Antonia Jabbour

S T Éphane

Zeina Yassine
email: zeina.yassine@u-pec.fr

A SYSTOLIC-LIKE EXTREMAL GENUS TWO SURFACE

Keywords: 2010 Mathematics Subject Classification. Primary 53C23; Secondary 58E10 systole, systolic-like inequalities, extremal metrics, calibration, genus two surface

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Sharp systolic bounds on negatively curved surfaces

Introduction

We are interested in optimal geometric inequalities relating the area of a closed Riemannian surface Σ to the lengths of the shortest loops in certain homotopy classes. A typical example is given by the systolic inequalities, which relate the area of the surface to the systole, that is, the length of the shortest noncontractible loop in Σ. The first known systolic inequality is due to C. Loewner in 1949, who proved that every Riemannian two-torus (T 2 , g) satisfies area(g)

√ 3 2 sys 2 (g),
where sys(g) denotes the systole of the torus. Furthermore, the equality holds if and only if the torus is endowed with a flat hexagonal metric. C. Loewner did not publish his result, however it was mentioned by his student P. Pu, cf. [START_REF] Pu | Some inequalities in certain non-orientable Riemannian manifolds[END_REF], who established a sharp systolic inequality on the projective plane RP 2 . In this case, the equality is attained precisely by the round metrics. More than thirty years later, C. Bavard, cf. [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF] (see also [START_REF] Sakai | A proof of the isosystolic inequality for the Klein bottle[END_REF]) proved a sharp systolic inequality on the Klein bottle K.

Here, the extremal metrics are not smooth. These are the only manifolds for which an optimal systolic inequality is known. All these systolic inequalities can be written as follows. If sys(g) sys(g 0 ) then area(g) area(g 0 ) (1.1) where g 0 is the extremal metric in each of these cases.

Optimal systolic-like inequalities, i.e., inequalities that relate the area to the product of the lengths of the shortest loops or arcs in different relative homotopy classes are known only for the two-torus T 2 , cf. [START_REF] Keen | An extremal length on a torus[END_REF], the Klein bottle K, cf. [START_REF] Bavard | Une remarque sur la géométrie systolique de la bouteille de Klein[END_REF][START_REF] El Mir | Conformal geometric inequalities on the Klein bottle[END_REF], and the Mobius band M, cf. [START_REF] Pu | Some inequalities in certain non-orientable Riemannian manifolds[END_REF][START_REF] Blatter | Zur Riemannschen Geometrie im Grossen auf dem Möbiusband[END_REF][START_REF] Bavard | Une remarque sur la géométrie systolique de la bouteille de Klein[END_REF][START_REF] El Mir | Conformal geometric inequalities on the Klein bottle[END_REF].

For instance, the following result follows as a particular case of L. Keen's optimal systolic-like inequality for the two-torus, cf. [START_REF] Keen | An extremal length on a torus[END_REF]. Let (T 2 , g 0 ) be a flat torus. Denote by sys 1 (g) and sys 2 (g) the lengths of the shortest loops on T 2 inducing a basis in homology If sys 1 (g) sys 1 (g 0 ) sys 2 (g) sys 2 (g 0 ) then area(g) area(g 0 ).

About ten years ago, M. Katz and the first author, cf. [START_REF] Katz | An optimal systolic inequality for CAT(0) metrics in genus two[END_REF], proved that there exists a piecewise flat metric g 0 on the genus two surface Σ 2 which is extremal among all nonpositively curved Riemannian metrics. This metric is composed of six identical flat regular octagons Ω k with the identifications given in Figure 1. It admits regions where only one systolic loop, i.e., a noncontractible loop of length the systole, passes through every point. Hence this metric cannot be extremal for the general systolic inequality, i.e., without restriction on the curvature. This follows from a classical result in the field which says that at least two systolic loops pass through every point of a systolically extremal surface. In fact, the non-extremality of the metric g 0 for the systolic inequality can also be deduced from a result of [START_REF] Sabourau | Systoles des surfaces plates singulières de genre deux[END_REF], which says that no flat metric with conical singularities is extremal for the systolic inequality in genus two.

We need to introduce a few definitions related to the structure of this piecewise flat surface.

Definition 1.1. Simple closed geodesics of (Σ 2 , g 0 ) parallel to the edges of the octagons Ω k fall into two categories of curves of the same length: curves passing through opposite sides of the octagons and curves not passing through opposite sides of the octagons. The simple closed geodesics of the former category of curves will be referred to as 1 -loops. Similarly, the simple closed geodesics of the latter category will be referred to as 2 -loops. Note that the 1 -loops are the systolic loops of g 0 . However the 2 -loops are not the second shortest noncontractible loops of g 0 since there exist isolated shorter noncontractible loops which are not systolic. Definition 1.2. Denote by Λ i the subset of the free homotopy classes generated by the i -loops. For a Riemannian metric g on Σ 2 , define

i (g) = inf γ ∈Λ i g (γ).
(1.2)

Note that 1 (g 0 ) = sys(g 0 ).

We observe that the piecewise flat metric g 0 defined on the genus two surface Σ 2 may be a potential extremal metric for another systolic problem on Σ 2 involving 1 and 2 . The reason for such a claim follows from some geometric properties of the surface (Σ 2 , g 0 ):

Figure 1. The critical surface (Σ 2 , g 0 ) is isometric to the to the piecewise flat genus two surface composed of six identical regular octagons with the identifications given in the figure .   • First, as mentioned before, the systolic loops of a systolically extremal surface cover the surface. In our case, the 1 -loops and the 2 -loops cover the surface Σ 2 . Here, by an i -loop we mean a loop of length i (g 0 ) in (Σ 2 , g 0 ) whose free homotopy class lies in Λ i . • Second, the unit tangent vectors of these i -loops are well distributed on each tangent plane of the surface. More precisely, their convex hull forms a regular octagon on these tangent planes. In general, a similar pattern occurs for systolically extremal surfaces. Indeed, the convex hull of the unit tangent vectors of the systolic loops on all the known systolically extremal surfaces is symmetric. • Finally, systolically extremal surfaces of genus at least two tend to have flat regions, cf. [START_REF] Calabi | Extremal isosystolic metrics for compact surfaces, Actes de la Table Ronde de Géométrie Différentielle[END_REF][START_REF] Bryant | On extremals with prescribed lagrangian densities, Manifolds and geometry[END_REF], and the same applies for systoliclike inequalities.

