Random flat bundles and equidistribution

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arXiv:2210.09547v2 [math.NT] 3 Dec 2023

RANDOM FLAT BUNDLES AND EQUIDISTRIBUTION

MASOUD ZARGAR

Abstract. Each signature λ(n) = (λ1(n),...,λn(n)), where λ1(n)≥ · · · ≥ λn(n) are integers, gives an irreducible

representation πλ(n):U(n)→GL(Vλ(n)) of the unitary group U(n). Suppose Xis a ﬁnite-area cusped hyperbolic

surface, χis a random surface representation in Hom(π1(X), U(n)) equipped with a Haar unitary probability

measure, and (λ(n))∞

n=1 is a sequence of signatures. Let |λ(n)|:= Pi|λi(n)|. We show that there is an absolute

constant c > 0 such that if 0 6=|λ(n)| ≤ clog n

log log nfor suﬃciently large n, then the Laplacians ∆χ,λ(n)acting on

sections of the ﬂat unitary bundles associated to the surface representations

π1(X)χ

−→ U(n)

πλ(n)

−−−−→ GL(Vλ(n))

have the property that for every ε > 0

Pχ: inf Spec(∆χ,λ(n))≥1

4−εn→∞

−−−−→ 1,

where Spec(∆χ,λ(n)) is the spectrum of ∆χ,λ(n). A special case of this is that ﬂat unitary bundles associated to

χ:π1(X)→U(n) asymptotically almost surely as n→ ∞ have least eigenvalue at least 1

4−ε, irrespective of the

spectral gap of Xitself. This is proved using the Hide–Magee method [HM21]. The general spectral theorem above

leads to a probabilistic equidistribution theorem for the images of closed geodesics under surface representations χ.

Using the spectral theorem above and proving a probabilistic prime geodesic theorem, we also obtain a probabilistic

equidistribution theorem for the images under χof geodesics of lengths dependent on the rank n.

1. Introduction 1

2. Preliminaries 5

3. Minimal eigenvalues of ﬂat unitary bundles 9

4. Random matrix theory estimates 14

5. Counting eigenvalues 16

6. Probabilistic prime geodesic theorem 18

7. Applications of the spectral theorem 25

8. Comments on the balanced case 26

References 29

1. Introduction

A topic of great interest in mathematics, physics, and computer science is the study of spectral gaps of graphs and

Riemannian manifolds. Those ﬁnite simple d-regular graphs whose eigenvalues diﬀerent from ±dare bounded in

absolute value from above by 2√d−1 are known as Ramanujan graphs. Inﬁnite families of d-regular Ramanujan

graphs are known to exist. For prime powers d−1, they have been explicitly constructed by Margulis [Mar88],

Lubotzky–Phillips–Sarnak [LPS88], and Morgenstern [Mor94] using number-theoretic methods. For other values of

d, using probabilistic methods combined with interlacing polynomials, Marcus–Spielman–Srivastava [MSS18] have

proved the existence of inﬁnite families of d-regular Ramanujan graphs.

2√d−1 is the spectral radius of the universal covering tree of a d-regular graph. For hyperbolic surfaces, their

universal cover Hplays an analogous role, with its Laplacian having least eigenvalue 1

4as the analogue of 2√d−1

for graphs. As of the writing of this paper, it is not known if inﬁnite families of closed connected hyperbolic surfaces

with enlarging genus exist each of whose Laplace-Beltrami operators have least positive eigenvalue, i.e. spectral gap,

at least 1

4. However, a recent result of Hide–Magee [HM21] shows that inﬁnite families of closed connected hyperbolic

surfaces with enlarging genera exist with spectral gaps converging to 1

4. Their proof is probabilistic and, in contrast to

graphs, no constructive proof is known as of the writing of this paper—even for this near-optimal spectral gap result.

The theorem proved by Hide–Magee [HM21] should be compared to a conjecture and theorem of Friedman [Fri03]

stating that random covers of any ﬁxed graph are almost Ramanujan with high probability. Recently, a new simpler

proof of Friedman’s conjecture was given Bordenave–Collins [BC19] using new methods. On the other hand, one of

the central conjectures in number theory, Selberg’s conjecture, states the following.

