Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus
QQUANTUM ERGODICITY FOR EISENSTEIN SERIES ONHYPERBOLIC SURFACES OF LARGE GENUS
ETIENNE LE MASSON AND TUOMAS SAHLSTEN
Abstract.
We give a quantitative estimate for the quantum variance on hyperbolicsurfaces in terms of geometric parameters such as the genus, number of cusps and injectivityradius. It implies a delocalisation result of quantum ergodicity type for eigenfunctionsof the Laplacian on hyperbolic surfaces of finite area that Benjamini-Schramm convergeto the hyperbolic plane. We show that this is generic for Mirzakhani’s model of randomsurfaces chosen uniformly with respect to the Weil-Petersson volume. Depending on theparticular sequence of surfaces considered this gives a result of delocalisation of most cuspforms or Eisenstein series. Introduction
Delocalisation of eigenfunctions and quantum variance.
The question of thedelocalisation of eigenfunctions is a widely studied topic in hyperbolic geometry. One ofthe main open problems is the quantum unique ergodicity (QUE) conjecture of Rudnickand Sarnak [24]. Let X be a compact hyperbolic surface (or more generally a compactmanifold of negative curvature). Denote by ∆ the Laplacian acting on L ( X ) and by λ j the non-decreasing sequence of eigenvalues of ∆. The QUE conjecture asserts that for anyorthonormal basis of eigenfunctions ψ j in L ( X ), the probability measures | ψ j ( z ) | d Vol( z )weakly converge to the normalised Riemannian volume measure X ) d Vol( z ) when λ j → + ∞ . This conjecture was motivated by the earlier work of Snirelman, Zelditch and Colin deVerdi`ere [26, 28, 6] who proved that the convergence to the uniform measure holds for asubsequence of density 1 on manifolds with ergodic geodesic flow. This weaker property calledQuantum Ergodicity can be alternatively formulated by studying the quantum variance forany continuous test function a : X → R { j : λ j ≤ λ } (cid:88) j : λ j ≤ λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) X a ( z ) (cid:18) | ψ j ( z ) | − X ) (cid:19) d Vol( z ) (cid:12)(cid:12)(cid:12)(cid:12) and showing that it tends to 0 when λ → + ∞ . The idea is that by this convergence weobtain an equidistribution of eigenfunctions on average over the spectrum. On hyperbolicmanifolds, this quantum ergodicity property has also been shown to hold by the authorswhen averaging on a bounded spectral interval, and making the volume of X tend to infinityinstead [11] (see also [2] for dimension > level aspect ,as opposed to the eigenvalue aspect , i.e. the limit λ j → + ∞ .In the eigenvalue aspect, Zelditch proved that the Quantum Ergodicity property extendsto non-compact hyperbolic surfaces of finite area [29], which will be the focus of this article.Since the surface X is non-compact there is both discrete and continuous spectra for theLaplacian. Let λ = 0 < λ ≤ λ ≤ . . . be the discrete spectrum and fix a corresponding Mathematics Subject Classification.
Key words and phrases.
Eisenstein series, Maass cusp forms, Quantum chaos, quantum ergodicity,Benjamini-Schramm convergence, ergodic theorem, lattice counting.E. Le Masson was partially supported by the Marie Sk(cid:32)lodowska-Curie Individual Fellowship grant (cid:93) (cid:93) a r X i v : . [ m a t h . SP ] J un ETIENNE LE MASSON AND TUOMAS SAHLSTEN orthonormal system { ψ j } j ∈ N ⊂ L ( X ) of eigenfunctions of the Laplacian. The continuousspectrum is the interval ( , + ∞ ). We denote by C ( X ) the set of cusps on X . Given r ∈ R and b ∈ C ( X ), there are (non- L ) eigenfunction of the Laplacian E b ( · , + ir ) : X → C called Eisenstein series, with eigenvalue τ ( r ) = + r , see for example [9] for background.We similarly parametrise the discrete eigenvalues τ ( r j ) = λ j , with r j possibly complex.Let now I ⊂ (0 , + ∞ ) be an arbitrary interval. We let N ( X, I ) be the number of (discrete)eigenvalues which are in I , and M ( X, I ) := 14 π (cid:90) I ϕ (cid:48) ϕ (cid:0) + ir (cid:1) ds where ϕ ( s ) is the determinant of the scattering matrix, see [9] for details. Then the sum N ( X, I ) + M ( X, I ) measures the total contribution of the discrete and continuous spectrain the interval I .We define the quantum variance over I of the eigenfunctions, for any a ∈ L ∞ ( X ) compactlysupported, byVar X,I ( a ) = 1 N ( X, I ) + M ( X, I ) (cid:88) r j ∈ I |(cid:104) ψ j , a ψ j (cid:105) − ¯ a | + 14 π (cid:90) I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) b ∈ C ( X ) (cid:104) E b ( · , + ir ) , a E b ( · , + ir ) (cid:105) + ϕ (cid:48) ϕ (cid:18)
12 + ir (cid:19) ¯ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dr , (1.1)where ¯ a = 1Vol( X ) (cid:90) X a ( z ) dµ ( z ) . The quantity Var
X,I ( a ) measures how far the L -mass (localised by a ) of typical eigenfunc-tions and Eisenstein series is from being uniformly distributed.Zelditch proved in [29] that when I r = [0 , r ], then for any smooth compactly supportedtest function a : X → C , δ X,I r ( a ) → r → + ∞ , where δ X,I r ( a ) is the mean absolutedeviation, similar to Var X,I ( a ) but without the squares on the absolute value terms. Inthis paper, we are interested in estimating Var X,I for a fixed bounded interval I , in termsof geometric parameters of X . We prove in particular that under natural assumptions,Var X,I ( a ) → X ) → + ∞ , providing the level aspect counterpart of Zelditch’sresult. Unlike the latter, we are able here to work on the quantum variance instead of theabsolute deviation.Before we state our estimate let us introduce some definitions. We see a hyperbolic surface X = Γ \ H as a quotient of the hyperbolic plane by a discrete group Γ of isometries. Wedenote by inj X the radius of injectivity of X , that is, and inj X = inf z ∈ X inj X ( z ), whereinj X ( z ) = 12 inf { d ( z, γz ) | γ ∈ Γ − { id }} , and by ( X ) ≤ R the R -thin part, i.e. the set( X ) ≤ R = { z ∈ X : inj X ( z ) ≤ R } . We say that a sequence of finite area hyperbolic surfaces X n Benjamini-Schramm convergesto H if Vol(( X n ) ≤ R )Vol( X n ) → n → + ∞ . Under Benjamini-Schramm converging X n → H , we obtain the followingequidistribution theorem for eigenfunctions: Theorem 1.1.
Let X n = Γ n \ H be a sequence of finite area hyperbolic surface that Benjamini-Schramm converge to H . Assume in addition that UANTUM ERGODICITY FOR EISENSTEIN SERIES 3 (1) X n has a uniform spectral gap (the first non-zero eigenvalue of the Laplacian isbounded away from uniformly in n ); (2) The systole (length of the shortest closed geodesic) of X n is bounded uniformly frombelow; (3) The number of cusps k n of X n satisfies that for any α > , k αn / Vol( X n ) → when n → + ∞ .Fix Y > and let ( a n ) n ∈ N be a uniformly bounded sequence of measurable functions suchthat spt a n ∈ X n ( Y ) . We have Var X n ,I ( a n ) → when n → + ∞ . This limit theorem is based on a quantitative estimate for the quantum variance thatwe prove for a fixed surface. As will become apparent from this estimate, the systole andthe spectral gap can actually shrink to 0 and the support of the test functions can expand,provided that all of this happens slowly enough when n → + ∞ . To state the quantitativetheorem we need to decompose the surface in the following way. Given Y > X into a compact part where the cusps are cut at a height Y , and a non-compactcuspidal part: the compact part is the complement of the cuspidal part and is given by X ( Y ) = X \ (cid:91) b X b ( Y ) , where X b ( Y ) is the cuspidal zone associated with the cusp b (See the background in Section2.2 for details or [9, Section 2.2]). All cuspidal zones are isometric. We also use the notation L ∞ Y ( X ) for test functions a ∈ L ∞ ( X ) such that spt a ⊂ X ( Y ). Theorem 1.2.
