# Determinants of Laplacians on random hyperbolic surfaces

By Frédéric Naud  Frédéric Naud
Institut Mathématique de Jussieu
Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05
France<EMAIL_ADDRESS>
###### Abstract.

For sequences $(X_{j})$ of random closed hyperbolic surfaces with volume
$\mathrm{Vol}(X_{j})$ tending to infinity, we prove that there exists a
universal constant $E>0$ such that for all $\epsilon>0$, the regularized
determinant of the Laplacian satisfies

$\frac{\log\det(\Delta_{X_{j}})}{\mathrm{Vol}(X_{j})}\in[E-\epsilon,E+\epsilon]$

with high probability as $j\rightarrow+\infty$. This result holds for various
models of random surfaces, including the Weil–Petersson model.

##  1\. Introduction and results

###  1.1. On determinants

Let $X=\Gamma\backslash\mathbb{H}^{2}$ be a compact connected hyperbolic
surface obtained as a quotient of the hyperbolic plane $\mathbb{H}^{2}$ by a
discrete co-compact torsion-free group of orientation-preserving isometries.
The hyperbolic Laplacian $\Delta_{X}$ on $L^{2}(X)$ has a pure point spectrum
which we denote by

$0=\lambda_{0}<\lambda_{1}\leq\ldots\leq\lambda_{j}\leq\ldots.$

For all $s\in\mathbb{C}$ with ${\rm Re}(s)$ large enough, we know by Weyl’s
law that the spectral zeta function

$\zeta_{X}(s)=\sum_{j=1}^{\infty}\frac{1}{\lambda_{j}^{s}}$

is well defined and holomorphic. The regularized determinant is then usually
defined by

$\log\det(\Delta_{X}):=-\zeta_{X}^{\prime}(0),$

provided one can prove an analytic extension to $s=0$ of $\zeta_{X}$.
Practically, one performs a meromorphic continuation by noticing that for
large ${\rm Re}(s)$ we have

$\zeta_{X}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}(\mathrm{Tr}(e^{-t\Delta_{X}})-1)dt,$

where $e^{-t\Delta_{X}}$ is the heat semi-group. The small time asymptotics at
$t\sim 0$ of the heat kernel is the main tool which allows to “renormalize”
the divergent behavior at $t=0$ and obtain the meromorphic continuation, see
for example Chavel [11, Page 156].

In the literature, Polyakov’s string theory [35, 14] has emphasized the role
of determinants on Riemann surfaces. In particular, the computation of
“partition functions” in perturbative string theory involves formal sums over
all genera of averages of determinants over the moduli space which have proved
since then to be divergent, see Wolpert [41]. Several authors have provided
[36, 6, 13] some explicit formulas for regularized determinants for various
Laplace-like operators on Riemann surfaces. In small genus, it is possible to
compute accurately such determinants by reducing to certain sums over closed
geodesics which provide a fast convergence, see [34, 38]. In variable
curvature, the behavior of determinants in a conformal class has been studied
by Osgood, Phillips and Sarnak [30], in particular the constant curvature
metric maximizes the determinant.

In higher dimensions, determinants of Laplacians on differential forms are
related to the so-called analytic torsion, which in turn is related to
important topological invariants by results of Cheeger and Müller [12, 28]. In
particular the work of Bergeron–Venkatesh [3] establishes exponential growth
of the analytic torsion for certain families of covers of arithmetic
manifolds.

In the case of Riemann surfaces, if $\Gamma$ is a co-compact arithmetic
Fuchsian group derived from a quaternion algebra, one can define congruence
covers

$X_{\mathscr{P}}:=\Gamma(\mathscr{P})\backslash\mathbb{H}^{2}$

of $X=\Gamma\backslash\mathbb{H}^{2}$ by looking at prime ideals $\mathscr{P}$
in the ring of integers of the corresponding number field. We denote by
$\|\mathscr{P}\|$ the norm of ideals. Using the uniform spectral gap of these
surfaces proved by Sarnak and Xue in [37], together with the fact that the
injectivity radius goes to infinity as $\|\mathscr{P}\|\rightarrow\infty$, see
in [19], one can readily show (for example by using the arguments from [2] or
as a direct application of Theorem 3.1) that

$\lim_{\|\mathscr{P}\|\rightarrow\infty}\frac{\log\det\Delta_{X_{\mathscr{P}}}}{\mathrm{Vol}(X_{\mathscr{P}})}=E,$

where $E>0$ is some universal constant. Arithmetic surfaces being highly non-
generic, it is therefore natural to ask if this behavior is typical among
larger families of surfaces whose volume (equivalently genus) goes to
infinity.

###  1.2. Models of random surfaces

In this paper we will focus on the behaviour of determinants of the Laplacian
in the large volume (equivalently large genus) regime, using probabilistic
tools. The first historical model of random compact Riemann surfaces in the
mathematics literature is perhaps the model of Brooks–Makover [9] which is
based on random $3$-regular graphs as follows. Consider $\mathcal{G}_{n}$ a
$3$-regular graph on $2n$ vertices, endowed with an orientation 111an
orientation on a graph is a function which assigns to each vertex $v$ of the
graph a cyclic ordering of the edges emanating from $v$. $\mathcal{O}$. On the
(finite) set of all possible pairs $(\mathcal{G}_{n},\mathcal{O})$, one can
put a probability measure (which is not the uniform measure) introduced first
by Bollobás [4], see [9, Section 5] for a summary on this construction which
allows tractable computations in the large $n$ regime. By glueing $2n$ ideal
hyperbolic triangles according to $(\mathcal{G}_{n},\mathcal{O})$ in such a
way that the feet of the altitudes in adjacent triangles match up, one then
obtains a random finite area hyperbolic surface
$S_{n}^{O}:=S^{O}(\mathcal{G}_{n},\mathcal{O})$ with $\mathrm{Vol}(S_{n})=2\pi
n$. It is possible to show, see [8], that all surfaces in $S_{n}^{O}$ are
actually (non ramified) covers of the modular surface
$\mathrm{PSL}_{2}(\mathbb{Z})\backslash\mathbb{H}^{2}$.

One can then conformally compactify $S_{n}^{O}$ by cutting cusps and filling
them with discs. Provided that $S_{n}^{O}$ has genus at least $2$, we then
denote by $S_{n}^{C}$ the unique hyperbolic surface in the conformal class of
this compactification, see [9, Section 3]. In $\S 4$ of the same paper they
also show that there exists a constant $C_{0}>0$ such that with high
probability as $n\rightarrow+\infty$,

$\mathrm{Vol}(S_{n}^{C})\geq C_{0}n.$

Most of the geometric properties of $S_{n}^{O}$ (and then $S_{n}^{C}$, after a
mild loss) can be read off from the combinatorics of $\mathcal{G}_{n}$.

Another more recent discrete model of random surfaces is the so-called random
cover model which has been studied and used recently in [26, 21, 29]. In what
follows, we fix a compact surface $X=\Gamma\backslash\mathbb{H}^{2}$, ”the
base surface”. Let $\phi_{n}:\Gamma\rightarrow\mathcal{S}_{n}$ be a group
homomorphism, where $\mathcal{S}_{n}$ is the symmetric group of permutations
of $[n]:=\\{1,\ldots,n\\}$. The discrete group $\Gamma$ acts on
$\mathbb{H}^{2}\times[n]$ by

$\gamma.(z,j):=(\gamma(z),\phi_{n}(\gamma)(j)).$

The resulting quotient $X_{n}:=\Gamma\backslash(\mathbb{H}^{2}\times[n])$ is
then a finite cover of degree $n$ of $X$, possibly not connected. By
considering the (finite) space of all homomorphism
$\phi_{n}:\Gamma\rightarrow\mathcal{S}_{n}$, endowed with the uniform
probability measure, we obtain a notion of random covering surfaces of degree
$n$, $X_{n}\rightarrow X$. Let us remark that we can also view (up to
isometry) the random cover $X_{n}$ as

$X_{n}=\bigsqcup_{k=1}^{p}\Gamma_{k}\backslash\mathbb{H}^{2},$

where each $\Gamma_{k}$ is the (a priori non-normal) subgroup of $\Gamma$
given by

$\Gamma_{k}=\mathrm{Stab}_{\Gamma}(i_{k})=\\{\gamma\in\Gamma\ :\
\phi_{n}(\gamma)(i_{k})=i_{k}\\},$

where $i_{1},\ldots,i_{p}\in[n]$ are representatives of the orbits of $\Gamma$
(acting on $[n]$ via $\phi_{n}$). In general, the cover $X_{n}$ is not
connected, but it follows directly from [20] that the probability that this
cover is connected tends to $1$ as $n$ goes to infinity. In this model, we
have $\mathrm{Vol}(X_{n})=n\mathrm{Vol}(X)$.

A smooth model of random hyperbolic surfaces is given by the moduli space
$\mathscr{M}_{g}$ of closed hyperbolic surfaces with genus $g$, up to
isometry. It is often defined as the quotient

$\mathscr{M}_{g}=\mathscr{T}_{g}/\mathrm{MCG},$

where $\mathscr{T}_{g}$ is the Teichmüller space of hyperbolic metrics on a
surface $S$ of genus $g$ and

$\mathrm{MCG}=\mathrm{Diff}(S)/\mathrm{Diff}_{0}(S)$

is the group of isotopy classes of diffeomorphisms on $S$, aka the mapping
class group. We refer the reader for example to [10], chapter 6 for more
details. A symplectic form $\omega_{WP}$ lives naturally on $\mathscr{T}_{g}$
and descends to the moduli space, endowing it with a natural notion of volume,
Weil–Petersson volume. The moduli space is a non-compact finite dimensional
orbifold, but as a consequence of Bers’ theorem on pants decomposition, see
[10, Theorem 5.1.2] it has a finite volume with respect to this Weil–Petersson
volume. We can therefore normalize this measure and obtain a probability
measure on $\mathscr{M}_{g}$. Notice that in this case if
$X\in\mathscr{M}_{g}$, $\mathrm{Vol}(X)=4\pi(g-1)$ by Gauss–Bonnet. The
calculation of Weil–Petersson volumes of the moduli space by Mirzakhani [24]
has made possible [25] the large genus asymptotic analysis of various
geometric and spectral quantities, see for example [27, 43, 42] for recent
works in that direction.

