Xavier Bekaert
email: xavier.bekaert@lmpt.univ-tours.fr

Johanna Erdmenger

Dmitry Ponomarev
email: dmitry.ponomarev@physik.uni-muenchen.de

Charlotte Sleight
email: csleight@mpp.mpg.de

Bulk quartic vertices from Boundary four-point correlators

We present arguments which suggest that the bulk higher-spin gravity duals of weaklycoupled conformal field theories obey some refined notion of locality. In particular, we discuss the Mellin amplitude programme in this context. We focus on the O(N ) vector model and minimal higher-spin gravity as a paradigmatic example of such holographic dual pairs. We restrict ourselves to three-and four-point functions of scalar primary operators, but the qualitative conclusions are expected to hold for the generic case.

Bulk locality and Mellin amplitudes

Bulk locality is one of the most remarkable features of the AdS/CFT correspondence. The concept of bulk locality refers to the fact that the usual conditions for locality in a quantum field theory also hold in the interior of anti de Sitter (AdS) space. It is expected to hold in the usual regime where the duality is tested, i.e. when the curvature radius R AdS is large compared to the string length s which, in turn, is large compared to the Planck length P . On the gravity side, this corresponds to (I) a semi-classical limit: R AdS P and (II) a higher-spin gap: all particles of spins s greater than two have large masses M s>2 M s ≤2 . On the conformal field theory (CFT) side, this translates into:

(I) a semi-classical limit: a perturbative expansion around a generalised free field theory when N 1,

(II) a higher-spin gap: all single-trace operators of higher-spins have large conformal dimensions ∆ s>2 ∆ s ≤2 .

These two properties were argued to provide necessary and sufficient conditions for a CFT to possess a semiclassical and local bulk dual [START_REF] Heemskerk | Holography from Conformal Field Theory[END_REF][START_REF] Fitzpatrick | Effective Conformal Theory and the Flat-Space Limit of AdS[END_REF]. Later, a third condition was added from the perspective of effective field theory [START_REF] Fitzpatrick | AdS Field Theory from Conformal Field Theory[END_REF]:

(III) polynomial boundedness of Mellin amplitudes.

Over the last years, significant progress has been made in the Mellin amplitude programme. Initiated in [START_REF] Penedones | Writing CFT correlation functions as AdS scattering amplitudes[END_REF], the goal is to interpret the Mellin amplitudes [START_REF] Mack | D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models[END_REF] associated to CFT correlation functions as AdS scattering amplitudes (the corresponding dictionary is summarised in Table 1). For a given correlation function, the corresponding Mellin amplitude is a suitably normalised Mellin transform of the factor that is not fixed by conformal symmetry. Consider, for example, the simplest case of a four-point correlator of a scalar primary operator O with scaling dimension ∆,

O (y 1 ) O (y 2 ) O (y 3 ) O (y 4 ) = 1 y 2 12 y 2 34 ∆ F (u, v) , (1)
with u and v the two cross ratios u = y , and y ij = y i -y j . Then the reduced Mellin amplitude [START_REF] Mack | D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models[END_REF] is the function M (s, t) of two scattering variables s and t defined via the relation

F (u, v) = ds 2πi dt 2πi u s v ∆-s-t M (s, t), (2)
providing a Mellin-Barnes representation of the conformal correlator [START_REF] Heemskerk | Holography from Conformal Field Theory[END_REF]. The Mellin amplitude itself is the function M(s, t) defined by

M (s, t) = Γ(∆ -s)Γ(∆ -t)Γ(s + t -∆) 2 M(s, t), (3)
where a numerical coefficient (irrelevant for the present discussion) has been dropped. The extraction of the corresponding gamma functions is important to ensure the following property: For a large-N CFT with a discrete set of primary operators, the Mellin amplitudes of single-trace primary-operator correlators are meromorphic functions with simple poles determined by the twists of the single-trace operators in the OPE [START_REF] Penedones | Writing CFT correlation functions as AdS scattering amplitudes[END_REF][START_REF] Mack | D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models[END_REF]. In particular, the reduced Mellin amplitude M (s, t) possesses extra poles with respect to the Mellin amplitude M(s, t), due to the gamma functions in [START_REF] Fitzpatrick | AdS Field Theory from Conformal Field Theory[END_REF]. These extra poles originate from the double-trace operators in the OPE. For individual (Feynman or Witten) diagrams, scattering amplitudes of elementary particles and Mellin amplitudes for single-trace primary operators thus share common properties as functions of Mandestam invariants vs Mellin variables [START_REF] Penedones | Writing CFT correlation functions as AdS scattering amplitudes[END_REF][START_REF] Mack | D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models[END_REF][START_REF] Paulos | Towards Feynman rules for Mellin amplitudes[END_REF][START_REF] Fitzpatrick | A Natural Language for AdS/CFT Correlators[END_REF] (c.f. Table 2) suggesting a common Boundary criterion of bulk locality: Interactions on flat/AdS spacetime are local iff the amplitudes of the corresponding contact Feynman/Witten diagram are polynomial functions of Mandelstam invariants / Mellin variables.

In this sense, the condition (III) can be seen as the criterion of bulk locality itself, in the same way that the condition (I) stands for the criterion of existence of a semiclassical limit. Nevertheless, the condition (II) remains of interest because the presence of a higher-spin gap sets a higher-spin symmetry breaking scale in the bulk. Even if the gravity theory may be nonlocal in the sense of being an effective field theory, the previous scale controls the low-energy expansion of the theory and the validity range of its local truncations.

The Mellin amplitude programme of rewriting CFT correlators as Witten diagrams seemingly applies to the large class of strongly-coupled CFTs obeying the criteria (I)-(III). Under these hypotheses, the corresponding bulk dual would possess a simultaneously weakly-coupled & weakly-curved regime. Indeed, the Mellin amplitude programme seems well adapted (but presently restricted) to the holographic reconstruction of bulk theories (or individual scattering processes) possessing a weakly-coupled & weakly-curved limit. However it does not apply directly to the simplest example of CFTs: free ones (or weakly coupled ones). In fact, the Mellin transform of correlators of free CFTs is often not even well defined (see section 4). The bulk duals of free CFTs are conjectured to be higher-spin gravity theories which are indeed non-local in the restricted sense of locality (see section 2).

What could be a mild replacement of the criteria (II) and (III), that could provide necessary and sufficient conditions for CFTs (including weakly-coupled ones) to possess (mildly non-local) bulk duals (including higher-spin gravity theories)? A tentative answer could be:

(II') a finite number of single-trace primary operators with conformal dimension below any fixed dimension, in the N → ∞ limit, (III') analyticity of Mellin amplitudes.

The criterion (II') is inspired from the second basic property in [START_REF] El-Showk | Emergent Spacetime and Holographic CFTs[END_REF]. In spirit, it is analogous to the second assumption of the Coleman-Mandula theorem [START_REF] Coleman | All Possible Symmetries of the S Matrix[END_REF] since the conformal dimension of a singletrace operator translates into the energy of the corresponding elementary particle in the bulk. In contrast to flat spacetime, in AdS this condition does not rule out a tower of massless higher spins as their dimensions grow with spin. Another motivation for criterion (II') is that bulk locality would be obscure if an infinite number of fields were relevant in a scattering process at a given energy. The criterion (III') relaxes strict locality and replaces it with the milder requirement that the coefficients in the Taylor series expansion of the amplitude decrease fast enough in order to have an infinite radius of convergence. In other words, the amplitude can be approximated, for any fixed accuracy, by a polynomial of sufficiently high degree, i.e. by a local interaction of sufficiently high order. More precisely, the criterion (III') amounts to the following [START_REF] Bekaert | Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory[END_REF] boundary criterion of weak locality: Interactions on flat/AdS spacetime are weakly local iff the amplitudes of the corresponding contact Feynman/Witten diagram are entire functions of Mandelstam invariants / Mellin variables.

2 Bulk locality and higher-spin holography

The higher-spin holographic duality arose shortly after the birth of AdS/CFT correspondence and was initially motivated by the semiclassical but stringy regime ( s R AdS P ) in the strong version of Maldacena conjecture. A more general picture progressively emerged in a long series of paper (see [START_REF] Sundborg | Stringy gravity, interacting tensionless strings and massless higher spins[END_REF][START_REF] Klebanov | AdS dual of the critical O(N) vector model[END_REF][START_REF] Sezgin | Holography in 4D (super) higher spin theories and a test via cubic scalar couplings[END_REF] for some early steps) but the basic idea underlying higher-spin holography is easy to summarise. Free (or integrable) CFTs have an infinite number of global higher (sometimes called "hidden") symmetries (including conformal symmetry). Applying Noether's theorem, one deduces that their spectrum must contain an infinite tower of traceless conserved currents with unbounded spin (including spin two). Therefore, the AdS/CFT dictionary suggests that free (or some integrable) CFTs should be dual to "higher-spin gravity" theories, in the sense of theories whose spectra contain an infinite tower of gauge fields with unbounded spin (including spin two). The simplest example of such a scenario states that the singlet sector of the free and critical vector models should be holographically dual to minimal higher-spin gravity [START_REF] Klebanov | AdS dual of the critical O(N) vector model[END_REF][START_REF] Sezgin | Holography in 4D (super) higher spin theories and a test via cubic scalar couplings[END_REF].

Turning back to the issue of bulk locality, higher-spin interactions in four (and higher) dimension appear to be generically:

(A) quasi-local in the sense that they possess a perturbative expansion (in powers of fields and their derivatives) where each individual term in the Lagrangian is local (effective field theories are typically quasilocal).

(B) non-local in the sense that the total number of derivatives in the complete Lagrangian is unbounded (as in string field theory). This is a corollary of:

(a) Metsaev bounds: The number of derivatives appearing in a non-trivial cubic vertex evaluated on the free mass-shell is bounded from [START_REF] Metsaev | Cubic interaction vertices of massive and massless higher spin fields[END_REF] i. below by the highest spin involved, ii. above by the sum of the spins involved. (b) Higher-spin algebra structure: The Jacobi identity requires a spectrum with an infinite tower of fields with unbounded spin [START_REF] Fradkin | Candidate to the Role of Higher Spin Symmetry[END_REF][START_REF] Boulanger | On the uniqueness of higher-spin symmetries in AdS and CFT[END_REF].

The number of derivatives is bounded from below by the highest spin involved by virtue of Metsaev's lower bound (a.i). However, there cannot be any upper bound on the spin in a consistent theory due to point (b). Hence, the number of derivatives is necessarily unbounded. The point (i) is expected; the coinciding point limit of two single-trace operators will produce double-trace operators as well. The point (ii) is AdS/CFT standard lore. The correspondence between single-trace conformal blocks and Witten exchange diagrams is one-to-one: It is essentially an identity up to double-trace conformal blocks (see e.g. the recent discussion in [START_REF] El-Showk | Emergent Spacetime and Holographic CFTs[END_REF]). The point (iii) holds because Mellin amplitudes are meromorphic functions with simple poles arising only from single-trace operators [START_REF] Penedones | Writing CFT correlation functions as AdS scattering amplitudes[END_REF].