In [START_REF] Calabi | Extremal isosystolic metrics for compact surfaces, Actes de la Table Ronde de Géométrie Différentielle[END_REF], E. Calabi described two piecewise flat genus three surfaces and conjectured that one of them is the global minimum for the systolic inequality. Later, the first author, cf. [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF], proved that these two genus three surfaces are critical for the systolic inequality with respect to slow metric variations. Of course, the systolic volume functional is not necessarily differentiable and an adequate notion of criticality needs to be introduced. The notion of criticality used in [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] was introduced in [Na96, EI00] to study the Riemannian surfaces that maximize the product of the area with the first nonzero eigenvalue of the Laplacian. Other notions of systolically critical metrics were used in [START_REF] Balacheff | Sur la systole de la sphère au voisinage de la métrique standard[END_REF][START_REF] Balacheff | A local optimal diastolic inequality on the two-sphere[END_REF][START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF][START_REF] Paiva | Contact geometry and isosystolic inequalities[END_REF].

In this article we show that the metric g 0 on the genus two surface Σ 2 is critical in the sense of [START_REF] Nadirashvili | Berger's isoperimetric problem and minimal immersions of surfaces[END_REF][START_REF] El Soufi | Riemannian manifolds admitting isometric immersions by their first eigenfunctions[END_REF][START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] for slow metric variations for some systolic-like inequality. More precisely, we prove Theorem 1.3. Let (g t ) be a slow metric variation of g 0 defined on the genus two surface Σ 2 . If 1 (g t ) 1 (g 0 ) and 2 (g t ) 2 (g 0 ), then area(g t ) area(g 0 ) + o(t).

The slow metric variations involved in Theorem 1.3 are analogous to those defined in [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF]. We refer to Definition 4.1 for a precise definition of these metric variations and to the last section of [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] for examples. Observe for instance that deforming the regular octagons composing the extremal surface (Σ 2 , g 0 ) into non-regular octagons gives rise to a slow metric variation.

With the exception of some recent proofs of systolic inequalities, cf. [START_REF] Ivanov | On two-dimensional minimal fillings[END_REF][START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF], and systolic-like inequalities, cf. [START_REF] Sabourau | Optimal systolic inequalities on Finsler Mobius bands[END_REF], on the projective plane RP 2 , the two-torus T 2 and the Mobius band M, all the other known proofs of the aforementioned optimal geometric inequalities require the uniformization theorem as a main tool. In our proof of Theorem 1.3, we do not make use of the uniformization theorem. Instead, we rely on recent calibrating methods, cf. [START_REF] Ivanov | On two-dimensional minimal fillings[END_REF][START_REF] Ivanov | Filling minimality of Finslerian 2-discs[END_REF][START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF]. Briefly, we embed the universal cover Σ 2 of the genus two surface Σ 2 into an infinite-dimensional space R ∞ as follows. Given that

i (g) i (g 0 ), we construct a 1-Lipschitz equivariant map Ψ g : Σ 2 → R ∞
using the Busemann functions induced by the lifts of the i -loops of Σ 2 , cf. Section 3. Then, we introduce an appropriate infinitesimally calibrating two-form ω on R ∞ whose pull-back passes to the quotient on the surface Σ 2 . This allows us to show that

Σ 2 (Ψ g ) * ω c 0 area(Σ 2 , g)
for every Riemannian metric g on Σ 2 , where c 0 is a sharp positive constant. Moreover, the equality holds if g = g 0 . Finally, we prove that if (g t ) t 0 is a slow metric variation then

Σ 2 (Ψ gt ) * ω - Σ 2 (Ψ g 0 ) * ω = o(t).
This completes the proof, cf. Section 4.

Acknowledgement. The authors are grateful to thank Barton Zwiebach for pointing out an inacurracy in the identifications of the edges of the octagons in the description of the critical metric.

An extremal piecewise flat metric in genus two

In this section we provide a description for the critical piecewise flat genus two surface and we introduce some notations and definitions.

The Bolza surface B is a genus two Riemann surface. It is the smooth completion of the smooth affine algebraic curve

y 2 = x 5 -x.
(2.1)

The set {-1, 0, 1, -i, i, ∞} of roots of the polynomial (2.1) (including the point at infinity) corresponds to the set of vertices of the regular octahedral triangulation of the Riemann sphere S 2 under the conformal stereographic projection. Hence, these six points can be identified with the ramification points of the ramified conformal double cover Q : B → S 2 , or equivalently, with the Weierstrass points of B.

The conformal class of the Bolza surface B admits a piecewise flat nonpositively curved Riemannian metric g 0 with 16 conical singularities whose orientation-preserving isometry group is isomorphic to the automorphism group of B. This metric g 0 has been introduced by M. Katz and the first author in [START_REF] Katz | An optimal systolic inequality for CAT(0) metrics in genus two[END_REF], where they prove that it is extremal for the systolic inequality among all nonpositively curved metrics on the genus two surface Σ 2 . The metric g 0 , defined up to homothety, on Σ 2 is isometric to the piecewise flat genus two surface composed of six identical flat regular octagons Ω k with the identifications given in Figure 1. The Weierstrass points of B correspond to the centers of the octagons and the 16 conical singularities are located at their vertices.

Denote by h the length of an edge of a regular octagon Ω k of g 0 . We have

area(g 0 ) = 12(1 + √ 2)h 2 . The systole 1 (g 0 ) of the surface (Σ 2 , g 0 ) is computed in [KS06, Lemma 3.2].
Namely,

1 (g 0 ) = 2(1 + √ 2)h. A direct computation shows that 2 (g 0 ) = 4(2 + √ 2)h.
Recall that an i -loop is a loop in (Σ 2 , g 0 ) of length i (g 0 ) and whose free homotopy class lies in Λ i = Λ i (g 0 ), cf. Definition 1.1 and Definition 1.2. We define an i -band as follows.

Definition 2.1. Let α be a free homotopy class in Λ i . The i -loops in α are parallel to each other and form a flat cylinder B α of height h

if i = 1 and √ 2 2 h if i = 2
. Such a cylinder B α will be called an i -band of (Σ 2 , g 0 ). The soul of an i -band B α is the i -loop of B α equidistant from the boundary components of the i -band.

