FORMULAE FOR ZETA FUNCTIONS FOR INFINITE GRAPHS

Mark Pollicott

University of Warwick

0. Introduction

Given a connected finite d-regular graph G (for d � 3) one can associate theIhara zeta function ⇣G(z), a complex function defined in terms of closed cycles inthe graph. It transpires that ⇣G(z) is the reciprocal of a polynomial, whose explicitform was given by Ihara [Ihara] (cf. also Bass [Bass] and [KS] for generalisations).For an interesting example of such a graph, consider the group PGL(2, Q

p

) whichhas a natural action on a (p + 1)-regular tree T . If � < PGL(2, Q

p

) is a torsionfree discrete subgroup (necessarily isomorphic to a free group) then the quotientG = T /� is a (p + 1)-regular tree.

Consider next an infinite graph G1, which is either the Cayley graph of an infi-nite group or associated to a sequence of d-regular graphs {G

n

} which converge (ina suitable sense) to the infinite graph. Following Grigorchuk and Zuk [GZ] we canthen associate to G1 the logarithm of its zeta function log ⇣G1(z) either by using anintegral with respect to the spectral measure or by taking suitable limits of the func-tions log ⇣G(z). In particular, log ⇣G1(z) has a formal power series which convergesto an analytic function in the region |z| < 1

d�1

. The approach for infinite graphs wasfurther studied by Guido-Isola-Lapidus, [GIL] and Clair-Mokhtan-Sharghi [CMS]using Fudge-Kadison determinants and traces on von Neumann algebras .

The purpose of this note it to consider new expressions for the zeta functionswhich naturally lead to the analytic extensions of log ⇣G1(z) as new explicit conver-gent series, which is particularly useful in numerical evaluation of the zeta functionin the region of its analytic extension. In the case that the associated spectra mea-sure is a Gibbs measure, there is an approach using classical Fredholm-Grothendieckdeterminants for transfer operators. In some other cases, the transfer operatorcan be applied directly. In either case, this viewpoint has significant advantages.Firstly, it gives an explicit convergent expression for log ⇣G1(z) within its domainof analyticity. Secondly, it leads to a simple approach to numerical computation oflog ⇣G1(z) within the domain of analyticity.

1. Zeta functions for graphs

In this section we begin by recalling the properties of the Ihara zeta function forfinite d-regular graphs and their extension to suitable infinite graphs.

1.1 Ihara zeta function. Let d � 3. Given a finite d-regular graph G (withvertices V) we first define the zeta function.

Typeset by AMS-TEX

1

2 MARK POLLICOTT

Definition. We can define the Ihara zeta function as

⇣G(z) = exp

�

1X

k=1

ck

zk

k

!

where cn

= Card {closed loops of length n}, which converges for |z| < 1

d�1

.

It is possible to say quite a lot about the analytic extension of the zeta function.We first denote by M the the |V|⇥ |V| adjacency matrix defined by

M(v, v0) =⇢

1 if v, v are adjacent0 otherwise.

The zeta function ⇣G(z) has the following properties.

Theorem (Ihara). For a d-regular graph G:(1) the Ihara zeta function has a closed form given by

1⇣G(z)

= (1� z2)|V|(d�2)/2 det((1 + (d� 1)z2)I � zM);

(2) the zeros of 1/⇣G(z) lie on the following union of a circle and two intervals:

X =⇢

z 2 C : |z| =1

(d� 1)1/2

�[

�1,�

1(d� 1)

�[

1

(d� 1), 1�

.