2 MASOUD ZARGAR

Conjecture 1.1 (Selberg’s conjecture).Arithmetic hyperbolic surfaces have spectral gaps λ1at least 1

It is known that Conjecture 1.1 implies that there should be an inﬁnite sequence of connected closed hyperbolic

surfaces with increasing genera each of which has a spectral gap of at least 1

4. However, Selberg’s conjecture 1.1

is beyond the reach of current methods. The best known spectral gap for arithmetic hyperbolic surfaces is due to

Kim–Sarnak [Kim03]:

λ1≥1

4−7

64 2

Spectral gaps are closely related to equidistribution results, for example by the prime geodesic theorem. A goal of this

paper is the proof of a probabilistic equidistribution theorem for images of geodesics on cusped hyperbolic surfaces

under unitary surface representations. This is intimately connected to the smallest eigenvalue of the Laplacian acting

on sections of families of ﬂat unitary bundles on such surfaces.

Our setup is as follows. Suppose Xis a non-compact, complete, connected, ﬁnite area hyperbolic surface.

Henceforth, we simply refer to such surfaces as cusped hyperbolic surfaces.

We consider surface representations χ:π1(X)→U(n) into the unitary group. Since Xis a non-compact surface,

π1(X) is a free group, and so the space Hom(π1(X), U (n)) of surface representations may be identiﬁed with a

product of copies of U(n), one copy for each generator of π1(X). Therefore, we naturally endow Hom(π1(X), U(n))

with a Haar unitary probability measure. Every sequence of signatures λ(n) = (λ1(n),...,λn(n)), nvarying, where

λ1(n)≥...≥λn(n) are integers, gives an inﬁnite family of irreducible unitary representations

πλ(n):U(n)→GL(Vλ(n)).

For each sequence of signatures λ(n), we have an inﬁnite family of normalized characters

χλ(n)(−) := tr(πλ(n)(−))

dim Vλ(n)

The surface representation

π1(X)χ

−→ U(n)πλ(n)

−−−−→ GL(Vλ(n))

gives rise to a ﬂat unitary vector bundle Eχ,λ(n)over Xof rank dim Vλ(n). The Laplacian ∆χ,λ(n)acts on sections

of Eχ,λ(n), whose smallest eigenvalue we want to study. We denote by Spec(∆χ,λ(n)) the spectrum of ∆χ,λ(n).

In this paper, we abuse language and speak of the eigenvalues of surface representations to mean the eigenvalues

of the Laplacian acting on sections of the associated ﬂat bundles. Our ﬁrst theorem is the following analogue

of [HM21] concerning minimal eigenvalues for ﬂat unitary vector bundles whose proof follows ideas of [HM21] with

the appropriate generalizations and modiﬁcations to the case of irreducible representations of the unitary groups. It

serves as a lemma in the proof of the equidistribution Theorem 1.7 below.

Theorem 1.2. There is an absolute constant c > 0such that if Xis any cusped hyperbolic surface and λ(n)is a

sequence of signatures with 06=|λ(n)| ≤ clog n

log log nfor suﬃciently large n, for every ε > 0we have

Pχ: inf Spec(∆χ,λ(n))≥1

4−εn→∞

−−−−→ 1.

In particular, if our irreducible representations are the standard representations U(n)id

−→ U(n) corresponding to the

signatures

λ(n) = (1,0,...,0

|{z }

n−1 times

Theorem 1.2 implies that asymptotically almost surely as n→ ∞, ﬂat unitary rank nbundles over Xhave smallest

eigenvalue at least 1

4−ε, irrespective of the spectral gap of Xitself. The following corollary to Theorem 1.2 follows

from the prime geodesic theorem for ﬂat unitary bundles.

Corollary 1.3. Suppose Xis a cusped hyperbolic surface, and let λ1(n),...,λk(n)be ksequences of signatures. Let

fn:= Pk

i=1 aiχλi(n),ai∈Cﬁxed. Then if for each iwe have |λi(n)| ≤ clog n

log log nfor nlarge enough (where c > 0is

the absolute constant in Theorem 1.2), then asymptotically almost surely as n→ ∞,χ:π1(X)→U(n)satisﬁes

πX(T)X

ℓ(γ)≤T

fn(χ(γ)) = ZU(n)

fn(g)dµ(g) + O(e−T /4)

as T→ ∞. Here, πX(T)is the number of closed oriented hyperbolic geodesics γon Xof length ℓ(γ)≤T, and dµ is

the probability Haar measure on U(n).