The intersections of the i -bands decompose the surface Σ 2 into 150 polygons, cf. Figure 2. These polygons define four regions R 1 , ..., R 4 depending on the number of i -loops that pass through every point in their interior. More precisely, such a polygon ∆ k lies in a region R k if through every point in its interior pass exactly 2k oriented 1 -loops and 8 -2k oriented 2 -loops, cf. Figure 2. Hence, every region R k with k = 4 is composed of exactly 48 identical polygons ∆ k (with 8 polygons in each octagon), while the region R 4 is composed of only 6 identical polygons ∆ 4 . Observe that the polygons ∆ 1 , ∆ 2 , ∆ 3 and ∆ 4 have the shape of a right isosceles triangle, a kite, a right isosceles triangle and a (small) octagon. Furthermore, the right isosceles triangles ∆ 1 are the only polygons which have an edge in common with the six regular flat octagons forming the critical surface (Σ 2 , g 0 ), namely their hypothenuses coincide with the edges of the regular octagons Ω k . Fix an orientation on Σ 2 . Let Ω be one of the six regular octagons composing the surface (Σ 2 , g 0 ). Denote by ( Σ 2 , g 0 ) the universal cover of (Σ 2 , g 0 ).

Let Ω be a lift of Ω in the universal cover Σ 2 and let E be an (oriented) edge of the regular octagon Ω. Moreover, let ∆ k be a lift of the polygon ∆ k in Σ 2 . The souls of the i -bands lift to Σ 2 as follows.

Definition 2.2. An 1 -direction of ( Σ 2 , g 0 ) is a g 0 -unit vector ζ = ζ E Ω based
at the center of Ω and pointing in the same direction as E, cf. Definition 2.3. An 2 -direction of ( Σ 2 , g 0 ) is a g 0 -unit vector ξ = ξ E Ω based at x E and pointing in the same direction as the edge E, cf. Figure 2. This unit vector ξ induces a geodesic line c ξ on ( Σ 2 , g 0 ) such that

c ξ (0) = ξ.
The projection c ξ of c ξ to (Σ 2 , g 0 ) is a closed geodesic curve which coincides with the soul of the 2 -band B c ξ .

Remark 2.4. In what follows, we will replace the symbols ζ and ξ by ν when there is no need to distinguish between an 1 -direction and an 2 -direction. Moreover, the indices Ω and E in ν E Ω will be omitted.

Busemann functions and calibrating forms

In this section, we introduce some notations and preliminary results that will be useful in our proof of Theorem 1.3. Definition 3.1. Let ν be an i -direction of ( Σ 2 , g 0 ). Extend ν to a map, still denoted by ν : R → U g 0 Σ 2 defined as ν(s) = c ν (s). In particular, ν(0) = ν.

Along with the metric g 0 , consider another Riemannian metric g defined on Σ 2 and denote by g its lift to Σ 2 . Throughout the article, we will assume that 1 (g) 1 (g 0 ) and 2 (g) 2 (g 0 ). Definition 3.2. Let ν be a g 0 -unit vector tangent to a geodesic line induced by an i -direction of ( Σ 2 , g 0 ). Define the Busemann function b

g ν : Σ 2 → R as b g ν (x) = lim sup t→∞ d g(x, c ν (t)) i (g 0 ) -t g ( c ν )
where c ν is the g 0 -geodesic line induced by ν and c ν is its projection to Σ 2 .

Here, g ( c ν ) denotes the smallest length of a noncontractible loop in the free homotopy class c ν of c ν with respect to the Riemannian metric g. Remark 3.3. Contrary to the original definition of a Busemann function, cf. [START_REF] Bridson | Metric Spaces of Non-Positive Curvature[END_REF], the line c ν is geodesic for g 0 but not for g. Moreover, the function

h : R → R t → d g(x, c ν (t)) i (g 0 ) -t g ( c ν )
which appears in Definition 3.2, is not necessarily non-increasing.

Proposition 3.4. The real function h is bounded.

Proof. By the triangle inequality, we obtain h(t) d g( c ν (0), c ν (t)) i (g 0 ) -t g ( c ν ) -d g(x, c ν (0)) i (g 0 ).

(3.1)

Let β be a length-minimizing loop of (Σ 2 , g) in the homotopy class of c ν , that is,

g (β) = g ( c ν ). Let [t] i be the integer part of t i (g 0 ) , that is, [t] i = max{k ∈ N | t k i (g 0 )}.

We have

[t] i i (g 0 ) -ti (g 0 ).

(3.2) Since the genus two surface is orientable, the [t] i -iterate of β is lengthminimizing in its homotopy class with respect to g, cf. [Gr99, Lemma 4.32]. Let γ in Σ 2 be a length-minimizing arc connecting c ν (0) to c ν ([t] i ). Its projection γ to Σ 2 is homotopic to c

[t] i ν and its length satisfies the two relations

g (γ) = d g( c ν (0), c ν ([t] i )) and g (γ) [t] i g ( c ν ).
(3.3) Combining (3.1), (3.2) and (3.3), we deduce that the map h is bounded from below. Now, we show that the map h is bounded from above. Indeed, by the triangle inequality, we derive that

h(t) d g (x, c ν (0)) + d g ( c ν (0), c ν ([t] i )) + d g ( c ν ([t] i ), c ν (t)) -[t] i g ( c ν ) i (g 0 ).
Moreover,

d g ( c ν (0), c ν ([t] i )) -[t] i g ( c ν ) 2 diam(Σ 2 , g). (3.4)
Indeed, let γ be a g-length minimizing loop in the free homotopy class c ν , that is, g (γ) = g ( c ν ). Let α be a length-minimizing arc joining c ν (0) to γ. We deduce that

d g ( c ν (0), c ν ([t] i )) g (α ∪ γ [t] i ∪ α -1 ) g (γ [t] i ) + diam(Σ 2 , g).
Then, the inequality (3.4) is satisfied. Furthermore,

d g ( c ν ([t] i ), c ν (t)) diam(Σ 2 , g).
This shows that h is bounded from above.

Definition 3.5. The map Ψ g ν : Σ 2 → R induced by an i -direction ν, cf. Definitions 2.2 and 2.3, is defined as

Ψ g ν (x) = 1 i (g 0 ) g ( c ν ) R/ i (g 0 )Z b g ν(s) (x) -b g -ν(s) (x) 2 ds,
where ν(s) = c ν (s), cf. Definition 3.1.

Example 3.6. If g = g 0 , the g 0 -gradient ∇ x Ψ g 0 ν is a g 0 -unit tangent vector parallel to and pointing in the same direction as the i -direction ν.

Proposition 3.7. Let ν be an i -direction of ( Σ 2 , g 0 ). The map Ψ g ν satisfies the following properties:

(1) It is 1-Lipschitz and differentiable almost everywhere.

(2) Ψ g -ν = -Ψ g ν .

(3) The differential of the map Ψ g ν depends only on the oriented i -band B generated by the vector ν.