1.2 The Laplacian. Given any finite d-regular graph G = (V, E) we can associatethe Laplacian �G : L2(V, R) ! L2(V, R) defined by

�Gh(v) =1d

X

v

0⇠v

h(v0)

where v 2 V and v ⇠ v0 denotes a neighbouring vertex. The operator is selfadjoint and thus the spectrum lies in the the interval [�1, 1]. As for any self-adjointoperator, one can write �G =

R1

�1

�dE(�), where E is a spectral measure takingvalues in projections on L2(V, R). For any v 2 V we can associate the spectralmeasure µ(B) = hE(B)�

v

, �v

i on Borel sets B (which is independent of the choiceof vertex v). Equivalently, consider the measure

µ =1

|spec(�G)|

X

�2spec(�G)

��

supported on the finite set of eigenvalues in the spectrum of �G .1 We can thenwrite

log ⇣G(z) = �

✓d� 2

2

◆log(1� z2)�

Z1

�1

log(1� zdx + (d� 1)z2)dµ(x)

where µ is the spectral measure associated to G.

1We could have definedP

v0⇠v(h(v) � h(v0)), giving a closer analogy with the laplacian onmanifolds. However, the present normalization is more convenient for taking limits of measures

FORMULAE FOR ZETA FUNCTIONS FOR INFINITE GRAPHS 3

1.3 Infinite graphs. For definiteness, consider the particular case of taking limitsof graphs and associating a zeta function. Let G

n

be a finite family of graphs andlet µ

n

be the associated measures. In this context, there are two natural ways toapproach this, which can be shown to be equivalent using the following result.

Proposition 1.2 (Serre ([Serre], cf. [GZ, Theorem 4.2]). The convergence ofµ

n

! µ1 in the weak star topology is equivalent to the formal power series

log ⇣Gn(z) = �

1X

k=1

c(n)

k

zk

k

converging termwise to a limit, i.e., ck

= limn!+1 c

(n)

k

exists for each k � 1.

When convergence in either of these equivalent senses holds then we can definethe zeta function for the limiting graph G by:

log ⇣G(z) := limn!+1

1|V

n

|

log ⇣Gn(z).

The uniqueness of the definition in the connected component V0

of C �X followsfrom the uniqueness of the analytic extension. Furthermore, given the explicit formof the Ihara zeta function for finite graphs (in Part (i) of Ihara’s theorem) we havethe following:

Proposition 1.3 (Grigochuk-Zuk) [GZ, Theorem 5.1]. When the limit of log ⇣Gn(z)exists it takes the form

log ⇣G1(z) := �

d� 22

log(1� z2)�Z

1

�1

log(1� zdx + (d� 1)z2)dµ1(x) (1.2)

where µ1 is the weak star limit of the associated spectral measures µn

.

In particular, the expression for log ⇣G1(z) converges to an analytic function on|z| < 1

d�1

and has an analytic extension to V0

.More generally, for the Cayley graph associated to an infinite graph we can

associate the logarithm of the zeta function log ⇣(z) by using the definition (1.2)where µ1 is taken to be the Kesten spectral measure for the random walk.

2. Gibbs measures and integrals

2.1 Gibbs measures. In order to calculate the analytic extension of log ⇣G1(z)using (1.2) we need to address the same problem for the integral. We can considera more general problem.

Assume that we have a piecewise real analytic expanding map T : X ! X for acube X ⇢ Rd, say, and a real analytic function g : X ! R (which has an analyticextension to a the neighbourhood U , say).

Definition. The T -invariant Gibbs measure µ associated to g is the unique T -invariant measure on [�1, 1] which realises the supremum in

P (g) = sup{h(m) +Z

gdm : m = T -invariant probability}.

Consider the pressure function t 7! P (t log(f(z, ·)) + g(·)) where z 2 V0

. Thefollowing lemma is now standard

4 MARK POLLICOTT

Lemma 2.1 (Ruelle). Assume that P (g) = 0. The map t 7! P (t log(f(z, ·))+g(·))is analytic in a neighbourhood of t = 0. Moreover, we can write

Z

X

log(f(z, x))dµ(x) =@

@tP (t log(f(z, ·)) + g(·))|

t=0

. (2.1)

2.2. Gibbs measures and transfer operators. We can consider the contractinglocal inverses T

i

: X ! X (i = 1, · · · , k) and the linear operator L : B ! Bdefined on the Banach space B of bounded analytic functions w : U ! C (with thesupremum nom k · k1) by

Lw(x) =kX

i=1

eg(Tix)w(Ti

x).