Fix I ⊂ (1 / , + ∞ ) a compact interval. Then there exists R I > such thatfor all R > R I , k ∈ N and Y > the following holds. Assume X is a finite area hyperbolicsurface with k cusps. For any a ∈ L ∞ Y ( X ) , we have ( N ( X, I ) + M ( X, I )) Var
X,I ( a ) (cid:46) I (cid:37) ( λ ( X )) R (cid:107) a (cid:107) + (cid:32) e R inj X ( Y ) Vol(( X ) ≤ R ) + (cid:0) k log Y + k e − πY (cid:1) (cid:33) (cid:107) a (cid:107) ∞ , where (cid:37) ( λ ( X )) is a function of the spectral gap of X To prove Theorem 1.1, we also need to study the spectral asymptotic behaviour of N ( X n , I ) + M ( X n , I ), which is given by the following theorem that we prove in Section 5. Theorem 1.3.
Let I ⊂ ( , + ∞ ) be a compact interval and X n a sequence of hyperbolicsurfaces of finite area that Benjamini-Schramm converges to the hyperbolic plane H . Then lim n →∞ N ( X n , I ) + M ( X n , I )Vol( X n ) = 1 . (1.3)Combining Theorem 1.2 and Theorem 1.3 we obtain the limit form Theorem 1.1. Remark . As showed in [2], the methods of [11] can be extended to higher dimensionalcompact hyperbolic manifolds. This is because Selberg’s theory (spectral side of the proof)and the quantitative ergodic theorem of Nevo (geometric side of the proof) extend naturallyto more general symmetric spaces. We expect that the new elements we introduce inthis paper can also be generalised to finite volume hyperbolic manifolds of any dimension.Concerning variable curvature cusp manifolds — for a level aspect analogue of [4] — the maindifficulties lie already in proving a version of the theorem for compact variable curvaturemanifolds. In this case indeed, we cannot use Selberg’s theory and Nevo’s ergodic theorem.We would need to use lower level tools such as estimates on wave propagation and exponentialmixing of the geodesic flow.
ETIENNE LE MASSON AND TUOMAS SAHLSTEN
Let us now discuss about some consequences of Theorems 1.1, 1.2 and 1.3.1.2.
Equidistribution of Maass forms in the level aspect.
On non-compact finitearea surfaces the existence of a discrete sequence of eigenvalues in L is not guaranteed andis in fact believed to rarely happen (See [21, 22]). Our general result therefore can mostly beseen as an equidistribution result for Eisenstein series. However, in the case of the modularsurface, Γ = SL(2 , Z ), a discrete spectrum is known to exist since the work of Selberg. Moregenerally, this is the case for any congruence subgroup defined byΓ ( N ) = (cid:26)(cid:18) a bc d (cid:19) ∈ SL(2 , Z ) : c ≡ N (cid:27) . (1.4)In this setting, relevant in number theory, eigenfunctions are usually called Maass forms .The arithmetic structure carries a family of operators called Hecke operators that commutewith the Laplacian and it was proved by Lindenstrauss [12] and Soundararajan [27] thatjoint eigenfunctions of these operators and the Laplacian satisfy quantum unique ergodicity.This property implies the equidistribution of all eigenfunctions in the large eigenvalue limit(see for example [3] for an introduction to these questions).The level aspect limit in the arithmetic setting was considered recently in a series of paperconcerning the equidistribution of holomorphic forms by Nelson [17] and Nelson, Pitale andSaha [19]. The results are analogous to the quantum unique ergodicity theory but they relyon the proof of the Ramanujan conjectures which is not available for Maass forms.It turns out that the surfaces Y ( N ) = Γ ( N ) \ H Benjamini-Schramm converge to H whenthe level N → ∞ ([1, 23]). Moreover, in the case of increasing congruence covers, Finis,Lapid and M¨uller showed that the discrete spectrum dominates the asymptotics as thelevel N → ∞ (see [7]). This means that M ( Y ( N ) , I ) (cid:28) N ( Y ( N ) , I ). Hence Theorem 1.2together with Theorem 1.3 implies a Quantum Ergodicity theorem for Maass cusp forms,which incidentally does not need to assume the cusp forms are Hecke eigenfunctions. ForHecke-Maass cusp forms, however, a quantum ergodicity theorem with a stronger rate ofconvergence has recently been obtained by Nelson [18].1.3. Quantum ergodicity on random surfaces of large genus.
The quantitative es-timate in Theorem 1.2 allows us to study the delocalisation of eigenfunctions on randomsurfaces of large genus, where the random model we will use is the uniform distributionwith respect to the Weil-Petersson volume on the moduli space of hyperbolic surfaces offinite volume. This probability model for hyperbolic surfaces was popularised by the workof Mirzakhani [15] and provides very effective tools to estimate the geometric parametersappearing in Theorem 1.2.The approach we use here for delocalisation of eigenfunctions was introduced in [8] in thestudy L p norms of eigenfunctions on random surfaces in Mirzakhani’s model. See also therecent work of Monk [16] on spectral theory of random surfaces in the Mirzakhani model andalso Magee-Naud [13] and Magee-Naud-Puder [14] who consider spectral theory in anothermodel based on random coverings.To fix some notation, let M g,k be the moduli space of genus g hyperbolic surfaces with k cusps and let P g,k be the Weil-Petersson probability on M g,k . For the compact case, wealso abbreviate P g = P g, and M g = M g, . We have the following: Theorem 1.5.
Fix a compact interval I ⊂ (1 / , + ∞ ) . (i) For a P g -random surface X ∈ M g the probability that for any a ∈ L ∞ ( X ) we have Var
X,I ( a ) (cid:46) I g (cid:107) a (cid:107) ∞ converges to as g → ∞ . UANTUM ERGODICITY FOR EISENSTEIN SERIES 5 (ii) If k ( g ) g ∈ N is such that for any α > , we have k ( g ) α /g → as g → ∞ , then thereexists ε g → such that for a P g,k ( g ) -random surface X ∈ M g,k ( g ) the probability thatfor any a ∈ L ∞ log g ( X ) we have Var
X,I ( a ) (cid:46) I ε g (cid:107) a (cid:107) ∞ converges to as g → ∞ . In other words, provided we control the number of cusps and the support of the testfunction, the quantum variance tends to 0 with high probability when g → + ∞ . Theproof of Theorem 1.5 follows from Theorem 1.2 together with Theorem 1.3 on the spectralconvergence by inspecting the geometric parameters and referring to known properties oftheir asymptotic behaviour in large genus, mainly after the works of Mirzakhani [15] andMonk [16]. We note that these works have been done only in the random model ( M g , P g )of compact surfaces. However, all the estimates needed in their proofs rely on boundingWeil-Petersson volumes V g,k with k independent of g , but these estimates stay unchangedwhen we allow a dependence of k on g of the type k ( g ) α /g → g → ∞ . See Section 6for more discussion on this.A non-compact version of the results of [16] on spectral convergence on random surfacescould provide a quantitative rate ε g → Conjecture 1.6.