Proof. For (1), the map Ψ g ν is 1-Lipschitz for the supremum norm as the Busemann function b g ν is i (g 0 )-Lipschitz and i (g) i (g 0 ). Hence, it is differentiable almost everywhere by the Rademacher theorem. Part (2) follows directly from Definition 3.5. For (3), let ν and η be two i -directions generating the same oriented i -band. That is, their basepoints lie in the same i -band and the vectors point in the same direction. Recall that ν(s) = c ν (s), cf. Definition 3.1, and the same with η. The definition of i -directions and the assumption on the vectors ν and η imply that

η(s) = ν s + k n i (g 0 )
for some integer k with n = 2 if i = 1 and n = 8 if i = 2. We derive easily from Definition 3.5 that dΨ g η = dΨ g ν . This completes the proof.

Let Γ be the deck transformation group of Σ 2 . The g 0 -geodesic line c ν induces a unique element γ ν in Γ that leaves c ν globally invariant and such that γ ν . c ν (s) = c ν (s + i (g 0 )).

Moreover, we define an action of Γ on the set of unit tangent vectors to the geodesic line c ν as γ ν(s) = (γ. c ν ) (s) where γ ∈ Γ.

We state now the following properties of the map Ψ g ν .

Proposition 3.8. Let ν be an i -direction of ( Σ 2 , g 0 ). The map Ψ g ν : Σ 2 → R satisfies the following properties:

(1) It is equivariant, that is, for every γ in Γ,

Ψ g γ ν (γ.x) = Ψ g ν (x).
(2) It passes to the quotient by the cyclic subgroup γ ν and induces a map

Ψ g ν : Σ 2 / γ ν → R/ i (g 0 )Z defined as Ψ g ν (γ ν .x) ≡ Ψ g ν (x) mod i (g 0 ).
Proof. Let x ∈ Σ 2 and γ ∈ Γ. Since γ is an isometry and γ. c ν = c γ.ν , we derive

g ( c γ ν ) = g ( c ν ). Moreover, since the distance d g induced by the Riemannian metric g is Γ- invariant, we derive that b g γ ν (γ.x) = b g ν (x). Hence, the desired result Ψ g γ * ν (γ.x) = Ψ g ν (x). Now, since γ -1 ν . c ν (t) = c ν (t -i (g 0 )), we derive from the Γ-invariance of d g that d g(γ ν .x, c ν (t)) = d g(x, γ -1 ν . c ν (t)) = d g(x, c ν (t -i (g 0 ))). Hence, b g ν (γ ν .x) = lim sup t→∞ i (g 0 )d g(x, c ν (t -i (g 0 ))) -t g ( c ν ) = b g ν (x) -i (g 0 ) g ( c ν ). We deduce that Ψ g ν (γ ν .x) = Ψ g ν (x) -i (g 0 ).
(3.5) Remark 3.9. As a consequence of Proposition 3.8 and the relation (3.5), we deduce the following couple of points:

(1) γ * (dΨ g γ ν ) = dΨ g ν for every γ ∈ Γ, (2) The map Ψ g ν -Ψ g 0 ν passes to the quotient by the cyclic subgroup γ ν and induces a real function which will be (improperly) denoted by

Ψ g ν -Ψ g 0 ν : Σ 2 / γ ν → R. By definition, this function satisfies (Ψ g ν -Ψ g 0 ν )(x) = Ψ g ν (x) -Ψ g 0 ν (x).
Actually, these two properties hold if we replace Ψ g ν with the Busemann function b g ν . Let Ω be a regular octagon of the universal cover ( Σ 2 , g 0 ) of the surface (Σ 2 , g 0 ). We order the (oriented) edges (E j ) 1 j 8 of Ω with respect to the cyclic order induced by the orientation of Ω by fixing an initial edge E 1 . Denote ν j = ν E j Ω . We state the following definition.

Definition 3.10. Let R ∞ = Π Ω R 8 be the infinite product of R 8 where Ω runs over all the regular octagons of ( Σ 2 , g 0 ). Let

Ψ g : Σ 2 → R ∞
be the map defined as

π Ω • Ψ g | Ω : Ω → R 8 x → (Ψ g ν j (x)) 8 j=1 where π Ω : R ∞ → R 8
is the canonical projection to the factor R 8 of R ∞ corresponding to Ω. The vector ν j is the i -direction that is parallel and point in the same direction as the g 0 -unit vector based at x and tangent to the lift of an oriented i -loop.

Example 3.11. Consider the regular octagon Ω in Figure 2. Suppose that the numbering of the edges of ∂ Ω follows a cyclic order starting at E, then we have Ψ g

| ∆ 2 = (Ψ g ζ 1 , Ψ g ζ 2 , Ψ g ξ 3 , Ψ g ξ 4 , Ψ g ζ 5 , Ψ g ζ 6 , Ψ g ξ 7 , Ψ g ξ 8 )
where ∆ 2 is the gray polygon in Figure 2.

Remark 3.12. Note that the map Ψ g is not well defined in the lift of the boundaries of the polygons ∆ since there is some ambiguity in the choice of ν j . Definition 3.13. Consider the two-form

ω 0 = 8 j=1 dx j ∧ dx j+2
in R 8 . We define a two-form ω on Σ 2 as the pull-back of ω 0 by the map

π Ω • Ψ g | Ω : Ω → R 8
on every regular octagon Ω. It can be expressed on Ω as

ω = 8 j=1 dΨ g ν j ∧ dΨ g ν j+2
where ν j = ν j Ω and all indices j are taken modulo 8. This defines a two-form on Σ 2 that we still denote by ω.

Note that the two-form ω in Definition 3.13 is defined almost everywhere since the maps Ψ g ν are almost everywhere differentiable, cf. Proposition 3.7.(1).

Proposition 3.14. The two-form ω passes to the quotient on Σ 2 .

Proof. Let γ ∈ Γ. Observe that γ preserves the octagonal decomposition of the universal cover ( Σ 2 , g 0 ) and its orientation. Therefore, for every regular octagon Ω, there exists a permutation σ = σ γ of {1, • • • , 8} such that

γ -1 (E j Ω ) = E σ(j) γ -1 ( Ω) . That is, γ -1 (ν j Ω ) = ν σ(j) γ -1 ( Ω)
.