For all of the previous models, we will denote by ${\mathbb{P}}$ the
associated probability measure, which depends either on $n$ or $g$, which are
both proportional to the volume. We say that an event $\mathcal{A}$ is
asymptoticaly almost sure (a.a.s.), or holds with high probability, if
$\mathbb{P}(\mathcal{A})$ tends to $1$ as the volume of surfaces tends to
infinity. The expectation of any relevant random variable will also be denoted
by $\mathbb{E}$.

###  1.3. Main result

######  Theorem 1.1.

There exists a universal constant $E>0$ such that for all the above models of
random surfaces, for all $\epsilon>0$ we have

$\frac{\log\det(\Delta_{X})}{\mathrm{Vol}(X)}\in[E-\epsilon,E+\epsilon],$

a.a.s as $\mathrm{Vol}(X)\rightarrow+\infty.$

The constant $E$ is actually explicit and is approximately $0.0538$, see $\S
2$ for an exact description. This result shows that exponential growth of the
determinant is typical when the genus goes to infinity. This low dimensional
result is consistent, in a much simpler setting, with the conjectures on the
exponential growth of the analytic torsion and the torsion homology for higher
dimensional hyperbolic manifolds, see for example the paper of
Bergeron–Venkatesh [3] and references therein.

Remark that the above statement says that the random variable
$\frac{\log\det(\Delta_{X})}{\mathrm{Vol}(X)}$ converges in probability to the
constant $E$. What about other modes of convergence? It is possible to derive
from Theorem 1.1 a convergence result for the expectation of
$|\log\det(\Delta)|^{\beta}$, see section $\S 5$, Theorem 5.1: for both models
of random covers and Weil–Petersson, we show the existence of exponents
$\beta>0$ such that

$\lim_{\mathrm{Vol}(X)\rightarrow\infty}\mathbb{E}\left(\frac{|\log\det(\Delta_{X})|^{\beta}}{\mathrm{Vol}(X)^{\beta}}\right)=E^{\beta}.$

The paper is organized as follows. In $\S 2$ we recall how one establishes an
identity for $\log\det(\Delta_{X})$ which involves infinite sums over closed
geodesics via the Heat trace formula. In $\S 3$ we prove an abstract Theorem
which guarantees the exponential growth of determinants as long as a certain
natural list of assumptions are satisfied. These hypotheses turn out to be
valid a.a.s. for the probabilistic models listed above, and this is
established in $\S 4$. In $\S 5$, we derive from Theorem 1.1 a convergence
result for the expectation, based on some moments estimates for the systole
and the smallest positive eigenvalue.

Acknowledgement. It is a pleasure to thank my neighbor Bram Petri for several
discussions around this work. Thanks to Yuhao Xue and an anonymous referee for
pointing out an improvement of Theorem 3.1. I also thank Zeev Rudnick for his
reading and comments. Finally, Yunhui Wu and Yuxin He have recently shown to
me that Theorem 5.1 can also be refined in the Weil–Petterson case, see in $\S
5$ for details.

##  2\. Heat kernels and determinants

In this section, we recall some standard calculations on regularized
determinants mostly taken from [6, Appendix B]. Our goal is to show how the
heat trace formula allows to derive an identity for $\log\det\Delta_{X}$.

On the hyperbolic plane $\mathbb{H}^{2}$, the heat kernel $p_{t}(x,y)$ (see
for example [10] chapter 7) has an explicit formula given by

$p_{t}(x,y)=\frac{\sqrt{2}e^{-t/4}}{(4\pi
t)^{3/2}}\int_{d(x,y)}^{\infty}\frac{re^{-r^{2}/4t}dr}{\sqrt{\cosh r-\cosh
d(x,y)}},$

where $d(x,y)$ denotes the hyperbolic distance in $\mathbb{H}^{2}$. On the
quotient $X=\Gamma\backslash\mathbb{H}^{2}$, we can recover the heat kernel by
summing over the group, i.e.,

$h_{t}^{X}(x,y)=\sum_{\gamma\in\Gamma}p_{t}(x,\gamma y).$

Convergence of the above series on any compact subset of $\mathbb{H}^{2}$ is
guaranteed by the lattice counting bound

$N_{\Gamma}(x,y,T):=\\#\\{\gamma\in\Gamma\ :\ d(\gamma x,y)\leq
T\\}=O(e^{T}),$ (1)

which is standard and follows from a basic volume argument. The semi-group of
operators $e^{-t\Delta_{X}}$ is then of trace class and one has the explicit
“heat trace formula”

$\mathrm{Tr}(e^{-t\Delta_{X}})=\sum_{j}e^{-t\lambda_{j}}=\mathrm{Vol(X)}\frac{e^{-t/4}}{(4\pi
t)^{3/2}}\int_{0}^{\infty}\frac{re^{-r^{2}/4t}}{\sinh(r/2)}dr\\\ $ (2)
$+\frac{e^{-t/4}}{(4\pi t)^{1/2}}\sum_{k\geq
1}\sum_{\gamma\in\mathcal{P}}\frac{\ell(\gamma)}{2\sinh(k\ell(\gamma)/2)}e^{-(k\ell(\gamma))^{2}/4t},$

where $\mathcal{P}$ stands for the set of primitive conjugacy classes in
$\Gamma$ (i.e. oriented primitive closed geodesics on $X$) and if
$\gamma\in\mathcal{P}$, $\ell(\gamma)$ is the length. For more details on the
calculation of this trace and more generally Selberg’s formula, see [17, 10].
Setting

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

it is easy to see from the spectral side of the trace formula that
$|S_{X}(t)-1|$ is exponentially small as $t\rightarrow+\infty$. We also
observe that $S_{X}(t)$ is exponentially small as $t\rightarrow 0$. We now
write (for ${\rm Re}(s)$ large)

$\zeta_{X}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}(\mathrm{Tr}(e^{-t\Delta_{X}})-1)dt=\zeta^{(1)}_{X}(s)+\zeta^{(2)}_{X}(s),$

where we have set

$\zeta^{(1)}_{X}(s)=:\mathrm{Vol(X)}\frac{1}{4\pi\Gamma(s)}\int_{0}^{\infty}t^{s-1}\frac{e^{-t/4}}{\sqrt{4\pi}t^{3/2}}\int_{0}^{\infty}\frac{re^{-r^{2}/4t}}{\sinh(r/2)}drdt,$

and

$\zeta^{(2)}_{X}(s)=:\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}(S_{X}(t)-1)dt.$

Writing (for ${\rm Re}(s)$ large)

$\zeta^{(2)}_{X}(s)=\frac{1}{\Gamma(s)}\int_{0}^{1}t^{s-1}(S_{X}(t)-1)dt+\frac{1}{\Gamma(s)}\int_{1}^{\infty}t^{s-1}(S_{X}(t)-1)dt$
$=\frac{-1}{\Gamma(s+1)}+\frac{1}{\Gamma(s)}\int_{0}^{1}t^{s-1}S_{X}(t)dt+\frac{1}{\Gamma(s)}\int_{1}^{\infty}t^{s-1}(S_{X}(t)-1)dt,$

we notice that the last two integrals make sense for all $s\in\mathbb{C}$.
Therefore $\zeta^{(2)}_{X}(s)$ has an analytic extension to $\mathbb{C}$ and
using elementary facts on the gamma function (in particular that it has a
simple pole at $s=0$ with residue $1$), we have that

$-\left.\frac{d}{ds}\right|_{s=0}\zeta^{(2)}_{X}(s)=-\Gamma^{\prime}(1)-\int_{0}^{1}\frac{S_{X}(t)}{t}dt-\int_{1}^{\infty}\frac{(S_{X}(t)-1)}{t}dt.$

On the other hand, for large ${\rm Re}(s)$ we use that 222for example one can
use the identity valid for $x\in\mathbb{R}$ $\tanh(\pi
x)=\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{2x}{x^{2}+(k+1/2)^{2}}=\frac{1}{\pi}\int_{0}^{\infty}\frac{\sin(ux)}{\sinh(u/2)}du.$

$\frac{e^{-t/4}}{\sqrt{4\pi}t^{3/2}}\int_{0}^{\infty}\frac{re^{-r^{2}/4t}}{\sinh(r/2)}dr=\int_{-\infty}^{+\infty}x\tanh(\pi
x)e^{-(x^{2}+1/4)t}dx,$

see [6, Page 593], and one can compute the Mellin transform to obtain (again
for ${\rm Re}(s)$ large)

$\zeta^{(1)}_{X}(s)=\frac{2\mathrm{Vol}(X)}{4\pi}\int_{0}^{\infty}\frac{u\tanh(\pi
u)}{(u^{2}+1/4)^{s}}du.$

This function can be analytically continued to $s=0$, see for example [6,
Appendix B], and the value can be actually computed as

$-\left.\frac{d}{ds}\right|_{s=0}\zeta^{(1)}_{X}(s)=\frac{\mathrm{Vol}(X)}{4\pi}\left(4\zeta^{\prime}(-1)-1/2+\log(2\pi)\right):=\mathrm{Vol(X)}E,$

with $\zeta^{\prime}(-1)=1/12-\log(A)$ and $A$ is the so-called
Glaisher–Kinkelin constant, which is for example defined by

$A=\lim_{n\rightarrow\infty}\frac{\prod_{k=1}^{n}k^{k}}{e^{-n^{2}/4}n^{n^{2}/2+n/2+1/12}}.$

We have $E\approx 0,0538$. Using in addition that
$\Gamma^{\prime}(1)=-\gamma_{0}$, where

$\gamma_{0}=\lim_{n\rightarrow+\infty}\sum_{k=1}^{n}\frac{1}{k}-\log(n),$

is the Euler constant, we have obtained the celebrated identity

$\log\det\Delta_{X}=\mathrm{Vol}(X)E+\gamma_{0}-\int_{0}^{1}\frac{S_{X}(t)}{t}dt-\int_{1}^{\infty}\frac{(S_{X}(t)-1)}{t}dt.$
(3)

This formula can be interpreted multiplicatively via Selberg zeta function at
$s=1$, see [36, 6, 13].

