Cédric Boutillier

WITH AN APPENDIX Kilian Raschel

Alin Bostan

Martin boundary of killed random walks on isoradial graphs

Keywords: Mathematics Subject Classification. Primary 31C35, 05C81, Secondary 82B20, 60J45 Martin boundary, Green function, isoradial graphs, killed random walk, discrete exponential function

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction and results

1.1. Ney and Spitzer theorems. Let (Z n ) n 0 be a random walk on Z d with finite support and non-zero drift. In the celebrated paper [START_REF] Ney | The Martin boundary for random walk[END_REF], Ney and Spitzer derive the asymptotics of the Green function

G(x, y ) = ∞ n=0 P x [Z n = y ] (1)
where the mapping r → ζ(r) is proved to define a homeomorphism between the ambient sphere S d-1 and the set

∂D = {ζ ∈ R d : φ(ζ) = 1}, (4)
see [START_REF] Ney | The Martin boundary for random walk[END_REF]Cor. 1.3] and [START_REF] Hennequin | Processus de Markoff en cascade[END_REF]; see also [START_REF] De Tilière | Scaling limit of isoradial dimer models and the case of triangular quadri-tilings[END_REF] for an expression of the above homeomorphism. Figure 1 contains a few examples of sets ∂D in [START_REF] Beffara | On the critical parameters of the q 4 random-cluster model on isoradial graphs[END_REF].

Introduce now the discrete Laplacian ∆ = Id -p, i.e., for x ∈ Z d ,

∆f (x) = y ∈Z d p(x, y )(f (x) -f (y )).
From a potential theory viewpoint it is obvious that the exponential functions f (x) = e ζ•x are discrete harmonic (meaning ∆f = 0) if and only if ζ ∈ ∂D. It is further known [START_REF] Choquet | Sur l'équation de convolution µ = µ * σ[END_REF][START_REF] Doob | Application of boundary theory to sums of independent random variables[END_REF] that they are (positive) minimal, in the sense that if h is harmonic and satisfies 0 h f on Z d , then h = cf for some 0 c 1.

A corollary of Ney and Spitzer result (3) is that the Martin compactification of the random walk coincides with the geometric compactification of Z d by the sphere S d-1 at infinity. More precisely, any harmonic function f for (Z n ) n 0 may be written as f (x) = ∂D e ζ•x µ(dζ), where µ is a positive Radon measure on ∂D, see [START_REF] Ney | The Martin boundary for random walk[END_REF]Rem. a]. For a general introduction on Martin boundary see [START_REF] Martin | Minimal positive harmonic functions[END_REF][START_REF] Doob | Discrete potential theory and boundaries[END_REF][START_REF] Hunt | Markoff chains and Martin boundaries[END_REF][START_REF] Sawyer | Harmonic functions on trees and buildings[END_REF][START_REF] Woess | Random walks on infinite graphs and groups[END_REF].

Let us mention a few strongly related results. First, the original Ney and Spitzer result (3) does not assume the boundedness of the support (it is only assumed that every point of ∂D has a neighborhood in which φ in (2) is finite), but throughout this article we shall only consider random walks with bounded increments. In absence of drift and if d 3 (so as to have a transient behavior), the Martin boundary is reduced to a single point, see [START_REF] Spitzer | Principles of random walk[END_REF]Sec. 26] and [56, (25.11) and (25.12)] for exact statements and proofs. Let us also point out the concept of t-Martin boundary (which probabilistically corresponds to a random walk with killing), which involves the Green function

G(x, y | 1 t ) = ∞ n=0 P x [Z n = y ] 1 t n . (5)
Obviously G(x, y |1) = G(x, y ) in our notation [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]. This t-Martin boundary is either empty (when t < ρ, where ρ is the spectral radius of the transition kernel), reduced to one point (t = ρ) or homeomorphic to S d-1 (when t > ρ), see Woess [START_REF] Woess | Random walks on infinite graphs and groups[END_REF]Sec. 25.B]. The ratio G(x 1 , y | 1 t )/G(x 0 , y | 1 t ) converges to the same quantity as in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF], but with ζ(r) ∈ ∂D t , where

∂D t = {ζ ∈ R d : φ(ζ) = t}. (6)
We finally mention the work [START_REF] Dussaule | The Martin boundary of a free product of abelian groups[END_REF], in which Dussaule obtains the Martin boundary of thickenings of Z d (i.e., Cartesian products of Z d by a finite set). In the reference [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF] Babillot considers non-lattice random walks and obtains the Martin boundary of thickenings of R d . 6 + e -ζ 2 6 and t = 2 √ 2/3 ≈ 0.9428, 0.943, 0.95, 0.97, 1, 1.1. It corresponds to a nearest neighbor walk in Z 2 with jump probability 1 3 to the North and East, and 1 6 to the South and West. The spectral radius of the transition kernel is 2 √ 2/3.

Spatial inhomogeneity.

A crucial ingredient in the proof of all above results is the spatial homogeneity (or invariance by translation) of Z d and of the random walk, meaning that p(x, y ) = p(0, yx) for all x, y ∈ Z d . The perturbation of the homogeneity at a finite number of sites of the grid Z d should not affect the structure of the Martin boundary, but already impacts on the expressions for harmonic functions. This is shown by Kurkova and Malyshev [42,Thm 2.1] in the case of planar random walks with jumps to the nearest neighbors; they obtain the asymptotics of the Green function and show that the Martin boundary is homeomorphic to the circle S 1 . On the other hand, in the case of a homogeneity perturbed at an infinite number of points, computing the Green function asymptotics or deriving the Martin boundary seems too difficult in general (even if some upper and lower bounds may exist for the transition probabilities or for the Green functions, see [START_REF] Mustapha | Gaussian estimates for spatially inhomogeneous random walks on Z d[END_REF]).

It is therefore natural to introduce a structure behind the inhomogeneities. A class of such models is composed of random walks killed or reflected when they exit a domain (inhomogeneities then appear on the boundary of the domain). In this context, Kurkova and Malyshev [START_REF] Kurkova | Martin boundary and elliptic curves[END_REF], Ignatiouk-Robert [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a half-space[END_REF][START_REF] Ignatiouk-Robert | Martin boundary of a reflected random walk on a half-space[END_REF][START_REF] Ignatiouk-Robert | t-Martin boundary of reflected random walks on a half-space[END_REF], Ignatiouk-Robert and Loree [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF] obtain the asymptotics of (ratios of) Green functions as in (3) and derive the Martin boundary when the domain is a quarter plane Z 2 + or a half-space Z + × Z d-1 . See also [START_REF] Godefroy | Martin boundary of polynomial random walks on the d-dimensional lattice of nonnegative integers[END_REF].

1.3. The case of isoradial graphs. In this article we explore another class of models having inhomogeneities in infinite number, namely, killed random walks on isoradial graphs. By definition, isoradial graphs (see Figure 2 for a few examples) are planar embedded graphs such that all faces are inscribable in a circle of radius 1; more details will be provided in Section 2. So the irregularities of such graphs are structured, but as it is illustrated by looking at the pictures, the behavior may be quite wild. In particular, with the exception of a few simple particular cases (the square lattice Z 2 or the triangular lattice), an isoradial graph is not a linear transform of a two-dimensional lattice, is not translation-invariant and can be actually highly non-homogeneous. Introduced by Duffin [START_REF] Duffin | Potential theory on a rhombic lattice[END_REF], Mercat [START_REF] Mercat | Discrete Riemann surfaces and the Ising model[END_REF] and Kenyon [START_REF] Kenyon | The Laplacian and Dirac operators on critical planar graphs[END_REF],

these graphs have become particularly popular, as they are well suited for discrete complex analysis [START_REF] Chelkak | Discrete complex analysis on isoradial graphs[END_REF]. Moreover, because the star-triangle transformation (see Figure 4), which plays an important role in integrable statistical mechanics, preserves isoradiality, many models of statistical mechanics are now studied [START_REF] Boutillier | Statistical mechanics on isoradial graphs[END_REF] on isoradial graphs: dimer model [START_REF] Kenyon | The Laplacian and Dirac operators on critical planar graphs[END_REF][START_REF] De Tilière | Scaling limit of isoradial dimer models and the case of triangular quadri-tilings[END_REF], two-dimensional Ising model [START_REF] Boutillier | The Z-invariant Ising model via dimers[END_REF], spanning trees, bond percolation [START_REF] Grimmett | Bond percolation on isoradial graphs: criticality and universality[END_REF], random cluster model [START_REF] Beffara | On the critical parameters of the q 4 random-cluster model on isoradial graphs[END_REF][START_REF] Duminil-Copin | Universality for the random-cluster model on isoradial graphs[END_REF], etc. Despite their intrinsic irregularity and the absence of group structure, isoradial graphs form a broad class of graphs on which one can define an exponential function in a uniform way; this is a first step towards a Ney and Spitzer theorem. Such an exponential function is only proved to exist when the isoradial graph G is equipped with certain natural conductances, involving trigonometric or elliptic functions and geometric angles of the embedding of the graph. The existence is obtained by Kenyon [START_REF] Kenyon | The Laplacian and Dirac operators on critical planar graphs[END_REF] for critical (trigonometric) weights and is extended to the non-critical (elliptic) case by de Tilière and the two present authors [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]. Whereas the classical exponential function between any two points x, y ∈ Z d may be written as e ζ•(y -x) with a parameter ζ ∈ C, see (3), the exponential function on G takes the form e (x,y ) (u|k),

where x, y ∈ G, k ∈ (0, 1) is an elliptic modulus naturally attached to the model and u lies on the torus T(k) = C/(4K(k)Z + 4i K (k)Z), K and K denoting the elliptic integrals of the first kind and its complementary, respectively, see Section 2.3 and [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Sec. 3.3]. In the same manner that f (x) = e ζ•x is harmonic on Z d for ζ ∈ ∂D, the exponential function e (x,y ) (u|k) is discrete harmonic as a function of x (y being fixed), in the sense that its massive Laplacian

∆f (x) = z∼x ρ(θ xz |k)(f (x) -f (z)) + m 2 (x|k)f (x) (8)
is zero by [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Prop. 11], the conductances ρ and (squared) masses m 2 being defined in [START_REF] Boutillier | The critical Z-invariant Ising model via dimers: locality property[END_REF] and [START_REF] Boutillier | Statistical mechanics on isoradial graphs[END_REF]. The work [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF] also provides exact and asymptotic expressions for the Green function G(x, y ) of the killed random walk described by [START_REF] Bostan | Differential equations for algebraic functions[END_REF], see in particular [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Thm 14].