For a P g,k ( g ) -random hyperbolic surface X ∈ M g,k ( g ) of large genus g ,assuming that for all α > , k ( g ) α /g → as g → ∞ , we have that for any compactinterval I : N ( X, I ) M ( X, I ) = o (1) when g → + ∞ , with high probability. As far as we know this problem is open.1.4.
Organisation of the article.
The paper is organised as follows. In Section 2 we givethe necessary background on harmonic analysis on finite volume hyperbolic surfaces andSelberg’s theory we use in the spectral side of the proof. In Section 3 we prove a version ofTheorem 1.2 for mean zero observables, which is similar to the compact case but requiresadditional steps to handle the presence of cusps. This is the first step of the proof of thegeneral quantitative estimate that we prove in Section 4 where we deal more specificallywith the continuous spectrum using Maass-Selberg estimates. In Section 5 we prove thespectral convergence (Theorem 1.3). Finally in Section 6 we give the argument for Theorem1.5 on random surfaces. 2.
Background
In this section, we give some definitions and introduce elements of harmonic analysis onhyperbolic surfaces that we will use in the proof. For more background on the geometryand spectral theory of hyperbolic surfaces we refer to the books [5, 9, 3].2.1.
Hyperbolic surfaces.
The hyperbolic plane is identified with the upper-half plane H = { z = x + iy ∈ C | y > } , ETIENNE LE MASSON AND TUOMAS SAHLSTEN equipped with the hyperbolic Riemannian metric ds = dx + dy y . We will denote by d ( z, z (cid:48) ) the distance between two points z, z (cid:48) ∈ H . The hyperbolic volumeis given by dµ ( z ) = dx dyy . The group of isometries of H is identified with PSL(2 , R ), the group of real 2 × ± id, acting by M¨obius transformations (cid:18)(cid:18) a bc d (cid:19) ∈ PSL(2 , R ) , z ∈ H (cid:19) (cid:55)→ (cid:18) a bc d (cid:19) · z = az + bcz + d . A hyperbolic surface can be seen as a quotient X = Γ \ H of H by a discrete subgroupΓ ⊂ PSL ( R ). We denote by F a fundamental domain associated to Γ. If we fix z ∈ H , anexample of a fundamental domain is given by the set F = { z ∈ H | d ( z , z ) < d ( z , γz ) for any γ ∈ Γ − {± id }} . The injectivity radius on the surface X = Γ \ H at a point z is given byinj X ( z ) = 12 min { d ( z, γz ) | γ ∈ Γ − { id }} . Thus inj X ( z ) gives the largest R > B X ( z, R ) is isometric to a ball of radius R inthe hyperbolic plane. It is also equal to half of the length of the largest geodesic loop at z .Let g ∈ PSL(2 , R ), we define the translation operator T g , such that for any function f on H T g f ( z ) = f ( g − · z ) . We will generally see a function f on a hyperbolic surface X = Γ \ H as a Γ-invariant function f : H → C , T γ f ( z ) = f ( γ − z ) = f ( z ) for all γ ∈ Γ . The integral of the function on the surface is then equal to the integral of the invariantfunction over any fundamental domain (cid:90) F f ( z ) dµ ( z ) . Cusps.
Let X be a finite area hyperbolic surface and Y > k = k ( X ) closed loops (horocycles) γ , . . . , γ k in X of equal length 1 /Y such that wecan decompose X as X = X ( Y ) ∪ X ( Y ) = X ( Y ) ∪ k (cid:91) j =1 Z j ( Y ) , where X ( Y ) is a compact manifold with the k closed horocycles γ , . . . , γ k ⊂ X as boundariesand X ( Y ) is the union of topological cylinders (cusps) Z ( Y ) , . . . , Z k ( Y ) cut along thehorocycles. All the cusps cut at height Y are isometric to C Y = Γ ∞ \ { z = x + iy ∈ H | ≤ x ≤ , y ≥ Y } , where Γ ∞ is the subgroup generated by the transformation z (cid:55)→ z + 1. In particular we seethat Vol( C Y ) = 1 Y .
UANTUM ERGODICITY FOR EISENSTEIN SERIES 7
For each cusp Z j ( Y ), using the isometry σ j : C Y → Z j ( Y ), we can see any function f of X = Γ \ H as a function f ( j ) ( x, y ) = f ( σ j ( x, y )) such that f ( j ) ( x, y ) = f ( j ) ( x + 1 , y ), whichallows to write a Fourier series decomposition f ( j ) ( x, y ) = (cid:88) n f ( j ) n ( y ) e inx , in any cusp.2.3. Geodesic flow.
The tangent bundle of H can be identified with H × C . The hyperbolicmetric gives the following inner product for two tangent vectors ( z, re iθ ) and ( z, r (cid:48) e iθ (cid:48) ) onthe tangent plane T z H (cid:104) re iθ , r (cid:48) e iθ (cid:48) (cid:105) z = r r (cid:48) Im ( z ) cos( θ (cid:48) − θ ) . As a consequence, the map( z, θ ) ∈ H × S (cid:55)→ ( z, Im ( z ) e iθ ) ∈ H × C , where S = R / π Z , identifies H × S with the unit tangent bundle.The group PSL(2 , R ) acts on the tangent bundle via the differential of its action on H . Itis well known (see for example [10]) that this action induces a homeomorphism betweenPSL(2 , R ) and the unit tangent bundle of H , such that the action of PSL(2 , R ) on itself byleft multiplication corresponds to the action of PSL(2 , R ) on the unit tangent bundle.We denote by ϕ t : H × S → H × S the geodesic flow associated to H . It is invariant withrespect to the Liouville measure dµ dθ , where dθ is the Lebesgue measure on S . Via theidentification H × S ∼ PSL(2 , R ), the geodesic flow is equal to the multiplication on theright by the diagonal subgroup ϕ t ( g ) = g (cid:18) e t/ e − t/ (cid:19) , g ∈ G, t ∈ R . For a hyperbolic surface Γ \ H , the unit tangent bundle is identified with Γ \ PSL(2 , R ),and via this identification the geodesic flow will be given simply by ϕ t (Γ g ) = Γ g (cid:18) e t/ e − t/ (cid:19) . Polar coordinates.
Let z ∈ H be an arbitrary point. For any point z ∈ H differentfrom z , there is a unique geodesic of length r going from z to z . Using the geodesic flow, itmeans that there is a unique θ ∈ S and r ∈ (0 , ∞ ) such that z is the projection of ϕ r ( z , θ )on the first coordinate. The change of variable z (cid:55)→ ( r, θ ) is called polar coordinates . Theinduced metric is ds = dr + sinh r dθ , and the hyperbolic volume in these coordinates is given by dµ ( r, θ ) = sinh r dr dθ. Spectrum of the Laplacian and Eisenstein series.
In the coordinates z = x + iy ,the Laplacian ∆ on H is the differential operator∆ = − y (cid:18) ∂ ∂x + ∂ ∂y (cid:19) . A fundamental property of the Laplacian is that it commutes with isometries. We have forany g ∈ PSL(2 , R ), T g ∆ = ∆ T g . The Laplacian can therefore be seen as a differential operator on any hyperbolic surfaceΓ \ H . ETIENNE LE MASSON AND TUOMAS SAHLSTEN
The spectrum of the Laplacian ∆ on a finite area surface X with k cusps can be decomposedinto the discrete part λ = 0 < λ ≤ . . . and the absolutely continuous part [1 / , + ∞ ).For each λ ∈ [1 / , + ∞ ) in the absolutely continuous part, and each cusp j = 1 , . . . , k , wecan associate a family of non- L eigenfunctions E j ( s λ , z ) with s λ (1 − s λ ) = λ , called the Eisenstein series . For each z ∈ X , the Eisenstein series have a meromorphic extension s (cid:55)→ E j ( s, z ) to the whole complex plane C .2.6. Scattering matrix and truncated Eisenstein series.
