2$, we observe that due to
time-reversal symmetry ($\chi$ is real-valued) the counting function (8.8) is
reduced by a factor of 2 to account for Kramer’s degeneracy. Hence we should
only take $\frac{1}{2}$ of (8.12), and $\frac{1}{4}$ of (8.14). This indeed
yields the formula for the GSE density, since
$\Sigma_{\mathrm{GSE}}^{2}=\frac{1}{4}\Sigma_{\mathrm{GOE}}^{2}$. ∎

## References

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# Securing Majority-Attack In Blockchain Using Machine Learning And
Algorithmic Game Theory: A Proof of Work
††thanks: * This work was supported in part by a grant from Codeepy Pvt. Ltd.
with reference CDPY/2018/1.

Somdip Dey  School of Computer Science and Electronic Engineering
University of Essex
Colchester, UK
<EMAIL_ADDRESS>
###### Abstract

Recently we could see several institutions coming together to create
consortium based blockchain networks such as Hyperledger. Although for
applications of blockchain such as Bitcoin, Litcoin, etc. the majority-attack
might not be a great threat but for consortium based blockchain networks where
we could see several institutions such as public, private, government, etc.
are collaborating, the majority-attack might just prove to be a prevalent
threat if collusion among these institutions takes place. This paper proposes
a methodology where we can use intelligent software agents to monitor the
activity of stakeholders in the blockchain networks to detect anomaly such as
collusion, using supervised machine learning algorithm and algorithmic game
theory and stop the majority-attack from taking place.

###### Index Terms:

Computer Security, network, blockchain, machine learning, algorithmic game
theory, majority attack, anomaly detection

##  I Introduction

When Satoshi Nakamoto [2] released the technology named Bitcoin, he
revolutionised the industry not because he has invented a new currency system,
which do not require intervention of institutional mediator while transferring
money from one entity to another, but because he has gifted one of the most
disruptive technology, which has come to life in decades. With the
introduction of Bitcoin, Blockchain got introduced to the world, which is a
digital ledger in which all transactions are recorded chronologically and
publicly. But the application of blockchain is not just limited to crypto-
currencies [3, 4] such as Bitcoin and have proved to be useful in tracking
ownership, provenance of documents, digital assets, physical assets, voting
rights, etc. As we can see the rise in use of blockchain technologies, we can
also see rise of security issues such as ’Double-Spending’, especially in the
Majority Attack [2, 5, 6, 7, 8]. The majority attack is carried out by a group
of individuals/entities in the decentralised environment, who colludes to take
control over the ledger to gain profit from it. In this proposed methodology,
a novel approach of using Algorithmic Game Theory concepts and Machine
Learning techniques is used to reduce the chances of collusion in the
decentralized system to gain advantage over other miners so that the system
can be as fair as possible. In section 2, some background theory regarding
blockchain and economy of double-spending is discussed. In section 3, the
proposed methodology is discussed. Finally the paper ends with some discussion
in section 4 and conclusion.

##  II Background Theory And Related Work

In this section we will visit the concepts of double-spending and the majority
attack in Blockchain. Later in this section we will also discuss the economy
of the attack being performed and how Game Theory is applicable to security
implications in blockchain as well.

###  II-A Blockchain, Double-Spending & Majority Attack

A double-spending attack [2, 5, 6, 7] in blockchain means the attacker has to
convince the merchant that a transaction has been confirmed and then convince
the entire network to approve some other transaction, which will lead to the
attacker keeping both the money and the service (goods) from the merchant
whereas the merchant would be left with neither the money or the service. This
problem in synchronization is solved by proof-of-work, which is a
computational effort consisting of hashes to acknowledge the groups of
transactions, also known as blocks. For a transaction to be valid, sufficient
work has been done to acknowledge that the block contains it. Since,
validation of blocks require computational effort to do so, this also gives
rise to another issue, what if the attacker has substantial computational
power at its disposal? All the attacker has to do is mine a blockchain
privately till the length of the chain becomes longer than the chain mined by
the honest network, and release this private blockchain for confirmation when
it is appropriate.