1.4. Main results. Building on the results of [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF], our main theorem is the asymptotics of the Martin kernel

lim y →∞ G(x 1 , y ) G(x 0 , y ) = e (x 0 ,x 1 ) (u 0 |k), (9)
where the exponential function e (x 0 ,x 1 ) in ( 7) is evaluated at a point u 0 ∈ 2i K + R which depends on the way that y goes to infinity in the graph. See Theorem 1 for the precise statement. Let us present the main features of (9):

• First, it gives the limit of the ratio of the Green function in any asymptotic direction in the graph. It also proves that the Martin boundary is homeomorphic to the sphere S1 , providing the announced generalization of Ney and Spitzer result (3) on isoradial graphs. The latter result follows from showing that u 0 describes a circle in the torus, as the asymptotic direction varies along all possible directions in the graph (see our Theorem 2). • Theorem 3 shows that for all u 0 ∈ 2i K +R and y ∈ G, e (x,y ) (u 0 |k) defines a minimal positive harmonic function (see Section 1.1 for the definition), thereby proving that the Martin boundary is minimal. Recall that positive harmonic functions are particularly important in potential theory, as they allow, via the Doob transformation, to construct conditioned processes (for example going to infinity along a given direction in the graph). • In the few particular cases which are both isoradial graphs and lattices (namely, the square and triangular lattices), we show that our results coincide with the classical Ney and Spitzer theorem and we unify the two points of view. • Beyond the square and triangular lattices, our results in particular apply to periodic isoradial graphs, which are constructed as lifts to the plane of a fundamental pattern on the torus. We prove that some tools developed for their study (characteristic polynomial, amoeba, quantity u 0 in (9), etc.) naturally correspond with objects in Z d (generating function of the transition probabilities, set ∂D t in (6), mapping ζ(r) in (3), etc.). The paper is organized as follows. Section 2 presents some crucial definitions and properties of isoradial graphs, Green functions and discrete exponential functions. Section 3 contains the statements of the main results and their proofs. After gathering and reinterpreting existing results on general periodic planar graphs, Section 4 presents a refined study of the case of periodic isoradial graphs. Section 5 focusses on minimal harmonic functions.

Acknowledgments. We warmly thank Béatrice de Tilière for interesting discussions. We also thank an anonymous referee for his/her reading and comments.

2. Isoradial graphs, Green functions and exponential functions 2.1. Isoradial graphs. Isoradial graphs, named after [START_REF] Kenyon | The Laplacian and Dirac operators on critical planar graphs[END_REF], are defined as follows 1 . An embedded planar graph G = (V, E) is isoradial if all internal faces are inscribable in a circle, with all circles having the same radius (fixed for example to 1), and such that all circumcenters are in the interior of the faces. Examples are provided in Figures 2 and7.

Let G be an infinite, isoradial graph, whose bounded faces fill the whole plane. Then the dual graph G * , embedded by placing dual vertices at the circumcenters of the corresponding faces, is also isoradial.

The diamond graph, denoted G , is constructed from an isoradial graph G and its dual G * . Vertices of G are those of G and those of G * . A dual vertex of G * is joined to all primal vertices on the boundary of the corresponding face. Since edges of the diamond graph G are radii of circles, they all have length 1 and can be assigned a direction ±e iα . Note that faces of G are side-length 1 rhombi.

A train-track is a bi-infinite path on the dual of the diamond graph (G ) * which does not turn: when entering in a rhombus, it exits through the opposite edge. See Figure 2. A train-track T thus crosses edges with the same direction ±e iα . We then say that the angle α is associated to the train-track. This angle is a priori well defined only modulo π. This ambiguity can be lifted by considering an orientation for the train-track and choosing a convention (i.e., the vector e iα crosses the oriented train-track from left to right), and then, the angle associated to the train-track oriented in the other direction is α + π.

A minimal path between two vertices x and y is a shortest path in G between these vertices. It is not unique, but the set of unit steps used is the same for all minimal paths.

Using the diamond graph, angles can naturally be assigned to edges of the graph G as follows. Every edge e of G is the diagonal of exactly one rhombus of G , and we let θ e be the half-angle at the vertex it has in common with e, see Figure 3. Note that we have Figure 2. Upper left: piece of an infinite graph G isoradially embedded in the plane with the circumcircles of the faces. Upper right: a periodic isoradial embedding of the square lattice. Lower left: an isoradial graph with multiple cone-like regions (see Section 3.3 for a related discussion). Lower right: a piece of the Penrose tiling with rhombi, which can be used as the diamond graph of an isoradial graph, and three train-tracks, as bi-infinite paths in the dual.

θ e ∈ (0, π 2 ), because circumcircles are assumed to be in the interior of the faces. From now on, we actually ask more and suppose that there exists ε > 0 such that θ e ∈ (ε, π 2 -ε). We also assign two rhombus vectors to the edge e, denoted e iαe and e iβ e , see Figure 3, and we assume that α e , β e satisfy β e -αe 2 = θ e . An isoradial graph G is said to be quasicrystalline if the number d of possible directions assigned to edges of its diamond graph G is finite. 2.2. Monotone surfaces and asymptotically flat isoradial graphs. In the quasicrystalline case, the diamond graph can be seen as the projection of a monotone surface in Z d onto the plane, and every edge of G is the projection of a unit vector of Z d , see [START_REF] De Bruijn | Algebraic theory of Penrose's nonperiodic tilings of the plane[END_REF][START_REF] Bobenko | Discrete differential geometry[END_REF]. Vertices (resp. faces) of G correspond then to even (resp. odd) vertices of Z d on that surface. For example, in the particular case when d = 3, the diamond graph appears naturally to the eye as the projection in the plane of a landscape made of unit cubes, see for example Figure 7. Also, the Penrose tiling (see Figure 2, lower right) is the projection of a monotone surface very close to a given plane with irrational slope in Z 5 [START_REF] De Bruijn | Algebraic theory of Penrose's nonperiodic tilings of the plane[END_REF].

Let (e 1 , e 2 , . . . , e d ) be the canonical basis of Z d and α 1 , α 2 , . . . , α d be the angles associated to the train-tracks such that e iα j is the projection of e j , and

α 1 < α 2 < • • • < α d < α 1 + π < • • • < α d + π < α 1 + 2π. (10)
Using this notion of surface and the notation above, we can give a sense to the difference of two vertices yx as the vector d j=1 N j e j in Z d joining the corresponding points on the surface. The quantity

N = |y -x| = d j=1 |N j |
is the graph distance between x and y seen as vertices of G , as well as the graph distance in Z d between the corresponding points on the monotone surface. The reduced coordinates of yx are the quantities n j = N j N , which define a point ř = d j=1 n j e j in the L 1 -unit ball of R d .

Monotone surfaces can be wild in the following sense: fix a reference vertex x 0 and a direction r ∈ S 1 , and look at the reduced coordinates n j in a minimal path from x 0 to y , as y tends to infinity in the direction r ∈ S 1 in the plane embedding. These quantities may not converge. This is what happens for example on Figure 7, because of the larger and larger "waves" causing oscillations for the n j 's.

We say that the isoradial graph is asymptotically flat if this does not happen, and if proportions converge for each directional limit. This property, and the values of the limits when they exist, do not depend on the choice of x 0 . Periodic isoradial graphs and multiple cone-like regions (see Figure 2 and Section 3.3) are asymptotically flat. Let us call n j (r) the limits for j = 1, . . . , d of the reduced coordinates in the direction r. In that case, the function

r → n(r) = (n j (r)) 1 j d
is automatically continuous and injective, as the map from the diamond graph to the monotone surface in Z d is a bi-Lipschitz bijection.