##  3\. An abstract deterministic statement

Theorem 1.1 actually follows from a more general deterministic result for
sequences of compact surfaces satisfying certain hypotheses denoted by
$\mathscr{H}_{1}$ and $\mathscr{H}_{2}$. More precisely, if $(X_{k})$ is a
sequence of compact connected hyperbolic surfaces with
$\mathrm{Vol}(X_{k})\rightarrow+\infty$, let $\mathcal{P}_{k}$ denote the set
of oriented primitive closed geodesics on $X_{k}$, and let $\Delta_{k}$ be the
hyperbolic Laplacian on $X_{k}$. We also denote by $\ell_{0}(X_{k})$ the
length of the shortest closed geodesic on $X_{k}$. Let $C>0$, $\eta>0$, $L>0$
and $0<\alpha<1/2$ be some constants.

We say that the sequence $(X_{k})$ satisfies hypothesis
$\mathscr{H}_{1}(\eta)$ if for all $k\in\mathbb{N}$ we have

$\lambda_{1}(\Delta_{k})\geq\eta.$ (4)

We say that the sequence $(X_{k})$ satisfies hypothesis
$\mathscr{H}_{2}(C,L,\alpha)$ if for all $k\in\mathbb{N}$ we have the
following bound on the number of closed geodesics:

$N_{k}(L):=N_{X_{k}}(L):=\\#\\{(\gamma,m)\in\mathcal{P}_{k}\times\mathbb{N}\ \
:\ m\ell(\gamma)\leq L\\}\leq C\mathrm{Vol}(X_{k})^{\alpha}.$ (5)

In this paper $\mathbb{N}=\\{1,2,\ldots\\}$ is the set of natural integers
starting at $1$. We point out that exponential growth of Laplace determinants
is established in the literature for families of covers for which a uniform
spectral gap holds and the injectivity radius of the manifolds goes to
infinity, see for example [3] and [2]. Typical examples are congruence covers
of arithmetic hyperbolic manifolds and Laplacians twisted by a ”strongly
acyclic” representation which ensures a uniform spectral gap. In random models
of surfaces, having the injectivity radius grow to infinity is atypical and we
establish the result under the weaker assumption of small growth of the number
of closed geodesics with bounded length.

Theorem 1.1 will follow from the following deterministic result.

######  Theorem 3.1.

Fix some $\eta>0$ and $0<\alpha<1$. Assume that $(X_{k})$ satisfies
$\mathscr{H}_{1}(\eta)$ and $\mathscr{H}_{2}(C_{0},L_{0},\alpha)$ with
$L_{0}=2\mathrm{arcsinh}(1)$ for some $C_{0}>0$. Then for all $\epsilon>0$,
there exists $L_{\epsilon}>0$ such that if in addition $(X_{k})$ satisfies
$\mathscr{H}_{2}(C_{\epsilon},L_{\epsilon},\alpha)$ for some $C_{\epsilon}>0$,
then uniformly for all $\mathrm{Vol}(X_{k})$ large,

$\frac{\log\det(\Delta_{X_{k}})}{\mathrm{Vol}(X_{k})}\in[E-\epsilon,E+\epsilon],$

where $E>0$ is the universal constant from above.

All the positive constants denoted by $C_{1},C_{2},\ldots,C_{j}$ below depend
only on $C_{0},L_{0}$. Before we give a proof of Theorem 3.1, we will need a
preliminary Lemma which is needed to control uniformly sums over closed
geodesics.

######  Lemma 3.2.

Under hypothesis $\mathscr{H}_{2}(C_{0},L_{0},\alpha)$ where
$L_{0}=2\mathrm{arcsinh}(1),$ there exists $C_{1}>0$ such that for all $k$ and
all $T\geq 0$,

$N_{k}(T)\leq C_{1}\mathrm{Vol}(X_{k})e^{T}.$

###### Proof.

A result of Buser ([10] Lemma 6.6.4) says that for any compact connected
hyperbolic surface of genus $g$, the number of oriented closed geodesics with
length $\leq T$ which are not iterates of primitive closed geodesics of length
$\leq 2\mathrm{arcsinh}(1)$ is bounded from above by

$(g-1)e^{T+6}.$

Therefore we have

$N_{k}(T)\leq\frac{\mathrm{Vol}(X_{k})}{4\pi}e^{T+6}+\\#\\{(\gamma,m)\in\mathcal{P}_{k}\times\mathbb{N}\
\ :\ m\ell(\gamma)\leq T\ \mathrm{and}\ \ell(\gamma)\leq
2\mathrm{arcsinh}(1)\\}.$

On the other hand we have

$\\#\\{(\gamma,m)\in\mathcal{P}_{k}\times\mathbb{N}\ \ :\ m\ell(\gamma)\leq T\
\mathrm{and}\ \ell(\gamma)\leq
2\mathrm{arcsinh}(1)\\}\leq\sum_{\ell(\gamma)\leq\mathrm{arcsinh}(1)}\frac{T}{\ell(\gamma)}.$

We can observe that by definition of $N_{k}(L)$, we have

$N_{k}(L)=\sum_{m\ell(\gamma)\leq L}1=\sum_{\ell(\gamma)\leq
L}\left[\frac{L}{\ell(\gamma)}\right],$

where $[.]$ is the integer part. Hence we can write

$\sum_{\ell(\gamma)\leq L}\frac{1}{\ell(\gamma)}\leq\frac{2}{L}N_{X_{k}}(L).$
(6)

Going back to the estimate of $N_{X_{k}}(T)$ and using
$\mathscr{H}_{2}(C_{0},L_{0},\alpha)$ with $L_{0}=\mathrm{arcsinh}(1)$, we get

$N_{k}(T)\leq\mathrm{Vol}(X_{k})e^{T}\frac{e^{6}}{4\pi}+\frac{2}{\mathrm{arcsinh}(1)}N_{X_{k}}(\mathrm{arcsinh}(1))$
$\leq C_{1}\mathrm{Vol}(X_{k})e^{T}$

the proof is done. ∎

######  Lemma 3.3.

Under hypothesis $\mathscr{H}_{1}(\eta)$ and
$\mathscr{H}_{2}(C_{0},L_{0},\alpha)$ where $L_{0}$ is as above, there exists
$C_{2}>0$ such that for all $k$ and all $t\geq 1$,

$\left|S_{X_{k}}(t)-1\right|\leq C_{2}\mathrm{Vol}(X_{k})e^{-\eta_{0}t},$

where $\eta_{0}=\min(\eta,1/4)$.

###### Proof.

In this proof we will use Vinogradov’s notation $A\ll B$ meaning that $A\leq
CB$ where $C>0$ is a universal constant. By formula (2) we have

$\mathrm{Tr}(e^{-t\Delta_{X_{k}}})-1=\mathrm{Vol(X_{k})}\frac{e^{-t/4}}{(4\pi
t)^{3/2}}\int_{0}^{\infty}\frac{re^{-r^{2}/4t}}{\sinh(r/2)}dr+S_{X_{k}}(t)-1,$

and therefore we get

$|S_{X_{k}}(t)-1|\ll\mathrm{Vol}(X_{k})e^{-t/4}+\sum_{j=1}^{\infty}e^{-t\lambda_{j}(X_{k})},$

for all $t\geq 1$. On the other hand using the uniform spectral gap we have,

$\sum_{j=1}^{\infty}e^{-t\lambda_{j}(X_{k})}=\sum_{j=1}^{\infty}e^{-(t-1)\lambda_{j}-\lambda_{j}}\leq
e^{-(t-1)\eta}\mathrm{tr}(e^{-\Delta_{X_{k}}}).$

Going back to formula (2) with $t=1$, we have also

$\mathrm{tr}(e^{-\Delta_{X_{k}}})\ll\mathrm{Vol}(X_{k})+S_{X_{k}}(1),$

and

$S_{X_{k}}(1)\ll\sum_{m\geq
1}\sum_{\gamma\in\mathcal{P}_{k}}\frac{\ell(\gamma)}{2\sinh(m\ell(\gamma)/2)}e^{-(m\ell(\gamma))^{2}/4}$
$\ll\int_{0}^{\infty}\frac{u}{2\sinh(u/2)}e^{-u^{2}/4}dN_{k}(u),$

where we have used the Stieltjes integral notation with the measure $dN_{k}$
associated to the counting function $N_{k}$. We can use Lemma 3.2 to bound
$N_{k}(u)$ as