The proof of (C.b) goes as follows: Consider a single-trace scalar primary operator 4-point function in a CFT with discrete spectrum and at large N . First, decompose this 4-point function into conformal blocks. Second, using facts (i)-(ii) associate to each single-trace conformal block its corresponding Witten exchange diagram. Third, substract from the 4-point function all these Witten exchange diagrams. By construction, the difference that remains is a 4-point function whose decomposition contains only double-trace conformal blocks. Fourth, conclude from (iii) that this remainder is an entire function of the Mellin variables since it does not contain any single-trace contribution. This allows to interpret the remainder as a 4-point contact Witten diagram associated to a weakly local quartic vertex.

As one can see, the boundary criterion of weak locality (see Section 1) tantamounts to the existence of a neat separation in the amplitude between the exchange and contact contributions: Poles in the scattering amplitude are accounted by particle exchanges and what remains is interpreted as the contact amplitude.

Remarks:

• It would be interesting to compare the criterion of weak locality for higher-spin gravity advocated here with the recent proposals [START_REF] Vasiliev | Star-Product Functions in Higher-Spin Theory and Locality[END_REF] and [START_REF] Skvortsov | On Locality, Holography and Unfolding[END_REF] based, respectively, on functional classes of starproduct elements and on classes of field redefinitions leaving Witten diagrams invariant.

• A caveat of our argument that the bulk dual of the O(N ) model is a weakly local higher-spin gravity is that it implicitly assumes that Mellin amplitudes are well defined functions while they actually require some regularisation. They may also be thought as distributions (c.f. Section 4) but then it is their analyticity properties which are somewhat elusive.

• Another subtlety is that the third step in the proof (substraction of all Witten exchange diagrams) can bring infinities in the double-trace conformal block decomposition of the remainder. Taking an optimistic standpoint, this step might actually regularise the Mellin amplitudes alluded above.

Holographic reconstruction of higher-spin gravity vertices

The line of arguments presented in section 2 is expected to generalise (to all spins and to higher numbers of points) and indicates that the higher-spin interactions for the bulk dual of the O(N ) model might be weakly local. This picture is very suggestive but remains somewhat qualitative. The explicit holographic reconstruction of some quartic vertices in higher-spin gravity, to which we now turn, provides a more concrete playground to test bulk locality.

More specifically, we will consider the holographic reconstruction of quartic AdS interations from a free CFT. In practice, this task amounts to:

• Compute the 3 and 4 point conformal correlators via Wick contraction,

• Write the most general ansatz for the relevant cubic and quartic vertices,

• Compute the corresponding exchange and contact amplitudes,

• Fix the coefficients of vertices by matching correlators with total amplitudes. A priori, it is not guaranteed that such a purely holographic reconstruction produces interactions compatible with the Noether procedure. However, it is natural to expect that these two perturbative procedures are compatible with each other since Ward identities of the boundary CFT should be dual to Noether identities of the AdS theory.

The holographic reconstruction was performed in [START_REF] Bekaert | Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory[END_REF][START_REF] Bekaert | Towards holographic higher-spin interactions: Four-point functions and higher-spin exchange[END_REF] for the simplest nontrivial case: The quartic self-interaction of the AdS 4 scalar field in the higher-spin multiplet dual to the d Among the important technical simplifications in this example are the structure of relevant vertices: On the one hand, the bulk cubic vertices s -0 -0 are of Noether type ϕ s J s (see [START_REF] Berends | Explicit Construction of Conserved Currents for Massless Fields of Arbitrary Spin[END_REF][START_REF] Bekaert | Higher spin interactions with scalar matter on constant curvature spacetimes: conserved current and cubic coupling generating functions[END_REF]), i.e. a gauge field ϕ s times a conserved current J s = ϕ 0 (∇) s ϕ 0 + ... bilinear in the scalar field ϕ 0 , and traceless on-shell. Therefore, the diagrams (a)-(c) correspond to current exchange J s P s J s , where P s is the spin-s gauge field propagator. On the other hand, all bulk quartic vertices 0 -0 -0 -0 are also of current exchange type J s m J s (see [START_REF] Heemskerk | Holography from Conformal Field Theory[END_REF][START_REF] Bekaert | Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory[END_REF]). Consequently, the exchange and contact Witten diagrams are of the same type and can be easily compared for each spin s and in each channel (see figure 2). The form of the quartic vertex can be summarised as follows:

V = s∈2N V s , V s = J µ 1 •••µs a s ( ) J µ 1 •••µs , (4)
where a s ( ) are generating functions

a s ( ) = ∞ m=0 a m,s m , (5)
for the individual couplings a m,s in front of the individual vertices

V m,s = J µ 1 •••µs m J µ 1 •••µs . (6)
The explicit expression of the quartic vertex V can be found in [START_REF] Bekaert | Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory[END_REF]. The generating functions are defined by

a s ν 2 + s + 9 4 ∝ 2 8-s ν 2 + (s -1 2 ) 2 π Γ 2s-2iν+1 4 2 Γ 2s+2iν+1 4 2 - 1 Γ (s) 2 - ( -1)
s 2 π 3 2 2 s+5 Γ s + 3 2 Γ s 2 + 1 2 √ 2 Γ s 2 + 1 Γ (s + 1) Γ 3 4 -iν 2 Γ 3 4 + iν 2 Γ s + 1 2 + iν Γ s + 1 2 -iν ,
which are entire functions, though it is not manifest. Heuristically, this indicates that each contact diagram associated to the cubic vertex V s and depicted in figure 2 looks weakly local.

A subtlety is that the quartic vertices (6) are linearly independent on the free mass shell only for 2m s. However, the generating function includes infinitely many terms with 2m > s. The latter should be reexpressed in terms of the previous independent ones. Since this involves infinitely many couplings and since we are on AdS 4 background where covariant derivatives do not commute, this procedure can a priori produce infinities, c.f. the examples in [START_REF] Skvortsov | On Locality, Holography and Unfolding[END_REF][START_REF] Boulanger | Higher Spin Interactions in Four Dimensions: Vasiliev vs. Fronsdal[END_REF] at cubic level (see also [START_REF] Giombi | Higher Spin Gauge Theory and Holography: The Three-Point Functions[END_REF][START_REF] Boulanger | On The Uniqueness of Minimal Coupling in Higher-Spin Gauge Theory[END_REF][START_REF] Bekaert | How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples[END_REF] where this possibility was discussed). In other words, the couplings in front of the independent quartic vertices might be infinite. However, one may expect this phenomenon not to happen in the present context of holographic reconstruction because infinite couplings in front of the independent quartic vertices would correspond to infinite coefficients in the double-trace conformal block decomposition.

Mellin amplitudes of the free and critical O(N ) model

As mentioned above, Mellin amplitudes of weakly-coupled theories are sometimes ill-defined as genuine functions, so they may either require some regularisation procedure or to be treated as generalised functions (i.e. distributions). We will investigate the latter option and focus on the O(N ) model.

The scalar single-trace primary operator O has conformal dimension ∆ = d-2 for the d-dimensional free O(N ) model. Its 4-point correlator (1) has a factor not fixed by conformal symmetry

F free (u, v) = F dis. (u, v) + 1 N F con. (u, v) (7)
where the first term F dis. (u, v) arise from disconnected diagrams and the second term F con. (u, v) from connected diagrams. Both terms are power functions of the two cross ratios. This is true for the disconnected piece

F dis. (u, v) = 1 + u ∆ + u v ∆ (8)
and for the connected piece

F con. (u, v) = u ∆ 2 + u v ∆ 2 + u ∆ 2 u v ∆ 2 , (9)
where, again, front numerical factors have been neglected for the sake of simplicity.

A power function, say

f (x) = x α (α ∈ R) , (10)
does not have a well defined Mellin transform [START_REF] Flajolet | Mellin transforms and asymptotics: Harmonic sums[END_REF]

M (z) = ∞ 0 x z f (x) dx x , (11)
or, rather, its fundamental strip of convergence degenerates to the vertical line z = -α + i R in the complex plane. Actually, the Mellin transform [START_REF] Sundborg | Stringy gravity, interacting tensionless strings and massless higher spins[END_REF] of the power function ( 10) may be understood * as a distribution [START_REF] Bertrand | The Mellin transform[END_REF]: M (z) = δ(z + α) . With this interpretation, the Mack ampitudes corresponding to (8) and ( 9) are, respectively,

M dis. (s, t) = δ(s) δ(t -∆) + δ(s -∆) δ(t) + δ(s -∆) δ(t -∆) (12)
and

M con. (s, t) = δ(s -∆ 2 ) δ(t -∆ 2 ) + δ(s -∆ 2 ) δ(t -∆) + δ(s -∆) δ(t -∆ 2 ) . (13)
Due to the distributional nature of these reduced Mellin amplitudes M dis. (s, t) and M con. (s, t), the definition (3) for the corresponding Mellin amplitudes M dis. (s, t) and M con. (s, t) is difficult to apply. The problem is that each value in the support of the left-hand-side in (3) corresponds to a pole at zero of one (even two for the disconnected piece) gamma function on the right-hand-side. Formally, treating the Mellin amplitudes M dis. (s, t) and M con. (s, t) as genuine functions, one would conclude that they must vanish identically. In other words, although the reduced Mellin amplitudes ( 12) and ( 13) are well defined distributions, the nature of the corresponding Mellin amplitudes remains to be clarified.

Let us stress that the above observations are not restricted to free CFTs but even holds for some integrable CFTs. A good example is the large-N limit of the O(N ) model at large N whose integrability has been recognised a long time ago [START_REF] Stanley | Spherical model as the limit of infinite spin dimensionality[END_REF]. From a modern perspective, the integrability of the large-N limit of the O(N ) model is intimately related to its equivalence to the free O(N ) model, up to a Legendre transformation. This latter fact lies at the heart of the Klebanov-Polyakov conjecture [START_REF] Klebanov | AdS dual of the critical O(N) vector model[END_REF]. In the N → ∞ limit, the scalar single-trace primary operator O for the critical O(N ) model has conformal dimension ∆ = 2 and its 4-point correlator (1) has a factor not fixed by conformal symmetry

F int (u, v) = G dis. (u, v) + 1 N G con. (u, v) + O 1 N 2 (14)
which is also a sum of power functions of the two cross ratios. More precisely, the disconnected part is given by formula ( 8) with ∆ = 2 and the connected part reads for d

= 3 [29] G con. (u, v) = u 2 u -3 2 (1 + u -v) + v -3 2 (1 -u + v) -(uv) -3 2 (1 -u -v) . (15)
The reduced Mellin amplitude of the connected part is thus

M con. (s, t) = δ(s -1 2 ) δ(t -3 2 ) + δ(s -3 2 ) δ(t -1 2 ) -δ(s -1 2 ) δ(t -1 2 ) + δ(s -2) δ(t -3 2 ) -δ(s -3) δ(t -1 2 ) + δ(s -2) δ(t -1 2 ) (16) -δ(s -1 2 ) δ(t -3) + δ(s -3 2 ) δ(t -2) + δ(s -1 2 ) δ(t -2) .
Again the support of the Dirac delta functions coincide with poles of the gamma functions in [START_REF] Fitzpatrick | AdS Field Theory from Conformal Field Theory[END_REF]. The presence of a product of two Dirac distributions in the reduced Mellin amplitudes ( 12), ( 13) and ( 16) seem to preclude their interpretation as scattering amplitudes since trivial scattering (i.e. alignment of pairs of momenta) should correspond to fixing a single Mandelstam variable (and not all of them). This may be a hint that Mellin amplitudes of free (or integrable) CFTs do not admit a sensible flat limit, a reasonable expectation since unbroken higher-spin gravity theories do not admit a weakly-coupled flat regime.