Given s ∈ C , the scatteringmatrix Φ( s ) is the k × k matrix with entries given by Φ (cid:96)j ( s ), where each Φ (cid:96)j is a meromorphicfunction coming from the zeroth Fourier mode of the (cid:96) -th Eisenstein series E (cid:96) in the j -thcusp Z j ( Y ), which can be shown to be of the form z = x + iy (cid:55)→ δ (cid:96)j y s + Φ (cid:96)j ( s ) y − s , x = ( y, θ ) ∈ Z j ( Y ) . The determinant of Φ( s ), denoted by ϕ ( s ), is called the scattering determinant . WhenRe s = 1 /
2, we have that Φ( s ) is a unitary matrix.Given s ∈ C , the Eisenstein series x (cid:55)→ E j ( s, z ) as functions are independent of the height Y chosen, but given the height Y , we can form a truncated version E Yj ( s, z ) by defining itfor each j = 1 , . . . , k , as the function obtained from E j ( s, · ) by removing the zeroth Fouriermode δ j(cid:96) y s + Φ j(cid:96) ( s ) y − s when entering the (cid:96) -th cusp Z l ( Y ), for each cusp (cid:96) .For any y > ∗ y the projector on functions whose zeroth Fourier modevanish in each cusp at height higher than y .2.7. Invariant integral operators and Selberg transform.
We say that a boundedmeasurable kernel K : H × H → C is invariant under the diagonal action of Γ if for any γ ∈ Γ we have K ( γ · z, γ · w ) = K ( z, w ) , ( z, w ) ∈ H × H . Assume for simplicity that K ( z, w ) = 0 whenever d ( z, w ) > C for some constant C >
0. Forany Γ-invariant function f , such a kernel defines an integral operator A on the surface X bythe formula Af ( z ) = (cid:90) H K ( z, w ) f ( w ) dµ ( w ) = (cid:90) D (cid:88) γ ∈ Γ K ( z, γw ) f ( w ) dµ ( w ) , z ∈ D. The function k : H × H → C given by k ( z, w ) = (cid:88) γ ∈ Γ K ( z, γw )is such that k ( γz, γ (cid:48) w ) = k ( z, w ) for any γ, γ (cid:48) ∈ Γ, which defines the kernel on the surface.A special case of invariant kernels is given by radial kernels. Let k : [0 , + ∞ ) → C be abounded measurable compactly supported function, then K ( z, w ) = k ( d ( z, w )) , ( z, w ) ∈ H × H is an invariant kernel.For k : [0 , + ∞ ) → C , the Selberg transform S ( k ) of k is obtained as the Fourier transform S ( k )( r ) = (cid:90) + ∞−∞ e iru g ( u ) du of g ( u ) = √ (cid:90) + ∞| u | k ( (cid:37) ) sinh (cid:37) √ cosh (cid:37) − cosh u d(cid:37). UANTUM ERGODICITY FOR EISENSTEIN SERIES 9
For a function h : R → C , the Selberg transform is inverted using the inverse Fouriertransform g ( u ) = 12 π (cid:90) + ∞−∞ e − isu h ( s ) ds and the formula k ( (cid:37) ) = − √ π (cid:90) + ∞ (cid:37) g (cid:48) ( u ) √ cosh u − cosh (cid:37) du. Eigenfunctions of the Laplacian are eigenfunctions of all operators of convolution by aradial kernel and the eigenvalues are given precisely by the Selberg transform.
Proposition 2.1 ([9] Theorem 1.14) . Let X = Γ \ H be a hyperbolic surface. Let k :[0 , + ∞ ) → C be a smooth function with compact support. If ψ λ is an eigenfunction of theLaplacian on X of eigenvalue λ , then it is an eigenfunction of the radial integral operator A associated to k . That is, Aψ λ ( z ) = (cid:90) k ( d ( z, w )) ψ λ ( w ) dµ ( w ) = h ( r λ ) ψ λ ( z ) , where the eigenvalue h ( r λ ) is given by the Selberg transform of the kernel k : h ( r λ ) = S ( k )( r λ ) , and r λ ∈ C is defined by the equation λ = + r λ . Note that this statement can be generalised to the case of k : [0 , + ∞ ) → C measurablebounded and compactly supported by approximation and dominated convergence.3. Mean zero case
We first consider the case where the test function is of mean 0. We will consider thegeneral case in Section 4. The proof of the mean zero case follows closely the proof forcompact surfaces in [11]. The main difference is the Hilbert-Schmidt norm estimate inLemma 3.5 that now takes into account the presence of cusps.
Proposition 3.1.
Fix I ⊂ (1 / , + ∞ ) a compact interval. Then there exists R I > suchthat for all R > R I and for all hyperbolic surface X with Vol( X ) < ∞ and for any compactlysupported measurable function a ∈ L ∞ ( X ) such that (cid:82) a ( x ) dµ ( x ) = 0 , we have (cid:103) Var
X,I ( a ) (cid:46) I (cid:37) ( λ ) R (cid:107) a (cid:107) + e R sys( X ( Y a )) Vol(( X ) ≤ R ) (cid:107) a (cid:107) ∞ . where (cid:103) Var
X,I ( a ) = ( N ( X, I ) + M ( X, I )) Var
X,I ( a ) and (cid:37) ( λ ) is a function of the spectralgap. The key idea proposed in [11] is to introduce a regularised wave propagation operator.For any bounded measurable function u : X → C we write P t u ( z ) = 1 e t/ (cid:90) B ( z,t ) u ( w ) dµ ( w ) . (3.1)The propagator P t is at the centre of our dynamical approach. Our goal is to show thatthe quantum variance Var X,I ( a ) satisfies some invariance under wave propagation so thatwe can replace it with Var X,I (cid:18) T (cid:90) T P t a P t dt (cid:19) , which in turn is controlled by the dynamics of the geodesic flow when T → + ∞ . In fact wejust need Var X,I ( a ) (cid:46) Var
X,I (cid:18) T (cid:90) T P t a P t dt (cid:19) where the implied constant is uniform in T and X . Invariance of the quantum variance (spectral side).
The first step is to under-stand the action of P t on eigenfunctions ψ λ of the Laplacian of eigenvalue λ . The propagator P t has the form of a radial integral operator: for u ∈ L ∞ ( X ), we have P t u ( z ) = (cid:90) K t ( z, w ) u ( w ) dµ ( w )with radial kernel K t ( z, w ) = k t ( d ( z, w )), where k t ( (cid:37) ) := e − t/ { (cid:37) ≤ t } . By Proposition 2.1 we have for any function ψ λ such that ∆ ψ λ = λψ λ P t ψ λ ( z ) = (cid:90) k t ( d ( z, w )) ψ λ ( w ) dµ ( w ) = S ( k t )( r λ ) ψ λ ( z ) , where S ( k t ) is the Selberg transform of the kernel k t and r λ ∈ C is defined by the equation λ = + r λ .The action of T (cid:82) T P t a P t dt on ψ λ will be understood through the following lemma,proved in [11, Proposition 4.2]. Lemma 3.2.