$N_{k}(u)\leq C_{1}\mathrm{Vol}(X_{k})e^{u},$

and a summation by parts shows that

$\int_{0}^{\infty}\frac{u}{2\sinh(u/2)}e^{-u^{2}/4}dN_{k}(u)=-\int_{0}^{\infty}N_{k}(u)\frac{d}{du}\left\\{\frac{u}{2\sinh(u/2)}e^{-u^{2}/4}\right\\}du$
$\ll C_{1}\mathrm{Vol}(X_{k}),$

which ends the proof. ∎

###### Proof of Theorem 3.1.

First notice that by formula (3), we have

$\left|\frac{\log\det(\Delta_{X_{k}})}{\mathrm{Vol}(X_{k})}-E\right|\leq
O(\mathrm{Vol}(X_{k})^{-1})+\mathcal{D}^{(1)}_{X_{k}}+\mathcal{D}^{(2)}_{X_{k}},$

where

$\mathcal{D}^{(1)}_{X_{k}}=\frac{1}{\mathrm{Vol}(X_{k})}\int_{1}^{\infty}\frac{|S_{X_{k}}(t)-1|}{t}dt,\
\mathcal{D}^{(2)}_{X_{k}}=\frac{1}{\mathrm{Vol}(X_{k})}\int_{0}^{1}\frac{S_{X_{k}}(t)}{t}dt.$

Let us fix $\epsilon>0$. We first investigate $\mathcal{D}^{(1)}_{X_{k}}$.
Using $\mathscr{H}_{1}(\eta)$ and $\mathscr{H}_{2}(C_{0},\alpha,L_{0})$, we
can use Lemma 3.3 and write

$\int_{1}^{\infty}\frac{|S_{X_{k}}(t)-1|}{t}dt\leq\int_{1}^{R}\frac{S_{X_{k}}(t)}{t}dt+\log(R)+C_{2}\mathrm{Vol}(X_{k})\int_{R}^{\infty}\frac{e^{-\eta_{0}t}}{t}dt,$

for any $R>1$. Fixing $R=R(\epsilon)$ so large that

$C_{2}\int_{R}^{\infty}\frac{e^{-\eta_{0}t}}{t}dt\leq\epsilon,$

we have

$\mathcal{D}^{(1)}_{X_{k}}\leq\frac{1}{\mathrm{Vol}(X_{k})}\int_{1}^{R}\frac{S_{X_{k}}(t)}{t}dt+\frac{\log(R)}{\mathrm{Vol}(X_{k})}+\epsilon.$

We now pick $L_{1}>1$ (to be adjusted later on) and write

$\int_{1}^{R}\frac{S_{X_{k}}(t)}{t}dt=\int_{1}^{R}\frac{S_{X_{k}}^{L_{1},-}(t)}{t}dt+\int_{1}^{R}\frac{S_{X_{k}}^{L_{1},+}(t)}{t}dt,$

where

$S_{X_{k}}^{L_{1},-}(t)=\frac{e^{-t/4}}{(4\pi t)^{1/2}}\sum_{m\ell(\gamma)\leq
L_{1}}\frac{\ell(\gamma)}{2\sinh(m\ell(\gamma)/2)}e^{-(m\ell(\gamma))^{2}/4t},\
$ $S_{X_{k}}^{L_{1},+}(t)=\frac{e^{-t/4}}{(4\pi
t)^{1/2}}\sum_{m\ell(\gamma)>L_{1}}\frac{\ell(\gamma)}{2\sinh(m\ell(\gamma)/2)}e^{-(m\ell(\gamma))^{2}/4t}.$

Clearly we have

$\int_{1}^{R}\frac{S_{X_{k}}^{L_{1},+}(t)}{t}dt\leq
C_{3}\sum_{m\ell(\gamma)>L_{1}}\frac{\ell(\gamma)}{2\sinh(m\ell(\gamma)/2)}e^{-(m\ell(\gamma))^{2}/4R},$

for some universal constant $C_{3}>0$. Using Lemma 3.2 and a summation by
parts, we deduce that

$\int_{1}^{R}\frac{S_{X_{k}}^{L_{1},+}(t)}{t}dt\leq
C_{4}\mathrm{Vol(X_{k})}\int_{L_{1}}^{\infty}\left|\frac{d}{du}\left(\frac{u}{\sinh(u/2)}e^{-u^{2}/(4R)}\right)\right|e^{u}du.$

We now take $L_{1}=L_{1}(\epsilon)$ so large that

$C_{4}\int_{L_{1}}^{\infty}\left|\frac{d}{du}\left(\frac{u}{\sinh(u/2)}e^{-u^{2}/(4R)}\right)\right|e^{u}du<\epsilon.$

We now observe that if $\mathscr{H}_{2}(C,L_{1},\alpha)$ holds, we have

$\int_{1}^{R}\frac{S_{X_{k}}^{L,-}(t)}{t}dt=\sum_{m\ell(\gamma)\leq
L_{1}}\frac{\ell(\gamma)}{2\sinh(m\ell(\gamma)/2)}\int_{1}^{R}e^{-(m\ell(\gamma))^{2}/4t}\frac{e^{-t/4}}{\sqrt{4\pi}t^{3/2}}dt$
$\leq C_{5}L_{1}N_{k}(L_{1})\leq C_{5}CL_{1}\mathrm{Vol}(X_{k})^{\alpha}\leq
C_{6}(L_{1},C)\mathrm{Vol}(X_{k})^{\alpha},$

for some possibly large constant $C_{6}(L_{1},C)>0$. In a nutshell, we have
obtained, provided that $\mathscr{H}_{2}(C,L_{1},\alpha)$ is satisfied with
$L_{1}=L_{1}(\epsilon)$ taken large enough,

$\limsup_{\mathrm{Vol}(X_{k})\rightarrow+\infty}\mathcal{D}^{(1)}_{X_{k}}\leq
2\epsilon.$

We now turn our attention to $\mathcal{D}^{(2)}_{X_{k}}$, and this is where a
good control of sums over short geodesics is required. We first use the same
idea as above by writing

$\mathcal{D}^{(2)}_{X_{k}}=\frac{1}{\mathrm{Vol}(X_{k})}\int_{0}^{1}\frac{S_{X_{k}}(t)}{t}dt=\frac{1}{\mathrm{Vol}(X_{k})}\int_{0}^{1}\frac{S_{X_{k}}^{L_{2},-}(t)}{t}dt+\frac{1}{\mathrm{Vol}(X_{k})}\int_{0}^{1}\frac{S_{X_{k}}^{L_{2},+}(t)}{t}dt.$

Writing for $t>0$,

$S_{X_{k}}^{L_{2},+}(t)\leq
C_{7}t^{-1/2}\sum_{m\ell(\gamma)>L_{2}}\frac{\ell(\gamma)}{2\sinh(m\ell(\gamma)/2)}e^{-(m\ell(\gamma))^{2}/4t},$

where $C_{7}>0$ is universal, we have by Fubini

$\int_{0}^{1}\frac{S_{X_{k}}^{L_{2},+}(t)}{t}dt\leq
C_{7}\sum_{m\ell(\gamma)>L_{2}}\frac{\ell(\gamma)}{2\sinh(m\ell(\gamma)/2)}G(m\ell(\gamma)),$

where for $u>0$,

$G(u)=\int_{0}^{1}t^{-3/2}e^{-u^{2}/4t}dt.$

Notice that $u\mapsto G(u)$ is a decreasing function and by a change of
variable we have actually for all $u>0$,

$G(u)=\frac{4}{u}\int_{u/2}^{\infty}e^{-x^{2}}dx.$

We have therefore the bound

$G(u)=\frac{4}{u}\int_{u/2}^{\infty}e^{-x^{2}/2-x^{2}/2}dx\leq\frac{4}{u}e^{-u^{2}/8}\int_{0}^{\infty}e^{-x^{2}/2}dx=\frac{2\sqrt{2\pi}}{u}e^{-u^{2}/8}.$

As a consequence we get for $L_{2}>1$

$\int_{0}^{1}\frac{S_{X_{k}}^{L_{2},+}(t)}{t}dt\leq
C_{8}\sum_{m\ell(\gamma)>L_{2}}\frac{\ell(\gamma)}{2\sinh(m\ell(\gamma)/2)}e^{-(m\ell(\gamma))^{2}/8},$

and by using Lemma 3.2 and a summation by parts, we can definitely fix
$L_{2}=L_{2}(\epsilon)$ large enough so that

$\int_{0}^{1}\frac{S_{X_{k}}^{L_{2},+}(t)}{t}dt\leq
C_{8}\sum_{m\ell(\gamma)>L_{2}}\frac{\ell(\gamma)}{2\sinh(m\ell(\gamma)/2)}e^{-(m\ell(\gamma))^{2}/8}\leq\mathrm{Vol}(X_{k})\epsilon.$

From the above bound on $G(u)$ we also deduce

$\int_{0}^{1}\frac{S_{X_{k}}^{L_{2},-}(t)}{t}dt\leq
C_{9}\sum_{m\ell(\gamma)\leq
L_{2}}\frac{\ell(\gamma)}{\sinh(m\ell(\gamma)/2)}\frac{1}{m\ell(\gamma)}$
$\leq C_{9}^{\prime}\sum_{m\ell(\gamma)\leq
L_{2}}\frac{1}{m^{2}\ell(\gamma)}.$

By writing

$\sum_{m\ell(\gamma)\leq
L_{2}}\frac{1}{m^{2}\ell(\gamma)}=\sum_{m=1}^{\infty}\frac{1}{m^{2}}\sum_{\ell(\gamma)\leq
L_{2}/m}\frac{1}{\ell(\gamma)}\leq\frac{\pi^{2}}{6}\sum_{\ell(\gamma)\leq
L_{2}}\frac{1}{\ell(\gamma)},$

we can use estimate (6) and $\mathscr{H}_{2}(C,L_{2}(\epsilon),\alpha)$ and we
have again as above

$\int_{0}^{1}\frac{S_{X_{k}}^{L_{2},-}(t)}{t}dt\leq
C_{10}(L_{2},C)\mathrm{Vol}(X_{k})^{\alpha},$

where $C_{10}(L_{2},C)>0$ is some (possibly very large) constant depending on
$L_{2},C$ and $0<\alpha<1$. We have therefore shown, that whenever $(X_{k})$
satisfies $\mathscr{H}_{2}(C,L(\epsilon),\alpha)$ for some $C>0$ and with
$L(\epsilon)=\max\\{L_{1},L_{2}\\}$ we have

$\limsup_{\mathrm{Vol}(X_{k})\rightarrow+\infty}\left|\frac{\log\det(\Delta_{X_{k}})}{\mathrm{Vol}(X_{k})}-E\right|\leq
3\epsilon.$

Theorem 3.1 is proved. ∎

##  4\. Hypotheses $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ hold with high
probability

Theorem 1.1 follows immediately from Theorem 3.1 if one can establish for the
three models of random hyperbolic surfaces considered here that there exists
$\eta>0$ and such that $\mathscr{H}_{1}(\eta)$ holds a.a.s. and also that
there exists $0<\alpha<1$ such that for all $L$ large, one can find $C>0$ such
that $\mathscr{H}_{2}(C,L,\alpha)$ also holds a.a.s.