The distributional nature of tree-level amplitudes of four scalar fields and their absence of scattering has also been recently observed in conformal higher-spin gravity [START_REF] Joung | Scalar scattering via conformal higher spin exchange[END_REF]. The triviality of scattering amplitudes may look surprising at first sight since higher-spin gravity theory has nonvanishing bulk interaction vertices. Nevertheless, retrospectively it seems reasonable to expect that the algebra of asymptotic higher-spin symmetries is so huge as to constrain the scattering of particles to be essentially trivial.

Conclusion and open directions

Characterising the degree of (non)locality of the bulk duals of weakly-coupled CFTs is an important issue for understanding holographic duality beyond the regime where the bulk spacetime is weakly curved. Looking at higher-spin gravity suggests to broaden the definition of locality in order to encompass interactions on flat/AdS spacetime for which the Feynman/Witten amplitudes of the corresponding contact diagrams are not polynomial but entire functions. At the level of four-point conformal correlators and quartic bulk vertices, this weak locality criterion is equivalent to the property that Witten exchange diagrams account for all single-trace conformal blocks in the decomposition of correlators. General arguments and explicit computations indicate that, for the higher-spin gravity dual of the free O(N ) model, the quartic self-interaction of the bulk scalar field is weakly (non)local.

Extending the holographic reconstruction of quartic vertices to spin s = 0 using twistor techniques, or to boundary dimension d = 3, are interesting challenges, as well as comparing the quartic vertices obtained purely from holographic reconstruction with the corresponding ones that can be extracted from Vasiliev equations.

  (C) weakly-local in the sense that the Mellin amplitude of contact Witten diagrams are entire functions of the Mellin variables. This weak-locality holds at cubic level due to Metsaev upper bound and should hold for the quartic self-interactions of the AdS scalar field due to general facts about Mellin amplitudes. (a) Locality of cubic vertices: Individual cubic higher-spin interactions are indeed local in the sense that any 3-point contact Witten diagram with fixed external legs is a polynomial function of the Mellin variables. This follows as a corollary from Metsaev upper bound (a.ii): For any triplet of spins, the number of derivatives in any relevant cubic vertex is bounded from above by the sum of the spins. (b) Weak-locality of quartic vertices: The quartic self-interactions of the AdS scalar field dual to the single-trace scalar primary operator in the O(N ) model (or, more generally, in a large-N CFT with discrete spectrum) appear to be weakly local, in the sense that the 4-point contact Witten diagram with four scalar external legs is an entire function of the Mellin variables. This follows from the following general facts about the decomposition of single-trace scalar primary operator 4-point function in conformal blocks at leading order in 1/N : (i) This decomposition contains both single-trace and double-trace conformal blocks. (ii) Any single-trace conformal block can be accounted by a Witten exchange diagram. (iii) The Mellin amplitude of a 4-point function is an entire function of the Mellin variables iff its conformal block decomposition does not contain any single-trace contribution.

  = 3 free O(N ) model. The relevant four-point Witten diagrams are displayed in figure 1. Diagrams (a)-(c) are exchanges of massless spin-s fields between two pairs of the real scalar. The contact diagram (d) is the amplitude associated to the quartic vertex.

Figure 1 :

 1 Figure 1: Four-point Witten diagrams contributing to the holographic reconstruction.

Figure 2 :

 2 Figure 2: Comparison of exchange and contact Witten diagrams in a single channel.

Table 1 :

 1 Mellin amplitude dictionaryMinkowski spacetime Anti-de Sitter spacetime

	Feynman diagram	Witten diagram
	Mandelstam invariants	Mellin variables
	Scattering amplitude	Mellin amplitude

Table 2

 2

	: Amplitude properties
	Bulk process	Boundary amplitude
	Local contact interaction	Polynomial
	Particle exchange	Simple pole

* We thank S. Rychkov and M. A. Vasiliev for pointing out to us this possibility, which we later traced in the mathematical litterature.

Acknowledgments

We thank E. Joung, E. Skvortsov, M. Taronna and M. A. Vasiliev for useful discussions. X.B. is very grateful to the Institute of Advanced Studies from Nanyang Technological University in Singapore for hospitality, and to the organisers of the workshop "Higher Spin Gauge Theories" for providing the opportunity to present this work and contribute to the proceedings. He also acknowledges the Asia Pacific Center for Theoretical Physics (APCTP) in Pohang and the Institute for Studies in Theoretical Physics and Mathematics (IPM) in Tehran for hospitality where part of this work was done during, respectively, the program "Duality and Novel Geometry in M-theory" and the "IPM school on Higher Spin Theory". D.P. thanks E. Joung and Seoul National University for their kind hospitality.

The research of X.B. was supported by the Russian Science Foundation grant 14-42-00047 in association with Lebedev Physical Institute. The work of J.E. and C.S. was partially supported by the European Science Foundation Holograv network (Holographic methods for strongly coupled systems). The work of D.P. was supported by the ERC Advanced grant No.290456.


Brahim Benmedjdoub

Éric Sopena

Isma Bouchemakh

2-Distance Colorings of Integer Distance Graphs

Keywords:

published or not. The documents may come

Introduction

All the graphs we consider in this paper are simple and loopless undirected graphs. We denote by V (G) and E(G) the set of vertices and the set of edges of a graph G, respectively. For any two vertices u and v of G, we denote by d G (u, v) the distance between u and v, that is the length of a shortest path joining u and v. We denote by ∆(G) the maximum degree of G.

A (proper) k-coloring of a graph G is a mapping from V (G) to the set of colors {1, . . . , k} such that every two adjacent vertices receive distinct colors. The smallest k for which G admits a k-coloring is the chromatic number of G, denoted χ(G). A 2-distance k-coloring of a graph G is a mapping from V (G) to the set of colors {1, . . . , k} such that every two vertices at distance at most 2 receive distinct colors. 2-distance colorings are sometimes called L(1,1)-labelings (see [START_REF] Calamoneri | The L(h, k)-labelling problem: An updated survey and annotated bibliography[END_REF] for a survey on L(h, k)-labelings) or square colorings in the literature. The smallest k for which G admits a 2-distance k-coloring is the 2-distance chromatic number of G, denoted χ 2 (G).

The square G 2 of a graph G is the graph defined by V (G 2 ) = V (G) and uv ∈ E(G 2 ) if and only if d G (u, v) ≤ 2. Clearly, a 2-distance coloring of a graph G is nothing but a proper coloring of G 2 and, therefore, χ 2 (G) = χ(G 2 ) for every graph G.

The study of 2-distance colorings was initiated by Kramer and Kramer [START_REF] Kramer | Un problème de coloration des sommets d'un graphe[END_REF] (see also their survey on general distance colorings in [START_REF] Kramer | A survey on the distance-coloring of graphs[END_REF]). The case of planar graphs has attracted a lot of attention in the literature (see e.g. [START_REF] Agnarsson | Coloring powers of planar graphs[END_REF]2,3,[START_REF] Borodin | 2-distance (∆ + 2)-coloring of planar graphs with girth six and ∆ ≥ 18[END_REF][START_REF] Dvořàk | Coloring squares of planar graphs with girth six[END_REF][START_REF] Lih | Coloring the square of an outerplanar graph[END_REF]12]), due to the conjecture of Wegner that suggests an upper bound on the 2-distance chromatic number of planar graphs depending on their maximum degree (see [START_REF] Wegner | Graphs with given diameter and a colouring problem[END_REF] for more details).

In this paper, we study 2-distance colorings of distance graphs. Although several coloring problems have been considered for distance graphs (see [START_REF] Liu | From rainbow to the lovely runner: A survey on coloring parameters of distance graphs[END_REF] for a survey), it seems that 2distance colorings have not been considered yet. We present in Section 2 a few basic results on the chromatic number of distance graphs. We then consider specific sets D, namely D = {1, a}, a ≥ 3 (in Section 3), D = {1, a, a + 1}, a ≥ 3 (in Section 4), and D = {1, . . . , m, a}, 2 ≤ m < a (in Section 5). We finally propose some open problems in Section 6. It is easy to observe that the square of the distance graph G(D) is also a distance graph, namely the distance graph G(D 2 ) where

Preliminaries

D 2 = D ∪ {d + d ′ / d, d ′ ∈ D} ∪ {d -d ′ / d, d ′ ∈ D, d > d ′ }.
For instance, for D = {1, 2, 5}, we get D 2 = {1, 2, 3, 4, 5, 6, 7, 10}. Note that if D has cardinality k, then D 2 has cardinality at most k(k + 1).

As observed in the previous section, χ 2 (G) = χ(G 2 ) for every graph G. Therefore, since (G(D)) 2 = G(D 2 ), determining the 2-distance chromatic number of the distance graph G(D) reduces to determining the chromatic number of the distance graph G(D 2 ). The problem of determining the chromatic number of distance graphs has been extensively studied in the literature. When |D| ≤ 2, this question is easily solved, thanks to the following general upper bounds:

Proposition 1 (folklore) For every finite set of positive integers D = {d 1 , . . . , d k } and every positive integer p such that d i ≡ 0 (mod p) for every i,

1 ≤ i ≤ k, χ(G(D)) ≤ p.
Proof. Let λ : V (G(D)) -→ {1, . . . , p} be the mapping defined by λ(x) = 1 + (x mod p), for every integer x ∈ Z. Since d i ≡ 0 (mod p) for every i, 1 ≤ i ≤ k, the mapping λ is clearly a proper coloring of G(D).

Theorem 2 (Walther [START_REF] Walther | Über eine spezielle Klasse unendlicher Graphen[END_REF]) For every finite set of positive integers D,

χ(G(D)) ≤ |D| + 1.
Proof. A (|D|+1)-coloring of G(D) can easily be produced using the First-Fit greedy algorithm, starting from vertex 0, from left to right and then from right to left, since every vertex has exactly |D| neighbors on its left and |D| neighbors on its right. Therefore, when |D| ≤ 2, χ(G(D)) = 2 if |D| = 1 or all elements in D are odd (since G(D) is then bipartite), and χ(G(D)) = 3 otherwise (since G(D) then contains cycles of odd length). The case |D| = 3 has been settled by Zhu [START_REF] Zhu | Circular chromatic number of distance graphs with distance sets of cardinality three[END_REF]. Whenever |D| ≥ 4, only partial results have been obtained, namely for sets D having specific properties.

A coloring λ of a distance graph G(D) is p-periodic, for some integer p ≥ 1, if λ(x+p) = λ(x) for every x ∈ Z. Walther also proved the following: Theorem 3 (Walther [START_REF] Walther | Über eine spezielle Klasse unendlicher Graphen[END_REF]) For every finite set of positive integers D, if χ(G(D)) ≤ k then G(D) admits a p-periodic k-coloring for some p.

The pattern of such a p-periodic coloring is defined as the sequence λ(x) . . . λ(x + p -1). In particular, the coloring defined in the proof of Proposition 1 was p-periodic with pattern 12 . . . p. In the following, we will describe such patterns using standard notation of Combinatorics on words. For instance, the pattern 121212345 will be denoted (12) 3 345.