In turn, we also obtain the following corollary by combining Corollary 1.3 with the main result of Diaconis–Shahshahani [DS94].

RANDOM FLAT BUNDLES AND EQUIDISTRIBUTION 3

Corollary 1.4. Suppose r≥1and a= (a1,...,ar)and b= (b1,...,br)are r-tuples of non-negative integers. Then

for Xany cusped hyperbolic surface, asymptotically almost surely as n→ ∞,χ:π1(X)→U(n)satisﬁes

πX(T)X

ℓ(γ)≤T

j=1

tr(χ(γj))ajtr(χ(γj))bj=δa,bZCr|z1|2a1...|zr|2ar

j=1

je−π|zj|2/j dxjdyj+O(e−T /4)

as T→ ∞, where δa,bis the Kronecker delta.

This says essentially that the (tr(χ(γ)),tr(χ(γ2)),...,tr(χ(γr))) distribute according to the law of rindependent com-

plex Gaussians with a good error term. The main term on the right hand side is the expectation of Qr

j=1 tr(gj)ajtr(gj)bj

over U(n). Explicitly, we have

ZCr|z1|2a1...|zr|2ar

j=1

je−π|zj|2/j dxjdyj=

j=1

jajaj!.

A more diﬃcult problem is the study of the rate at which equidistribution happens. This could be made precise by

bounding the lengths of the geodesic in terms of n. We could ask the following question.

Question 1.5.Given signature λ6= 0, is there a constant C(X, λ)>0 such that

P

χ:π1(X)→U(n) : 

πX(T(n)) X

ℓ(γ)<T (n)

χλ(χ(γ))≤C(X, λ)e−c(n)

n→∞

−−−−→ 1

for some functions T(n), c(n)>0 going to inﬁnity as n→ ∞? What are the optimal such functions?

Note that given a signature λof length m, we may extend it by zeros to obtain signatures of lengths nfor every

n≥m. This is how a ﬁxed signature λgives rise to an inﬁnite family of irreducible unitary representations of U(n)

for large enough n.

Deﬁnition 1.6. A signature λis unbalanced if the sum of its entries is nonzero. Otherwise, λis balanced.

In relation to Question 1.5, we prove the following theorem whose proof occupies the bulk of this paper.

Theorem 1.7. Suppose Xis a cusped hyperbolic surface, and suppose λ(n)is sequence of unbalanced signatures such

that |λ(n)| ≤ clog n

log log nfor large enough n, where c > 0is the universal constant in Theorem 1.2. Suppose T(n)is a

function of nsuch that T(n)→ ∞ as n→ ∞. For every ε > 0,

(1) P

χ:

πX(log T(n)) X

ℓ(γ)≤log T(n)

χλ(n)(χ(γ))≤n2dim Vλ(n)T(n)−1

4+ε

n→∞

−−−−→ 1.

A reason we impose the condition that the signatures λ(n) are unbalanced is the following. For unbalanced signatures

λ(n), we show that almost surely, πλ(n)◦χgives a ﬂat bundle Eχ,λ(n)that is non-singular, that is, whose spectrum has

no continuous part. For balanced signatures λ(n) whose sum of entries is zero, the irreducible representations πλ(n)

are such that for every A∈U(n), πλ(n)(A) has 1 as an eigenvalue. Since Xis cusped hyperbolic, this means that

for such representations, we always have a continuous part in the spectrum of Eχ,λ(n). The existence of a continuous

spectrum requires a study of the main term of the hyperbolic scattering determinants of such ﬂat bundles. The study

of the hyperbolic scattering determinant in this setting is closely related to currently intractable problems in random

matrix theory. Furthermore, in order to obtain Theorem 1.7 for balanced λ(n), we also need a bound in terms of

Xand λ(n) on the real parts of the resonances of Eχ,λ(n), at least asymptotically almost surely as n→ ∞. This

is also related to understanding the main term of the hyperbolic scattering determinants. We brieﬂy discuss these

complications in Section 8.