2.3. Exponential functions. We will be using definitions and results on the Z-invariant massive Laplacian ∆ :

C V → C V introduced in [13] 2 . Let x be a vertex of G of degree n;
denote by e 1 , . . . , e n the edges incident to x and by θ 1 , . . . , θ n the corresponding rhombus half-angles, then the Laplacian operator is given by ( 8) (see [13, Eq. ( 1)] for the original statement), where the conductances ρ and (squared) masses m2 are defined by

ρ e = ρ(θ e |k) = sc(θ e |k), (11)
m 2 (x|k) = n j=1 (A(θ j |k) -sc(θ j |k)), ( 12
)
where sc is one of the twelve Jacobi elliptic functions (see [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]Chap. 16] or [START_REF] Lawden | Elliptic functions and applications[END_REF]Chap. 2] for extensive presentations, or [13, App. A] for a selection of key properties) and

A(u|k) = 1 k Dc(u|k) + E -K K u ,
where

k = √ 1 -k 2 is the complementary elliptic modulus, Dc(u|k) = u 0 dc 2 (v |k)dv
, and E = E(k) is the complete elliptic integral of the second kind. Since k is fixed once and for all, most of the time we will drop the elliptic modulus from our notation, writing for example A(u) instead of A(u|k).

We also need the definition of the complex-valued discrete k-massive exponential function e (x,y ) (u) of [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Sec. 3.3], briefly introduced in ( 7), depending on a pair of vertices (x, y ) and of a complex parameter u. Consider an edge-path x = x 1 , . . . , x n = y of the diamond graph G from x to y and let e iα j be the vector corresponding to the edge x j x j+1 . Then the exponential function is defined inductively along the edges of the path:

∀ u ∈ C, e (x j ,x j+1 ) (u) = i √ k sc u -α j 2 , e (x,y ) (u) = n-1 j=1 e (x j ,x j+1 ) (u), (13)
where α j = α j 2K π . These functions do not depend on the path chosen for their definition, and are harmonic for the Laplacian [START_REF] Bostan | Differential equations for algebraic functions[END_REF], see [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Prop. 11].

2.4. Exact and asymptotic expressions for the Green function. The massive Green function, denoted G, is the inverse of the massive Laplacian operator [START_REF] Bostan | Differential equations for algebraic functions[END_REF]. The following formula is proved in [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Thm 12]:

G(x, y ) = k 4i π Γx,y e (x,y ) (u)du, (14)
where Γ x,y is a vertical contour on the torus T(k), whose abscissa can be identified with the angle of the ray R -→ xy . Let us remark that the discrete massive exponential function e (x,y ) (u) in ( 13) and ( 14) is defined using a path of the embedded graph from x to y . This implies that the expression [START_REF] Boutillier | The Z-invariant Ising model via dimers[END_REF] for G(x, y ) is local, meaning that it remains unchanged if the isoradial graph G is modified away from a path from x to y . This is far from being a general situation: when computing the inverse of a discrete operator, one expects the geometry of the whole graph to be involved. The idea of the proof of the local formula [START_REF] Boutillier | The Z-invariant Ising model via dimers[END_REF] is the following: find a one-parameter family of local, complex-valued functions in the kernel of the massive Laplacian (8) (the exponential functions), define its inverse (minus the Green function) as a contour integral of these functions, and adjust (if possible) the contour of integration in such a way that ∆G = -Id.

Let us also state [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Thm 14], which contains the asymptotic behavior of the Green function. Let G be a quasicrystalline isoradial graph. Let χ the function defined (for x and y fixed) by

χ(v ) = χ(v |k) = 1 |x -y | log e (x,y ) (v + 2i K ), (15)
where |x -y | is the graph distance between x and y . By [START_REF] Boutillier | The critical Z-invariant Ising model via dimers: locality property[END_REF]Lem. 17] the set of zeros of e (x,y ) (u) is contained in an interval of length 2K -2ε, for some ε > 0. Let us denote by α the midpoint of this interval. When the distance |x 0 -y | between vertices x 0 and y of G is large, we have

G(x 0 , y ) = k e (x 0 ,y ) (2i K + v 0 ) 2 2π|x 0 -y |χ (v 0 ) (1 + o(1)), (16)
where

v 0 is the unique v ∈ α + (-K + ε, K -ε) such that χ (v ) = 0, see [13, Lem. 15],
and χ(v 0 ) < 0. Let us briefly recall the main idea in [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF] to derive the asymptotics [START_REF] Chelkak | Discrete complex analysis on isoradial graphs[END_REF].

Starting from (15), we may reformulate [START_REF] Boutillier | The Z-invariant Ising model via dimers[END_REF] as

G(x 0 , y ) = k 4i π Γx 0 ,y e |x 0 -y |χ(v ) dv .
Typically, as |x 0 -y | → ∞, one may analyse the above integral using the saddle-point method.

Classically, the saddle point is a critical point, therefore χ (v ) = 0.

2.5.

Going to infinity in an isoradial graph and 3D-consistency. Let us first recall that by construction [START_REF] Doob | Discrete potential theory and boundaries[END_REF][START_REF] Hunt | Markoff chains and Martin boundaries[END_REF][START_REF] Sawyer | Harmonic functions on trees and buildings[END_REF][START_REF] Woess | Random walks on infinite graphs and groups[END_REF], a sequence of points y converges to a point in the Martin boundary if it exits any finite subset of the graph and if the ratio of Green functions G(x 1 , y ) G(x 0 , y ) converges pointwise. While [START_REF] Chelkak | Discrete complex analysis on isoradial graphs[END_REF] provides the main term in the Green function asymptotics and is independent on how y goes to infinity in the graph, we thus need here a different, somehow more precise information.

In the classical Ney and Spitzer theorem (3), y goes to infinity along an angular direction, namely y |y | → r ∈ S 1 . However, this simple geometric description will not work in the isoradial setting. Indeed, it can be shown (see the proof of [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Thm 14]) that the convergence of v 0 appearing in ( 16) is equivalent to the convergence of the reduced coordinates, which as already discussed (see Section 2.2 and Figure 7) is not guaranteed by the angular convergence of y .

In our opinion, the best way to characterize the convergence of the Martin kernel is to use the concept of 3D-consistency [START_REF] Bobenko | Discrete differential geometry[END_REF], that we now introduce. The notion of harmonicity for our massive Laplacian (8) is compatible with the star-triangle transformation (see Figure 4), which corresponds in the monotone surface picture to pushing the surface along a 3-dimensional cube: let G Y and G ∇ be two isoradial graphs differing by a star-triangle transformation, such that G Y has an extra vertex x 0 of degree 3. If f is a harmonic function on G ∇ , there is a unique way to extend it at x 0 to make it harmonic on G Y . This operation, together with its inverse corresponding to forgetting the value at x 0 , realize a bijection between harmonic functions on G Y and G ∇ ; see [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Prop. 8]. Using the 3D-consistency and the interpretation of quasicrystalline isoradial graphs as monotone surfaces on Z d (Section 2.2), the exponential function and the Green function can be uniquely lifted to vertices of Z d , keeping their harmonicity properties. More precisely, if we assume that the isoradial graph G has enough train-tracks associated to each direction of Z d , namely that each coordinate on the monotone surface is not bounded either from above or below 3 , one can generate any monotone surface of Z d from G by a (maybe infinite) number of star-triangle transformations, and can extend the exponential and Green functions to all even vertices of Z d . Otherwise, one may get only half-spaces or slabs [START_REF] Bodini | A characterization of flip-accessibility for rhombus tilings of the whole plane[END_REF], but which are bi-infinite in d 2 directions. We will often assume that we are in the first situation, in which G spans all Z d . To extend our results to the second case, replacing d by d in the proofs will often be enough.

This lift to a lattice of higher dimension allows us to reformulate the asymptotics of the Green function as a limit as y tends to infinity in a certain direction, thereby showing the strong parallel with the classical statement of Ney and Spitzer theorem. Indeed, in Z d the convergence of y along a direction is equivalent to the convergence of the reduced coordinates. These directions to infinity may not be obtained by staying on the surface corresponding to the graph G (unless for some directions, if the graph is asymptotically flat).

In the rest of the manuscript, we will always use this notion of convergence when we will compute the asymptotics of ratios of Green functions.

Main results and proofs

Statements.

Theorem 1 (asymptotics of the Martin kernel). Let G be the massive Green function on a quasicrystalline isoradial graph, which can be lifted to Z d via the star-triangle transformation. Then

lim y →∞ G(x 1 , y ) G(x 0 , y ) = e (x 0 ,x 1 ) (u 0 ), (17)
where u 0 = 2i K + v 0 , with v 0 defined as in [START_REF] Chelkak | Discrete complex analysis on isoradial graphs[END_REF].

Theorem 2 (structure of the Martin boundary). As the asymptotic direction of y → ∞ in [START_REF] Choquet | Sur l'équation de convolution µ = µ * σ[END_REF] varies in S d-1 , the point u 0 describes entirely the circle 2i K + R/4KZ in T(k). If moreover G is asymptotically flat, the Martin boundary is homeomorphic to the circle S 1 .

Theorem 3 (minimality of the Martin boundary). Let x 0 ∈ G be any fixed point in the graph and v ∈ R/4KZ. The positive harmonic functions h v (x) = e (x 0 ,x) (2i K + v ) are minimal.

3 With the n j 's denoting the reduced coordinates of y -x, where x is some reference vertex, this is equivalent to asking that for each j and ±, there is at least a direction r along which lim sup ±n j is strictly positive as y tends to infinity along r, or in the planar embedding picture, asking that for any train-track T associated to a direction α, the two half-planes obtained by cutting along the parallel edges of T have both an infinite number of train-tracks with this direction α.

Proofs.