Finally, note that in any 2-distance coloring of a graph G, all vertices in the closed neighborhood of any vertex must be assigned distinct colors. Therefore, we have the following:

Observation 4 For every graph G, χ 2 (G) ≥ ∆(G) + 1.
In particular, this bound is attained by the distance graph G(D) with D = {1, . . . , k}, k ≥ 2:

Proposition 5 For every k ≥ 2, χ 2 (G({1, . . . , k})) = 2k + 1 = ∆(G({1, . . . , k})) + 1.
Proof. It is easy to check that the mapping λ given by λ(x) = 1 + (x mod 2k + 1) for every x ∈ Z is a 2-distance (2k + 1)-coloring of G({1, . . . , k}). Equality then follows from Observation 4.

3 The case D = {1, a}, a ≥ 3

We study in this section the 2-distance chromatic number of distance graphs G(D) with D = {1, a}, a ≥ 3 (note that the case a = 2 is already solved by Proposition 5).

When D = {1, a}, a ≥ 3, we have ∆(G(D)) = 4 and

D 2 = {1, 2, a -1, a, a + 1, 2a}.
The following theorem gives the 2-distance chromatic number of any such graph:

Theorem 6 For every integer a ≥ 3,

χ 2 (G({1, a})) = 5 if a ≡ 2 (mod 5), or a ≡ 3 (mod 5), 6 otherwise.
Proof. Since {1, a} 2 = {1, 2, a -1, a, a + 1, 2a}, we get d ≡ 0 (mod 5) for every d ∈ {1, a} 2 whenever a ≡ 2 (mod 5) or a ≡ 3 (mod 5) and thus, by Proposition 1 and Observation 4,

χ 2 (G({1, a})) = 5.
Note that for every x ∈ Z, the set of vertices

C(x) = {x -a, x -1, x, x + 1, x + a}
induces a clique in G({1, a} 2 ) (see Figure 1). We now claim that every 2-distance 5-coloring λ of G({1, a}) is necessarily 5-periodic, that is λ(x + 5) = λ(x) for every x ∈ Z. To show that, it suffices to prove that any five consecutive vertices x, . . . , x + 4 must be assigned distinct colors. Assume to the contrary that this is not the case and, without loss of generality, let x = 0. Since vertices 0, 1 and 2 necessarily get distinct colors, we only have two cases to consider:

0 1 2 3 4 a 1+a 2+a 3+a 4+a -a 1-a 2-a 3-a 4-a . . . . . . Figure 1: The distance graph G({1, a}), a ≥ 3 1. λ(0) = λ(3) = 1, λ(1) = 2, λ(2) = 3.
Since C(1) induces a clique in G({1, a} 2 ) (depicted in bold in Figure 1), we have

{λ(1 -a), λ(1 + a)} = {4, 5}, which implies {λ(2 -a), λ(2 + a)} = {4, 5}. (More precisely, λ(2 -a) = 9 -λ(1 -a) and λ(2 + a) = 9 -λ(1 + a)). This implies λ(3 -a) = λ(3 + a) = 2, a contradiction since d(3 -a, 3 + a) = 2. 2. λ(0) = λ(4) = 1, λ(1) = 2, λ(2) = 3, λ(3) = 4.
As in the previous case we have Therefore, χ 2 (G({1, a})) = 5 if and only if 5 do not divide any element of {1, a} 2 = {1, 2, a -1, a, a + 1, 2a}. This is clearly the case if and only if a ≡ 2 (mod 5) or a ≡ 3 (mod 5).

We finally prove that there exists a 2-distance 6-coloring of G({1, a}) for any value of a. We consider three cases, according to the value of (a mod 3):

1. a = 3k, k ≥ 1.
Let λ be the (2a -1)-periodic mapping defined by the pattern

(123) k (456) k-1 45. If λ(x) = λ(y) = c, 1 ≤ c ≤ 5, then d(x, y) ∈ {3q, 0 ≤ q ≤ k -1} ∪ {(2a -1)p + 3q, p ≥ 1, 1 -k ≤ q ≤ k -1}. If λ(x) = λ(y) = 6, then d(x, y) ∈ {3q, 0 ≤ q ≤ k -2} ∪ {(2a -1)p + 3q, p ≥ 1, 2 -k ≤ q ≤ k -2}.
Therefore, in both cases, d(x, y) / ∈ {1, 2, a -1, a, a + 1, 2a}, and thus λ is a 2-distance 6-coloring of G({1, a}).

2. a = 3k + 1, k ≥ 1.
Let λ be the (2a -2)-periodic mapping defined by the pattern

(123) k (456) k . If λ(x) = λ(y) = c, 1 ≤ c ≤ 6, then d(x, y) ∈ {3q, 0 ≤ q ≤ k -1} ∪ {(2a -2)p + 3q, p ≥ 1, 1 -k ≤ q ≤ k -1}.
Therefore, d(x, y) / ∈ {1, 2, a -1, a, a + 1, 2a}, and thus λ is a 2-distance 6-coloring of G({1, a}).

3. a = 3k + 2, k ≥ 1.
Let λ be the (2a + 1)-periodic mapping defined by the pattern

(123) k+1 (456) k 45. If λ(x) = λ(y) = c, 1 ≤ c ≤ 5, then d(x, y) ∈ {3q, 0 ≤ q ≤ k} ∪ {(2a + 1)p + 3q, p ≥ 1, -k ≤ q ≤ k}. If λ(x) = λ(y) = 6, then d(x, y) ∈ {3q, 0 ≤ q ≤ k -1} ∪ {(2a + 1)p + 3q, p ≥ 1, 1 -k ≤ q ≤ k -1}.
Therefore, in both cases, d(x, y) / ∈ {1, 2, a -1, a, a + 1, 2a}, and thus λ is a 2-distance 6-coloring of G({1, a}). This concludes the proof. [START_REF] Borodin | 2-distance (∆ + 2)-coloring of planar graphs with girth six and ∆ ≥ 18[END_REF] The case D = {1, a, a + 1}, a ≥ 3

We study in this section the 2-distance chromatic number of distance graphs G(D) with D = {1, a, a + 1}, a ≥ 3 (note that the case a = 2 is already solved by Proposition 5).

When D = {1, a, a + 1}, a ≥ 3, we have ∆(G(D)) = 6 and

D 2 = {1, 2, a -1, a, a + 1, a + 2, 2a, 2a + 1, 2a + 2}.
We first prove the following:

Theorem 7 For every integer a, a ≥ 3,

χ 2 (G({1, a, a + 1})) = 7 = ∆(G({1, a, a + 1})) + 1
if and only if a ≡ 2 (mod 7) or a ≡ 4 (mod 7).

Proof. Since {1, a, a + 1} 2 = {1, 2, a -1, a, a + 1, a + 2, 2a, 2a + 1, 2a + 2}, we get d ≡ 0 (mod 7) for every d ∈ {1, a, a+1} 2 whenever a ≡ 2 (mod 7) or a ≡ 4 (mod 7) and thus, by Proposition 1 and Observation 4, χ 2 (G({1, a, a + 1})) = 7. Note that for every x ∈ Z, the set of vertices

C(x) = {x -a -1, x -a, x -1, x, x + 1, x + a, x + a + 1}
induces a clique in G({1, a, a + 1} 2 ). We now claim that every 2-distance 7-coloring λ of G({1, a, a + 1}) is necessarily 7-periodic, that is λ(x + 7) = λ(x) for every x ∈ Z. To show that, it suffices to prove that any 7 consecutive vertices x, . . . , x + 6 must be assigned distinct colors. Assume to the contrary that this is not the case and, without loss of generality, let x = 0. Since vertices 0, 1 and 2 necessarily get distinct colors, we only have four cases to consider (see Figure 2): We thus get a contradiction since we only have three available colors for the clique induced by the four vertices 2 -a, 3 -a, a + 3 and a + 4 in G({1, a, a + 1} 2 ). Therefore,

{λ(4 -a), λ(5 + a)} ⊆ {3, λ(2 -a), λ(3 + a)} \ {1, 6} = {3}, a contradiction since d G({1,a,a+1}) (4 -a, 5 + a) = 2.
Therefore, every 2-distance 7-coloring λ of G({1, a, a + 1}) is necessarily 7-periodic, and thus χ 2 (G({1, a, a + 1})) = 7 if and only if 7 do not divide any element of {1, 2, a -1, a, a + 1, a + 2, 2a, 2a + 1, 2a + 2}. This is clearly the case if and only if a ≡ 2 (mod 7) or a ≡ 4 (mod 7).

The following result provides an upper bound on χ 2 (G({1, a, a + 1})) for any value of a.

Theorem 8 For every integer a, a ≥ 3, χ 2 (G({1, a, a + 1})) ≤ 9 = ∆(G({1, a, a + 1})) + 3.

Proof. We consider three cases, according to the value of (a mod 3):

1. a = 3k, k ≥ 1.
Let λ be the 3a-periodic mapping defined by the pattern

(123) k (456) k (789) k . If λ(x) = λ(y) = c, 1 ≤ c ≤ 9, then d(x, y) ∈ {3q, 0 ≤ q ≤ k -1} ∪ {3ap + 3q, p ≥ 1, 1 -k ≤ q ≤ k -1}. Therefore, d(x, y) ∈ {1, 2, a -1, a, a + 1, a + 2, 2a, 2a + 1, 2a + 2}, and thus λ is a 2-distance 9-coloring of G({1, a, a + 1}). 2. a = 3k + 1, k ≥ 1.
Let λ be the (3a + 2)-periodic mapping defined by the pattern

(123) k (456) k 7123(789) k-1 4568. If λ(x) = λ(y) = c, 1 ≤ c ≤ 6, then d(x, y) ∈ {3q, 0 ≤ q ≤ k -1} ∪ {3q + 2a -1, 1 -k ≤ q ≤ 0} ∪ {(3a + 2)p + 2a -1, p > 0} ∪ {(3a + 2)p -2a + 1, p > 0} ∪ {(3a + 2)p + 3q, p > 0, 1 -k ≤ q < 0} ∪ {(3a + 2)p + 3q + 2a -1, p > 0, 1 -k ≤ q < 0} ∪ {(3a + 2)p + 3q, p > 0, 0 < q ≤ k -1} ∪ {(3a + 2)p + 3q -2a + 1, p > 0, 0 < q ≤ k -1}. If λ(x) = λ(y) = 7, then d(x, y) ∈ {3q, 0 ≤ q ≤ k -2} ∪ {3q + 4, 0 ≤ q ≤ k -2} ∪ {(3a + 2)p + 3q -4, p > 0, 2 -k ≤ q ≤ 0} ∪ {(3a + 2)p + 3q + 4, p > 0, 0 ≤ q ≤ k -2} ∪ {(3a + 2)p + 3q, p > 0, 2 -k ≤ q ≤ k -2}. If λ(x) = λ(y) = 8, then d(x, y) ∈ {3q, 0 ≤ q ≤ k -2} ∪ {3q + a -2, 2 -k ≤ q ≤ 0} ∪ {(3a + 2)p + a -2, p > 0} ∪ {(3a + 2)p -a + 2, p > 0} ∪ {(3a + 2)p + 3q, p > 0, 2 -k ≤ q < 0} ∪ {(3a + 2)p + 3q + a -2, p > 0, 2 -k ≤ q < 0} ∪ {(3a + 2)p + 3q, p > 0, 0 < q ≤ k -2} ∪ {(3a + 2)p + 3q -a + 2, p > 0, 0 < q ≤ k -2}. If λ(x) = λ(y) = 9, then d(x, y) ∈ {3q, 0 ≤ q ≤ k -2} ∪ {(3a + 2)p + 3q, p ≥ 1, 2 -k ≤ q ≤ k -2}.
Therefore, in all these cases, d(x, y) ∈ {1, 2, a -1, a, a + 1, a + 2, 2a, 2a + 1, 2a + 2}, and thus λ is a 2-distance 9-coloring of G({1, a, a + 1}).