Let I ⊂ (0 , ∞ ) be a compact interval. Then there exists a constant C I > such that for all T > T I we have inf r ∈ I T (cid:90) T |S ( k t )( r ) | dt ≥ C I Applying Lemma 3.2 to cusp forms and Eisenstein series, we obtain (cid:88) r j ∈ I |(cid:104) ψ j , a ψ j (cid:105)| ≤ C I (cid:88) r j ∈ I (cid:12)(cid:12)(cid:12)(cid:68) ψ j , (cid:16) T (cid:90) T P t aP t dt (cid:17) ψ j (cid:69)(cid:12)(cid:12)(cid:12) ; (3.2)and (cid:90) I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) b ∈ C ( X ) (cid:104) E b ( · , + ir ) , a E b ( · , + ir ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dr ≤ C I (cid:90) I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) b ∈ C ( X ) (cid:68) E b ( · , + ir ) , (cid:16) T (cid:90) T P t aP t dt (cid:17) E b ( · , + ir ) (cid:69)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dr. (3.3)Summing the above estimates, we get a formal bound for the quantum variance (1.1). Itis not obvious that this bound is in fact finite. We will show it by establishing in Proposition3.4 that T (cid:82) T P t aP t dt is a Hilbert-Schmidt operator. By the spectral theorem (See [9,Theorem 7.3 and Theorem 7.4]) the estimates (3.2) and (3.3) imply: Proposition 3.3.
Under the assumptions of Theorem 1.2, there exists T I > such that forall T > T I and any a ∈ L ∞ ( X ) with (cid:82) X a d Vol = 0 , we have
Var
X,I ( a ) (cid:46) I (cid:13)(cid:13)(cid:13) T (cid:90) T P t aP t dt (cid:13)(cid:13)(cid:13) HS . Here the quantity (cid:13)(cid:13)(cid:13) T (cid:90) T P t aP t dt (cid:13)(cid:13)(cid:13) HS is the Hilbert-Schmidt norm of the operator T (cid:82) T P t aP t dt , that we bound in the nextsection. UANTUM ERGODICITY FOR EISENSTEIN SERIES 11
Bounding the Hilbert-Schmidt norm.
By Proposition 3.3, to prove Theorem 1.2it suffices to obtain the required bound for (cid:13)(cid:13)(cid:13) T (cid:90) T P t aP t dt (cid:13)(cid:13)(cid:13) HS . Since the test function a ∈ L ∞ ( X ) has compact support in X , we can choose Y = Y a > a ⊂ X ( Y )where X ( Y ) = X \ (cid:91) b X b ( Y ) , and X b ( Y ) is the cuspidal zone associated with b . This means that the support of a doesnot go beyond height Y into the cusps.We can prove that T (cid:82) T P t aP t dt is Hilbert-Schmidt and has the following quantitativebound: Proposition 3.4 (Geometric bound) . For every a ∈ L ∞ ( X ) compactly supported and every T > the operator T (cid:82) T P t aP t dt is Hilbert-Schmidt with norm (cid:13)(cid:13)(cid:13) T (cid:90) T P t aP t dt (cid:13)(cid:13)(cid:13) HS (cid:46) (cid:107) a (cid:107) T (cid:37) ( λ ) + e T inj X ( Y a ) Vol(( X ) < T ) (cid:107) a (cid:107) ∞ , for Y a > large enough such that spt a ⊂ X ( Y a ) . We will work with a fundamental domain F of X that we decompose such that: F ( Y ) = F \ (cid:91) b F b ( Y ) , and F b ( Y ) represent the cuspidal zone associated with the cusp b . Having fixed T > a ∈ L ∞ ( X ), we write for ( z, w ) ∈ F × F K T ( z, w ) = (cid:104) T (cid:90) T P t aP t dt (cid:105) ( z, w ) = 1 T (cid:90) T [ P t aP t ]( z, w ) dt where we use the bracket notation [ A ] for the kernel of an integral operator A . Then wehave (cid:13)(cid:13)(cid:13)(cid:13) T (cid:90) T P t aP t dt (cid:13)(cid:13)(cid:13)(cid:13) = (cid:90) F (cid:90) F | K T ( z, w ) | dµ ( z ) dµ ( w ) . Now the kernel K T : X × X → R on X can be represented as an invariant kernel K T : H × H → R under the diagonal action of Γ on H × H as follows: K T ( z, w ) = (cid:88) γ ∈ Γ K T ( z, γ · w )for any ( z, w ) ∈ F × F . In our case, seeing a as a Γ-invariant function on H , we can writein the above K T ( z, w ) = 1 T (cid:90) T e − t (cid:90) B ( z,t ) ∩ B ( w,t ) a ( x ) dµ ( x ) dt. Thus in particular, we have that the Hilbert-Schmidt norm can be written as (cid:13)(cid:13)(cid:13)(cid:13) T (cid:90) T P t aP t dt (cid:13)(cid:13)(cid:13)(cid:13) HS = (cid:90) F (cid:90) F (cid:12)(cid:12)(cid:12) (cid:88) γ ∈ Γ K T ( z, γ · w ) (cid:12)(cid:12)(cid:12) dµ ( z ) dµ ( w ) . Hence to prove Proposition 3.4, we need to estimate Hilbert-Schmidt norms of integraloperators with invariant kernels, which we do in the following lemma.
Fix
R >
0. Recall that ( F ) ≤ R denotes the points in the fundamental domain F withradius of injectivity less than R :( F ) ≤ R = { z ∈ F : inj X ( z ) ≤ R } , and we denote by ( F ) >R the complement of this set in F . We write H ( Y ) = Γ · F ( Y ) for allthe images of the compact part of F by the action of Γ. Lemma 3.5.
Let A be an integral operator on X such that (cid:107) A (cid:107) = (cid:90) F (cid:90) F (cid:12)(cid:12)(cid:12) (cid:88) γ ∈ Γ K ( z, γ · w ) (cid:12)(cid:12)(cid:12) dµ ( z ) dµ ( w ) . (3.4) for a kernel K : H × H → R invariant under the diagonal action of Γ on H × H . Fix R, Y > . We assume that the kernel K satisfies K ( z, w ) (cid:54) = 0 only when d ( z, w ) ≤ R and z, w ∈ H ( Y ) .Then we have: (cid:107) A (cid:107) ≤ (cid:90) F (cid:90) H | K ( z, w ) | dµ ( z ) dµ ( w ) + e R inj X ( Y ) Vol (( F ) ≤ R ) sup ( z,w ) ∈ F × H | K ( z, w ) | . Proof.