Proof of Theorem 1. The starting point is the asymptotics (16) obtained in [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF], that we first apply to the Green function G(x 0 , y ). To derive the asymptotics of G(x 1 , y ) we begin by rewriting [START_REF] Boutillier | The Z-invariant Ising model via dimers[END_REF] as

G(x 1 , y ) = k 4i π Γx 1 ,y e (x 1 ,x 0 ) (u) e (x 0 ,y ) (u)du.
We then perform an asymptotic analysis of the above integral. The only difference with the proof of [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Thm 14] is the presence of the term e (x 1 ,x 0 ) (u), compare with [START_REF] Boutillier | The Z-invariant Ising model via dimers[END_REF]. Using the saddle point method and similar ideas as in [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF], we obtain that this factor will actually appear as a prefactor in the final asymptotics, evaluated at u 0 . In other words,

G(x 1 , y ) = e (x 1 ,x 0 ) (u 0 ) k e (x 0 ,y ) (u 0 ) 2 2π|x 0 -y |χ (v 0 ) (1 + o(1)),
with the exact same u 0 = 2i K + v 0 as in [START_REF] Chelkak | Discrete complex analysis on isoradial graphs[END_REF]. We then easily reach the conclusion that the ratio of Green function behaves as in [START_REF] Choquet | Sur l'équation de convolution µ = µ * σ[END_REF], which completes the proof.

Proof of Theorem 2.

There is correspondence between the direction along which y goes to infinity and u 0 = 2i K + v 0 , see [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Lem. 15] and its proof. This v 0 can be anywhere on the circle R/4KZ: the interval from the asymptotics of the Green function moves with the set of poles of the exponential function, which depends on the direction of y . Now we assume G to be asymptotically flat, so the application r → n(r) = (n j (r)) 1 j d is well defined and continuous. It is even differentiable everywhere, except for r for which at least one of the n j (r) is zero (meaning that the corresponding minimal path seen on the monotone surface is moving to another orthant of Z d ), but even at these points it has left and right derivatives.

We want to prove that r → u 0 (r) is differentiable (has one-sided derivatives) when r → n j (r) is differentiable (has one-sided derivatives), and that the derivative is strictly positive, in the sense that the points move in the same direction around their respective circles.

Up to a possible cyclic relabelling of the coordinates, we can assume that all the n j (r) are non-negative, and we look at a variation of r in the positive direction. Looking at how frequencies of steps along an infinite path should vary as the asymptotic direction in the plane moves slightly in the positive direction while the path stays in the same orthant, we see that the (one-sided) derivative d n dr of the reduced coordinates is a linear combination with positive coefficients of vectors of the form δn k,l = (0, . . . , 0, -1, 0, . . . , 0, 1, 0), where the 1 is at position l and the -1 at position k < l, namely,

d n dr = k<l a k,l (r)δn k,l ,
with a k,l (r) 0, and not all equal to zero. Rewrite the condition that v 0 = u 0 -2i K is a critical point as in the proof of [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Lem. 15]:

χ (v 0 ) = d j=1 n j sn • cn dn v 0 -α j 2 = n • F (v 0 ) = 0, with F (v ) = ( sn • cn dn ( v -α j 2
)) 1 j d . The implicit function theorem implies that v 0 , as a function of r, has a (one-sided) derivative dv 0 dr , and it satisfies:

dv 0 dr χ (v 0 ) + d n dr • F (v 0 ) = 0.
Now, writing down explicitly the scalar product

d n dr • F (v 0 ) = k<l a k,l sn • cn dn v 0 -α l 2 - sn • cn dn v 0 -α k 2 ,
we see that this quantity is strictly negative, as all the quantities

v 0 -α j 2
are in the interval [-K, K], on which the function sn • cn dn is strictly increasing (recall that α k < α l ). Since v 0 corresponds to a simple critical point of χ, which is actually a local minimum, χ (v 0 ) is strictly positive. As a consequence,

dv 0 dr > 0.
This implies that the application

T : r ∈ S 1 → u 0 ∈ 2i K + R/4KZ
is continuous, and lifts to a strictly increasing function on the universal cover. Moreover, we know [11, Lem. 17] that the interval of length 2K -2ε in which the real part of u 0 lies is winding once around the circle. Indeed, it should contain the zeros of the exponential function from the base vertex x to y . When y switches to the next orthant, while going around x, the sliding interval is dropping the α j closer to its right boundary, and swallows α j + π. Therefore, after a full turn around x, the left and right boundaries of this sliding interval made also exactly one full turn. Since the total length between the left end of this interval at the beginning of the full turn and its right end at the end is strictly less than twice the length of the circle, it means that T winds also exactly once around the circle, therefore T is bijective. Since the image is compact, T is automatically open, so it defines a homeomorphism.

Proof of Theorem 3. It will be given in Section 5.

3.3.

An example: cones of homogeneities. The simplest inhomogeneous extension of the fully homogeneous case of Z d consists in modifying the jumps at a finite number of sites, as explained in our introduction (see also [START_REF] Kurkova | Martin boundary and elliptic curves[END_REF]). Another simple extension is to split the grid in a finite union of connected domains, for example two half-planes, or more generally a finite number of cone-like regions as on Figure 2, and to assign to each region a (different) set of transition probabilities. Up to our knowledge, a Ney and Spitzer theorem is not known in this simple setting, even in the simplest instance of two half-planes. It is worth mentioning that our approach allows to study such cases, provided the transition probabilities follow the construction of elliptic conductances as introduced in Section 2.

The case of periodic isoradial graphs

In Section 4.1, we first study general periodic isoradial graphs and relate important quantities needed in their analysis (characteristic polynomial, amoeba, degeneracy of the amoeba, quantity u 0 in (9), etc.) to natural objects in Z d (generating function of the transition probabilities, set ∂D t in (6), spectral radius, mapping u(p) in (3), etc.). Then we show in Section 4.2 how our results give another proof of the classical Ney and Spitzer in two cases (the square and triangular lattices). 4.1. Amoeba of the characteristic polynomial and jump generating function. Using the theory of analytic combinatorics in several variables [START_REF] Pemantle | Analytic Combinatorics in Several Variables[END_REF], we analyze the exponential decay of the Green function through Fourier analysis. We denote a vertex of G by a triple (v , n 1 , n 2 ), where v is the copy of the vertex in the fundamental domain and (n 1 , n 2 ) is the element of Z 2 corresponding to the translation moving v to the vertex.

A Ney-Spitzer theorem for periodic planar graphs. The first part of this section is not specific to isoradial graphs and holds for all planar periodic graphs (having periodic masses and conductances) with at least one positive mass.

The Laplacian can be seen as a periodic convolution operator on vector-valued functions of Z 2 . It acts in Fourier space by multiplication by a matrix ∆(z, w ), with |z| = |w | = 1. The square matrix ∆(z , w ) has rows and columns indexed by vertices of the fundamental domain, and the coefficient between v and w is the sum of the conductances of the edges between (v , 0, 0) and (w , n 1 , n 2 ) multiplied by z n 1 w n 2 .

If (v , n 1 , n 2 ) and (w , n 1 , n 2 ) are two vertices of G, the Green function can be expressed as the Fourier inverse transform of ∆(z , w ) -1 = Q(z,w ) P (z,w ) , where P (z , w ) = det ∆(z , w ) and Q(z, w ) is the adjugate matrix of ∆(z , w ):

G((v , n 1 , n 2 ), (w , n 1 , n 2 )) = |z|=|w |=1 z n 1 -n 1 w n 2 -n 2 Q(z , w ) v ,w P (z , w ) dz 2i πz dw 2i πw .
The asymptotic behavior of the Green function above is encoded in the singularities of the integrand, namely, the zeros of P . The zero-set of P defines an algebraic curve, called the spectral curve of the Laplacian. The amoeba of P (see [START_REF] Gelfand | Resultants, and Multidimensional Determinants[END_REF]Chap. 6]) is the image of the spectral curve by the application

(z , w ) → (log |z|, log |w |)
and plays an important role in the discussion; see Figure 5 for an example. In the massive case, there is no zero of P on the unit torus |z| = |w | = 1, and thus we can deform continuously the contour of integration defining G to increase or decrease the radii of the torus, until we touch the spectral curve, in order to obtain exponential decay.

We can say even more, because of geometric properties of spectral curves of Laplacians. The spectral curve is always a simple Harnack curve [START_REF] Kenyon | Determinantal spanning forests on planar graphs[END_REF]. In particular, the boundary of the amoeba is the image of the real locus of the spectral curve. The bounded connected component of the complementary of the amoeba containing the origin (called the oval O in the sequel, see Figure 5) is a convex set whose boundary corresponds to a (subset) of positive real roots of P :

P (e ζ 1 , e ζ 2 ) = 0,
which in the classical theorem by Ney and Spitzer is the generalization of the level set ∂D t defined in [START_REF] Bobenko | Discrete differential geometry[END_REF]. All conditions are met to be in the smooth point situation described in [START_REF] Pemantle | Analytic Combinatorics in Several Variables[END_REF]Chap. 9], allowing us to readily obtain the asymptotics of the ratio of Green functions for copies of the same vertex v in the fundamental domain. ). Let ρ denote the spectral radius of the transition kernel. Assume t ρ, with t = ρ if the random walk has zero drift. Then the set

D t = {ζ ∈ R d : φ(ζ)
t} is compact and convex, the gradient

grad φ(ζ) = x∈Z d xe ζ•x P (0, x)
exists everywhere on D t and does not vanish on its boundary ∂D t . Furthermore, the mapping

ζ → r = grad φ(ζ) |grad φ(ζ)| (18)
determines a homeomorphism between ∂D t and S d-1 .