3. a = 3k + 2, k ≥ 1.

Let λ be the (3a + 1)-periodic mapping defined by the pattern (123) k+1 (456) k+1 (789) k 7.

If λ(x) = λ(y) = c, 1 ≤ c ≤ 7, then d(x, y) ∈ {3q, 0 ≤ q ≤ k} ∪ {(3a + 1)p + 3q, p ≥ 1, -k ≤ q ≤ k}.

If λ(x) = λ(y) = c, 8 ≤ c ≤ 9, then d(x, y) ∈ {3q, 0 ≤ q ≤ k -1} ∪ {(3a + 1)p + 3q, p ≥ 1, 1 -k ≤ q ≤ k -1}.
Therefore, in both cases, d(x, y) ∈ {1, 2, a -1, a, a + 1, a + 2, 2a, 2a + 1, 2a + 2}, and thus λ is a 2-distance 9-coloring of G({1, a, a + 1}).

This concludes the proof.

From Theorems 7 and 8, we thus get: We now claim that every 2-distance (2m + 3)-coloring λ of G({1, . . . , m, a}) is necessarily (2m + 3)-periodic, that is λ(x + 2m + 3) = λ(x) for every x ∈ Z. To show that, it suffices to prove that any 2m + 3 consecutive vertices x, . . . , x + 2m + 2 must be assigned distinct colors. Assume to the contrary that this is not the case and, without loss of generality, let x = 0. Since vertices 0, 1, . . . , 2m necessarily get distinct colors, we only have two cases to consider: Proof. Let a = (2m + 1)k + r, 0 ≤ r < 2m + 1. We consider four cases, depending on the value of r. In each case, we will provide a periodic 2-distance (4m + 2)-coloring of the distance graph G({1, . . . , m, a}).

This concludes the proof.

From Theorems 10 and 11, we thus get:

Corollary 12 For all integers m and a, 2 ≤ m < a, a ≡ m + 1, m + 2 (mod 2m + 3), 2m + 4 ≤ χ 2 (G({1, . . . , m, a})) ≤ 4m + 2.

Discussion

In this paper, we studied 2-distance colorings of several types of distance graphs. In each case, we characterized those distance graphs that admit an optimal 2-distance coloring, that is distance graphs G(D) with χ 2 (G(D)) = ∆(G(D)) + 1. We also provided general upper bounds for the 2-distance chromatic number of the considered graphs.

We leave as open problems the question of completely determining the 2-distance chromatic number of distance graphs G(D) when D = {1, a, a+1}, a ≥ 3, or D = {1, . . . , m, a}, 2 ≤ m < a.

Considering other types of sets D would certainly be also an interesting direction for future research.

Let D = {d 1

 1 , . . . , d k } be a finite set of positive integers. The integer distance graph (simply called distance graph in the following) G = G(D) is the infinite graph defined by V (G) = Z and uv ∈ E(G) if and only if |v -u| ∈ D. If gcd({d 1 , . . . , d k }) = p > 1, the distance graph G(D) has p connected components, each of them being isomorphic to the distance graph G(D ′ ) with D ′ = {d 1 /p, . . . , d k /p}. In that case, we thus have χ 2 (G(D)) = χ 2 (G(D ′ )) so that we can always assume gcd(D) = 1.

{λ( 1 -

 1 a), λ(1 + a)} = {4, 5}, which implies {λ(2 -a), λ(2 + a)} = {1, 5}. We then get λ(3 -a) = λ(3 + a) = 2, again a contradiction.

Figure 2 : 3 1.

 23 Figure 2: The distance graph G({1, a, a + 1}), a ≥ 3

2 .

 2 Vertices 0, 1, 2, 3, 4 are colored with the colors 1, 2, 3, 4 and 1, respectively. Again considering cliques C(2) and C(3) in G({1, a, a + 1} 2 ), we get {λ(1 -a), λ(2 + a)} ⊆ {5, 6, 7}, and {λ(2 -a), λ(3 + a)} ⊆ {5, 6, 7}, a contradiction since vertices 1-a, 2-a, a+2 and a+3 induce a clique in G({1, a, a+1} 2 ).

3 . 4 .

 34 Vertices 0, 1, 2, 3, 4, 5 are colored with the colors 1, 2, 3, 4, 5 and 1, respectively. Considering the cliques C(1), C(2) and C(3) in G({1, a, a + 1} 2 ), we get {λ(-a), λ(1 -a), λ(1 + a), λ(2 + a)} = {4, 5, 6, 7}, {λ(2 -a), λ(3 + a)} ⊆ {1, λ(-a), λ(1 + a)} \ {4, 5}, {λ(3 -a), λ(4 + a)} ⊆ {2, λ(1 -a), λ(2 + a)} \ {4, 5}, and thus {λ(2 -a), λ(3 + a)} ⊆ {1, 6, 7} and {λ(3 -a), λ(4 + a)} ⊆ {2, 6, 7}. Assuming that none of cases 1 or 2 occurs, we have two subcases to consider: (a) λ(6) = 2. Considering the clique C(4) in G({1, a, a + 1} 2 ), we get {λ(4 -a), λ(5 + a)} ⊆ {3, λ(2 -a), λ(3 + a)} \ {1, 2} = {3, 6, 7}. If {λ(4 -a), λ(5 + a)} = {3, 6}, then {λ(3 -a), λ(4 + a)} = {2, 7}, {λ(2 -a), λ(3 + a)} = {1, 6}, {λ(1 -a), λ(2 + a)} = {5, 7} and {λ(-a), λ(1 + a)} = {4, 6}. If λ(-a) = 6 then λ(2 -a) = 1 and thus λ(4 -a) = λ(-a) = 6 which corresponds to subcase 2. If λ(1 + a) = 6 then λ(3 + a) = 1 and thus λ(5 + a) = λ(1 + a) = 6 which again corresponds to subcase 2. The case {λ(4 -a), λ(5 + a)} = {3, 7} is similar and leads to the same conclusion. Finally, if {λ(4 -a), λ(5 + a)} = {6, 7} then λ(3 -a) = λ(4 + a) = 1, a contradiction since d G({1,a,a+1}) (4 -a, 5 + a) = 2. (b) λ(6) = 6. Considering the clique C(4) in G({1, a, a + 1} 2 ), we get {λ(4 -a), λ(5 + a)} ⊆ {3, λ(2 -a), λ(3 + a)} \ {1, 6} = {3, 7}. This implies {λ(3 -a), λ(4 + a)} = {2, 6}, {λ(2 -a), λ(3 + a)} = {1, 7}, {λ(1 -a), λ(2 + a)} = {5, 6} and {λ(-a), λ(1 + a)} = {4, 7}. If λ(-a) = 7 then λ(2 -a) = 1 and thus λ(4 -a) = λ(-a) = 7 which corresponds to subcase 2. If λ(1 + a) = 7 then λ(3 + a) = 1 and thus λ(5 + a) = λ(1 + a) = 7 which again corresponds to subcase 2. Vertices 0, 1, 2, 3, 4, 5, 6 are colored with the colors 1, 2, 3, 4, 5, 6 and 1, respectively. Again considering the cliques C(1), C(2) and C(3) in G({1, a, a + 1} 2 ), we get {λ(-a), λ(1 -a), λ(1 + a), λ(2 + a)} = {4, 5, 6, 7}, {λ(2 -a), λ(3 + a)} ⊆ {1, λ(-a), λ(1 + a)} \ {4, 5}, and thus {λ(3 -a), λ(4 + a)} ⊆ {2, λ(1 -a), λ(2 + a)} \ {4, 5, 6} = {2, 7}. This implies {λ(2 -a), λ(3 + a)} = {1, 6}, {λ(1 -a), λ(2 + a)} = {5, 7} and {λ(-a), λ(1 + a)} = {4, 6}.

Corollary 9

 9 For every integer a, a ≥ 3, a ≡ 2, 4 (mod 7), 8 ≤ χ 2 (G({1, a, a + 1})) ≤ 9. 5 The case D = {1, . . . , m, a}, 2 ≤ m < a We study in this section the 2-distance chromatic number of distance graphs G(D) with D = {1, . . . , m, a}, 2 ≤ m < a (note that the case a = m + 1 is already solved by Proposition 5). When D = {1, . . . , m, a}, we have ∆(G(D)) = 2m + 2 and D 2 = {1, 2, . . . , 2m} ∪ {a -m, a -m + 1, . . . , a + m} ∪ {2a}. We first prove the following: Theorem 10 For all integers m and a, 2 ≤ m < a, χ 2 (G({1, . . . , m, a})) = 2m + 3 = ∆(G({1, . . . , m, a})) + 1 if and only if a ≡ m + 1 (mod 2m + 3) or a ≡ m + 2 (mod 2m + 3). Proof. Since {1, . . . , m, a} 2 = {1, . . . , 2m} ∪ {a -m, a -m + 1, . . . , a + m} ∪ {2a}, d ≡ 0 (mod 2m + 3) for every d ∈ {1, . . . , m, a} 2 whenever a ≡ m + 1 (mod 2m + 3) or a ≡ m + 2 (mod 2m + 3), and thus, by Proposition 1 and Observation 4, χ 2 (G({1, . . . , m, a})) = 2m + 3.

1 . 2 .

 12 Vertices 0, 1, . . . , 2m + 1 are colored with the colors 1, 2, . . . , 2m + 1 and 1, respectively. Note that vertices m -a and m + a are both adjacent to all vertices 0, 1, . . . , 2m. Hence, {λ(m -a), λ(m + a)} = {2m + 2, 2m + 3}, which implies {λ(m + 1 -a), λ(m + 1 + a)} = {2m + 2, 2m + 3} (more precisely, λ(m+1-a) = 4m+5-λ(m-a) and λ(m+1+a) = 4m+5-λ(m+a)). This implies λ(m + 2 -a) = λ(m + 2 + a) = 2, a contradiction since d(m + 2 -a, m + 2 + a) = 2. Vertices 0, 1, . . . , 2m + 1, 2m + 2 are colored with the colors 1, 2, . . . , 2m + 1, 2m + 2 and 1, respectively. As in the previous case we have {λ(m -a), λ(m + a)} = {2m + 2, 2m + 3}, which implies {λ(m + 1 -a), λ(m + 1 + a)} = {1, 2m + 3}. We thus get λ(m + 2 -a) = λ(m + 2 + a) = 2, again a contradiction. Therefore, every 2-distance (2m + 3)-coloring λ of G({1, . . . , m, a}) is necessarily (2m + 3)periodic, and thus χ 2 (G({1, . . . , m, a})) = 2m+3 if and only if 2m+3 do not divide any element of {1, 2, . . . , 2m} ∪ {a -m, a -m + 1, . . . , a + m} ∪ {2a}. This is clearly the case if and only if a ≡ m + 1 (mod 2m + 3) or a ≡ m + 2 (mod 2m + 3). For other values of a, we propose the following general upper bound on Theorem 11 For all integers m and a, 2 ≤ m < a, χ 2 (G({1, . . . , m, a})) ≤ 4m + 2 = 2∆(G({1, . . . , m, a})) -2.