Hence, dΨ g γ -1 ν j = dΨ g ν σ(j) . By the equivariance property of the map Ψ ν j , cf. Remark 3.9, we derive

γ * ω = 8 j=1 dΨ g γ -1 ν j ∧ dΨ g γ -1 ν j+2 = 8 j=1 dΨ g ν σ(j) ∧ dΨ g ν σ(j+2) .
Finally, since the permutation σ arises from an orientation-preserving isometry γ, the angle between ∇ x Ψ g ν σ(j) and ∇ x Ψ g ν σ(j+2) is equal to the angle between ∇ γ(x) Ψ g ν j and ∇ γ(x) Ψ g ν j+2 for every x ∈ Σ 2 at which ω is defined. Therefore, σ(j + 2) = σ(j) + 2. Reindexing the sum in the expression of the two-forms ω 0 and ω, we derive that γ * ω = ω.

Remark 3.15. The induced two-form on Σ 2 will be denoted by ω.

Proof of Theorem 1.3

We briefly restate the statement of Theorem 1.3. Let (g t ) be a slow metric variation of g 0 defined on Σ 2 . If 1 (g t ) 1 (g 0 ) and 2 (g t ) 2 (g 0 ), then area(g t ) area(g 0 ) + o(t).

Before proceeding to the proof, let us first define what we mean by a slow metric variation. Definition 4.1. A slow metric variation (g t ) t 0 of g 0 is a smooth deformation of g 0 (more precisely, g t is a Riemannian metric with conical singularities on M smoothly varying with t on M \{conical singularities of g 0 }) such that for every i -direction ν, the following condition is satisfied

||d(b gt ν -b gt -ν ) -d(b g 0 ν -b g 0 -ν )|| L 2 (Bν ) = o( √ t), (4.1)
where B ν is a fundamental domain in ( Σ 2 , g 0 ) of the band B ν = B cν .

Note that the map b gt ν -b g 0 ν passes to the quotient by the cyclic subgroup c ν , cf. Remark 3.9.(2). Remark 4.2. A metric variation (g t ) should have conical singularities as the Riemannian metric g 0 has conical singularities, cf. Section 2.

Definition 4.3. The F-functional of the map

Ψ g : Σ 2 → R ∞
introduced in Definition 3.10 is defined as

F(Ψ g ) = Σ 2 ω,
where ω is the quotient two-form on Σ 2 introduced in Section 3.

The proof of Theorem 1.3 follows directly from the next two propositions.

Proposition 4.4. The two-form ω is a calibrating form for the map Ψ g . That is, F(Ψ g ) 8 area(g)

with equality for g = g 0 .

Proof. Let x ∈ ∆ be a point at which the maps Ψ g ν j are differentiable. Let v be a g-unit vector in the tangent plane T x Σ 2 . Since the map Ψ g ν j is 1-Lipschitz, cf. Proposition 3.7, we derive that

| d x Ψ g ν j (v) | 1. Hence || ω || g = || 8 j=1 dΨ g ν j ∧ dΨ g ν j+2 || g 8.
The desired inequality directly follows.

Let D denote a fundamental domain of the surface (Σ 2 , g 0 ) in its universal cover made of regular octagons Ω. By Definition 4.3, the F-functional of the almost everywhere differentiable map Ψ g is defined as

F(Ψ g ) = Σ 2 ω = Ω∈D Ω 8 j=1 dΨ g ν j ∧ dΨ g ν j+2 .
When g = g 0 , the vectors ν j and ν j+2 form an oriented orthonormal basis. The same holds for the covectors d x Ψ g 0 ν j and d x Ψ g 0 ν j+2 from Example 3.6. Hence, d x Ψ g 0 ν j ∧ d x Ψ g 0 ν j+2 = d g 0 where d g 0 represents the area form of the Riemannian metric g 0 . In particular, we derive F(Ψ g 0 ) = 8 area(g 0 ) after passing to the quotient.

Remark 4.5. An analogous two-form ω along systolic directions could be defined on the genus three surface to recover the result of [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] without using the Euclidean metrics.

Proposition 4.6. Let (g t ) be a slow metric variation of g 0 . Then F(Ψ gt ) -F(Ψ g 0 ) = o(t).

Proof. Let ν = ν E

Ω

, where Ω is a regular octagon of ( Σ 2 , g 0 ) of side length h. By definition, an i -band B ν = B cν is the quotient of

B ν = {x ∈ Σ 2 | d g 0 (x, c ν ) λ}
by the cyclic subgroup c ν , where λ is equal to 1 2 h if i = 1 and equal to √ 2 4 h if i = 2. We still denote by ν the vector given by the systolic direction after passing to the quotient.

Recall that the differential dΨ g ν of the map Ψ g ν depends only on the oriented i -band B generated by the vector ν (or rather the oriented geodesic line c ν ), but not on ν, cf. Proposition 3.7.(3). Moreover, by passing to the quotient, we can rearrange the sum in Definition 4.3 to be taken over all the oriented i -bands B of (Σ 2 , g 0 ) as follows

F(Ψ g ) = B B dΨ g ν B ∧ dΨ g (ν B ) ⊥
where ν B is an i -direction that generates the i -band B and the vector (ν B ) ⊥ is an 1 -direction or an 2 -direction orthogonal to the boundary ∂B and whose basepoint lies in the same regular octagon Ω as the vector ν B . More precisely, if ν B points in the direction of an edge E j of Ω, then (ν B ) ⊥ points in the direction of the edge E j+2 , where the edges follow the cyclic order on ∂Ω.

In particular, we have

F(Ψ gt ) -F(Ψ g 0 ) = I 1 + I 2 ,
where

I 1 = B B d(Ψ gt ν B -Ψ g 0 ν B ) ∧ dΨ g 0 (ν B ) ⊥ -d(Ψ gt (ν B ) ⊥ -Ψ g 0 (ν B ) ⊥ ) ∧ dΨ g 0 ν B (4.2)
and

I 2 = B B d(Ψ gt ν B -Ψ g 0 ν B ) ∧ d(Ψ gt (ν B ) ⊥ -Ψ g 0 (ν B ) ⊥ ). (4.3)
First, we show that I 1 = 0. Recall that from Proposition 3.7.(2), we have

-Ψ g (ν B ) ⊥ = Ψ g -(ν B ) ⊥
and the same holds for Ψ. Moreover, since the oriented angle between the vectors ν B and (ν B ) ⊥ is the same as the one between the vectors -(ν B ) ⊥ and ν B , we can write

I 1 = 2 B B d(Ψ gt ν B -Ψ g 0 ν B ) ∧ dΨ g 0 (ν B ) ⊥

Figure 2 .

 2 Figure 2. A flat regular octagon Ω of the metric g 0 .

  Figure 2. The unit vector ζ induces a geodesic line c ζ on ( Σ 2 , g 0 ) such that c ζ (0) = ζ. The projection c ζ of c ζ to (Σ 2 , g 0 ) is a closed geodesic curve which coincides with the soul of the 1 -band B c ζ . Now, denote by x E the point of Ω lying in the perpendicular bisector of E and at distance √ 2 4 h from E. (Recall that h is the side length of Ω.) Note that the point x E lies in the right isosceles triangle ∆ 1 of Ω with hypotenuse E.