This is a more general version of Lemma 5.1 of [11] on Hilbert-Schmidt norm estimatesin terms of the injectivity radius, that allows us to treat the case when X is not compact. Wesplit the integral (3.4) into two parts over points with small and large radius of injectivity,and use that in the first part, the sum over Γ is reduced to one term. (cid:107) A (cid:107) = (cid:90) ( F ) >R (cid:90) F (cid:88) γ ∈ Γ | K ( z, γ · w ) | dµ ( z ) dµ ( w ) + (cid:90) ( F ) ≤ R (cid:90) F | (cid:88) γ ∈ Γ K ( z, γ · w ) | dµ ( z ) dµ ( w ) . We get using the Cauchy-Schwarz inequality that | (cid:88) γ ∈ Γ K ( z, γ · w ) | ≤ N Γ ( R ; z, w ) (cid:88) γ ∈ Γ | K ( z, γ · w ) | , with the lattice counting parameter N Γ ( R ; z, w ) = (cid:93) { γ ∈ Γ : d ( z, γw ) ≤ R } . Since z ∈ H ( Y ), we know that for any γ ∈ Γ, d ( z, γz ) ≥ inj X ( Y ) , and similarly for w . Wededuce the following bound by a standard counting argument of the number of fundamentaldomains in a ball of radius R | N Γ ( R ; z, w ) | ≤ e R inj X ( Y ) . The rest is similar to the proof of Lemma 5.1 in [11]. We have (cid:107) A (cid:107) ≤ (cid:90) F (cid:90) H | K ( z, w ) | dµ ( z ) dµ ( w ) + e R inj X ( Y ) (cid:90) ( F ) ≤ R (cid:90) H | K ( z, w ) | dµ ( z ) dµ ( w ) . The second term on the right-hand side is bounded by e R inj X ( Y ) Vol( B ( R )) Vol (( F ) ≤ R ) sup ( z,w ) ∈ F × H | K ( z, w ) | , and Vol( B ( R )) (cid:46) e R , which concludes the proof. (cid:3) We are interested in the invariant kernel K T ( z, w ) = 1 T (cid:90) T e − t (cid:90) B ( z,t ) ∩ B ( w,t ) a ( x ) dµ ( x ) dt UANTUM ERGODICITY FOR EISENSTEIN SERIES 13 associated with T (cid:82) T P t aP t dt . We see that K ( z, w ) = 0 whenever d ( z, w ) ≥ T so Lemma3.5 can be applied with R = 2 T . Hence in order to prove Proposition 3.4 we are left withproving L and L ∞ estimates for our invariant kernel.The L ∞ bound is straightforward, we havesup ( z,w ) ∈ F × H | K T ( z, w ) | (cid:46) (cid:107) a (cid:107) ∞ , (3.5)since Vol( B ( z, t ) ∩ B ( w, t )) (cid:46) Vol( B ( t )) (cid:46) e t for all ( z, w ) ∈ F × H .The L bound is at the core of our analysis. Lemma 3.6.
We have (cid:90) F (cid:90) H | K T ( z, w ) | dµ ( z ) dµ ( w ) (cid:46) (cid:107) a (cid:107) T (cid:37) ( β ) . The proof of this follows from a quantitative ergodic theorem by Nevo published in [20](see also [11] for more explanations on the application to our setting).Let (
Y, ν ) be a probability space, and G a group equipped with its left-invariant Haarmeasure dg , and a measure-preserving action on Y . For a collection of measurable sets A t ⊂ G we define the averaging operators π Y ( A t ) f ( x ) = 1 | A t | (cid:90) A t f ( g − x ) dg, f ∈ L ( Y ) , x ∈ Y. Theorem 3.7 (Nevo [20]) . If G is a connected simple Lie group equipped with a measure-preserving action on the probability space ( Y, ν ) that has a spectral gap, then there exist C, θ > such that for any family A t ⊂ G , t ≥ , of measurable sets of positive measure, wehave (cid:13)(cid:13)(cid:13)(cid:13) π Y ( A t ) f − (cid:90) Y f dµ (cid:13)(cid:13)(cid:13)(cid:13) L ( Y,ν ) ≤ C | A t | − θ (cid:107) f (cid:107) L ( Y,ν ) for any f ∈ L ( Y, ν ) , where we denote by | A t | the measure of the set A t . The constant C depends only on G and θ depends only on the spectral gap. Theorem 3.7 assumes nothing on the uniformity of the lattice Γ, in particular, it applieswhen G = PSL(2 , R ) and Y = Γ \ PSL(2 , R ) for Γ co-finite as in our setting. The importantpoint is then that the spectral gap of the Laplacian implies that the action of G on Y has aspectral gap, and that θ depends on the spectral gap of the Laplacian.In order to use Theorem 3.7, we need to use a change of variable and lift the kernels K T to SL(2 , R ). Proof of Lemma 3.6.
The proof is identical to the one in Section 7 of [11]. We brieflyreproduce its main steps for the convenience of the reader. We identify PSL(2 , R ) with theunit tangent bundle { ( z, θ ) ∈ H × S } of H (see Section 2). We define A t ( s ) ⊂ PSL(2 , R )to be a set such that A t ( s ) − · ( z, θ ) is the lift in the unit tangent bundle of two balls ofradius t with centres given by the projections z and z onto X of the points ϕ − s/ ( z, θ )and ϕ s/ ( z, θ ) of the unit tangent bundle. Here ϕ t is the geodesic flow on the unit tangentbundle of X . We notice that we have a natural change of variable ( z, θ, s ) (cid:55)→ ( z , z ), and π ( A t ( s )) f ( z, θ ) = 1 | A t ( s ) | (cid:90) A t ( s ) f ( g − · ( z, θ ) , s ) dg = 1 | B t ( z , t ) ∩ B t ( z , t ) | (cid:90) B t ( z ,t ) ∩ B t ( z ,t ) f ( z , z ) dg Using this change of variable (see [11, Lemma 7.1] for details) we have (cid:90) F (cid:90) H | K T ( z , z ) | dµ ( z ) dµ ( z )= (cid:90) T sinh s (cid:90) F (cid:90) S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T (cid:90) Ts/ e − t | A t ( r ) | π ( A t ( r )) a ( z, θ ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( z ) dθ ds. Then Minkowski’s integral inequality yields (cid:90) T sinh s (cid:90) F (cid:90) S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T (cid:90) Ts/ e − t | A t ( s ) | π ( A t ( s )) a ( z, θ ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( z ) dθ ds, ≤ (cid:90) T sinh s (cid:32) T (cid:90) Ts/ e − t | A t ( s ) | (cid:107) π ( A t ( s )) a (cid:107) L ( F × S ) dt (cid:33) ds. By Theorem 3.7 with G = PSL(2 , R ) and Y = Γ \ PSL(2 , R ), there is a constant (cid:37) ( β ) > β of the Laplacian such that for some uniform (Γ-independent) constant C > (cid:107) π ( A t ( s )) a (cid:107) L ( F × S ) ≤ C | A t ( s ) | − (cid:37) ( β ) (cid:107) a (cid:107) . Hence the previous integral is bounded by (cid:90) T sinh s (cid:32) T (cid:90) Ts/ e − t | A t ( s ) | − (cid:37) ( β ) (cid:107) a (cid:107) dt (cid:33) ds. (3.6)Note that as A t ( s ) is given by the lift of intersections of two balls of radius t such that theircentres is at a distance s from each other, we know that for some uniform constant | A t ( s ) | (cid:46) e t − s/ . We can thus bound (3.6) with a uniform constant times (cid:90) T sinh s (cid:32) T (cid:90) Ts/ e − s/ e − (cid:37) ( β )( t − s/ (cid:107) a (cid:107) dt (cid:33) ds (cid:46) T (cid:90) T (cid:107) a (cid:107) (cid:37) ( β ) ds (cid:46) (cid:107) a (cid:107) T (cid:37) ( β ) , so the proof of the claim is complete (cid:3) Combining (3.5) and Lemma 3.6 with Lemma 3.5 we thus proved the desired boundclaimed in Proposition 3.4. Together with the spectral side of Section 3.1, this completesthe proof of Proposition 3.1.4.