1.1. Relation to other works. In Hide–Magee [HM21], the authors show that a cusped hyperbolic surface is such

that, for any ε > 0, if one takes n-sheeted covers of it, as n→ ∞, asymptotically almost surely they do not introduce

new eigenvalues less than 1

4−ε. In particular, if one starts with a cusped hyperbolic surface with spectral gap at

least 1

4, then one obtains that n-sheeted covers asymptotically almost surely have spectral gaps at least 1

4−ε. A

compactiﬁcation argument allows them to compactify their sequence of hyperbolic surfaces to show the existence of

an inﬁnite family of connected closed hyperbolic surfaces {Xn}nwith increasing genera and spectral gaps λ1(Xn)

satisfying

lim sup

n→∞

λ1(Xn) = 1

4 MASOUD ZARGAR

The existence of such a sequence of hyperbolic surfaces was an important open question in the spectral geometry of

hyperbolic surface. Note that by a theorem of Huber [Hub74], any sequence {Xn}nof closed connected hyperbolic

surfaces with increasing genera satisﬁes

lim sup

n→∞

λ1(Xn)≤1

Note that n-sheeted covers of a surface Xare parametrized by surface representations

π1(X)→Sn

into the symmetric group Snon nelements. One could instead consider the larger family of rank nﬂat bundles over

X, essentially parametrized by surface representations

χ:π1(X)→U(n)

into the unitary group U(n) of rank n. If one wants to investigate how the images of (homotopy classes of oriented)

closed geodesics γon Xunder surface representations χdistribute, we are led to considering class functions f:

U(n)→Cevaluated at χ(γ). Since class functions are generated by irreducible characters, we need to understand

the minimal eigenvalue of the Laplacian acting on sections of the ﬂat bundle associated to

π1(X)χ

−→ U(n)πλ(n)

−−−−→ GL(Vλ(n)),

where πλ(n)are irreducible representations. Our ﬁrst theorem, Theorem 1.2, addresses this minimal eigenvalue prob-

lem. Its proof follows that of the main theorem of [HM21] with the necessary modiﬁcations and generalizations to

our unitary setting. The proofs of the other results occupy the bulk of this paper and use other ideas.

The problem considered in this paper is also related to the expected Wilson loops. Indeed, suppose γis a closed

oriented geodesic on a hyperbolic surface X. One could consider the ﬁnite space of surface representations

χ:π1(X)→Sn

given the uniform probability measure. Then, one could ask about the quantity

Eχ[ﬁxχ(γ)] := 1

|Hom(π1(X), Sn)|X

ﬁxχ(γ),

where ﬁxχ(γ) is the number of ﬁxed points of the permutation χ(γ)∈Sn. In the case of closed orientable surfaces,

Magee–Puder [MP20] proved the following result.

Theorem 1.8. [MP20, Thm. 1.2] Suppose Xis a connected closed orientable surface. If γ∈π1(X)is non-trivial

and q∈Nis maximal such that γ=γq

0for some γ0∈π1(X), then as n→ ∞,

Eχ[ﬁxχ(γ)] = d(q) + O(1/n),

where d(q)is the number of divisors of q.

This is a special case of a more general result they proved. This result was preceeded by a result in the case when X

is not closed, that is, π1(X) is a free group. In this case, Parzanchevski–Puder [PP15] proved the following.

Theorem 1.9. [PP15, Thm. 1.8]

Eχ[ﬁxχ(γ)] = 1 + c(γ)

nπ(γ)−1+O(1/nπ(γ)),

where π(γ)∈ {0,...,r}∪{∞},rthe rank of the free group π1(X), is an algebraic invariant of γcalled the primitivity

rank, and c(γ)∈N.

In Magee–Naud–Puder [MNP22], the following theorem was used in combination with Selberg’s trace formula to

prove that asymptotically almost surely as n→ ∞,n-sheeted covers of a closed hyperbolic surface introduce no new

eigenvalues below 3

16 −ε.

Theorem 1.10. [MNP22, Thm. 1.11]Suppose Xis a closed orientable surface of genus g≥2. For each g≥2, there

is a constant A=A(g)such that for any c > 0, if 16=γ∈π1(X)is not a proper power of another element in π1(X)

with word length ℓw(γ)≤clog nthen

Eχ[ﬁxχ(γ)] = 1 + Oc,g (log n)A

n.