Acknowledgement. Most of this work has been done while the first author was visiting LaBRI, thanks to a grant from University of Sciences and Technology Houari Boumediene (USTHB). The second author was partially supported by the Cluster of excellence CPU, from the Investments for the future Programme IdEx Bordeaux (ANR-10-IDEX-03-02).


Arthur Charpentier

Mathieu Pigeon

February

Arthur Charpentier

MACRO VS. MICRO METHODS IN NON-LIFE CLAIMS RESERVING (AN ECONOMETRIC PERSPECTIVE)

1. Introduction 1.1. Macro and Micro Methods. For more than a century, actuaries have been using run-off triangles to project future payments, in nonlife insurance. In the 30's, [START_REF] Astesan | Les réserves techniques des sociétés d'assurances contre les accidents automobiles[END_REF] formalized this technique that originated the popular chain ladder estimate. In the 90's, [START_REF] Mack | Distribution-free calculation of the standard error of chain ladder reserve estimates[END_REF] proved that the chain ladder estimate can be motivated by a simple stochastic model, and later on [START_REF] England | Stochastic claims reserving in general insurance[END_REF] provided a comprehensive overview on stochastic models that can be connected with the chain ladder method, included regression models, that could be seen as extension of the so-called "factor" methods used in the 70's.

But using the terminology of [START_REF] Van Eeghen | Loss reserving methods[END_REF] and [START_REF] Taylor | Claims reserving in non-life insurance[END_REF], those were macro-level models for reserving. In the 70's, [START_REF] Karlsson | The expected value of IBNR claims[END_REF] suggested to used some marked point process of claims to project future payments, and quantify the reserves. More recently, [START_REF] Arjas | The claims reserving problem in nonlife insurance -some structural ideas[END_REF], [START_REF] Jewell | Predicting IBNYR events and delays I. Continuous time[END_REF], [START_REF] Norberg | Prediction of outstanding liabilities in non-life insurance[END_REF], [START_REF] Hesselager | A Markov model for loss reserving[END_REF] or [START_REF] Norberg | Prediction of outstanding liabilities II: model variations and extensions[END_REF] (among many others) investigated further some probabilistic micro-level models. These models handle claims related data on an individual basis, rather than aggregating by underwriting year and development period. As mentioned in [START_REF] Hesselager | Reserving in Non-Life Insurance[END_REF], these methods have not (yet) found great popularity in practice, since they are more difficult to apply.

All macro-level models are based on aggregate data found in a runoff triangle, which is their strength, but also probably their weakness. They are easy to understand, and can be mentioned in financial communication, without disclosing too much information. From a computational perspective, those models can also be implemented in a single spreadsheet. But recently, [START_REF] England | Stochastic claims reserving in general insurance[END_REF] started questioning actuaries about possible use of detailed micro-level information. Those models can incorporate heterogeneity, structural changes, etc (see [START_REF] Friedland | Estimating Unpaid Claims Using Basic Techniques; Estimating Unpaid Claims Using Basic Techniques[END_REF] for a discussion).

1.2. Best estimates and Variability. In the context of macro-level models, [START_REF] England | Stochastic claims reserving in general insurance[END_REF] mention that prediction errors can be large, because of the small number of observations used in run-off triangles. Quantifying uncertainty in claim reserving methods is not only important in actuarial practice and to assess accuracy of predictive models, it is also a regulatory issue. [START_REF] Antonio | Micro-level stochastic loss reserving for general insurance[END_REF] and [START_REF] Pigeon | Individual Loss Reserving using Paid-Incurred Data[END_REF] obtained, on real data analysis, lower variance on the total amount of reserves with "micro" models than with "macro" ones. A natural question is about the generality of such result. Should "micro" model generate less variability than standard "macro" ones? That is the question that initiated that paper. 1.3. Agenda. In section 2, we detail intuitive results we expect when aggregating data by clusters, moving from micro-level models to macrolevel ones. More precisely, we explain why with a linear model and a Poisson regression, macro-and micro-level models are equivalent. We also discuss the case of the Poisson regression model with random intercept. In section 3, we study "micro" and "macro" models in the context of claims reserving, on real data, as well as simulated ones.

Clustering in Generalized Linear Mixed Models

In the economic literature, several papers discuss the use of "micro" vs. "macro" data, for instance in the context of unemployment duration in [START_REF] Van Den Berga | Combining micro and macro unemployment duration data[END_REF] or in the context of inflation in [START_REF] Altissimo | Fast micro and slow macro: can aggregation explain the persistence of inflation[END_REF]. In [START_REF] Van Den Berga | Combining micro and macro unemployment duration data[END_REF], it is mentioned that both models are interesting, since "micro" data can be used to capture heterogeneity while "macro" data can capture cycle and more structural patterns. In [START_REF] Altissimo | Fast micro and slow macro: can aggregation explain the persistence of inflation[END_REF], it is demonstrated that both heterogeneity and aggregation might explain the persistence of inflation at the macroeconomic level.

In order to clarify notation, and make sure that objects are well defined, we use small letters for sample values, e.g. y i , and capital letters for underlying random variables, e.g. Y i in the sense that y i is a realisation of random variable Y i . Hence, in the case of the linear model (see Section 2.1), we usually assume that Y i ∼ N (x T i b, σ 2 ), and then b is the estimated model, in the sense that b = (xx T ) -1 xy while B = (xx T ) -1 xY (here covariates x are given, and non stochastic). Since B is seen as a random variable, we can write

E[ B] = b. With a Poisson regression, Y i ∼ P(λ i ) with λ i = exp[x T i b]. In that case, Var[Y i ] = E[Y i ] = λ i .
Y i ] = ϕE[Y i ] = ϕλ i .
And for convenience, we will denote Y i ∼ qP(λ i ), with an abuse of notation.

In this section, we will derive some theoretical results regarding aggregation in econometric models.

2.1.

The multiple linear regression model. Consider a (multiple) linear regression model,

y i,g = x T g a + ε i,g , (1)
a = a 1 . . . a k+1 T x g = x g,1 . . . x g,k+1 T ,
where observations belong to a cluster g and are indexed by i within a cluster g, i = 1, . . . , n g , g = 1, . . . , m. Assume further assumptions of the classical linear regression model [START_REF] Greene | Econometric Analysis[END_REF], i.e., (LRM1) no multicollinearity in the data matrix; (LRM2) exogeneity of the independent variables E [ε i,g |x g ] = 0, i = 1, . . . , n g , g = 1, . . . , m; and (LRM3) homoscedasticity and nonautocorrelation of error terms with Var [ε i,g ] = σ 2 .

Stacking observations within a cluster yield the following model

y g = x T g b + e g , (2)
where Proof.

y g = 1 n g i y i,
(i) The ordinary least-squares estimator for a -from model (1) -is defined as

a = argmin a i,g y i,g -x T g a 2 (3)
which can also be written

a = argmin a i,g y i,g -y g + y g -x T g a 2 . (4)
Now, observe that i,g

y i,g -y g + y g -x T g a 2 = i,g (y i,g -y g ) 2 + (y g -X T g a) 2 + 2(y i,g -y g )(y g -x T g a),
where the first term is independent of a (and can be removed from the optimization program), and the term with cross-elements sums to 0. Hence,

a = argmin a i,g (y g -x T g a) 2 = argmin a g n g (y g -x T g a) 2 = b, (5)
where b is the least square estimator of b from model (2), when weights n g are considered.

(ii) If we consider the sum of predicted values, observe that i,g

y i,g = g n g x T g a = g n g x T g b y g = g y g . (6)
Hence, the sum of predictions obtained from model (1) is the same as the sum of predictions obtained from model (2), even if partial sums are considered.

In the proposition above, the equality should be understood as the equality between estimators. Hence we have the following corollary.

Corollary 2.2. We define the following matrices

Y g = Y 1,g . . . Y ng,g T Y = Y T 1 . . . Y T m T Y = Y 1 . . . Y m T ε g = ε 1,g . . . ε ng,g T ε = ε 1 . . . ε m T e = e 1 . . . e m T x ng g = x g . . . x g ng times x = x n 1 1 . . . x nm m x = x 1 1 . . . x 1 m ,
the (1 × n g ) vectors 1 ng = 1 . . . 1 and 0 ng = 0 . . . 0 , and the matrix

1 =     1 n 1 0 n 2 . . . 0 nm 0 n 1 1 n 2 . . . 0 nm . . . . . . . . . . . . 0 n 1 0 n 2 . . . 1 nm     .
The OLS estimators are given by

A OLS = argmin a { Y -x T a T Y -x T a } = xx T -1 xY B OLS = argmin b { ( 11 T )Y -11 T x T b T ( 11
T )Y -( 11 T )x T b } = x11 T x T -1 x11 T Y .
Model (1) on a micro level and model ( 2) on a macro level are equivalent, in the sense that

(i) E[ A OLS ] = E[ B OLS ] and Var[ A OLS ] = Var[ B OLS ],
when weights n g are used in model [START_REF] Mack | Distribution-free calculation of the standard error of chain ladder reserve estimates[END_REF]; and

(ii) E i,g Y i,g = E g Y g and Var i,g Y i,g = Var g Y g .
Proof. Straightforward calculations lead to (11

T ) -1 1Y = Y and x1 = x. (i) Let E B OLS = x11 T x T -1 x11 T E Y = x11 T x T -1 x11 T (11 T ) -1 1E [Y ] = xx T -1 xE [Y ] = E A OLS .
For the equality of variances, we have

Var B OLS = x11 T x T -1 x11 T Var Y x11 T x T -1 x11 T T = x11 T x T -1 x11 T (11 T ) -1 1Var [Y ] 1 T (11 T ) -1 T x11 T x T -1 x11 T T = (xx T ) -1 xVar [Y ] x T (xx T ) -1 T = Var A OLS . (ii) Let E g Y g = E 1 m 11 T Y = E 1 m 11 T x T B = E 1 m 11 T x T A = E 1 n x T A = E 1 n Y = E i,g Y i,g .
The proof of the equality of variances is similar.

The quasi-Poisson regression.