The following result is stated under Corollary 1.3 in [START_REF] Ney | The Martin boundary for random walk[END_REF] for t = 1 and random walks with drift; it is extended to other values of t and to the zero drift case in [56, (25.21)].

Theorem 5. Under the same assumptions as in Lemma 4, let r be a point in S d-1 and ζ(r) be the unique solution in ∂D t of [START_REF] De Tilière | Scaling limit of isoradial dimer models and the case of triangular quadri-tilings[END_REF]. Let y denote any sequence in Z d such that |y | → ∞ and y |y | → r. Then the ratio G(x 1 , y

| 1 t )/G(x 0 , y | 1 t ) converges to e ζ(r)•(x 1 -x 0 )
. The isoradial periodic case. When in addition the periodic graph G is isoradial, with the elliptic conductances and masses from ( 11) and ( 12), the spectral curve has geometric genus 1 and there is a unique bounded connected component of the complementary of the amoeba [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]. Furthermore, we have an explicit parameterization of its boundary via the exponential function:

ξ : u → log |e x,x+(1,0) (u)|, log |e x,x+(0,1) (u)| , (19)
where u ∈ 2i K + R/4KZ. The composition ξ -1 • ψ is the application giving the u 0 from the asymptotics of Green function of Equation ( 16) for every direction r along which y can go to infinity.

4.2.

Examples. Among all periodic isoradial graphs, two of them (namely, the square and triangular lattices) are of special interest, as they correspond to models of homogeneous random walks in Z 2 . In this section we show how, in these two cases, our results match with the classical Ney and Spitzer theorem.

In order to state precisely this connection, it is important to notice that in Ney and Spitzer framework, a set of transition probabilities is fixed (through the function φ in (2)) and then the t-Martin boundary is computed for any fixed t larger than the spectral radius. On the other hand, in the isoradial setting the conductances (related to the transition probabilities) and the mass (related to the variable t) both depend on the same elliptic modulus k, and it is a priori unclear that we can construct a random walk model with transition probabilities independent of the t variable. This will be shown in the two above-mentioned examples, using the degrees of freedom that we have on the conductances and on the elliptic modulus.

The square lattice. The first example is the square lattice with angles α and β associated to the two families of parallel train-tracks, with α < β < α + 2K < β + 2K, see Figure 2. Because the definition of conductances and masses is invariant by rotation of the whole embedding, one can further assume that β = -α = θ (see Figure 3). There are thus two types of edges: horizontal with opening angle θ = π 2K θ ∈ (0, π 2 ) and vertical with opening angle π 2 -θ. The characteristic polynomial of the model is given by [13, Sec. 5.2]

P (z , w ) = m + 2(c 1 + c 2 ) -c 1 z + 1 z -c 2 w + 1 w , (20)
with c 1 = sc(θ), c 2 = sc(Kθ) and m as in [START_REF] Boutillier | Statistical mechanics on isoradial graphs[END_REF]. It is convenient to normalize (20) so as to have transition probabilities summing to one:

P (z , w ) = m + 2(c 1 + c 2 ) 2(c 1 + c 2 ) -p 1 z + 1 z -p 2 w + 1 w , with p 1 = c 1 2(c 1 + c 2 )
and

p 2 = c 2 2(c 1 + c 2 ) = 1 2 -p 1 . (21)
It is natural to associate to this model the classical simple random walk on Z 2 , with transition probabilities Laplace transform

φ(ζ 1 , ζ 2 ) = p 1 (e ζ 1 + e -ζ 1 ) + p 2 (e ζ 2 + e -ζ 2 ).
Notice that horizontal (resp. vertical) jumps have the same weight, due to the constraints on the conductances in (20) (in particular, the random walk has zero drift). We now state precisely the connection between the two frameworks.

Transition probabilities generating function, characteristic polynomial and their zero-sets. It is clear that P (z , w ) = 0 if and only if φ(log z , log w ) = t, with t = 1 + m/(2c 1 + 2c 2 ). In particular, using the notation of Section 4.1, the oval of the amoeba is in correspondence with the set ∂D t .

Exponential functions. As said in the introduction, extremal positive harmonic functions in group structures are exponential functions [START_REF] Choquet | Sur l'équation de convolution µ = µ * σ[END_REF]. In our context of the planar lattice, they are given by (writing

ζ = (ζ 1 , ζ 2 ) and n = (n 1 , n 2 )) f (x) = e ζ•n = e ζ 1 n 1 e ζ 2 n 2 , (22)
for ζ ∈ ∂D t . By definition of ∂D t , see [START_REF] Bobenko | Discrete differential geometry[END_REF], the pairs (e ζ 1 , e ζ 2 ) parametrize the zero-set of φt.

To compare the elementary expression [START_REF] Duminil-Copin | Universality for the random-cluster model on isoradial graphs[END_REF] for the exponential function to its elliptic analogue [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF], we should first introduce an elliptic uniformisation of the zero-set of the characteristic polynomial [START_REF] Doob | Application of boundary theory to sums of independent random variables[END_REF]. As shown in [13, Eq. ( 32)] (see also [START_REF] Doob | Discrete potential theory and boundaries[END_REF]),

{(z , w ) ∈ (C ∪ {∞}) 2 : P (z , w ) = 0} = {(z(u), w (u)) : u ∈ T(k)}, with z(u) = -k sc u -α 2 sc u -β 2 and w (u) = sc( u-β 2 ) sc( u-α 2 )
.

Moreover, using [START_REF] Doob | Application of boundary theory to sums of independent random variables[END_REF] one obtains

e (n 1 ,n 2 ) (u) = i √ k sc u -α 2 n 1 -n 2 i √ k sc u -β 2 n 1 +n 2 = z(u) n 1 w (u) n 2 . ( 23
)
The comparison between ( 22) and ( 23) is now clear: in both cases they are products of the coordinates (raised to some power) of parameterizations of the characteristic polynomial.

Choice of the parameters. Our main result is stated below; it shows that given arbitrary transition probabilities and any value of t in the spectral interval, we can adjust the values of θ and k so as to have the equivalence: P (z , w ) = 0 if and only if φ(log z , log w ) = t. Proposition 6. Let p 1 and p 2 be defined by [START_REF] Duffin | Potential theory on a rhombic lattice[END_REF]. For any q 1 , q 2 , t such that q 1 , q 2 > 0, q 1 + q 2 = 1 2 and t 1, there exist θ ∈ (0, K) and k ∈ [0, 1) such that p 1 = q 1 , p 2 = q 2 and m+2(c 1 +c 2 )

2(c 1 +c 2 ) = t.

Proof. We first look at the equation

c 1 2(c 1 +c 2 ) = q 1 . Since c 2 = sc(K -θ) = 1 k sc(θ) by [1, 16.8.9], it is equivalent to sc(θ) sc(θ) + 1 k sc(θ) = 2q 1 , which results in sc(θ) = 2q 1 1 -2q 1 1 √ k , (24)
which for any k ∈ [0, 1) has a (unique) solution, see [START_REF] George | Spectra of biperiodic planar networks[END_REF] for an explicit expression. Now we turn to the equation involving the mass. Note first that by [START_REF] Boutillier | Statistical mechanics on isoradial graphs[END_REF],

m = 2(A(θ) + A(K -θ)) -2(c 1 + c 2 ),
and by [13, Eq. ( 60)], A(θ) + A(Kθ) = ns(θ) dc(θ) k , so that

m + 2(c 1 + c 2 ) 2(c 1 + c 2 ) = nc(θ) dc(θ) 1 + k sc(θ) 2 .
Let us now reproduce [1, 16.9.3]: k 2 sc 2 +k 2 = k 2 nc 2 = dc 2 -k 2 . These equalities allow us to derive the values of nc(θ) and dc(θ), starting from the value (24) of sc(θ). After some computations, we conclude that

m + 2(c 1 + c 2 ) 2(c 1 + c 2 ) = 1 + 2q 1 (1 -2q 1 ) k -2 + 1 k .
As k varies in [0, 1), the latter function increases from 1 to ∞. In conclusion, k is determined by solving the equation

1 + 2q 1 (1 -2q 1 ) k -2 + 1 k = t, see (25)
, and then θ is found with [START_REF] Flajolet | Analytic combinatorics[END_REF]. We can even be more explicit:

k = t 2 -8q 2 1 + 4q 1 -1 -(t -1)(t + 1)(t + 1 -4q 1 )(t -1 + 4q 1 ) 4q 1 (1 -2q 1 ) , (25)
θ = F 2q 1 k + 2q 1 (1 -k ) , k , (26)
where F is the incomplete elliptic integral of the first kind. The proof is completed.