General case: Proof of Theorem 1.2
In this section, we treat the general case of observables with non-zero mean, and proveTheorem 1.2. If a is a test function that does not have mean 0, i.e. a := 1Vol( X ) (cid:90) a ( z ) dz (cid:54) = 0we fix an arbitrary Y ≥ Y a where Y a is defined as the smallest height such that the supportof a is in X ( Y a ). We then define b ( z ) := a ( z ) − aχ ( z ) , x ∈ X where χ ( z ) = (cid:40) Vol( X )Vol( X ( Y )) , if z ∈ X ( Y );0 , otherwise . This idea to use such a symbol is similar to what is done in [4], albeit simplified by the factwe do not need b smooth, as our proof works for L ∞ test functions. UANTUM ERGODICITY FOR EISENSTEIN SERIES 15
By this choice of χ we have that (cid:90) X b ( z ) dz = (cid:90) X a ( z ) dz − a (cid:90) χ ( z ) dz = 0 . It is straightforward to obtain (cid:103)
Var
X,I ( a ) (cid:46) (cid:103) Var
X,I ( b ) + a (cid:88) r j ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( χ ( z ) − | ψ j ( z ) | dz (cid:12)(cid:12)(cid:12)(cid:12) + E X,I , where E X,I = (cid:90) I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) X χ ( z ) k (cid:88) j =1 | E j ( r, z ) | dz + ϕ (cid:48) ( + ir ) ϕ ( + ir ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dr. By Proposition 3.1 we have (cid:103)
Var
X,I ( b ) (cid:46) I (cid:37) ( λ ) R (cid:107) b (cid:107) + e R sys( X ) Vol(( X ) ≤ R ) (cid:107) b (cid:107) ∞ , and using that a is supported inside X ( Y ) we can compute that (cid:107) b (cid:107) ∞ ≤ (cid:107) a (cid:107) ∞ , and (cid:107) b (cid:107) ≤ (cid:107) a (cid:107) . Moreover, z (cid:55)→ χ ( z ) − (cid:88) r j ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( χ ( z ) − | ψ j ( z ) | dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:103) Var
X,I ( χ − (cid:46) I (cid:37) ( λ ) R (cid:107) χ − (cid:107) + e R sys( X ) Vol(( X ) ≤ R ) (cid:107) χ − (cid:107) ∞ . Now we have that a (cid:107) χ − (cid:107) ≤ (cid:107) a (cid:107) and a (cid:107) χ − (cid:107) ∞ ≤ (cid:107) a (cid:107) ∞ , so in the end (cid:103) Var
X,I ( a ) (cid:46) I (cid:37) ( λ ) R (cid:107) a (cid:107) + e R sys( X ) Vol(( X ) ≤ R ) (cid:107) a (cid:107) ∞ + a E X,I . Therefore we just need to estimate a E X,I . We have a E X,I ≤ (cid:107) a (cid:107) ∞ (cid:90) I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) X ( Y ) k (cid:88) j =1 | E j ( r, z ) | dz + ϕ (cid:48) ( + ir ) ϕ ( + ir ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dr. We use the Maass-Selberg relations (see [25, Section 2]).
Lemma 4.1 (Maass-Selberg relations) . Let s = + ir . Then for any y we have (cid:90) X ( Y ) k (cid:88) j =1 | E j ( s, z ) | dz = 2 k log Y − ϕ (cid:48) ( s ) ϕ ( s ) + Tr (cid:16) Y ir Φ ∗ ( s ) − Y − ir Φ( s )2 ir (cid:17) + (cid:90) X \ X ( Y ) k (cid:88) j =1 | Π ∗ Y E j ( s, z ) | dz As the scattering matrix Φ( s ) is unitary when Re ( s ) = 1 /
2, we have in this case | Tr Φ( s ) | ≤ k . Hence by the linearity of the trace (cid:12)(cid:12)(cid:12)(cid:12) Tr (cid:16) Y ir Φ ∗ ( s ) − Y − ir Φ( s )2 ir (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) = | sin(2 r log Y ) | r | Tr Φ( s ) | ≤ | sin(2 r log Y ) | r k, using that Tr Φ( s ) ∗ = Tr Φ( s ). Moreover, as s = + ir , we have for all z = x + iy ∈ X \ X ( Y )that | Π ∗ Y E j ( s, z ) | (cid:46) I e − πy where the implied constant depends on the spectral interval I (see for example Iwaniec [9,(6.20)]).We thus have for Y ≥ Y a a E X,I (cid:46) I (cid:107) a (cid:107) ∞ (cid:18) k log Y + sin(2 r log Y ) r k + k e − πY (cid:19) (cid:46) I (cid:107) a (cid:107) ∞ (cid:0) k log Y + k e − πY (cid:1) where we used that Vol( X \ X ( Y )) ≤ k .5. Proof of the spectral convergence
We show in this section the following level aspect analogue of the Weyl law:
Theorem 5.1.
Let X n be a sequence of finite area hyperbolic surfaces BS-converging to theplane H and such that the length of the shortest closed geodesic (the systole) is uniformlybounded from below by a constant. Then N ( X n , I ) + M ( X n , I ) ∼ Vol( X n ) , when n → + ∞ . Proposition 5.2.
Let t > . Then lim n →∞ X n ) ∞ (cid:88) j =0 e − tλ ( n ) j + 14 π (cid:90) + ∞−∞ − ϕ (cid:48) n ϕ n (cid:18)
12 + ir (cid:19) e − t ( + r ) dr = 14 π (cid:90) R e − t (1 / (cid:37) ) tanh( π(cid:37) ) (cid:37) d(cid:37), where ϕ n is the determinant of the scattering matrix associated with X n .Proof of Proposition 5.2. The proof is based on Selberg trace formula for finite area hyper-bolic surfaces (See [9, Chapter 10]). One of the main difficulties is to deal with conjugacyclasses of parabolic elements, corresponding to cusps. The idea is to use a cut-off at a height Y in the cusps and to compute the truncated trace spectrally and geometrically. Divergingterms in Y then cancel each other and what remains is the final trace formula. The divergingterms only come from the parabolic classes and so we will use the final form of the traceformula [9, Theorem 10.2] for every term apart from the ones corresponding to hyperbolicelements in Γ − { id } (denoted by H n ), whose treatment does not require any cut-off. Forthe hyperbolic terms instead of using the final form as a sum over closed geodesics, we revertto the integral of a kernel, to which we can apply BS-convergence.We have the formula (cid:88) j h t ( r j ) + 14 π (cid:90) ∞−∞ h t ( r ) − ϕ (cid:48) n ϕ n ( 12 + ir ) dr = | F n | π (cid:90) ∞−∞ h t ( r ) r tanh( πr ) dr + (cid:88) γ ∈H n (cid:90) F n k t ( d ( z, γz )) dµ ( z )+ h t (0)4 Tr (cid:18) I − Φ n (cid:0) (cid:1)(cid:19) − | C n | g t (0) log 2 − | C n | π (cid:90) ∞−∞ h t ( r ) ψ (1 + ir ) dr. Here k t is the heat kernel, h t ( r ) = e − t ( + r ) its Selberg transform and g t = (cid:98) h t theFourier transform, F n is a fundamental domain and | C n | is the number of inequivalentcusps, ψ ( s ) = Γ (cid:48) ( s ) / Γ( s ), and Φ n ( s ) is the scattering matrix (See [9] for background). ForRe( s ) = the scattering matrix Φ n ( s ) is unitary (see [9, Theorem 6.6]) and its rank is equalto the number of cusps, so the term Tr (cid:0) I − Φ n (cid:0) (cid:1)(cid:1) is controlled by the number of cusps UANTUM ERGODICITY FOR EISENSTEIN SERIES 17 | C n | . By BS-convergence we have | C n || F n | → n → + ∞ . Indeed recall that all cusps areisometric and let R >
0, then there exists a Y = Y ( R ) such that the cuspidal zones areincluded in the R -thin part of X n , i.e. F n ( Y ) c ⊂ ( F n ) ≤ R and by BS-convergence we thus have | F n ( Y ) c || F n | ≤ | ( F n ) ≤ R || F n | → n → + ∞ . On the other hand | F n ( Y ) c | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:91) b ∈ C n F b ( Y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | C n | | F c ( Y ) | , where on the right-hand side we can take any arbitrary cuspidal zone as they all have thesame area. We deduce that | C n || F n | (cid:46) R | ( F n ) ≤ R || F n | → . The treatment of the hyperbolic terms follows exactly the proof of the compact case:Proposition 9.5 in [11]. Using the heat kernel estimate k t ( (cid:37) ) (cid:46) t e − (cid:37)/ (8 t ) we can show thatfor any R > | F n | (cid:88) γ ∈H n (cid:90) F n k n ( d ( z, γz )) dµ ( z ) = O (cid:32) e − R sys( X n ) (cid:33) + O (cid:18) X n ) Vol(( X n )
Let us now discuss in detail how we can prove quantum ergodicity for eigenfunctions ofthe Laplacian on random surfaces in large genus limit. We will focus here on the compactsetting as the theorems we use have been done for this setting. However, the ideas translateto the non-compact setting of genus g surfaces with k ( g ) cusps, as long as there is a limiton the growth of the number of cusps: k ( g ) = o ( g ), as g → ∞ , which we will discuss at theend.6.1. Compact case.