A similar result can be obtained in the context of Poisson regressions. A generalized linear model [START_REF] Mccullagh | Generalized Linear Models[END_REF] is made up of a linear predictor x T b, a link function that describes how the expected value depends on this linear predictor and a variance function that describes how the variance, depends on the expected value Var

[Y ] = ϕV (E [Y ]
), where ϕ denotes the dispersion parameter. For the Poisson model, the variance is equal to the mean, i.e., ϕ = 1 and

V (E [Y ]) = E [Y ]
. This may be too restrictive for many actuarial illustrations, which often show more variation than given by expected values. We use the term over-dispersed for a model where the variance exceeds the expected value. A common way to deal with overdispersion is a quasi-likelihood approach (see [START_REF] Mccullagh | Generalized Linear Models[END_REF] for further discussion) where a model is characterized by its first two moments. Consider either a Poisson regression model, or a quasi-Poisson one,

Y i,g ∼ P(λ i,g ) or Y i,g ∼ qP(λ i,g ). (7)
In the case of a Poisson regression,

E [Y i,g ] = λ i,g = exp[x T g a + ln(1/n g )] and Var [Y i,g ] = λ i,g , (8)
and in the context of a quasi-Poisson regression,

E [Y i,g ] = λ i,g = exp[x T g a + ln(1/n g )] and Var [Y i,g ] = ϕ micro λ i,g , (9
) with ϕ micro > 0 for a quasi-Poisson regression (ϕ micro > 1 for overdispersion). Here again, stacking observations within a cluster yield the following model (on the sum and not the average value, to have a valid interpretation with a Poisson distribution)

Y g = i Y i,g ∼ P(λ g ) or Y g ∼ qP(λ g ). ( 10
)
In the context of a Poisson regression,

E [Y g ] = λ g = exp[x T g b] and Var [Y g ] = λ g , and in the context of a quasi-Poisson regression, E [Y g ] = λ g = exp[x T g b] and Var [Y g ] = ϕ macro λ g , (11)
with ϕ macro > 0 for a quasi-Poisson regression. Here again, those two models ("micro" and "macro") are equivalent, in the sense that the following proposition holds. Proposition 2.3. Model ( 7) on a micro level and model [START_REF] Hesselager | A Markov model for loss reserving[END_REF] on a macro level are equivalent in the sense that

(i) a M LE = b M LE ; and (ii)
i,g

y i,g = g y g .
Proof.

(i) Maximum likelihood estimator of a is the solution of i,g

y i,g -exp[x T g a] ϕ micro x g = 0 or equivalently i,g y i,g -exp[x T g a] x g = 0
With offsets λ * g = exp[x T g b+log(n g )], g = 1, . . . , m, maximum likelihood estimator of b is the solution (as previously, we can remove ϕ macro ) of

g y g -n g exp[x T g b] x g = 0 i,g y i,g -exp[x T g b] x g = 0.
Hence, a = b, as (unique) solutions of the same system of equations.

(ii) The sum of predicted values is i,g

y i,g = g n g λ i,g = g n g exp[x T g a] = g n g exp[x T g b] = g exp[x T g b + log(n g )] = g λ * g = g y g .
-Nevertheless, as we will see later on, the Corollary obtained in the context of a Gaussian linear model does not hold in the context of a quasi-Poisson regression.

Corollary 2.4. Model ( 7) on a micro level and model [START_REF] Hesselager | A Markov model for loss reserving[END_REF] on a macro level are asymptotically equivalent for Poisson regressions, in the sense that

(i) E[ A M LE ] = E[ B M LE ] and Var[ A M LE ] = Var[ B M LE ],
when n goes to infinity; and

(ii) E i,g Y i,g = E g Y g and Var i,g Y i,g = Var g Y g , when
n goes to infinity.

Proof.

(i) A classical result of asymptotic theory for maximum likelihood estimators indicates that, under mild regularity conditions,

E A M LE → a and E B M LE → b as n → ∞. Without "micro" covariates in the model, a = b and so, E B M LE = E A M LE when n → ∞.
For model [START_REF] Arjas | The claims reserving problem in nonlife insurance -some structural ideas[END_REF], the Fisher information matrix is

I(A) = xW x T and, when n → ∞, Var A → xW x T -1 , where W = diag((λ 1 /n 1 )1 n 1 , . . . , (λ m /n m )1 nm ).
For model [START_REF] Hesselager | A Markov model for loss reserving[END_REF], we have

I(B) = x1W 1 T x T = xW x T and, when n → ∞, Var B → xW x T -1 .
(ii) By using a similar argument, we have when n goes to infinity

E g Y g = E 1 m 11 T Y = 1 m 11 T M B x T = 1 m 11 T M A x T = E 1 n 1 T e x T A = E 1 n e x T A = E 1 n Y = E i,g Y g,i .
-In small or moderate-sized samples, it should be noted that A and B may be biased for A and B, respectively. Generally, this bias is negligible compared with the standard errors (see [START_REF] Cordeiro | Bias correction in generalized linear models[END_REF] and [START_REF] Wüthrich | Stochastic claims reserving methods[END_REF]).

In the quasi-Poisson micro-level model (from model ( 7)), as discussed above, the estimator of a is the solution of the quasi-score function i,g

y i,g -λ i,g ϕ micro x g = 0,
which implies a QLE = a M LE . The classical Pearson estimator for the dispersion parameter ϕ micro is

ϕ micro = i,g (y i,g -y i,g ) 2 / y i,g g n g -(k + 1) = i,g (n g y i,g -y g ) 2 /n g y g g n g -(k + 1)
.

Empirical evidence (see [START_REF] Ruoyan | Estimation of dispersion parameters in GLMs with and without random effects[END_REF]) support the use of the Pearson estimator for estimating ϕ because it is the most robust against the distributional assumption. In a similar way, the quasi-Poisson macro-level model (from model [START_REF] Hesselager | A Markov model for loss reserving[END_REF]), the estimator of b is the solution of

g y g -λ g ϕ macro x g = 0,
which implies here also b QLE = b M LE . The dispersion parameter ϕ is estimated by

ϕ macro = g (y g -y g ) 2 / y g m -(k + 1)
.

Clearly, ϕ micro = ϕ macro involving the following results.

Corollary 2.5. Model ( 7) on a micro level and model ( 10) on a macro level are not asymptotically equivalent for quasi-Poisson regressions, in the sense that

(i) E[ A QLE ] = E[ B QLE ] but Var[ A QLE ] = Var[ B QLE ],
when n goes to infinity; and

(ii) E i,g Y i,g = E g Y g but Var i,g Y i,g = Var g Y g , when
n goes to infinity.

Proof.

(i) The property that variances are not equal is a direct consequence of classical results from the theory of generalized linear models (see [START_REF] Mccullagh | Generalized Linear Models[END_REF]), since the covariance matrices of estimators are given by

Var B → ϕ macro xW x T -1 and Var A → ϕ micro xW x T -1 , (12)
when n goes to infinity. Thus, covariance matrices of estimators are asymptotically equal for the Poisson regression model but differ for the quasi-Poisson model because ϕ micro = ϕ macro . (ii) Since the MLE and the QLE share the same asymptotic distribution (see [START_REF] Mccullagh | Generalized Linear Models[END_REF]), the proof is similar to 2.4(ii).

Poisson regression with random effect.

In the micro-level model described by Equation ( 7), observations made for the same event (subject) at different periods are supposed to be independent. Withinsubject correlation can be included in the model by adding random, or subject-specific, effects in the linear predictor. In the Poisson regression model with random intercept, the between-subject variation is modeled by a random intercept γ which represents the combined effects of all omitted covariates. Let Y (t) g represent the sum of all observations from subject t, in the cluster g and

E Y (t) g |γ t = exp[x T g a + γ t ] Var Y (t) g |γ t = E Y (t) g |γ t γ =   γ 1 . . . γ T   ∼ N T 0, σ 2 I ,
where I is the (T ×T ) identity matrix, and N T (µ, Σ) the T -dimensional Gaussian distribution with mean µ and covariance matrix Σ. Straightforward calculations lead to

E Y (t) g = exp[x T g a + σ 2 /2] Var Y (t) g = E Y (t) g 1 + E Y (t) g exp[σ 2 ] -1 . Hence, Var Y (t) g > E Y (t) g , σ 2 > 0.
(13) This last equation shows that the Poisson regression model with random intercept leads to an over-dispersed marginal distribution for the variable Y (t) g . The maximum likelihood estimation for parameters requires Laplace approximation and numerical integration (see the Chapter 4 of [START_REF] Snijders | Multilevel analysis: an introduction to basic and advanced multilevel modeling[END_REF] for more details). This model is a special case of multilevel Poisson regression model and estimation can be performed with various statistical softwares such as HLM, SAS, Stata and R (with package lme4).

One may be interested to verify the need of a source of betweensubject variation. Statistically, it is equivalent to testing the variance of γ to be zero. In this particular case, the null hypothesis places σ 2 on the boundary of the model parameter space which complicates the evaluation of the asymptotic distribution of the classical likelihood ratio test (LRT) statistic. From the very general result of [START_REF] Self | Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions[END_REF], it can be demonstrated (see [START_REF] Zhang | Random Effect and Latent Variable Model Selection[END_REF]) that the asymptotic null distribution of the LRT statistic is a 50/50 mixture of χ 2 0 and χ 2 1 as g n g → ∞. In this case, obtaining an equivalent macro-level model is of little practical interest since the construction of the variance-covariance matrix would require knowledge of the individual ("micro") data.

Clustering and loss reserving models

A loss reserving macro-level model is constructed from data summarized in a table called run-off triangle. Aggregation is performed by occurrence and development periods (typically years). For occurrence period i, i = 1, 2, . . . , I, and for development period j, j = 1, 2, . . . I, let C i,j and Y i,j represent the total cumulative paid amount and the incremental paid amount, respectively with Y i,j = C i,j -C i,j-1 , i = 1, . . . , I, j = 2, . . . , I.

    C 1,1 C 1,2 . . . C 1,I-1 C 1,I C 2,1 C 2,2 . . . C 2,I-1 . . . . . . . . . C I,1         Y 1,1 = C 1,1 Y 1,2 . . . Y 1,I-1 Y 1,I Y 2,1 = C 2,1 Y 2,2 . . . Y 2,I-1 . . . . . . . . . Y I,1 = C I,1    
where columns, rows and diagonals represent development, occurrence and calendar periods, respectively. Each incremental cell Y i,j can be seen as a cluster stacking n i,j amounts paid in the same development period j for the occurrence period i. These payments come from M claims and let Y (k) i,j represent the sum of all observations from claims k in the cluster (i, j). It should be noted that all claims are not necessarily represented in each of the clusters.

To calculate a best estimate for the reserve, the lower part of the triangle must be predicted and the total reserve amount is

R = I t=2 C t,I - I t=2 C t,I-t+1 = I t=2 I s=I+2-t Y t,s .
To quantify uncertainty in estimated claims reserve, we consider the mean square error of prediction (MSEP). Let R be a Y-mesurable estimator for E [R|Y] and a Y-mesurable predictor for R where Y represents the set of observed clusters. The MSEP is

M SEP R|Y ( R) = E R -R 2 |Y = Var [R|Y] + R -E [R|Y] 2 .
Independence between R and Y is assumed, so the equation is simplified as follows

M SEP R|Y ( R) = Var [R] + R -E [R] 2
and the unconditional MSEP is

M SEP R ( R) = Var [R] + E R -E [R] 2 .
3.1. The quasi-Poisson model for reserves.

3.1.1. Construction. From the theory presented in Subsection 2.2, we construct quasi-Poisson macro-and micro-level models for reserves. For both models, constitutive elements are defined in Table 1.

Table 1. Quasi-Poisson macro-and micro-level models for reserve (i, j = 1, . . . , I). All clusters and all payments are independent.