Indeed we then have for all $\epsilon>0$, and all $\mathrm{Vol}(X)$ large
enough

$\mathbb{P}\left(\frac{\log\det(\Delta_{X})}{\mathrm{Vol}(X)}\in[E-\epsilon,E+\epsilon]\right)\geq$
$\mathbb{P}\left(X\in\mathscr{H}_{1}(\eta)\cap\mathscr{H}_{2}(C_{0},L_{0},\alpha)\cap\mathscr{H}_{2}(C,L(\epsilon),\alpha)\right),$

with

$\lim_{\mathrm{Vol}(X)\rightarrow\infty}\mathbb{P}\left(X\in\mathscr{H}_{1}(\eta)\cap\mathscr{H}_{2}(C_{0},L_{0},\alpha)\cap\mathscr{H}_{2}(C,L(\epsilon),\alpha)\right)=1.$

###  4.1. Random covers

First we recall that the cover $X_{n}\rightarrow X$ may not be connected but
we know from [20] that

$\lim_{n\rightarrow+\infty}\mathbb{P}(X_{n}\ \mathrm{connected})=1.$

We can therefore either restrict ourselves to connected surfaces $X_{n}$ or
modify the definition of the regularized determinant by setting

$\zeta_{X_{n}}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}(\mathrm{Tr}(e^{-t\Delta_{X}})-d_{n})dt,$

where $d_{n}$ is the number of connected component of $X_{n}$ and use the fact
that $d_{n}=1$ with high probability.

It was recently shown in [21] that there exists a uniform spectral gap for
random covers $X_{n}$ a.a.s. as $n\rightarrow+\infty$. More precisely we have
for all $0<\eta<\min\\{3/16,\lambda_{1}(X)\\}$,

$\lim_{n\rightarrow+\infty}\mathbb{P}(\lambda_{1}(X_{n})\geq\eta)=1.$

Therefore $\mathscr{H}_{1}(\eta)$ holds a.a.s. provided $\eta$ is taken small
enough. On the other hand property $\mathscr{H}_{2}$ is less obvious from the
existing litterature and will require some explanations. We recall that given
a random homomorphism $\phi_{n}:\Gamma\rightarrow\mathcal{S}_{n}$, one can
define a unitary representation $\rho_{n}$ of $\Gamma$ by setting

$\rho_{n}(\gamma)(f):=f\circ\phi_{n}(\gamma)^{-1},$

where $f\in L^{2}([n])$, and the representation space is
$L^{2}([n])\simeq\mathbb{C}^{n}$. The main interest of this representation is
the following fact, often called “Venkov–Zograf induction formula”. For all
${\rm Re}(s)>1$, one can define the Selberg zeta function of $X_{n}$ by

$Z_{X_{n}}(s):=\prod_{m\geq
0}\prod_{\gamma\in\mathcal{P}_{X_{n}}}\left(1-e^{-(s+m)\ell(\gamma)}\right).$

One can also look at the twisted Selberg zeta function of the base
$X=\Gamma\backslash\mathbb{H}^{2}$ defined for ${\rm Re}(s)>1$ by

$Z_{X,\rho_{n}}(s):=\prod_{m\geq
0}\prod_{\gamma\in\mathcal{P}_{X}}\det\left(I-\rho_{n}(\gamma)e^{-(s+m)\ell(\gamma)}\right).$

It turns out that we have for all $s$, $Z_{X_{n}}(s)=Z_{X,\rho_{n}}(s)$, see
[40, Page 51]. By computing logarithmic derivatives we have for all ${\rm
Re}(s)>1$,

$\frac{Z^{\prime}_{X_{n}}(s)}{Z_{X_{n}}(s)}=\sum_{\gamma\in\mathcal{P}_{X_{n}}}\sum_{q\geq
1}\frac{\ell(\gamma)e^{-sq\ell(\gamma)}}{1-e^{-q\ell(\gamma)}}=\sum_{\gamma\in\mathcal{P}_{X}}\sum_{q\geq
1}\frac{\ell(\gamma)\mathrm{tr}(\rho_{n}(\gamma^{q}))e^{-sq\ell(\gamma)}}{1-e^{-q\ell(\gamma)}}.$

Let $\phi\in C_{0}^{\infty}(\mathbb{R}^{+})$ be a compactly supported smooth
test function, and set

$\psi(s):=\int_{0}^{\infty}e^{sx}\phi(x)dx.$

One can check that $\psi(s)$ is actually analytic on $\mathbb{C}$ and by
Fourier inversion formula we have for all $A>1$,

$\frac{1}{2i\pi}\int_{A-i\infty}^{A+i\infty}\frac{Z^{\prime}_{X_{n}}(s)}{Z_{X_{n}}(s)}\psi(s)ds=\sum_{\gamma\in\mathcal{P}_{X_{n}}}\sum_{q\geq
1}\frac{\ell(\gamma)}{1-e^{-q\ell(\gamma)}}\phi(q\ell(\gamma))$
$=\sum_{\gamma\in\mathcal{P}_{X}}\sum_{q\geq
1}\frac{\ell(\gamma)\mathrm{tr}(\rho_{n}(\gamma^{q}))}{1-e^{-q\ell(\gamma)}}\phi(q\ell(\gamma)).$

See for example [18] $\S 3$ for more details on the derivation of this
formula. By carefully choosing the test function $\phi$ we deduce that for all
$\mathcal{L}\in\mathbb{R}^{+}$, we recover the identity

$\sum_{\gamma\in\mathcal{P}_{X_{n}}}\sum_{q\geq 1\atop
q\ell(\gamma)=\mathcal{L}}\ell(\gamma)=\sum_{\gamma\in\mathcal{P}_{X}}\sum_{q\geq
1\atop
q\ell(\gamma)=\mathcal{L}}\ell(\gamma)\mathrm{tr}(\rho_{n}(\gamma^{q})).$

Notice that this formula can be proved directly by group theoretic arguments,
see for example in [16], in the proof of theorem 7.1. From this identity we
deduce that for all $L>0$, we have

$\sum_{\gamma\in\mathcal{P}_{X_{n}}}\sum_{q\geq 1\atop q\ell(\gamma)\leq
L}\ell(\gamma)=\sum_{\gamma\in\mathcal{P}_{X}}\sum_{q\geq 1\atop
q\ell(\gamma)\leq L}\ell(\gamma)\mathrm{tr}(\rho_{n}(\gamma^{q})).$

In particular we have

$\ell_{0}(X_{n})N_{X_{n}}(L)\leq L\sum_{\gamma\in\mathcal{P}_{X}}\sum_{q\geq
1\atop q\ell(\gamma)\leq L}\mathrm{tr}(\rho_{n}(\gamma^{q})),$

where $\ell_{0}(X_{n})$ denotes the shortest closed geodesic length on
$X_{n}$. We point out that we have actually
$\mathrm{tr}(\rho_{n}(\gamma^{q}))=\mathrm{Fix}(\phi_{n}(\gamma^{q}))$, where
$\mathrm{Fix}(\sigma)$ denotes the number of fixed points of the permutation
$\sigma$ acting on $[n]$. From the combinatorial analysis of Magee-Puder [22,
21], we know that for all primitive $\gamma\in\Gamma$ and $q\geq 1$, we have

$\lim_{n\rightarrow\infty}\mathbb{E}(\mathrm{Fix}(\phi_{n}(\gamma^{q})))=d(q),$

where $d(q)$ stands for the number of divisors of $q$. Noticing that in the
random cover model, we have always $\ell_{0}(X_{n})\geq\ell_{0}(X)$, this his
is enough to conclude that for all $L$, we have

$\lim_{n\rightarrow\infty}\mathbb{E}(N_{X_{n}}(L))\leq C(\Gamma,L),$

where $C(\Gamma,L)>0$ is some (possibly large) constant. Applying Markov’s
inequality, we get that for all $\varepsilon>0$ and $L$ fixed,

$\lim_{n\rightarrow\infty}\mathbb{P}\left(N_{X_{n}}(L)\leq\mathrm{Vol}(X_{n})^{\varepsilon}\right)=1.$

As a conclusion, in the random cover model, $\mathscr{H}_{2}(C,L,\alpha)$ is
satisfied a.a.s. for all $L$ large and all $\alpha>0$.

###  4.2. Brooks–Makover model

In the paper [9, Theorem 2.2], they show that there exists a constant
$C_{1}>0$ such that as $n\rightarrow\infty$,

$\mathbb{P}(\lambda_{1}(S_{n}^{C})\geq C_{1})\rightarrow 1,$

In other words, property $\mathscr{H}_{1}(\eta)$ is satisfied a.a.s. for some
$\eta>0$.

We point out that contrary to the model of random covers, the systole of
$S_{n}^{C}$ can be arbitrarily small, but we actually know that there exists
$C_{2}>0$ such that as $n\rightarrow+\infty$,

$\mathbb{P}(\ell_{0}(S_{n}^{C})\geq C_{2})\rightarrow 1.$

Counting results for closed geodesics follow from the later work of Petri
[32]. More precisely, one can derive from [32] the following fact.