In Rosenfeld’s paper [6] the probability of the attacker succeeding in his
attack is discussed. If we consider z as the number of blocks by which the
honest network has advantage over the attacker then z = n - m, where n is the
number of blocks in the chain on top of the one where fork started for the
honest network, whereas m is the number of blocks in the chain on top of the
fork which the attacker has built. Before we discuss the probability of having
advantage over the attacker, let us consider the following assumptions:

  * •

The total hashrate of the attacker and honest network is constant. They have a
hashrate of H combined, of which pH belongs to honest network and qH belongs
to the attacker, where p + q = 1.

  * •

The mining difficulty is constant, such that the time taken to find a block
with H hashrate is To.

There are two possibilities of double-spending attack, which is either the
attack succeeds or it fails, as follows:

$z_{i+1}=\begin{cases}z_{i}+1,&\text{{with probability p}}.\\\
z_{i}-1,&\text{{with probability q}}.\end{cases}$

If we consider az to be the probability of the attacker succeeding in the
attack then we can arrive at the following equation:

$a_{z}=pa_{z+1}+qa_{z-1}$ (1)

And if we solve this using the boundary condition and the notion p + q = 1
then we can conclude:

$\begin{split}a_{z}=\min({\frac{q}{p},1})^{\max({z+1,0})}\\\
=\begin{cases}1,&ifz<0||q>p\\\
〖(\frac{q}{p})^{z+1},&ifz▁(>)0||q▁(<)p\end{cases}\end{split}$ (2)

If we assume n number of blocks are found by the honest network and m + 1
number of blocks are found by the attacker during this time period then the
probability (r) of double-spending to succeed when the merchant waits for n
confirmations using the equation (2) is:

$\begin{split}r=\sum_{m=0}^{\infty}P〖(m)a〗_{(n-m-1)}\\\
=\begin{cases}1-\sum_{m=0}^{n}(\begin{array}[]{c}m+n-1\\\
m\end{array})(p^{n}q^{m}-p^{m}q^{n})&ifq<p\\\ 1&ifq>p\end{cases}\end{split}$
(3)

In the study [6], it is proved that as the number of confirmations by the
honest network increased, the success rate of the attack decreased but no
matter how many confirmations by the honest network has succeeded, the attack
will always succeed if the hashrate of the attacker approached 50% of the
total network hashrate, which means q $\geq 0.5$.

This proves that an attacker with more computing power at its disposal might
prove to be a key factor in succeeding in the attack. This particularly raises
security concerns in Consortium Blockchain [5, 7, 8] such as Hyperledger,
where we can see involvement of several companies or business entities.
Whoever in the Hyperledger network holds the maximum computing power, can
always get a competitive advantage over its competitors while performing
business transaction over the network.

With Proof of Work, more CPU/GPU power is required in checking hashes of each
block in the blockchain. Because of this mechanism, more and more business
entities would like to join in this mining process, which would create ?mining
pools?, and once the mining pool holds 51% computing power, then it would take
control of the blockchain. Therefore, by taking control what it can do is [5,
8]:

  * •

Modify the transaction data, which can lead to double spending attack

  * •

To stop the block verification transaction

  * •

To stop miners mining any available block

###  II-B Economy of Double-Spending

In the study by Rosenfeld [6], it was found that the number of confirmations
required to keep the success rate of the attacker (double-spending) below 10%,
1% and 0.1%, are 2, 4 and 6 respectively. In addition, we have already seen
that once the attacker’s hashrate reaches 50% of the total network hashrate
then the number of confirmations required reaches infinity, which means no
amount of confirmation can defeat the attack. Taking this into account, we
also have to consider the likelihood of the attack being performed in reality.
If value of the commodity being exchanged is assumed to have a value of v and
the attacker has mined o number of blocks where each block has a value of B,
then if the attack succeeds the attacker will gain v, where if the attack
fails then the attacker will loose v + oB. Therefore, if we consider the two
possibilities, the payoff (s) for the attacker is as follows:

$\begin{split}s=\begin{cases}v,&ifq=>0.5\\\
-(v+oB),&ifq<0.5\end{cases}\end{split}$ (4)

where q is the hashrate of the attacker

And in order to carry on with the attack the value of v has to be significant.
This payoff (s) will prove to be useful in portraying the security implication
in the light of Game Theory, and how decisions can be made to classify whether
an attack is taking place or not.