Components Macro Micro

Exp. value

E [Y i,j ] = λ i,j E Y (k) i,j = λ i,j Inv. link func. λ i,j = exp[x T i,j b] λ i,j = exp[x T i,j a + log(1/n i,j )] = exp[b i + b I+j ] = exp[a i + a I+j + log(1/n i,j )] with b I+1 = 0 with a I+1 = 0 Variance Var [Y i,j ] = ϕ macro λ i,j Var Y (k) i,j = ϕ micro λ i,j
Pred. value

Y i,j = exp[ b i + b I+j ] Y (k) i,j = exp[ a i + a I+j + log(1/n i,j )] Known values Y macro Y micro
As a direct consequence of Proposition 2.3, the best estimate for the total reserve amount is

R = (i,j)∈K n i,j k=1 Y (k) i,j = (i,j)∈K Y i,j ,
where K represents unobserved clusters. For both models, the Proposition 3.1 gives results for the unconditional MSEP. Proposition 3.1. In the quasi-Poisson macro-level model, the unconditional MSEP is given by

M SEP R ( R) ≈ (i,j)∈K ϕ macro y i,j + (i,j),(n,m)∈K ϕ macro y i,j y m,n x T i,j xW x T -1 x n,m ,
where x and W are defined by Equation [START_REF] Hesselager | Reserving in Non-Life Insurance[END_REF]. The unconditional MSEP for the quasi-Poisson micro-level model is similar with ϕ macro replaced by ϕ micro .

Proof. The proof for the macro-level model is done in [START_REF] Wüthrich | Stochastic claims reserving methods[END_REF]. For the micro-level model, we have

M SEP R ( R) = Var [R] + E R -E [R] 2 = (i,j)∈K n i,j k=1 ϕ micro exp[x T i,j a + log(1/n i,j )] + (i,j)∈K (m,n)∈K n i,j k=1 nm,n t=1 Cov Y (k) i,j , Y (t) m,n =
(i,j)∈K ϕ micro exp[x T i,j a] + (i,j)∈K (m,n)∈K n i,j k=1 nm,n t=1 exp[x T i,j a + log(1/n i,j )] exp[x T m,n a + log(1/n m,n )] × Cov exp[x T i,j a -x T i,j a], exp[x T m,n a -x T m,n a] Although Y (k) i,j is not an unbiased estimator of E Y (k)
i,j , the bias is generally of small order and by using the approximation exp[x] ≈ 1 + x for x ≈ 0, we obtain

= (i,j)∈K ϕ micro exp[x T i,j a] + (i,j),(m,n)∈K exp[x T i,j a + x T m,n a]Cov x T i,j a, x T m,n a .
By using the fact that b = a and the remark at the end of subsection 2.2, we obtain

= (i,j)∈K ϕ micro y i,j + (i,j),(m,n)∈K ϕ micro y i,j y m,n x T i,j xW x T -1 x m,n .
Thus, the difference between the variability in macro-and micro-level models results from the difference between dispersion parameters. Define standardized residuals for both models r i,g = (y i,g -y i,g ) y i,g and r g = (y g -y g ) y g .

Direct calculations lead to

Ψ = i,g r 2 i,g g r 2 g ≤ g n g -(k + 1) m -(k + 1) → ϕ micro ≤ ϕ macro . (14)
Thus, if the total number of payments ( g n g ) is greater than the value Ψ(m -(k + 1)) + k + 1, then the micro-level model [START_REF] Arjas | The claims reserving problem in nonlife insurance -some structural ideas[END_REF] will lead to a greater precision for the best estimate of the total reserve amount and conversely. Adding one or more covariate(s) at the micro level will decrease the numerator of Ψ and will increase the interest of the micro-level model.

Illustration and Discussion.

To illustrate these results, we consider the incremental run-off triangle from UK Motor Non-Comprehensive account (published by [START_REF] Christofides | Regression models based on log-incremental payments[END_REF]) presented in Table 2 where each cell (i, j), i + j ≤ 7, is assumed to be a cluster g, i.e., the value Y g is the sum of n g independent payments.

Table 2. Incremental run-off triangle for macro-level model (in 000's). We construct 2 macro-level models

Model A: Y g ∼ P(λ g ) λ g = exp[x T g a] Model B: Y g ∼ qP(λ g )
and 2 micro-level models

Model C: Y i,g ∼ P(λ i,g ) λ i,g = exp[x T g a -log(n g )] Model D: Y i,g ∼ qP(λ i,g ).
The final reserve amount obtained from the Mack's model ( [START_REF] Mack | Distribution-free calculation of the standard error of chain ladder reserve estimates[END_REF]) is 28 655 773$. To create micro-level datasets from the "macro" one, we perform the following procedure:

(1) simulate the number of payments for each cluster assuming N g ∼ P(θ), g = 1, . . . , m; (2) for each cluster, simulate a (n g × 1) vector of proportions assuming ω g = ω 1 . . . ω ng T ∼ Dirichlet(1), g = 1, . . . , m;

(3) for each cluster, define 1 shows √ M SEP as a function of a expected total number of payments, for the portfolio. Above a certain level, (close to 3 400 here), accuracy of the "micro" approach exceed the "macro". In order to illustrate the impact of adding a covariate at the microlevel, we define a quasi-Poisson micro-level model with a weakly correlated covariate (Model E) and with a strongly correlated covariate (Model F). Following a similar procedure, we obtain results presented in Table 3 and Figure 2. 2 As opposed to standard classical results on hierarchical models, the average of explanatory variable within a cluster ((1/n g ) i x ig ) has not been added to the macro-level model (Model B), for several reasons, With an explanatory variable highly correlated with the response variable, results obtained with Model D and E are very close. As claimed by Proposition 3.1 and equation ( 14), an explanatory variable highly correlated with the response variable will decrease the value of √ M SEP , and lowers the threshold above which the micro-level model is more accurate than the macro-level one.

  Y 1,g . . . Y ng,g   = ω g Y g , g = 1, . . . , m; (4
The quasi-Poisson macro-level model (Model B) with maximum likelihood estimators leads to the same reserves as the chain-ladder algorithm and the Mack's model (see [START_REF] Mack | A Comparison of Stochastic Models that Reproduce Chain Ladder Reserve Estimates[END_REF]), assuming the clusters exposure, for (i, j) ∈ K, is one. To obtain similar results with a quasi-Poisson micro-level model (Model D), a similar assumption is necessary: exposure of each claim within cluster (i, j) is 1/n i,j . That assmption implies, on a micro level, that predicted individual payments Y (k) ij are proportional to 1/n ij . That assumption has unfortunately no foundation.

In the Poisson and quasi-Poisson micro-level models (Model C and D), payments related to the same claim, in two different clusters are supposed to be non-correlated. As discussed in the previous Section, it is possible to include dependencies among payments for a given claim using a Poisson regression with random effects. Simulations and computations were performed in R, using packages ChainLadder and gtools.

The Mixed Poisson model for reserves.

3.2.1. Construction. From the results obtained in Section 2.3, it is possible to construct a micro-model for the reserves that includes a random intercept.The later will allow to model dependence between payments from a given claim. Note that it is hard to find an aggregated model with random effects that could be compared with individual ones. In the context of claims reserves, Y (t) g represents the sum of paids made for claim t within cluster g. The assumptions of that model (called model G) are

Y (t) g |γ t ∼ P(λ g e γt ), λ g = exp[x T g c + ln(1/n g )] γ t ∼ N (0, σ 2 ).
Because of those two random variables in the model, two kinds of predictions can be derived: un-conditional ones, where

Y (t)
g |γ t ∼ P( λ g e γt )

λ g = exp[x T g c + ln(1/n g )] so that E Y (t) g = λ g e σ 2 /2 ; and conditional ones, where the unknown magnitude of claim t is predicted by the so-called best linear estimate (that minimizes the MSEP) γt (see [START_REF] Wüthrich | Stochastic claims reserving methods[END_REF])

Ỹ (t)
g |γ t ∼ P( λ g e γt )

so that E Ỹ (t) g = λ g e γt .

It is then possible to compute the overall best estimate for the total amount of reserves.

3.2.2. Illustration and Discussion. In order to construct a micro-level model from triangle 2, we follow a procedure to the one described in the provious section, with steps 1-3 (that are not mentioned here) 4. for each accident year, allocate randomly the source of each payment 5. fit model G; and 6. compute the best estimate and the MSEP of the reserve.

For a fixed value of θ, the procedure is repeated 1000 times. Various values were considered for θ (10, 25, 50, 100 and 250), and results were similar. In order to avoid heavy tables, only the case where θ = 10 is mentioned here. Simulations and computations were performed with R, relying on package lme4. On Figure 3 we can see predictions of the model on observed data, while on Figure 4 we can see predictions of the model for non-observed cells. Finaly, results are reported in Table 4. At each step, a LRT is performed (see section 2.3) and each time, the variance at origin was significant non-null, meaning that correlation among payments (related to the same claim) is positive. Observe that with the random model, the log-likelihood is approximated using numerical intergration, which might bias computed p-values of the test. Here, p have been confirmed using a bootstrap procedure (using package glmmML). q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 5 10

  g and b = b 1 . . . b k+1 T with similar assumptions except for Var [e g ] = σ 2 /n g . Those two models are equivalent, in the sense that the following proposition holds. Proposition 2.1. Model (1) on a micro level and model (2) on a macro level are equivalent, in the sense that (i) a OLS = b OLS when weights n g are used in model (2); and (ii) i,g y i,g = g y g where y g = n g y g .

Figure 1 .

 1 Figure 1. Square root of the mean square error of prediction obtained for Model D (solid line) and the Model C (broken line) from simulated values for increasing expected number of payments for the portfolio.

Figure 2 .

 2 Figure 2. Mean square error of prediction (±2σ) obtained from simulated values as a function of the expected number of payments for Model E (red lines) and Model F (blue lines). For comparison purposes, the MSEP obtained for the Model D (solid black line) and the Model B (broken black line) are added.

Figure 3 .

 3 Figure 3. Observed data (circles) with conditional prediction (plain red lines) and un-conditional (plain blue lines) from model G (θ = 10).

Figure 4 .

 4 Figure 4. Predictions with the quasi-Poisson macrolevel model (strong black line), with conditional prediction (plain red lines) and un-conditional (plain blue lines) from model G (θ = 10).

  The estimate parameter b is a function of the observations, (x i , y i )'s, while B is a function of the observations, (x i , Y i )'s. In the context of the Poisson regression, recall that E[ B] → b as n goes to infinity. With a quasi-Poisson regression, Y i does not have, per se, a proper distribution. Nevertheless, its moments are well defined, in the sense that Var[

  ) adjust Model C and Model D; and (5) calculate the best estimate and the MSEP of the reserve.

	For each value of θ, we repeat this procedure 1 000 times and we cal-
	culate the average best estimate and the average MSEP. Those results
	are consistent with Corollaries 2.4, 2.5 and Proposition 3.1. For Pois-
	son regression (Model A and C), results are similar. For micro-level models, convergence of √ M SEP fowards (11 622) is fast. For quasi-
	Poisson regression (Model B and D), Figure

Table 3 .

 3 Results.

	Method	E [Reserve]	√ M SEP
	Mack's model	28 655 773 1 417 267
	Poisson reg.
	Model A	28 655 773 11 622
	Model C	28 655 773 11 622
	quasi-Poisson reg.
	Model B	28 655 773 1 708 196
	Model D	28 655 773 see Figure 1
	quasi-Poisson reg.
	Model E (ρ ≈ 0)	28 657 364 see Figure 2
	Model F (ρ ≈ 0.8) 20 514 566 see Figure

Table 4 .

 4 Numerical Results for θ = 10. Results for different values of θ are similar.

	Modèle	E [Reserve]	Var(Reserve)
	coll. quasi-Pois.	28 656 423 1 708 216
	mixed Poisson non-cond. 27 930 624 3 297 401
	mixed Poisson cond.	25 972 947 2 280 902