######  Proposition 4.1.

For all $L>0$ fixed, we can find an integer $N_{L}$ and a finite set of words

$\mathcal{W}_{L}\subset\\{l,r\\}^{N_{L}},$

such that with high probability as $n\rightarrow\infty$,

$N_{n}(L):=\\#\\{(\gamma,m)\in\mathcal{P}_{S_{n}^{C}}\times\mathbb{N}^{*}\ :\
m\ell(\gamma)\leq L\\}\leq\sum_{w\in\mathcal{W}_{L}}Z_{n,w},$

where $Z_{n,w}$ are integer-valued random variables. In addition each
$Z_{n,w}$ converges in the sense of moments (and hence in distribution) as
$n\rightarrow\infty$ to a Poisson variable with expectation $\lambda_{w}>0$.

By applying Markov’s inequality, one deduces readily that for all
$\varepsilon>0$ we have a.a.s.

$\sum_{w\in\mathcal{W}_{L}}Z_{n,w}\leq n^{\epsilon}\leq
C_{L}\mathrm{Vol}(S_{n}^{C})^{\varepsilon}.$

This is enough to conclude that for all $\varepsilon>0$, with high probability
as $n\rightarrow\infty$, we have

$N_{n}(L)\leq C\mathrm{Vol}(S_{n}^{C})^{\varepsilon},$

and therefore property $\mathscr{H}_{2}(C,\alpha,L)$ holds for any choice of
$\alpha>0$, just like in the previous model.

Let us now give some details on the proof of Proposition 4.1. The first step
is to reduce the problem to a counting bound for $S_{n}^{O}$. In [9, Section
3], they introduce the notion of “large cusps” condition for $S_{n}^{O}$. This
condition is satisfied a.a.s for $S_{n}^{O}$ as $n\rightarrow\infty$ see [9,
Theorem 2.1]. The main interest of this condition is [9, Theorem 3.2], see
also [32, Lemma 2.5], which allows to show that provided this “large cusps”
condition is satisfied, one can bound

$N_{n}(L)\leq\\#\\{(\gamma,m)\in\mathcal{P}_{S_{n}^{0}}\times\mathbb{N}^{*}\
:\ m\ell(\gamma)\leq 2L\\}=:N_{n}^{O}(2L).$

As explained in [9], $\S 4$, closed geodesics and their length in $S_{n}^{O}$
can be described via the combinatorial data of
$(\mathcal{G}_{n},\mathcal{O})$: any closed geodesic in $S_{n}^{O}$
corresponds to a word $w\in\\{l,r\\}^{N}$, for some $N>0$. To this word one
can associate a matrix $M_{w}$ in $SL_{2}(\mathbb{N})$ via the rule

$M_{w}=W_{1}\ldots W_{N},$

where $W_{j}=\mathcal{L}$ if $w_{j}=l$ and $W_{j}=\mathcal{R}$ if $w_{j}=r$,
where

$\mathcal{L}=\left(\begin{array}[]{cc}1&1\\\ 0&1\end{array}\right),\
\mathcal{R}=\left(\begin{array}[]{cc}1&0\\\ 1&1\end{array}\right).$

The length $\ell_{w}$ of the corresponding geodesic on $S_{n}^{O}$ is then
given by

$\mathrm{tr}(M_{w})=2\cosh(\ell_{w}/2).$

All we need to check is that fixing $L$ implies finiteness of the
corresponding set of words (by bounding their word length). This is done in
[33, Lemma 3.1]. One obtains therefore that $L$ being fixed, there exists a
finite subset $\mathcal{W}_{L}\subset\\{l,r\\}^{N_{L}}$ for some large
$N_{L}>0$, such that for all $n$, all closed geodesics with length $\leq 2L$
on $S_{n}^{O}$ are given by words $w\in W_{L}$. Proposition 4.1 now follows
directly from [32, Theorem B].

###  4.3. Weil–Petersson model

If $f$ is a measurable non-negative function on moduli space
$\mathscr{M}_{g}$, we will denote by $\int_{\mathscr{M}_{g}}f(X)dX$ the
corresponding integral with respect to Weil–Petersson volume. In the latter
$V_{g}$ will denote the Weil–Petersson volume of $\mathscr{M}_{g}$ so that the
expectation of $f$ is given by

$\mathbb{E}_{g}(f):=\frac{1}{V_{g}}\int_{\mathscr{M}_{g}}f(X)dX.$

In [25], the following fact was proved. There exists $\eta>0$ such that as
$g\rightarrow+\infty$, we have

$\mathbb{P}(\lambda_{1}(X)\geq\eta)\rightarrow 1.$

The constant $\eta$ given by Mirzakhani follows from Cheeger’s inequality and
an estimate a.a.s. of Cheeger’s isoperimetric constant. It was shown
independently in [43, 42] that one can actually take $\eta=3/16-\epsilon$, and
more recently $\eta=2/9-\epsilon$ by Anantharaman and Monk [1]. This shows
that $\mathscr{H}_{1}(\eta)$ holds with high probability as
$g\rightarrow+\infty$ for some universal $\eta>0$. In [25, Theorem 4.2] and
the remark after, Mirzakhani proved that there exists a universal
$\epsilon_{0}>0$ such that for all $0<\epsilon\leq\epsilon_{0}$, one has for
all $g$ large

$\mathbb{P}(\ell_{0}(X)\leq\epsilon)\leq C\epsilon^{2},$

where $C$ is uniform in $g$. In particular for all $\varepsilon>0$, we get for
all $g$ large

$\mathbb{P}\left(\ell_{0}(X)\geq\mathrm{Vol}(X)^{-\varepsilon}\right)\geq
1-O(\mathrm{Vol}(X)^{-2\varepsilon}).$

On the other hand, in the paper [26], Mirzakhani and Petri showed that for all
$L>0$ fixed, the random variable

$N_{g}^{0}(L):=\\#\\{\gamma\in\mathcal{P}_{X}\ :\ \ell(\gamma)\leq L\\}$

converges in distribution as $g\rightarrow+\infty$ to a Poisson variable
$Z_{\lambda_{L}}$ with parameter

$\lambda_{L}:=\int_{0}^{L}\frac{e^{t}+e^{-t}-2}{2t}dt.$

Moreover, we have also convergence of all moments with $p\in\mathbb{N}$

$\lim_{g\rightarrow\infty}\mathbb{E}(\left(N_{g}^{0}(L)\right)^{p})=\mathbb{E}(Z_{\lambda_{L}}^{p}).$

An application of Markov’s inequality then shows that for all $\varepsilon>0$,
for all $g$ large enough

$\mathbb{P}(N_{g}^{0}(L)\leq\mathrm{Vol}(X)^{\varepsilon})\geq
1-C_{L}\mathrm{Vol}(X)^{-\varepsilon}.$

To control the counting function

$N_{g}(L):=\\#\\{(\gamma,m)\in\mathcal{P}_{X}\times\mathbb{N}^{*}\ :\
m\ell(\gamma)\leq L\\},$

we write

$N_{g}(L)\leq\sum_{m=1}^{[L/\ell_{0}(X)]+1}N_{g}^{0}(L/m)\leq
N_{g}^{0}(L)\left(1+\frac{L}{\ell_{0}(X)}\right).$

For all $\varepsilon>0$, we have with high probability as
$g\rightarrow+\infty$

$N_{g}(L)\leq\mathrm{Vol}(X)^{\varepsilon}+L\mathrm{Vol}(X)^{2\varepsilon}=O_{L}(\mathrm{Vol}(X)^{2\varepsilon}),$

and thus for all $L$ large, there exists $C_{L}>0$ such that
$\mathscr{H}_{2}(C_{L},L,\alpha)$ holds with high probability for all
$\alpha>0$ on $\mathscr{M}_{g}$ as $g\rightarrow+\infty$.

##  5\. Convergence results for moments of $\log\det(\Delta_{X})$.

In this last section, we give the proof of the following fact.

######  Theorem 5.1.

In the Weil–Petersson model, for all $0<\beta<1$ we have

$\lim_{g\rightarrow\infty}\frac{1}{V_{g}(4\pi(g-1))^{\beta}}\int_{\mathscr{M}_{g}}\left|\log\det(\Delta_{X})\right|^{\beta}dX=E^{\beta}.$

In the random cover model, Let $\chi_{0}=\mathbf{1}_{\\{X_{n}\
\mathrm{connected}\\}}$. Then as $n\rightarrow+\infty$, for all $\beta>0$, we
have

$\lim_{n\rightarrow\infty}\mathbb{E}\left(\frac{|\log\det(\Delta_{X_{n}})|^{\beta}}{\mathrm{Vol}(X_{n})^{\beta}}\chi_{0}\right)=E^{\beta}.$

###### Proof for the Weil–Petterson model.

The proof for the Weil–Petersson model is a rather direct consequence of
Theorem 1.1 and some estimates of Mirzakhani [24]. We first need an a priori
estimate for $|\log\det(\Delta_{X})|$ which follows from similar ideas as in
Theorem 3.1, without the probabilistic input. We use Vinogradov’s notation
$\ll$ where the implied constant is universal.