The triangular lattice. The characteristic polynomial is (again by [START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF]Sec. 5.2])

P (z, w ) = m + 2(c 1 + c 2 + c 3 ) -c 1 z + 1 z -c 2 w + 1 w -c 3 z w + 1 z w ,
and our aim is to show an analogue of Proposition 6. However, as the computations become much more complex in this case we will only consider here the case of uniform probabilities, corresponding to three equal conductances c 1 = c 2 = c 3 = sc( K 3 ). By [START_REF] Boutillier | Statistical mechanics on isoradial graphs[END_REF], the normalized characteristic polynomial takes the form

P (z , w ) = A( K 3 ) sc( K 3 ) - 1 6 z + 1 z - 1 6 w + 1 w - 1 6 z w + 1 z w . Proposition 7. As k varies in [0, 1), A( K 3 )/ sc( K 3 ) varies continuously in [1, ∞). Proof. Let us first prove that t = A( K 3 )/ sc( K 3 ) is an algebraic function of k, solution to -27(1 -k 2 )t 4 + 18(1 -k 2 )t 2 + 2(2 -k 2 ) 2 t + 1 -k 2 + k 4 = 0. ( 27
)
The first terms of its series expansion are The coefficients of s are again non-negative (see Appendix A for a proof).

t =
To prove [START_REF] Grimmett | Bond percolation on isoradial graphs: criticality and universality[END_REF] we start from the equality sn(K) = 1 that we rewrite as sn( K 3 + 2K 3 ) = 1, and then we apply the classical addition formula [1, 16.17] together with some identities in Exercise 22 in [START_REF] Lawden | Elliptic functions and applications[END_REF]Chap. 2].

The second step is to show that both A( K 3 ) and sc( K 3 ) may be algebraically expressed in terms of s. By definition, sc 2 = (1/ sn 2 -1) -1 , so that sc( K 3 ) = (1/s 2 -1) -1/2 . Now, using [13, (60) and ( 61)], we obtain that

A K 3 = √ 1 -k 2 s 2 3s 2 √ 1 -k 2 1 - 2(1 -k 2 )s 4 1 -(2 -k 2 s 2 )s 2 .
Some computations (which we do not reproduce here) finally lead to [START_REF] Godefroy | Martin boundary of polynomial random walks on the d-dimensional lattice of nonnegative integers[END_REF].

We prove Proposition 7. Computing the discriminant of the polynomial in [START_REF] Godefroy | Martin boundary of polynomial random walks on the d-dimensional lattice of nonnegative integers[END_REF], it is easy to see that for any k ∈ (0, 1), two roots are real and the other two are non-real, complex conjugate. Moreover, the four roots are obviously non-zero. So the solution t = t(k) is such that t(0) = 1 (evaluate ( 27) at k = 0). Doing the change of variable t → t (1-k 2 ) 1/3 in ( 27), we deduce that t(k) behaves as 1 (1-k) 1/3 as k → 1, and in particular goes to ∞.

Minimal positive harmonic functions

From the construction of Martin boundary, any positive harmonic function f can be written as an integral of the exponential functions against some Radon measure µ supported on 2iK + R/4KZ:

∀x ∈ G, f (x) = e (x,x 0 ) (u)µ(du). (29)
Let us recall that positive harmonic functions are particularly important from a potential theory point of view, in relation with the concept of Doob transform. The function f is then said to be minimal if we cannot find a non-trivial measure µ for this decomposition, i.e., whose support is not just a singleton; see Section 1.1.

In this section, we prove that the exponential functions x → e (x,x 0 ) (u) with u = 2i K + v and v ∈ R are minimal, which provides a proof of Theorem 3. This implies in particular that these exponential functions form the minimal Martin boundary of the associated killed random walk.

Recall that any isoradial graph can be viewed as a step surface in a hypercubic lattice Z d , where d represents the number of possible orientations for the unit vectors representing edges of the diamond graph, and that the exponential function is naturally extended to Z d ; see Sections 2.1 and 2.2.

To each point ř = j n j e j in the L 1 -unit ball of R d , we associate a probability measure ν ř on the circle R/4KZ:

ν ř = d j=1 n + j δ α j + n - j δ α j +2K ,
where n ± j = max{±n j , 0} represents the asymptotic proportion of steps ±e j in a minimal path from a reference vertex x to y , as y tends to infinity in the direction ř. Because of the monotonicity of the surface, n + j and n - j cannot be both strictly positive. Therefore, the Figure 6. The unit sphere intersected by the kernel of τ in dimension 3.

limiting measures ν have a support included in half of the circle: there is an interval of length not smaller than 2K with ν-measure 0 (see also Section 2.4). Fix u = 2i K + v with v real. An important quantity of interest is the rate of growth of the massive exponential function in the asymptotic direction ř:

τ (ř, u) = lim y →ř∞ 1 N log | e (x 0 ,y ) (u)| = R/4KZ log √ k sc u -α 2 ν ř(dα) = χ(v ), see (15). Because √ k sc(u + K) = 1 √
k sc(u) , the function τ can be rewritten as

τ (ř, u) = R/4KZ log √ k sc u -α 2 νř (dα),
where νř = d j=1 n j δ α j is now a signed measure supported on the angles associated to the basis vectors e 1 , . . . , e d . It is then clear that τ (-ř, u) = -τ (ř, u). Even more, if we extend by homogeneity the definition of ν to all vectors of R d by νλř = λν ř for λ ∈ R, then τ (•, u) becomes a non-degenerate linear form, whose kernel is a hyperplane separating the unit sphere in two parts; see Figure 6 for an example. Lemma 8. The L 1 -unit sphere is separated in two (connected) hemispheres, corresponding to positive and negative values of τ (•, u).

Informally, this lemma says that for any u in the torus, e (x 0 ,y ) tends exponentially fast to 0 for half of the directions to infinity. This is in particular true for the positive exponential functions for which u = 2i K + v , with v ∈ R. We claim that this property is characterizing the massive exponential functions for u = 2i K + v with v real, among all positive massive harmonic functions: Proposition 9. If h is a positive harmonic function which tends to 0 in at least half of the directions, then it is proportional to an exponential function.

Figure 7. A piece of the diamond graph of an isoradial graph which is not asymptotically flat. From the center, there are "waves" of size growing exponentially. When going to infinity in a direction in the plane, the proportions of the different types of steps do not converge, but instead oscillate.

Proof. Write h as an integral over the positive exponential function for some positive measure µ as in [START_REF] Imamoglu | Computing hypergeometric solutions of second order linear differential equations using quotients of formal solutions and integral bases[END_REF]. Therefore, the set of directions along which h goes to infinity is the union of the half-spaces for which e (x 0 ,y ) (u) goes to infinity, for u in the support of µ. If µ is non-trivial then this is strictly more than a half-space, which is impossible by hypothesis. Therefore µ should have a support reduced to a singleton, and h is proportional to an exponential function.

Proof of Theorem 3. If h is less or equal to h v , then it goes to 0 at least in half of the directions (the ones where h v itself goes to 0). We can therefore apply Proposition 9 to conclude.

A shorter proof of equality [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a half-space[END_REF], with a different (more geometric) flavor, goes as follows: on the one hand, the hypergeometric function in [START_REF] Ignatiouk-Robert | Martin boundary of a reflected random walk on a half-space[END_REF] satisfies the algebraic transformation 4 2 F 1 1 2 5 6 5 3

;

x (x + 2) 3

(2 x + 1) 3 = 4

(2 x + 1)

3 2 (x + 2) 2 .
On the other hand, it is easy to prove, starting from the polynomial [START_REF] Hennequin | Processus de Markoff en cascade[END_REF], that the algebraic function s(k) satisfies s x

1 2 (x + 2) 3 2
(2 x + 1)

3 2 = 2 x + 1 x + 2 .
Denoting ϕ(x) = x (x + 2) 3 /(2 x + 1) 3 , identity [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a half-space[END_REF] becomes trivial in the new variable x:

2 x + 1 x + 2 = 1 2 + 3 2 • ϕ(x) • (2 x + 1) 3/2 (x + 2) 2 2 .
Now that ( 33) is proved, the positivity of the coefficients s 2n of s(k) is a direct consequence of the obvious fact that the hypergeometric power series in [START_REF] Ignatiouk-Robert | Martin boundary of a reflected random walk on a half-space[END_REF] has positive coefficients. The second proof is based on the so-called Stieltjes inversion formula [48, Lecture 2], that allows to express in some cases the n-th term of a sequence as the n-th moment of a positive density. Using this formula, one can deduce the following expression:

s 2n = √ 3 π • 1 0 k n • H 1 (k) • 1 2 4/3 • H 2 (k) k 2/3 - 1 2 8/3 • H 1 (k) k 1/3 dk,
where H 1 (k) and H 2 (k) are the hypergeometric functions At this point, positivity is not yet apparent. However, using the change of variables k = ϕ(x), and using the fact that H 1 (ϕ(x)) and H 2 (ϕ(x)) become the simple algebraic functions 2 (x + 1) 1/3 (2 x + 1) 1/2 /(x + 2) and (2 x + 1) -1/2 , the previous expression simplifies to:

H 1 = 2 F 1
s 2n = √ 3 • 3 √ 2 π • 1 0 (x -1) 2 3 √ 2 -3 √ x 2 + x 3 √ x + 1 (x + 2) (2 x + 1) 2 x 2/3 ϕ(x) n dx. (35)
Once found, equality [START_REF] Ignatiouk-Robert | t-Martin boundary of reflected random walks on a half-space[END_REF] can again be proved automatically using algorithmic tools, notably the method of "creative telescoping" [START_REF] Almkvist | The method of differentiating under the integral sign[END_REF][START_REF] Koutschan | Creative telescoping for holonomic functions[END_REF][START_REF] Bostan | Efficient algorithms for mixed creative telescoping[END_REF]. Note that Lagrange inversion is a tempting alternative to the previous moment approach; however, in our case, it gives that for all n > 0,

s 2n = 1 2n • [z -1 ] (z + 1) 3 (3 -z) 16 z n = (-1) n n • 2 4n+1 • n i=0 3 n i n i + 1 (-3) i+1 ,
and positivity is not clear on any of these expressions. In contrast, identity [START_REF] Ignatiouk-Robert | t-Martin boundary of reflected random walks on a half-space[END_REF] obviously proves that the sequence (s 2n ) n 0 is positive. And it actually proves much more, namely that (s 2n ) n 0 is a Stieltjes-Hausdorff moment sequence, and in particular that it is log-convex, i.e., s 2n+2 s 2n-2 s 2 2n for all n 1 [51, §2] and ∆ k s 2n 0 for all k 0 [52, p. 338], where ∆ is the difference operator ∆(c n ) = (c n -c n+1 ). 4 This is inspired by the Darboux covering for tetrahedral hypergeometric equations of the Schwarz type (1/3, 1/3, 2/3) [54, §6.1], see also [START_REF] Vidūnas | Algebraic transformations of Gauss hypergeometric functions[END_REF]. Note that the same change of variables has been used in [9, §4].

  as |y | → ∞ in any given direction y |y | → r ∈ S d-1 , see [47, Thm 2.2]. Introduce the one-step transition probabilities generating function (with ζ • y denoting the standard scalar product of ζ and y in R d ) φ(ζ) = y ∈Z d p(0, y )e ζ•y , (2) defined in terms of the transition kernel p(x, y ) = P x [Z 1 = y ] of the random walk. Ney and Spitzer deduce from their asymptotics of (1) that lim |y |→∞ y /|y |→r G(x 1 , y ) G(x 0 , y ) = e ζ(r)•(x 1 -x 0 ) ,

Figure 1 .

 1 Figure 1. The curve φ(ζ 1 , ζ 2 ) = t, for φ(ζ1 , ζ 2 ) = e ζ 1 3 + e ζ 2 3 + e -ζ 1 6 + e -ζ 2 6 and t = 2 √ 2/3 ≈ 0.9428, 0.943, 0.95, 0.97, 1, 1.1. It corresponds to a nearest neighbor walk in Z 2 with jump probability 1 3 to the North and East, and 1 6 to the South and West. The spectral radius of the transition kernel is 2 √ 2/3.

Figure 3 . 2 =

 32 Figure 3. An edge e of G is the diagonal of a rhombus of G , defining the angle θ e and the rhombus vectors e iαe and e iβ e . One has β e -αe 2 = θ e .

Figure 4 .

 4 Figure 4. A star-triangle transformation on isoradial graphs. G ∇ is on the left, G Y on the right.

Figure 5 .

 5 Figure5. Amoeba of the spectral curve of a Laplacian with mass on a periodic graph. The amoeba itself is the region between the yellow and blue curves. The curve here has geometric genus one, and the (unique in this case) bounded connected component of the complementary of the amoeba containing the origin is called the oval.

  Remark that the algebraicity of t is not obvious, as both functions A and sc are transcendental.We start by showing that s = sn( K 3 ) is the unique power series solution to k 2 s 4 -2k 2 s 3 + 2s -1 = 0.

	The first terms in its expansion are
	s =	1 2	+	3 32	k 2 +	3 64	k 4 +	123 4096	k 6 +	177 8192	k 8 +	34887 2097152	k 10 + O(k 12 ).
				1 +	3 64	k 4 +	3 64	k 6 +	711 16384	k 8 +	327 8192	k 10 + O(k 12 ),
	and further computations suggest that all non-zero coefficients are positive (see Appendix A
	for a proof). (28)

The presentation of Section

2.1 follows that of [13, Sec. 2].

The presentation of Section 2.3 follows that of[START_REF] Boutillier | The Z-invariant massive Laplacian on isoradial graphs[END_REF] Sec.

3].

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under the Grant Agreement No. 759702 and from the ANR DIMERS (ANR-18-CE40-0033).

Appendix A. Some thoughts on a positivity question (by Alin Bostan) We consider the following question: let s(k) = n 0 s n k n , starting

We want to prove that all the coefficients of s(k) are non-negative, and more precisely that all the coefficients s 2n are positive. Note that the sequence (s n ) n 0 satisfies various types of recurrences, either non-linear, obtained by extracting the coefficient of k n in both sides of [START_REF] Hennequin | Processus de Markoff en cascade[END_REF], or even linear, obtained by using a linear differential equation (with polynomial coefficients in k) satisfied by s(k). However, these recurrences cannot be used directly to solve our positivity question. For instance, from the algebraic equation ( 30), Cockle's algorithm (see [START_REF] Bostan | Differential equations for algebraic functions[END_REF] and the references therein) proves that s(k) satisfies the 3rd order linear inhomogeneous differential equation

from which a coefficient extraction proves that (s n ) n 0 satisfies the linear recurrence

It is not obvious from this recurrence how to derive a non-negativity proof because of the plus sign in front of the coefficient of s n . In principle, two quite general methods might be applied to this kind of question (alone, or combined): one relying on (effective) asymptotics for the coefficients of algebraic functions [24, p. 504-505], the other relying on the approach of [38, §4.2]. Instead, we present two different proofs.

The first one is based on the following hypergeometric expression for s(k):

where the Gauss' hypergeometric series with parameters 1 2 , 5 6 and 5 3 is defined by

and (x) n denotes the Pochhammer symbol (x

Once found (e.g., using algorithmic tools for automated guessing [START_REF] Kauers | The Guess-and-Prove Paradigm in Action[END_REF], or for differential equation solving [START_REF] Imamoglu | Computing hypergeometric solutions of second order linear differential equations using quotients of formal solutions and integral bases[END_REF]), equality [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a half-space[END_REF] can easily be proved using closure properties of D-finite functions [START_REF] Stanley | Differentiably finite power series[END_REF]: from the 2nd order linear differential equation satisfied by 2 F 1 ; k , one computes a 3rd order differential equation satisfied by the right-hand side of (33), which appears to coincide with the differential equation [START_REF] Hunt | Markoff chains and Martin boundaries[END_REF] satisfied by the left-hand side of [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a half-space[END_REF]. Therefore, the coefficients sequences of both right-hand side and left-hand side of (33) satisfy the recurrence relation [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF], hence they are equal since their initial terms coincide.

We also consider the following related question: let t(k) = n 0 t n k n , starting

be the unique power series solution of the algebraic equation

We want to prove that all the coefficients t n are non-negative. First, we remark that t(ϕ(x)

2 ) = (2x + 1)/(x + 2), this implies that t(k) = R(s(k)), where R is the rational function

As a consequence of this and of identity [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a half-space[END_REF], we get that

where H is the hypergeometric function with non-negative coefficients


Keywords: manuscriptcentral, com/, manuscriptcentral, com/

2 Introduction: Streptogramins (pristinamycin and quinupristin-dalfopristin) can be interesting 3 options for the treatment of infections due to Gram-positive cocci, especially multidrug-4 resistant isolates. 5 Areas covered: This review provides an updated overview on structural and activity 6 characteristics, mechanisms of action and resistance, pharmacokinetic/pharmacodynamic and 7 clinical use of streptogramins. 8 Expert opinion: The streptogramin antibiotics act by inhibition of the bacterial protein 9 synthesis. They are composed of two chemically distinct compounds, namely type A and type 10 B streptogramins, which exert a rapid bactericidal activity against a wide range of Gram-11 positive bacteria (including methicillin-resistant staphylococci and vancomycin-resistant 12 enterococci). Several mechanisms of resistance have been identified in staphylococci and 13 enterococci but the prevalence of streptogramin resistance among clinical isolates remains 14 very low. Even if only a few randomized clinical trials have been conducted, the efficacy of 15 pristinamycin has been largely demonstrated with an extensive use for 50 years in France and 16 some African countries. Despite its effectiveness in the treatment of severe Gram-positive 17 bacterial infections demonstrated in several studies and the low rate of reported resistance, the 18 clinical use quinupristin-dalfopristin has remained limited, mainly due to its its poor 19 tolerance. Altogether, streptogramins (especially pristinamycin) can be considered as 20 potential alternatives for the treatment of Gram-positive infections.

A c c e p t e d M a n u s c r i p t 71 macrolactones (ca. 500 Da) containing an unusual oxazole ring and a dienyl amide fragment, 72 whereas type B streptogramins are cyclic hepta-or hexadepsipeptides (ca. 800 Da) with a 73 skeleton of amino acids with variations depending on the compound (Figure 1) [START_REF] Mukhtar | Streptogramins, oxazolidinones, and other inhibitors of 530 bacterial protein synthesis[END_REF][START_REF] Barrière | Recent 532 developments in streptogramin research[END_REF][START_REF] Canu | Overcoming bacterial resistance by dual target inhibition: the 534 case of streptogramins[END_REF].