We then have the following useful theorem on the spectrum of the operator [Ruelle].

Lemma 2.2 (Ruelle). The operator L : B ! B is compact. In particular, forany 0 < ✓ < 1 on can associate:

(1) a positive eigenfunction h 2 B and eigenmeasure ⌫ 2 B⇤; and(2) a finite rank operator Q : B ! B with ⇢(Q) < ✓,

such that

L

nw = enP (g)

✓Zwd⌫

◆h + Qn(g) + O(✓n).

for w 2 B.

In the particular case that L1 = 1 we see that h = 1 and ⌫ = µ is the Gibbsmeasure associated to g.

2.3. Gibbs measures and determinants. We can use the following result togive another expression for (2.1)

Lemma 2.3.(1) The function

d(w, z, t) = exp

�

1X

n=1

wn

nZ

n

(t log f(z, ·) + g(·))

!(2.2)

where, for n � 1, we write

Zn

(t log f(z, ·) + g(·)) =X

T

nx=x

exp((t log(f(z, ·)) + g(·))n(x))det(I �D(Tn)(x)�1)

, (2.3)

is analytic w 2 C, z 2 V0

and t is su�ciently small. Furthermore, we canwrite

d(w, z, t) = 1 +1X

n=1

an

(z, t)wn

where |an

| = O(✓n

2) for some 0 < ✓ < 1;

(2) Given w, t the value e�P (t log f(z,·)+g(·)) is a simple zero for d(z, w, t);(3) We can write

Z1

�1

log(f(z, x))dµ(x) =@d(e

�P (g),t)

@t

|

t=0

@d(z,t)

@z

|

z=e

�P (g)

. (2.4)

FORMULAE FOR ZETA FUNCTIONS FOR INFINITE GRAPHS 5

Proof of Lemma 2.3. Part (1) and Part (2) are a direct application of the work ofRuelle, after Grothendieck. For Part (3) we observe that d(z, e�P (t log f(z,·)+g(·)), t) =0, thus we can use Lemma 2.1 and the Implicit Function Theorem to derive (2.4). ⇤Remark. The operator L in §2.2 is actually of trace class and the complex functionin (2.2) corresponds to the determinant of L.

3. Two Theorems

We can now state two theorems which can often be used to provide analyticextensions of the logarithm of the zeta function. They are both rather technical,but in the next section we shall apply them to some interesting example.

3.1. The first theorem. We begin with the following hypothesis which appliesin many examples.

Hypothesis I. Assume that there is a Gibbs measure µ, such that

log ⇣G1(z) = �

d� 22

log(1� z2)�Z

X

f(z, x)dµ(z)

where f(x, ·) : [�1, 1] ! R are real analytic (with an analytic extension to a commonopen neighbourhood U � [�1, 1]).

This brings us to our main result:

Theorem 1. Under the above hypothesis, for any z 2 V0

we can write

log ⇣G1(z) = �

d� 22

log(1� z2)�1X

k=1

dk

(z) (3.1)

where:(1) d

k

(z) are explicit polynomials in {log f(z, x) : T lx = x, 1 l k}; and(2) |d

k

(z)| = O(✓n

2).

Proof. Using (2.4) we can writeZ

1

�1

log(f(z, x))dµ(x) =@d(e

�P,t)

@t

|

t=0

@d(z,t)

@z

|

z=e

�P

=P1

k=1

@ak(z,t)

@t

|

t=0

e�kP

P1k=1

kak

(z, 0)e�(k�1)P

(3.2)

where@d(z, e�P , t)

@t|

t=0

=1X

k=1

@ak

(z, t)@t

|

t=0

e�kP and

@d(z, w, 0)@w

|

w=e

�P =1X

k=1

kak

(z, 0)e�(k�1)P .