##  III Proposed Methodology

In section 2.B we have already seen that payoff (s) for the attacker can only
have two possibilities: succeed or fail. This is where Game Theory [1] comes
into account. But before we get into the concept let us define few
terminologies of Game Theory in this context as follows:

  * •

Self-Interested Agents: This can be any entity such as a person, business or
any other institution in the blockchain network with their own preferences and
utility. This also includes honest entities and attacker(s).

  * •

Player: Each Self-Interested Agent who are participating in the blockchain
network. Let us assume that there are N players where N = (1,?.., n) is a
finite set of n, indexed by i.

  * •

Action: Action taken by each player based on their preferences and utility.
And let us assume that set of actions taken by the player i is Ai where Ai =
(a1,?., an)

  * •

Payoff: The reward, which each player receives

Now, if we consider the equation (4) then we can see the attacker would want
to maximize the probability of getting a payoff of v instead of loosing v +
oB. Therefore, we can extend the same equation (4) to derive the utility
function/ payoff function as follows:

$\begin{split}u(a)=\begin{cases}v,&ifq=>0.5\\\
-(v+oB),&ifq<0.5\end{cases}\end{split}$ (5)

where u is utility, a is the action taken by the attacker, q is the hashrate
of the attacker, v is value of commodity/service by the merchant, o is number
of blocks mined, B is value of each block.

This utility function (v) will govern the decision on whether an attack is
bound to happen or not by the attacker based on the value of the
commodity/service. And in order to keep the blockchain network safe from the
Majority Attack we should focus on this function.

We can feed this utility function to Supervised Machine Learning algorithms to
classify whether an attack is likely to take place or not. If the attack is
likely to take place then set of rules should be implemented by the system to
either prevent the blockchain confirmation from the attacker(s) or to prevent
confirmation of the whole transaction till a new fair transaction is performed
again i.e. no payoffs for anyone, in order to ensure fairness and legitimate
transactions being confirmed in the network.

In order to achieve this, an intelligent agent is implemented in the
application layer of the blockchain network system, which would have two
distinct parts:

  1. 1.

Based on the past transactions of the stakeholders the probability of each
stakeholder to defect

  2. 2.

Based on the current value of the commodity/service being sold in the current
transaction the probability of the stakeholder(s) to attack through majority
attack

Figure 1: Proof-of-Work (The Proposed Methodology)

In Fig.1, we can see the workflow of the proposed methodology where after the
new block is released in the network for the consensus purpose by the
stakeholders (including attacker and other players), the intelligent software
agent in the application layer of the network uses the utility function (v) to
classify the motive of the stakeholders and the value of the current service
being sold in the transaction. If the motive of the stakeholders is deemed to
be malicious in nature with the intent of collusion to perform a majority
attack then the transaction is cancelled and all the stakeholders are
requested for a new transaction instead.

##  IV Discussion

In the proposed methodology we have discussed about the utility function (v)
based on the value of the service or commodity being sold in the current
transaction. Here, the commodity does not have to be something that has a
tangible value in the network, rather it can have some personal attachment or
importance to the stakeholder(s). In that case the intelligent agent needs to
deduce the level of attachment or importance of the commodity or service being
handled in the transaction in order to calculate the utility function and then
the probability of the majority attack from taking place.

##  V Conclusion

As blockchain technology becomes more and more popular, we can see emergence
of several variations of such consensus based distributed ledger systems where
majority-attack can become more proficient. In order to prevent such malicious
activity in the consensus based distributed ledger systems we can utilise some
variations of the Proof-of-Work proposed in this paper. Although this is a
work in progress and in it?s preliminary stage, the proposed Proof-of-Work
will be extended to provide more holistic approach to such issues faced in the
system.

## Acknowledgment

The author wishes to thank his colleagues at ReMe Basket Ltd. and Codeepy UK
Pvt. Ltd. for their support. He would also like to thank his parents, Soma Dey
and Sudip Dey, for their continued support and faith on the author?s
capabilities. This work was supported in part by a grant from Codeepy Pvt.
Ltd. with reference CDPY/2018/1.

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