By using Buser’s counting bound [10], as in the proof of Lemma 3.2, we have
the following universal bound for the number of closed geodesics of a surface
with genus $g$:

$N_{X}(L)\leq(g-1)e^{L+6}+\frac{L}{\ell_{0}(X)}(3g-3)$
$\ll\mathrm{Vol}(X)e^{L}\left(1+\frac{1}{\ell_{0}(X)}\right),$ (7)

for some universal constant $A_{1}>0$. We now prove an a priori upper bound
for $|\log\det(\Delta_{X})|$. Following ideas of Wolpert [41], it is
convenient to write for all ${\rm Re}(s)$ large,

$\zeta_{X}(s)=\sum_{0<\lambda_{j}<1/4}\lambda_{j}^{-s}+\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}\left(\mathrm{tr}(e^{-t\Delta_{X}})-\sum_{0\leq\lambda_{j}<1/4}e^{-t\lambda_{j}}\right)dt,$

which can be rewritten as

$\zeta_{X}(s)=\sum_{0<\lambda_{j}<1/4}\lambda_{j}^{-s}-\sum_{0\leq\lambda_{j}<1/4}H(s,\lambda_{j})+\zeta^{(1)}_{X}(s)+$
$\frac{1}{\Gamma(s)}\int_{1}^{\infty}t^{s-1}\left(S_{X}(t)-\sum_{0\leq\lambda_{j}<1/4}e^{-t\lambda_{j}}\right)dt+\frac{1}{\Gamma(s)}\int_{0}^{1}t^{s-1}S_{X}(t)dt,$

where we have set for ${\rm Re}(s)$ large,

$H(s,\lambda):=\frac{1}{\Gamma(s)}\int_{0}^{1}t^{s-1}e^{-\lambda t}dt.$

By integration by parts and elementary properties of the Euler gamma function,
we then observe that for all $\lambda\in[0,1/4]$, $s\mapsto H(s,\lambda)$ has
an analytic extension to $s=0$. Morevover, if we set

$C(\lambda):=\left.\frac{d}{ds}\right|_{s=0}H(\lambda,s),$

then there exists a universal constant $A_{1}>0$ such that for all
$\lambda\in[0,1/4]$,

$|C(\lambda)|\leq A_{1}.$

This formula leads to the identity

$\log\det(\Delta_{X})=\sum_{0<\lambda_{j}<1/4}\log(\lambda_{j})+\sum_{0<\lambda_{j}<1/4}C(\lambda_{j})-\left.\frac{d}{ds}\right|_{s=0}\zeta^{(1)}_{X}(s)$
$-\int_{0}^{1}\frac{S_{X}(t)}{t}dt-\int_{1}^{\infty}\frac{(S_{X}(t)-\sum_{0\leq\lambda_{j}<1/4}e^{-\lambda_{j}t})}{t}dt.$

By mimicking the proof of Lemma 3.3 and using the above counting bound, we
deduce that for all $t\geq 1$,

$\left|S_{X}(t)-\sum_{0\leq\lambda_{j}<1/4}e^{-t\lambda_{j}}\right|\ll\mathrm{Vol}(X)\left(1+\frac{1}{\ell_{0}(X)}\right)e^{-t/4}.$

By Fubini and the estimate on $u\mapsto G(u)$ we have also

$\int_{0}^{1}\frac{S_{X}(t)}{t}dt\ll\sum_{m,\gamma}\frac{e^{-(m\ell(\gamma))^{2}/8}}{m\ell(\gamma)}\ll\sum_{m\ell(\gamma)\leq
1}\frac{1}{m\ell(\gamma)}+\sum_{m\ell(\gamma)>1}e^{-(m\ell(\gamma))^{2}/8}$
$\ll\sum_{m\ell(\gamma)\leq
1}\frac{1}{m\ell(\gamma)}+\mathrm{Vol}(X)\left(1+\frac{1}{\ell_{0}(X)}\right).$

By noticing that we have

$\sum_{m\ell(\gamma)\leq
1}\frac{1}{m\ell(\gamma)}\ll\frac{\log^{+}\ell_{0}^{-1}(X)}{\ell_{0}(X)}N_{X}^{0}(1),$

where $\log^{+}(x)=\max\\{0,\log(x)\\}$ and $N_{X}^{0}(L)$ is the counting
function for primitive closed geodesics, we have obtained the estimate:

$|\log\det(\Delta_{X})|\ll\mathrm{Vol}(X)\left(1+|\log(\lambda_{*}(X))|+\frac{1}{\ell_{0}(X)}+\right)+\frac{\log^{+}\ell_{0}^{-1}(X)}{\ell_{0}(X)}N_{X}^{0}(1),$

where we have set $\lambda_{*}(X)=\min\\{\lambda_{1}(X),1/4\\}$, and we have
used the rough bound of Buser [10]:

$\\#\\{\lambda_{j}<1/4\\}\leq 4g-3=1+\frac{\mathrm{Vol}(X)}{\pi}.$

Notice that the optimal bound of Otal–Rosas [31]

$\\#\\{\lambda_{j}<1/4\\}\leq 2g-2,$

won’t make any difference here.

This estimate of $|\log\det(\Delta_{X})|$ is consistent with the fact that
$\det(\Delta_{X})$ has exponential growth when $X$ approaches certain boundary
points of the (compactified) moduli space, a fact that was rigourously
established by Wolpert in [41]. In particular the so-called “bosonic Polyakov
integral” involving the determinant over the moduli space is indeed infinite,
see [41].

Using the inequality for all $\beta>0$ and all $a_{j}\geq 0$,

$\left(\sum_{j=1}^{4}a_{j}\right)^{\beta}\leq
4^{\beta}(\max_{j}a_{j})^{\beta}\leq
4^{\beta}\left(\sum_{j=1}^{4}a_{j}^{\beta}\right),$

we end up with the estimate

$\frac{|\log\det(\Delta_{X})|^{\beta}}{\mathrm{Vol}(X)^{\beta}}\ll\left(1+|\log(\lambda_{*}(X))|^{\beta}+\frac{1}{\ell_{0}(X)^{\beta}}+\mathrm{Vol}(X)^{-\beta}\left(\frac{\log^{+}\ell_{0}^{-1}(X)}{\ell_{0}(X)}N_{X}^{0}(1)\right)^{\beta}\right).$

From Mirzakhani [24, Corollary 4.3], we know that

$\int_{\mathscr{M}_{g}}\frac{1}{\ell_{0}(X)}dX\leq CV_{g},$

where $C>0$ is independent of $g$, from which we can deduce easily that for
all $0<\alpha<1$, we have

$\int_{\mathscr{M}_{g}}\left(\frac{\log^{+}\ell_{0}^{-1}(X)}{\ell_{0}(X)}\right)^{\alpha}dX\leq
CV_{g},$

for some $C>0$ uniform in $g$. Indeed, for all $0<\alpha<1$, there exists a
universal consant $r_{0}>0$ such that $0<x\leq r_{0}$ implies

$\left(\frac{|\log x|}{x}\right)^{\alpha}\leq\frac{1}{x}.$

By integrating over $\\{X\in\mathscr{M}_{g}\ :\ \ell_{0}(X)\leq r_{0}\\}$ and
using Mirzakhani’s bound we get the desired bound while on the complementary
set

$X\mapsto\left(\frac{\log^{+}\ell_{0}^{-1}(X)}{\ell_{0}(X)}\right)^{\alpha}$

is uniformly bounded.

On the other hand, Cheeger’s inequality says that

$\lambda_{1}(X)\geq\frac{h_{X}^{2}}{4},$

where $h_{X}$ is the so-called Cheeger constant of $X$, which is defined by an
isoperimetric quantity, see for example in [10], chapter 8. Again By
Mirzakhani [24, Theorem 4.8], we know that for all $0\leq\alpha<2$, we have

$\int_{\mathscr{M}_{g}}\frac{1}{(h_{X})^{\alpha}}dX\leq CV_{g},$

where $C>0$ is again uniform with respect to $g$. In particular we deduce that
for all $\beta>0$,

$\int_{\mathscr{M}_{g}}|\log(\lambda_{*}(X))|^{\beta}dX\leq CV_{g}.$

Assuming that $\beta<1$, we choose $p>1$ such that $\beta<p\beta<1$ and let
$q$ be such that $1/p+1/q=1$. By Hölder’s inequality we get

$\mathbb{E}_{g}\left(\left(\frac{\log^{+}\ell_{0}^{-1}(X)}{\ell_{0}(X)}N_{X}^{0}(1)\right)^{\beta}\right)\leq\left[\mathbb{E}_{g}\left(\left(\frac{\log^{+}\ell_{0}^{-1}(X)}{\ell_{0}(X)}\right)^{\beta
p}\right)\right]^{1/p}\left[\mathbb{E}_{g}\left(N_{X}^{0}(1)^{\beta
q}\right)\right]^{1/q},$

which by the convergence of moments in Mirzakhani-Petri [26] and the above
remarks is uniformly bounded as $g\rightarrow+\infty$. We therefore have shown
that for all $\beta<1$, there exists $C>0$ independent of $g$ such that

$\int_{\mathscr{M}_{g}}\frac{|\log\det(\Delta_{X})|^{\beta}}{\mathrm{Vol}(X)^{\beta}}dX\leq
CV_{g}.$ (8)

We now fix $\epsilon>0$, and $0<\beta<1$. By Theorem 1.1, there exists a
subset $\mathcal{A}_{g}(\epsilon)\subset\mathscr{M}_{g}$, with
$\mathbb{P}(\mathcal{A}_{g}(\epsilon))\rightarrow 1$ as $g\rightarrow+\infty$,
such that for all $X\in\mathcal{A}_{g}(\epsilon)$,

$(E-\epsilon)^{\beta}\leq\frac{|\log\det(\Delta_{X})|^{\beta}}{\mathrm{Vol}(X)^{\beta}}\leq(E+\epsilon)^{\beta}.$