RANDOM FLAT BUNDLES AND EQUIDISTRIBUTION 5

One could ask for analogues of the above results for surface representations

χ:π1(X)→U(n).

In this case, when π1(X) is free, one uses a uniform Haar probability measure. On the other hand, when Xis closed

orientable, then one uses the Atiyah–Bott–Goldman symplectic volume form on the character variety of surface

representations χ. One could then study quantities of the form

Eχ[fn(χ(γ))],

where we are averaging over the surface representations χand fn:U(n)→Care class functions. Magee [Mag21] has

studied this question in the case of closed orientable surfaces of genus g≥2 for the trace function. In particular, he

proved the following.

Theorem 1.11. [Mag21, Cor. 1.2]Suppose Xis a closed orientable surface of genus g≥2. If γ∈π1(X), then the

limit

lim

n→∞

Eχ[tr(χ(γ))]

exists.

The conjecture is that if γ6= 1, then this limit is actually 0. However, this is not known. In the case of cusped

surfaces, however, a result of Voiculescu [Voi91] states the following.

Theorem 1.12. [Voi91] Suppose π1(X)is free and 16=γ∈π1(X). Then as n→ ∞,

Eχ[tr(χ(γ))] = oγ(n).

In the case of non-closed surfaces, Magee–Puder [MP19] have related the asymptotic expression for such quantities to

the topology of the surface. If we consider more general families of class functions fn:U(n)→C, then the problem

is much more diﬃcult and very little is known.

In the above results, the curve γ∈π1(X) was ﬁxed and the average was taken over the diﬀerent spaces of sur-

face representations. However, one could instead ﬁx the surface representation and average over a family of closed

geodesics on a hyperbolic surface X. The theorems proved in this paper are the generalized analogues of some of the

above theorems in this orthogonal regime.

1.2. Outline of the paper. In Section 2, we discuss preliminaries on representation theory, random matrix theory,

ﬂat bundles, and Selberg’s trace formula. In Section 3, we prove Theorem 1.2 on the minimal eigenvalue of the

Laplacian acting on sections of ﬂat unitary bundles. The proof mostly follows that of the main theorem of [HM21]

with the necessary modiﬁcations and minor deviations. In Section 4, we prove results in random matrix theory that

allow us to restrict to ﬂat bundles whose spectra do not have continuous parts. In Section 5, we prove probabilistic

Weyl-law type results counting eigenvalues of the Laplacian for ﬂat bundles. In Section 6, we prove Theorem 1.7, a

probabilistic prime geodesic theorem whose error term is explicit in terms of the ranks of the bundles. In Section 7, we

deduce some equidistribution consequences of our spectral Theorem 1.2. In Section 8, we comment on the balanced

case and discuss some of its relations to very diﬃcult problems in random matrix theory.

1.3. Acknowledgments. This project was supported by Max Planck Institute for Mathematics in Bonn and Uni-

versity of Southern California.

2. Preliminaries

In this section, we brieﬂy discuss preliminaries on ﬂat bundles, representation theory of U(n), random matrix theory,

and Selberg’s trace formula.

2.1. Flat bundles. Suppose Xis a surface with universal cover the upper half plane

H:= {z∈C|ℑ(z)>0}

which may be equipped with the hyperbolic metric

ds2=dx2+dy2

y2.

Xalso inherits this metric, thus making it into a hyperbolic surface. Throughout this paper, our surfaces Xare

non-compact, connected, hyperbolic, complete, and of ﬁnite area. As previously mentioned, we refer to such surfaces

simply as cusped hyperbolic surfaces. For such X,π1(X) is a ﬁnitely generated free group. Given any surface

representation

χ:π1(X)→U(n),

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arXiv:2210.09547v2[math.NT]3Dec2023RANDOMFLATBUNDLESANDEQUIDISTRIBUTIONMASOUDZARGARAbstract.Eachsignatureλ(n)=(λ1(n),...,λn(n)),whereλ1(n)≥···≥λn(n)areintegers,givesanirreduciblerepresentationπλ(n):U(n)→GL(Vλ(n))oftheunitarygroupU(n).SupposeXisaﬁnite-areacuspedhyperbolicsurface,χisarandomsurfacerepr...

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