Partially supported by the French ANR project FINSLER .

after arranging the oriented bands B in the sum (4.2). Moreover, the oneforms dΨ g 0 (ν B ) ⊥ are constant over the bands B. Therefore, we derive

Now, let v be a tangent vector to the boundary ∂B. The vector v is parallel to the i -direction ν B . Moreover, by definition, the unit vectors ν B and (ν B ) ⊥ form an orthonormal basis. Since the g 0 -gradient of Ψ

for every i -direction ν B , cf. Example 3.6, and the same holds with (ν B ) ⊥ , we have dΨ

Then by Stokes' formula, we derive that

ν B is well defined as a real function and not merely as a map with values in the circle R/ i (g 0 )Z, cf. Remark 3.9.(2). Hence, the integral I 1 vanishes. Now, we need to show that I 2 = o(t). Note that this is the only place where we will make use of the slow metric variation assumption.

Let ν be an i -direction ν. From the definition of the function Ψ gt ν , cf. Definition 3.5, and after differentiation, we obtain

where the error term O(t) is uniformly bounded with respect to x. Hence,

. By definition of a slow metric variation, cf. Definition 4.1, we have

ν after some quotient identifications, cf. Remark 3.9.(2), we deduce

By the Cauchy-Schwartz inequality and after rearranging the terms in the sum as previously, the term I 2 defined in (4.3) satisfies the following upper bound

Combined with the relation (4.4), this implies that I 2 = o(t).

From Proposition 4.4 and Proposition 4.6, we can write area(g t ) -area(g 0 ) 1 8 (F(Ψ gt ) -F(Ψ g 0 )) = o(t)

which finishes the proof of Theorem 1.3.


Alexis Rosuel
email: alexis.rosuel@univ-eiffel.fr

Pascal Vallet
email: pascal.vallet@bordeaux-inp.fr

Philippe Loubaton
email: philippe.loubaton@univ-eiffel.fr

Cohérence spectrale et test de corrélation en grande dimension

Introduction

Détecter la présence de corrélation entre plusieurs séries temporelles est un problème fondamental en traitement du signal, dont les applications courantes concernent par exemple le traitement multi-capteurs, le radar ou encore les télécommunications.

Dans ce travail, nous considérons le cas où l'on observe M séries temporelles (y 1,n ) n∈Z , . . . , (y M,n ) n∈Z , supposées conjointement gaussiennes complexes circulaires et stationnaires. En notant r i,j (h) = E[y i,n+h y j,n ], on s'intéresse ici à déterminer la présence de corrélations entre les M séries, qui se formalise comme le test binaire dont l'hypothèse nulle est donnée par :

H 0 : r i,j (h) = 0 ∀ h ∈ Z, 1 ≤ i < j ≤ M. (1)
Le test (1) a fait l'objet de nombreux travaux dans le cadre des approches dites « temporelles », pour lesquelles des statistiques de test basées sur des estimées empiriques de la fonction de covariance r i,j ont été proposées, voir par exemple [START_REF] Haugh | Checking the independence of two covariancestationary time series : a univariate residual cross-correlation approach[END_REF][START_REF] Hong | Testing for independence between two covariance stationary time series[END_REF][START_REF] Ramírez | Detection of spatially correlated gaussian time series[END_REF]. De manière équivalente, le test (1) peut être reformulé dans le domaine spectral ; définissons à ce titre pour tout ν ∈ [0, 1], s i,j (ν) = h∈Z r i,j (h)e -2iπνh , la densité spectrale et

c i,j (ν) = s i,j (ν) s i,i (ν)s j,j (ν)
, la cohérence spectrale associées aux séries (y i,n ) n∈Z et (y j,n ) n∈Z . On aboutit alors au test équivalent d'hypothèse nulle :

H 0 : c i,j (ν) = 0 ∀ν ∈ [0, 1], 1 ≤ i < j ≤ M. (2)
Les approches dites « fréquentielles » pour résoudre (2) utilisent quant à elles des statistiques de test basées sur des estimées empiriques de s i,j ou c i,j , voir par exemple [START_REF] Wahba | Some tests of independence for stationary multivariate time series[END_REF][START_REF] Eichler | Testing nonparametric and semiparametric hypotheses in vector stationary processes[END_REF][START_REF] Wu | Asymptotic theory for spectral density estimates of general multivariate time series[END_REF]. Le cadre d'étude de ces travaux se place dans le régime asymptotique dit des « petites dimensions » où M est supposé fixe et où le nombre d'observations N de chaque série tend vers l'infini, et les méthodes développées sont pertinentes dans un contexte pratique où M ≪ N . De nombreuses applications actuelles mettant en jeu des données de grande dimension, il apparaît nécessaire de considérer un régime asymptotique des « grandes dimensions » où M tend également vers l'infini. Dans ce contexte, les travaux autour du test (2) sont encore peu nombreux. Dans [START_REF] Loubaton | Properties of linear spectral statistics of frequency-smoothed estimated spectral coherence matrix of high-dimensional gaussian time series[END_REF], une statistique de test basée sur un périodogramme lissé est proposée, et étudiée dans le régime des grandes dimensions. Notons ξ i la transformée de Fourier normalisée de (y i,n ) n=1,...,N donnée par 1) , ainsi que ŝi,j l'estimateur lissé de s i,j donné par

ξ i (ν) = 1 √ N N n=1 y i,n e -i2πν(n-
ŝi,j (ν) = 1 B + 1 B 2 b=-B 2 ξ i ν + b N ξ j ν + b N ,
où B est un entier pair représentant la largeur de la fenêtre de lissage fréquentiel. Notons également ĉi,j (ν) = ŝi,j (ν) ŝi,i (ν)ŝ j,j (ν) , l'estimateur associé de c i,j (ν) et Ĉ(ν) = (ĉ i,j (ν)) i,j=1,...,M l'estimateur de la matrice de cohérence spectrale C(ν) = (c i,j (ν)) i,j=1,...,M . Les travaux de [START_REF] Loubaton | Properties of linear spectral statistics of frequency-smoothed estimated spectral coherence matrix of high-dimensional gaussian time series[END_REF] montrent alors que dans le régime asymptotique où M, B, N → ∞ tel que M B → c > 0 et B N → 0, sous l'hypothèse nulle H 0 , la distribution empirique des valeurs propres λ 1 ( Ĉ(ν)) m=1,...,M de Ĉ(ν) converge vers la loi de Marcenko-Pastur µ, i.e.