For |w| su�ciently small we can also expand

d(w, z, t) = 1 +1X

k=1

1k!

�

1X

n=1

wn

nZ

n

(t log f(z, ·) + g(·))

!k

= 1 +1X

N=1

wN

X

n1+···+nk=N

1k!

Zn1 · · ·Znk

n1

· · ·nk

!

| {z }=aN

(3.3)

6 MARK POLLICOTT

One sees from the definition of Zn

that it depends only on the values of thefunctions at {x : Tnx = x}, and thus by (3.3) that a

n

(z, t) depends only on thevalues of the functions at [n

l=1

{x : T lx = x}. On the other hand, when t = 0 thend(w, z, 0), and its derivative @d(z,w,0)

@w

|

e

�P is clearly independent of log f(z, ·). Thusthe expression (3.0) follows with

dk

(z) =@

kak(z,t)

@t

|

t=0

@d(z,w,0)

@w

|

e

�P

.

and part (1) follows from (3.1).It is easy (using Cauchy’s theorem) that since |a

n

|O(e�ck

2), for suitable c > 0,

we can estimate |

@ak@t

(z, t)|t=0

| = O(e�c

0k

2) and |ka

k

(z, 0)| = O(e�c

0k

2), for any

c > c0 > 0. This proves part (2). ⇤A simple special case. In several examples µ can be taken merely to be the nor-malised Lebesgue measure on [�1, 1] then we can let T : [�1, 1] ! [�1, 1] be thedoubling map

T (x) =

8><

>:

2x� 1 if 1

2

x 12x if � 1

2

x 1

2

2x + 1 if � 1 x � 1

2

and g = � log 2. Then we can write

Zn

=1

2n

� 1

X

T

nx=x

exp

0

@tn�1X

j=0

log f(z, T jx)

1

A .

We can then construct d(w, z, t) from the definition and the integral 1

2

R1

�1

log(f(z, x))dx

by (3.1).

3.2. The second theorem. We now consider a di↵erent hypothesis which appliesin some other examples.

Hypothesis II. Assume that we can find a suitable choice of transfer operator andwrite

log ⇣G1(z) = � log(1� z2)�1X

n=0

L

nf(z, ·)(x0

) (3.4)

where f(x, ·) : [�1, 1] ! R are real analytic (with an analytic extension to a commonopen neighbourhood U � [�1, 1]) and x

0

2 [�1, 1].

This brings us to our second theorem:

Theorem 2. Under the above hypothesis, for any z 2 V0

we can write the analyticextension in the form

log ⇣G1(z) = � log(1� z2)� (I � L)�1f(z, ·)(x0

)

Proof. This is a direct application of the spectral properties of L in Lemma 2.2.

Remark. Theorem 3.1 is appears slightly more complicated to implement, but hasthe advantage that it gives a method for explicitly constructing the analytic exten-sion whenever it exists. Theorem 3.2 is easier to use, when applicable, but the sizeof the extension depends on the size of the second eigenvalue of the operator L.

FORMULAE FOR ZETA FUNCTIONS FOR INFINITE GRAPHS 7

4. Examples

Example 1 (Cayley graph of Z2). The Cayley graphs G of Z (or the free groups Fm

)satisfy ⇣G(z) = 1 [GZ, §6]. For k = 2 the Cayley graph of Zk the zeta function isgiven by

log ⇣G(z) = � log(1� z2)

�

Z1

�1

Z1

�1

log(f(z, x1

+ x2

))dx

1

dx2

⇡2

p1� x2

1

p1� x2

2

(4.1)

where f(z, x) = 1 � 4zx + 3z2. However, we can simply change variables in (4.1)using x

1

= sin�

⇡y12

�and x

2

= sin�

⇡y22

�to write the integral in (4.1) as

14

Z ⇡2

�⇡2

Z ⇡2

�⇡2

log⇣1� 4z

⇣sin⇣⇡y

1

2

⌘+ sin

⇣⇡y2

2

⌘⌘+ 3z2

⌘dy

1

dy2

.