Let us first address the compact case of Theorem 1.5. Here we use thenotation M g = M g, for the moduli space of compact hyperbolic surfaces and P g = P g, theassociated probability measure arising from the Weil-Petersson volume on the Teichm¨ullerspace of genus g compact Riemann surfaces. Theorem 6.1 (Theorem 1.5, compact surfaces) . Fix a compact interval I ⊂ (1 / , + ∞ ) .Then for P g random surface X ∈ M g the probability that for any a ∈ L ∞ ( X ) we have Var
X,I ( a ) (cid:46) I g (cid:107) a (cid:107) ∞ converges to as g → ∞ . The proof of this follows from Theorem 1.2 together with the proof of Theorem 1.3 on thespectral convergence by inspecting the geometric parameters and referring to the followingknown properties of their asymptotic behaviour in large genus. These are captured in thefollowing Lemma:
Lemma 6.2 (Mirzakhani [15], Monk [16]) . Let A g ⊂ M g be the set of surfaces satisfyingthe following three conditions: (i) There exists a constant c > such that Vol(( X ) ≤ log( g ) / )Vol( X ) ≤ cg − / (ii) The injectivity radius of X satisfies: inj X ≥ g − (log g ) (iii) The spectral gap of X satisfies λ ( X ) > (cid:16) log 2 π + log 2 (cid:17) / . Then P g ( A g ) → as g → ∞ . We remark that (i) was proved by Mirzakhani [15, Section 4.4], but Monk [16] correctedan error in the proof and gave a rate, (ii) is done by Mirzakhani [15] and Monk [16] with therate stated here. Result (iii) was proved by Mirzakhani [15, Theorem 4.8]. The cited resultfor (iii) is on Cheeger isoperimetric constant X in large genus and (iii) follows directly fromCheeger’s inequality. There is no explicit rate written in [15] but it should be possible toextract one from the proof.Then for any surface in A g Monk established the following:
Lemma 6.3 (Monk [16], Theorem 5) . Let I ⊂ (1 / , + ∞ ) be a compact interval. Then forany large enough g and for any X ∈ A g in Lemma 6.2 we have N ( X, I )Vol( X ) = 14 π (cid:90) ∞ / I ( λ ) tanh( π (cid:112) λ − / dλ + R ( X, I ) , where − C I (log g ) − / ≤ R ( X, I ) ≤ C I (cid:16) log log g log g (cid:17) / . Let us now look at how to prove Theorem 6.1 using Lemmas 6.2 and Lemma 6.3.
Proof of Theorem 6.1.
Fix I ⊂ (1 / , + ∞ ) a compact interval and g ∈ N . It is enough tojust fix X ∈ A g and prove Var X,I ( a ) ≤ ε g (cid:107) a (cid:107) ∞ by Lemma 6.2 as P g ( A g ) → g → ∞ .As there is no continuous spectrum in the compact case, by Theorem 1.2 (or already bythe main result of [11] done in the compact setting) there exists R I > R > R I and any a ∈ L ∞ ( X ), we have N ( X, I ) Var
X,I ( a ) (cid:46) I (cid:37) ( λ ( X )) R (cid:107) a (cid:107) + (cid:18) e R inj X Vol(( X ) ≤ R ) (cid:19) (cid:107) a (cid:107) ∞ , UANTUM ERGODICITY FOR EISENSTEIN SERIES 19 where (cid:37) ( λ ( X )) is a continuous function of the spectral gap of X . Now as X ∈ A g and λ ( X ) ≥ c for a constant c > X , and (cid:37) is a continuous function, we canbound (cid:37) ( λ ( X )) ≥ C uniformly over g . Hence using the properties of every X ∈ A g withboth Lemma 6.2 on the geometric parameters, and Lemma 6.3 on N ( X, I ) with R g := log( g ) / c I = π (cid:82) ∞ / I ( λ ) tanh( π (cid:112) λ − / dλ thatVar X,I ( a ) (cid:46) I (cid:37) ( λ ( X )) R g (cid:107) a (cid:107) ∞ c I + R ( X, I ) + (cid:18) e R g inj X Vol(( X ) ≤ R g )( c I + R ( X, I )) Vol( X ) (cid:19) (cid:107) a (cid:107) ∞ (cid:46) I (cid:107) a (cid:107) ∞ log( g ) + (cid:32) g / g − (log g ) (log g ) / g − / (cid:33) (cid:107) a (cid:107) ∞ = O (cid:16) g ) (cid:107) a (cid:107) ∞ (cid:17) as claimed. (cid:3) Non-compact case.
Theorem 6.1 translates to a version for X ∈ M g,k ( g ) if weassume that for any α > k ( g ) α /g → g → ∞ . The main difference with the proofof the compact case (Theorem 6.1) for P g,k ( g ) random X ∈ M g,k ( g ) comes in handling thecomponents N ( X, I ) + M ( X, I ) in Theorem 1.3 and K ( g ) := (2 k ( g ) log Y g + k ( g ) e − πY g ) from the estimate of Theorem 1.2:( N ( X, I ) + M ( X, I )) Var
X,I ( a ) (cid:46) I (cid:37) ( λ ( X )) R g (cid:107) a (cid:107) + (cid:32) e R g inj X ( Y g ) Vol(( X ) ≤ R g ) + K ( g ) (cid:33) (cid:107) a (cid:107) ∞ , for a ∈ L ∞ Y g ( X ), where Y g = log g .Firstly, we need an analogue of Lemma 6.3 that gives N ( X, I ) + M ( X, I ) ∼ I Vol( X )when g → ∞ . This will allow us to deal with the terms K ( g ) as K ( g )Vol( X ) = o (1)as g → ∞ by the assumptions on the growth of k ( g ) in Theorem 1.5(ii) and that Vol( X ) isof the order g + k ( g ) when g → ∞ .Secondly, to deal with the terms1 (cid:37) ( λ ( X )) R g and e R g inj X ( Y g ) Vol(( X ) ≤ R g )Vol( X )we need an analogue of the geometric Lemma 6.2 for non-compact surfaces. Here we notethat the proofs of these properties in [15] and [16] rely on estimates on the Weil-Peterssonvolumes V g,n . Here the estimates are independent of k ( g ) as long as we have the growthcondition on k ( g ) we assumed in Theorem 1.5. Acknowledgements
We thank Laura Monk, Paul Nelson, Jean Raimbault, and Abhishek Saha for usefuldiscussion related to the topic.
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UANTUM ERGODICITY FOR EISENSTEIN SERIES 21
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