1 M M k=1 φ λ k Ĉ(ν) p.s. --→ R φ(x)dµ(x),
pour toute fonction continue bornée φ. En exploitant le fait que sous certaines hypothèses alternatives, le comportement asymptotique de la distribution empirique des valeurs propres de Ĉ(ν) dévie de la loi de Marcenko-Pastur, une statistique de test basée sur les valeurs propres de Ĉ(ν) et la loi µ est proposée.

Dans cet article, nous explorons une approche alternative, non plus basée sur les valeurs propres de Ĉ(ν), mais sur l'étude du maximum des entrées non diagonales de Ĉ(ν), i.e. max |ĉ i,j (ν)| 2 : 1 ≤ i < j ≤ M, ν ∈ V , où V est un sous-ensemble fini de [0, 1] que nous détaillons en section suivante. En particulier, après un recentrage et une renormalisation convenables, nous montrons 1 que cette statistique converge en loi, sous l'hypothèse nulle H 0 et dans le régime où M, B, N → ∞ décrit précédemment, vers la distribution de Gumbel, et exploitons ce résultat pour construire une statistique de test dont le risque de 1ère espèce asymptotique est contrôlé. Notons qu'une étude similaire a été conduite dans [START_REF] Wu | Asymptotic theory for spectral density estimates of general multivariate time series[END_REF], dans le régime des petites dimensions (M = 2), en utilisant une estimée différente de ĉi,j (ν).

Hypothèses et résultats

Commençons par présenter formellement les hypothèses utilisées dans notre étude. On considère une suite de séries temporelles (y m,n ) n∈Z , m ≥ 1, mutuellement indépendantes et de loi gaussienne complexe circulaire. Notons r m (au lieu de r m,m ) la fonction de covariance de (y m,n ) n∈Z , pour laquelle on formule l'hypothèse de « mémoire courte » suivante.

Hypothèse 1. Les fonctions de covariance (r m ) m≥1 vérifient

sup m≥1 h∈Z (1 + |h|) |r m (h)| < ∞.
1. Les détails de la preuve sont disponibles dans [START_REF] Loubaton | On the asymptotic distribution of the maximum sample spectral coherence of gaussian time series in the high dimensional regime[END_REF].

En notant s m (au lieu de s m,m ) la densité spectrale de (y m,n ) n∈Z , l'hypothèse 1 assure en particulier que s m est uniformément bornée au sens où

sup m≥1 max ν∈[0,1] s m (ν) < ∞.
Afin que la cohérence spectrale c i,j soit bien définie, il est également nécessaire de garantir que les densités spectrales (s m ) m≥1 ne s'annulent pas.

s m (ν) > 0.
Notons que les hypothèses 1 et 2 sont standards et sont notamment vérifiées par les processus ARMA dont le filtre générateur n'admet pas de pôle ou zéro sur le cercle unité. L'hypothèse suivante donne le régime des grandes dimensions adopté dans ce travail, en précisant les rythmes de croissance des paramètres M, N, B.

Hypothèse 3. B = B(N ), M = M (N ) sont fonctions de N tel qu'il existe C 1 , C 2 > 0, ρ ∈]0, 1[ et c > 0 tel que C 1 N ρ ≤ B, M ≤ C 2 N ρ et M B → c > 0 quand N → ∞.
L'hypothèse 3 appelle à quelques justifications. Dans un régime asymptotique où M, N → ∞ de telle sorte que M N → 0, on peut montrer que Ĉ(ν) est un estimateur consistant de C(ν) (pour la norme spectrale) sous réserve que B M → 0 et moyennant quelques hypothèses supplémentaires. En pratique, ce régime asymptotique traduit des situations où l'on peut choisir B tel que M ≪ B ≪ N . Néanmoins, lorsque la dimension M est potentiellement grande et que la taille N de l'échantillon est limitée, un tel choix de B devient complexe, et il apparaît alors plus raisonnable de choisir que B du même ordre de grandeur que M , ce que modélise l'hypothèse 3.

Muni des hypothèses précédentes, nous sommes à présent en mesure de présenter le résultat principal de cet article.

Définissons V = {k B+1 N : k ∈ N, k ≤ N B+1 } comme le sous-ensemble des fréquences de Fourier { k N : k ∈ N, k ≤ N -1} espacées de B+1 N . Théorème 1. Sous les hypothèses 1, 2 et 3, on a P   max 1≤i<j≤M ν∈V |ĉ i,j (ν)| 2 ≤ t + γ B + 1   ----→ N →∞
e -e -t pour tout t ∈ R, où γ = log N B+1 + log M (M -1)

2

.

Le théorème 1 montre donc que le maximum de la cohérence spectrale max |ĉ i,j (ν)| 2 : 1 ≤ i < j ≤ M, ν ∈ V , après renormalisation par B + 1 et recentrage par le terme γ, converge en loi vers la distribution de Gumbel. Le théorème 1 implique directement via l'hypothèse 3 que

max 1≤i<j≤M ν∈V |ĉ i,j (ν)| 2 = O P log(N ) B .
Cette erreur d'estimation est à comparer directement au résultat classique |ĉ i,j (ν)| 2 = O P (B -1 ) pour i, j, ν fixés (voir [START_REF] David R Brillinger | Time series : data analysis and theory[END_REF]), et on remarque que le maximum sur i, j, ν induit une perte logarithmique en log(N ). A ce titre, notons que les 2 termes logarithmiques contenus dans γ sont directement liés aux ensembles de maximisation V et {(i, j) : 1 ≤ i < j ≤ M }, dont l'ordre de grandeur de leurs cardinaux sont O( N B ) et O(M 2 ) respectivement. De plus, le théorème 1 peut être exploité pour construire une statistique de test dont le risque de 1ère espèce est contrôlé dans le régime asymptotique des grandes dimensions. En effet, considérons le seuil

η(α) = γ -log log 1 1-α B + 1 ,
obtenu par inversion de la f.d.r. de la loi de Gumbel, ainsi que la statistique

T = 1 ]η(α),∞[   max 1≤i<j≤M ν∈V |ĉ i,j (ν)| 2   . (3)
Le théorème 1 montre alors que sous l'hypothèse H 0 ,

P (T = 1) ----→ N →∞ α,
i.e. la statistique de test T est de niveau asymptotique α.