Therefore we have

$\frac{1}{V_{g}}\int_{\mathscr{M}_{g}}\frac{|\log\det(\Delta_{X})|^{\beta}}{\mathrm{Vol}(X)^{\beta}}dX\leq(E+\epsilon)^{\beta}+\frac{1}{V_{g}}\int_{\mathcal{A}_{g}(\epsilon)^{c}}\frac{|\log\det(\Delta_{X})|^{\beta}}{\mathrm{Vol}(X)^{\beta}}dX,$

while

$(E-\epsilon)^{\beta}\mathbb{P}(\mathcal{A}_{g}(\epsilon))\leq\frac{1}{V_{g}}\int_{\mathscr{M}_{g}}\frac{|\log\det(\Delta_{X})|^{\beta}}{\mathrm{Vol}(X)^{\beta}}dX.$

We now apply Hölder’s inequality and estimate (8). Since $0<\beta<1$, let
$q>1$ be chosen such that $0<q\beta<1$ and let $0<p<\infty$ be such that
$1/p+1/q=1$, we have

$\frac{1}{V_{g}}\int_{\mathcal{A}_{g}(\epsilon)^{c}}\frac{|\log\det(\Delta_{X})|^{\beta}}{\mathrm{Vol}(X)^{\beta}}dX\leq\left(\mathbb{P}(\mathcal{A}_{g}(\epsilon)^{c})\right)^{1/p}\left(\frac{1}{V_{g}}\int_{\mathscr{M}_{g}}\frac{|\log\det(\Delta_{X})|^{q\beta}}{\mathrm{Vol}(X)^{q\beta}}dX\right)^{1/q}$
$\leq C\left(\mathbb{P}(\mathcal{A}_{g}(\epsilon)^{c}\right)^{1/p}.$

Since $\lim_{g\rightarrow+\infty}\mathbb{P}(\mathcal{A}_{g}(\epsilon)^{c})=0$,
we definitely have for all $g$ large enough

$(E-\epsilon)^{\beta}-\epsilon\leq\frac{1}{V_{g}}\int_{\mathscr{M}_{g}}\frac{|\log\det(\Delta_{X})|^{\beta}}{\mathrm{Vol}(X)^{\beta}}dX\leq(E+\epsilon)^{\beta}+\epsilon,$

and the proof is done for the Weil–Petterson case. ∎

We would like to mention that in this smooth Weil–Petterson model, it is very
likely that by using finer estimates one can improve the result to the
following statement: for any $\beta\in(0,2)$ we have

$\lim_{g\rightarrow\infty}\frac{1}{V_{g}(4\pi(g-1))^{\beta}}\int_{\mathscr{M}_{g}}\left|\log\det(\Delta_{X})\right|^{\beta}dX=E^{\beta},$

while for all $\beta\geq 2$,

$\int_{\mathscr{M}_{g}}|\log\det(\Delta_{X})|^{\beta}dX=+\infty.$

This fact was pointed out to the author by Yunhui Wu and Yuxin He in a private
communication and will be published in a separate paper.

###### Proof for the random covers model.

We recall that we have $\chi_{0}=\mathbf{1}_{\\{X_{n}\
\mathrm{connected}\\}}$. In this model, the systole is bounded uniformly from
below, so the a priori bound for $\log\det(\Delta_{X_{n}})$ whenever $X_{n}$
is connected, is actually

$\frac{|\log\det(\Delta_{X_{n}})|^{\beta}}{\mathrm{Vol}(X_{n})^{\beta}}\leq
A\left(1+|\log(\lambda_{*}(X))|^{\beta}\right),$

for some constant $A>0$ independent of $n$. Using the same arguments as above
based on Hölder’s inequality, the result follows directly from the next fact.

######  Proposition 5.2.

Assuming that $X_{n}$ is connected, then we have for all $n$ large,

$\lambda_{1}(X_{n})\geq\frac{C_{\Gamma}}{n^{3/2}},$

where $C_{\Gamma}$ depends only on the base surface
$X=\Gamma\backslash\mathbb{H}^{2}$. Consequently for all exponent $\beta$ with
$0<\beta$,

$\mathbb{E}\left(|\log(\lambda_{*}(X))|^{\beta}\chi_{0}\right)\leq C,$

where $C>0$ is uniform with respect to $n$.

We first need to prove a deterministic lower bound on $\lambda_{1}(X_{n})$,
provided that $X_{n}$ is connected. We know that the spectrum of
$\Delta_{X_{n}}$ coincides, with multiplicity, with the spectrum of
$\Delta_{\rho_{n}}$, which is the Laplacian on the base surface twisted by the
unitary representation $\rho_{n}$ of $\Gamma$ defined previously. See for
example [40, Page 51]. If $X_{n}$ is connected, then

$\lambda_{1}(X_{n})=\min\\{\lambda_{1}(X),\lambda_{0}(\Delta_{\rho_{n}^{0}})\\},$

where $\rho_{n}^{0}$ is the representation given by

$\rho_{n}^{0}(\gamma)U:=U_{\phi_{n}^{-1}(\gamma)},$

where $U\in V_{n}^{0}:=\\{U\in\mathbb{C}^{n}\ :\ \sum_{j=1}^{n}U(j)=0\\}$ and
$U_{\phi_{n}^{-1}(\gamma)}(j)=U(\phi_{n}^{-1}(\gamma)(j))$. Notice that
$\rho_{n}^{0}$ is just the orthogonal complement to the trivial representation
in $\rho_{n}$. Let us fix a system of generators of $\Gamma$, denoted by $S$.
A result of Sunada [39] then says that there exists $C_{S}$ depending only on
$X$ and $S$ such that

$\lambda_{0}(\Delta_{\rho_{n}^{0}})\geq C_{S}\inf_{U\in
V_{n}^{0}\atop\|U\|=1}\max_{\gamma\in S}\|\rho_{n}^{0}(\gamma)U-U\|.$

Let us take $U\in V_{n}^{0}$ such that $\|U\|=1$. We therefore have

$1=\|U\|^{2}\leq n\max_{j}({\rm Re}(U(j))^{2}+{\rm Im}(U(j))^{2}),$

and we can assume without loss of generality that we have ${\rm
Re}(U(j_{0}))\geq\frac{1}{\sqrt{2n}}$ for some $j_{0}\in[n]$. Because we have
in addition

$\sum_{j}{\rm Re}(U(j))=0,$

there exists also $j_{1}\in[n]$ such that ${\rm Re}(U(j_{1}))\leq 0$. If
$X_{n}$ is connected, then $\Gamma$ acts transitively on $[n]$ via $\phi_{n}$
and there exists $\gamma_{0}\in\Gamma$ such that
$\phi_{n}(\gamma_{0})^{-1}(j_{0})=j_{1}$. To bound the word length of
$\gamma_{0}$, consider the graph with set of vertices $[n]$ and define edges
by connecting $i$ to $j$ if there exists $g\in S$ such that
$\phi_{n}(g)(i)=j$. By transitivity of the action, this graph is connected and
thus has diameter less than $n-1$. Therefore we can choose $\gamma_{0}$ with
word length (with respect to $S$) less than $n-1$. We now have

$\frac{1}{\sqrt{2n}}\leq|{\rm Re}(U(j_{0}))-{\rm
Re}(U(j_{1}))|\leq\|\rho_{n}^{0}(\gamma_{0})U-U\|.$

Writing

$\gamma_{0}=g_{1}g_{2}\ldots g_{m},$

with $g_{j}\in S$ and $m\leq n-1$, we have therefore

$\|\rho_{n}^{0}(\gamma_{0})U-U\|\leq\sum_{j=1}^{m}\|\rho_{n}^{0}(g_{j})U-U\|\leq(n-1)\max_{g\in
S}\|\rho_{n}^{0}(g)U-U\|,$

which yields

$\max_{g\in S}\|\rho_{n}^{0}(g)U-U\|\geq\frac{1}{(n-1)\sqrt{2n}}.$

The first claim of the proposition is proved. Alternatively, one can use
directly a result of Brooks [7] which relates $\lambda_{1}(X_{n})$ to the
Cheeger constants of Schreier graphs of the covers (with a choice of
generators of $\Gamma$) to obtain a similar lower bound
$\lambda_{1}(X_{n})\geq C_{\Gamma}n^{-2}$ which is slightly worse but good
enough for our purpose.

From the proof of the uniform spectral gap in [21] one can directly derive
that for all $1/4>r>0$, for all $\epsilon>0$, as $n\rightarrow\infty$,

$\mathbb{P}(\lambda_{1}(X_{n})\leq
r)\leq\frac{C_{\epsilon}}{n^{4\sqrt{(1/4-r)}-1-\epsilon}}.$

From that we deduce that for all $\alpha<1$ we can find $r_{\alpha}>0$ such
that

$\mathbb{P}(\lambda_{1}(X_{n})\leq
r_{\alpha})\leq\frac{C_{\alpha}}{n^{\alpha}}.$

We now fix any $\beta>0$ and fix some $0<\alpha<1$. We can use the fact that
we have (by the lower bound on $\lambda_{*}(X)$)

$|\log(\lambda_{*}(X))|^{\beta}=O((\log(n))^{\beta}),$

where the implied constant is uniform with respect to $n$. We therefore get as
$n\rightarrow\infty$,

$\mathbb{E}(|\log(\lambda_{*}(X))|^{\beta}\chi_{0})\leq|\log(r_{\alpha})|^{\beta}(1+O(n^{-\alpha}))+O\left(n^{-\alpha}|\log(n)|^{\beta}\right)=O(1)$

and the proof is done. ∎

We conclude with some comments.

  * •

It would be interesting to know if a similar type of result can be proved for
the Brooks–Makover model, in particular can one show that

$\mathbb{E}\left(\frac{1}{\ell_{0}(S_{n}^{C})}\right)$

is finite and uniformly bounded with respect to $n$ ? This will require an
effective version of the compactification procedure, see the paper of Mangoubi
[23].

  * •

It is likely that our results can be extended to finite area surfaces, see
Efrat [15] for the definition and properties of determinants in this context,
or even geometrically finite surfaces where a notion of determinant also
holds, see [5].

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