Since integral is with respect to normalised Lebesgue measure on [�1, 1] which is aspecial case of a Gibbs measure, we can apply Theorem 3.1

Figure 1. The absolute value of the integral term of log ⇣G1(z)

Example 2 (Fabikovski-Gupta). Following [GZ], Corollary 10.2, we can write thezeta function for for the Fabikowski-Gupta (described in [BG]) takes the form:

log ⇣G

(z) = � log(1� z2)�13

log(1� z � 3z2)

�

1X

n=1

log

1� z

1 ±

r

6 ±q± · · · ±

p

6

!!1

3n+1

8 MARK POLLICOTT

In particular, the last two terms can be written as�(I � L)�1f(z, ·)

�(1) where

f(z, x) = log(1� zx + 3z2) and

Lw(x) =13w(T

1

z) +13w(T

2

z)

where T1

(x) = 1�p

x + 5 and T2

(x) = 1 +p

x + 5.

Figure 2. The absolute value of the integral term of log ⇣G1(z) in Ex-

ample 2.

Example 3 (cf. [GZ], Proposition 9.4). Grigorchuk and Zuk consider an exampleof a graph G1 with d = 8 associated to an automatic group for which the zetafunction satisfies

log ⇣G1(z) = �3 log(1� z2)�12

Z1

�1

log(F (z, x))dx

⇡p

1� x2

where F (z, x) = 1 � 4z � 2z2

� 28z2 + 49z4 + 16z2x. We can write x = sin�

⇡y

2

�

then dx

dy

= ⇡

2

cos�

⇡y

2

�. This converges for |z| < 1

7

and has an analytic extension toV

0

. Thus we want to evaluate

Z1

�1

log(F (z, sin⇣⇡y

2

⌘))dy

We can let z = 1

10

and we have the following approximations In

to the value ofthe integral:

FORMULAE FOR ZETA FUNCTIONS FOR INFINITE GRAPHS 9

Figure 3. The absolute value of the integral term of log ⇣G1(z) in Ex-

ample 3.

n In

1 -0.22 -0.273 -0.2921074 -0.3029795 -0.3108136 -0.3157287 -0.318678 -0.32049 -0.321421

10 -0.322035

Table 1. Values of p(c)

References

[BG] L. Bartholdi and R. Grigorchuk, ON the spectrum of Hecke type operators related to somefractal groups, Tr. Mat. Inst. Steklova 231 (2000), 5-45.

[Bass] D. Guido, T. Isola, and M. Lapidus,, Ihara’s zeta function for periodic graphs and itsapproximation in the amenable case, J. Funct. Anal. 255 (2008), 13391361.

[CM] B. Clair and S. Mokhtari-Sharghi,, Convergence of zeta functions of graphs., Proc. Amer.Math. Soc. 130 (2002), 18811886.

[GZ] R. Grigorchuk and A. Zuk, The Ihara zeta function and spectral measure for infinitegraphs, the KNS spectral measure and integrable maps,, Random walks and geometry,,Walter de Gruyter GmbH Co. KG,, Berlin, 2004., pp. 141-180.

[GIL] D. Guido, T. Isola, and M. Lapidus, Ihara’s zeta function for periodic graphs and itsapproximation in the amenable case, J. Funct. Anal. 255 (2008), 13391361.

10 MARK POLLICOTT

[Ihara] Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adicfields., J. Math. Soc. Japan. 18 (1966), 219-235.

[KS] M. Kotani and T. Sunada, Zeta functions of finite graphs, J. Math. Sci. Univ. Tokyo 7(2000), 7-25.

[Serre J.-P. Serre]Repartition asymptotique des valeurs propres de l’operateur de Hecke Tp, J. Amer. Math.Soc. 10 (1997), 75-102.

Mark Pollicott, Department of Mathematics, University of Warwick, Coventry

CV4 7AL

E-mail address: M.Pollicott@warwick.ac.uk