Illustrations numériques

Nous présentons ici quelques simulations autour du théorème 1 et des performances de la statistique de test T définie en (3), en considérant divers scénarios pour les hypothèses nulle H 0 et alternative H 1 . On considère pour la suite que les séries temporelles suivent un modèle AR(1) multivarié, i.e. en notant y n = (y 1,n , . . . , y M,n ) T ,

y n = Ay n-1 + ϵ n (4)
où (ϵ n ) n∈Z est un bruit blanc N C M (0, I) et où A sera ajustée suivant des scénarios précisés ci-dessous. Pour illustrer le théorème 1, nous considérons dans un premier temps A = diag(θ 1 , . . . , θ M ) avec θ 1 , . . . , θ M des variables distribuées uniformément sur le disque centré de rayon 0.9. La figure 1 représente le tracé de la fonction de répartition (f.d.r.) empirique de max i,j,ν (B + 1)|ĉ i,j (ν)| 2γ pour N = 20000, M = 500, B = 1000, évaluée sur 10000 tirages du modèle (4), et montre une bonne adéquation avec la loi de Gumbel. La table 1 représente quant à elle, toujours pour la même matrice A, la quantité P(T = 1) (évaluée sur 30000 tirages) pour un niveau α fixé à 5% et un jeu de valeurs pour (M, N, B) tel que ρ = 0.7 et c = 0.5 dans l'hypothèse 3. On constate que les valeurs de P(T = 1) sont effectivement proches de α à partir de M = 130.

Illustrons à présent les performances du test en considérant les scénarios suivants sous H 0 et sous deux hypothèses alternatives dites « locale » et « globale » définies comme suit. En notant e 1 , . . . , e M la base canonique de R M , -sous H 0 , on considère A = θ I, de telle sorte que les séries (y m,n ) n∈Z sont indépendantes ; -sous H 1,loc , on considère A = θ I + β e 2 e T 1 , de telle sorte que (y 1,n ) n∈Z et (y 2,n ) n∈Z constituent l'unique couple corrélé parmi les M séries ; -sous H 1,glob , on considère A = θ I + β i-j=1 e i e T j , de sorte que les M séries sont corrélées deux à deux. On choisira θ = 0.5 sous les trois hypothèses et β = 0.1 sous H 1,loc . Afin que les hypothèses H 0 et H 1,glob ne soient pas trivialement séparables quand M → ∞, on choisit sous H 1,glob le paramètre β fonction de M de sorte que le ratio r ci-dessous reste constant et égal à 0.1 :

r = i̸ =j 1 0 |s i,j (ν)| 2 dν i,j 1 0 |s i,j (ν)| 2 dν .
Nous prenons comme point de comparaison la statistique de test développée dans [START_REF] Loubaton | Properties of linear spectral statistics of frequency-smoothed estimated spectral coherence matrix of high-dimensional gaussian time series[END_REF] donnée par

S = 1 ]κ,+∞[ (∆) , où ∆ = max ν∈[0,1] 1 M M k=1 φ λ k Ĉ(ν) - R φ(x)dµ(x) ,
(x) = 1 - 1 c + δ 0 (dx) + (x + -x)(x -x -) 2πcx 1 [x -,x + ] (x)dx, avec x ± = (1 ± √ c) 2 . On considèrera les fonctions φ(x) = log(x) et φ(x) = (x -1)

Hypothèse 2 .

 2 Les densités spectrales (s m ) m≥1 vérifient inf m≥1 min [0,1]

Figure 2 -Figure 3 -

 23 Figure 2 -Courbes ROC de T, S log , S frob pour H 1,loc

Table 1 -

 1 Risque de 1ère espèce pour α = 0.05 en fonction de M, B, N

	1.0	max i, j, |cij( )| 2 renormalisé Gumbel
	0.8
	0.6
	0.4
	0.2
	0.0
	2	0	2	4	6	8
	Figure 1 -Comparaison entre la f.d.r. empirique du
	maximum de la cohérence spectrale et la f.d.r. de la loi
	de Gumbel.
		N	B	M	P(T = 1)
		42	20	10	0.012
		316	100	50	0.035
		659	180	90	0.037
		1044	260	130	0.045
		1459	340	170	0.046
		1901	420	210	0.048
		5623 1000 500	0.048
		13374 2000 1000	0.049

Table 2 -

 2 2 , et les statistiques de test associées seront notées S log et S frob . Comme la loi asymptotique de S sous H 0 n'est pas connue, on fixera le seuil κ de telle sorte à contrôler le risque de 1ère espèce empirique évalué par simulation Monte-Carlo (10000 tirages). Par mesure d'équité, on procédera de même pour fixer le seuil η de la statistique T définie en[START_REF] Haugh | Checking the independence of two covariancestationary time series : a univariate residual cross-correlation approach[END_REF].Les Figures 2 et 3 fournissent les courbes ROC empiriques des statistiques T, S log , S frob où M = 290, N = 2846, B = 580. Comme attendu, on observe que sous H 1,loc , la statistique T présente les meilleures performances, puisque l'opération de maximum utilisée dans T est spécialement indiquée pour détecter les entrées non nulles de la cohérence spectrale C(ν) dans un contexte parcimonieux. A contrario, sous H 1,glob où les corrélations sont réparties sur tous les couples de séries, les statistiques basées sur les valeurs propres de Ĉ(ν) sont plus adaptées. Nous présentons également en Tables 2 et 3 la puissance des tests sous H 1,loc et H 1,glob (évaluées sur 10000 tirages) Puissance des statistiques T, S log , S frob sous H 1,loc à un niveau 5%

	N	M	B	S frob	S log	T
	42	10	20 0.049 0.049 0.061
	316	50 100 0.038 0.044 0.352
	659	90 180 0.038 0.041 0.881
	1044 130 260 0.034 0.038 0.999
	1459 170 340 0.034 0.038 1.000
	1901 210 420 0.035 0.039 1.000
	2364 250 500 0.031 0.039 1.000
	2846 290 580 0.032 0.036 1.000
	N	M	B	S frob	S log	T
	42	10	20 0.050 0.049 0.052
	316	50 100 0.036 0.042 0.067
	659	90 180 0.067 0.065 0.086
	1044 130 260 0.142 0.122 0.133
	1459 170 340 0.339 0.255 0.214
	1901 210 420 0.601 0.462 0.328
	2364 250 500 0.836 0.682 0.503
	2846 290 580 0.960 0.852 0.672

Table 3 -

 3 Puissance des statistiques T, S log , S frob sous H 1,glob à un niveau 5% pour différentes valeurs de M, N, B et un niveau fixé à 5 %. Des conclusions similaires peuvent être tirées quant aux performances des statistiques sur les deux scénarios.