74 Originally, streptogramins are natural mixtures produced by different members of 75 Streptomyces or related genera (Table 1) [START_REF] Barrière | Recent 532 developments in streptogramin research[END_REF][START_REF] Johnston | Streptogramin antibiotics: mode of action 536 and resistance[END_REF]. Every antibiotic producer synthesizes a 76 mixture of various A and B components in different proportions. For instance, Streptomyces 77 pristinaespiralis produces a mixture of group B compounds called pristinamycins I 78 (pristinamycin I A , pristinamycin I B , and pristinamycin I C with a ratio of 80-90%, 3-5%, and 2-79 5%, respectively) and a mixture of group A compounds called pristinamycins II 80 (pristinamycin II A and pristinamycin II B ) [START_REF] Barrière | Recent 532 developments in streptogramin research[END_REF]. Note that pristinamycin II A is predominant in 81 the pristinamycin II mixture. The relative proportion of the different pristinamycin II 82 fermentation products also depends on culture conditions [START_REF] Canu | Overcoming bacterial resistance by dual target inhibition: the 534 case of streptogramins[END_REF]. Finally, the pristinamycin 83 produced by S. pristinaespiralis is a mixture of pristinamycin I A and pristinamycin II A in a 84 30:70 ratio by weight (Table 1 & Figure 1) [START_REF] Barrière | Recent 532 developments in streptogramin research[END_REF][START_REF] Canu | Overcoming bacterial resistance by dual target inhibition: the 534 case of streptogramins[END_REF].

85

Since a limitation in the clinical use of streptogramins was the poor water solubility, which 86 prevented the availability of an injectable form for the treatment of severe infections, semi-87 synthetic derivatives were developed in the 1990s from natural pristinamycins (Table 1) [START_REF] Barrière | Recent 532 developments in streptogramin research[END_REF].

88 Quinupristin (RP 57669) and dalfopristin (RP 54476) were obtained after several steps of 89 chemical modifications from pristinamycin II A and pristinamycin I A , respectively [START_REF] Barrière | Recent 532 developments in streptogramin research[END_REF]. The 90 streptogramin association selected, comprised quinupristin and dalfopristin combined in a 91 30:70 ratio (w/w) of methane sulfonate salts as RP 59500 (Figure 1) [START_REF] Barrière | Recent 532 developments in streptogramin research[END_REF][START_REF] Canu | Overcoming bacterial resistance by dual target inhibition: the 534 case of streptogramins[END_REF].

3. Mechanism of action 93

Streptogramins A and B act by interfering with bacterial protein synthesis [22,[START_REF] Vannuffel | Mechanism of action of streptogramins and macrolides[END_REF]. Their 94 bindings to adjacent but distinct regions within the P site of 23S rRNA of the 50S ribosomal 95 subunit result in synergic inhibiting activity [START_REF] Poehlsgaard | The bacterial ribosome as a target for antibiotics[END_REF]. Due to cooperative binding, streptogramin Cutibacterium acnes [START_REF] Devasahayam | Newer antibacterial drugs for a new 506 century[END_REF] 0.12-0.5 0.12-1 ≤0.06-0.12 1-32 4 4-32 0.5-2 0.5-32 0.25-16 ≤0.06-1 0.12-1 0.12-1 ≤0.06-1 0.25-1 0.25-1 0.5-2 0.12-4 0.5-1 ≤0.06-1 0.12-0.5 0.12-0.5 0.25-2 0.12-0. 2-4 2-8 0.12-2 0.25-32 0.12-4 0.12-0.5 0.12-1 0.12-1 0.12-0.5 0.12-1 ≤0.06-0.12 0.12-0.25 0.25-1 0.25-0.5 0.12-0.5 ≤0.03-0.12 ≤0.03-0.12 0.12-16 0.12 0.06-0.12 ≤0.03-0.12 0.25 1 0.12 2 4 8 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.12 0.25 0.5 0.5 0.12 0.12 ≤0.03 0.12 0.12 0.12 ≤0.03 0.5 2 1 4 4 8 0.5 4 2 0.25 0.5 0.5 0.25 0.5 0.12 0.25 0.5 0.5 0.5 0.12 0.12 4 0.12 0.12 ≤0.03 BLN Haemophilus influenzae (50) BLP Haemophilus influenzae (79) BLNAR Haemophilus influenzae (21) BLN Haemophilus parainfluenzae [START_REF] Mukhtar | Streptogramins, oxazolidinones, and other inhibitors of 530 bacterial protein synthesis[END_REF] BLP Haemophilus parainfluenzae (8) Moraxella catarrhalis [START_REF] Speciale | Antimicrobial activity of 558 quinupristin/dalfopristin, a new injectable streptogramin with a wide Gram-559 positive spectrum[END_REF] Bordetella pertussis [START_REF] Devasahayam | Newer antibacterial drugs for a new 506 century[END_REF] Neisseria meniingitidis [START_REF] Canu | Overcoming bacterial resistance by dual target inhibition: the 534 case of streptogramins[END_REF] Neisseria gonorrhoeae [START_REF] Speciale | Antimicrobial activity of 558 quinupristin/dalfopristin, a new injectable streptogramin with a wide Gram-559 positive spectrum[END_REF] Bacteroides fragilis group (100) Bacteroides spp. (50) Fusobacterium spp. (40) Mycoplasma pneumoniae [START_REF] Canu | Overcoming bacterial resistance by dual target inhibition: the 534 case of streptogramins[END_REF] Mycoplasma hominis [START_REF] Canu | Overcoming bacterial resistance by dual target inhibition: the 534 case of streptogramins[END_REF] Ureaplasma urealyticum [START_REF] Barrière | Recent 532 developments in streptogramin research[END_REF] Legionella spp. [START_REF] Canu | Overcoming bacterial resistance by dual target inhibition: the 534 case of streptogramins[END_REF] Chlamydophila pneumoniae (5) Chlamydia trachomatis ( [START_REF] Low | A review of in-vitro antibacterial activity of 572 quinupristin/dalfopristin against methicillin-susceptible and -resistant 573 Staphylococcus aureus[END_REF], erm( 43)- [START_REF] Fuchs | Bactericidal activity of quinupristin-dalfopristin 606 against Staphylococcus aureus: clindamycin susceptibility as a surrogate 607 indicator[END_REF], erm [START_REF] Schwarz | Lincosamides, Streptogramins, phenicols, and 614 pleuromutilins: mode of action and mechanisms of resistance[END_REF], erm [START_REF] Roberts | Environmental macrolide-lincosamide-streptogramin and 620 tetracycline resistant bacteria[END_REF] c lsa(C) is associated with a "LS A P" resistance phenotype Bernard et al. [START_REF] Bernard | Oral 778 pristinamycin versus standard penicillin regimen to treat erysipelas in adults: 779 randomised, non-inferiority, open trial[END_REF] Bernard et al. [START_REF] Bernard | Pristinamycin in the treatment of acute 781 bacterial dermohypodermitis in adults. An open study of 42 patients[END_REF] Chosidow et al. [

VRE infections

Linden et al. [START_REF] Linden | Treatment of vancomycin-resistant 813 Enterococcus faecium infections with quinupristin/dalfopristin[END_REF] Moellering et al. [START_REF] Moellering | The 819 efficacy and safety of quinupristin/dalfopristin for the treatment of infections 820 caused by vancomycin-resistant Enterococcus faecium. Synercid Emergency-Use 821 Study Group[END_REF] Raad et al. [ [START_REF] Nichols | Treatment of hospitalized patients 823 with complicated gram-positive skin and skin structure infections: two 824 randomized, multicentre studies of quinupristin/dalfopristin versus cefazolin, 825 oxacillin or vancomycin. Synercid Skin and Skin Structure Infection Group[END_REF] 450 443 QD 7.5 mg/kg q12h OXA 2 g q6h, VAN

A c c e p t e d M a n u s c r i p t 841 Legends of the figures 842 Figure 1 .Figure 2 .Figure 3 .

 842123 Figure 1 161x150mm (300 x 300 DPI)

A c c e p t e d M a n u s c r i p t

A c c e p t e d M a n u s c r i p t

A c c e p t e d M a n u s c r i p t

A c c e p t e d M a n u s c r i p t

Table 2 )

 2 [START_REF] Speciale | Antimicrobial activity of 558 quinupristin/dalfopristin, a new injectable streptogramin with a wide Gram-559 positive spectrum[END_REF][START_REF] Low | A review of in-vitro antibacterial activity of 572 quinupristin/dalfopristin against methicillin-susceptible and -resistant 573 Staphylococcus aureus[END_REF]. Noteworthy, Enterococcus faecalis is a gap in the antimicrobial spectrum 125 since this Gram-positive species is intrinsically resistant to type A streptogramins (phenotype 126 LS A, which results in lincosamide and streptogramins A resistance) and to the A plus B 127 streptogramin combinations, due to the presence of the lsa(A) gene[START_REF] Singh | An Enterococcus faecalis ABC homologue 575 (Lsa) is required for the resistance of this species to clindamycin and 576 quinupristin-dalfopristin[END_REF][START_REF] Singh | Differences in the Enterococcus faecalis lsa locus that 578 influence susceptibility to quinupristin-dalfopristin and clindamycin[END_REF][START_REF] Dina | Nonsense mutations in the lsa-like gene in 581 Enterococcus faecalis isolates susceptible to lincosamides and Streptogramins A[END_REF]. Streptogramins 128 are also active against S. pneumoniae (regardless of resistance to -lactams and macrolides), 396 dalfopristin are inhibitors of CYP3A4 and consequently drug interaction is observed with 397 ciclosporin, midazolam, nifedipine, tamoxifen, and terfenadine[39,91].401 Although the streptogramin compounds are separately bacteriostatic, the combination usually 402 becomes bactericidal. Their spectrum of activity includes a broad range of aerobic and 403 anaerobic Gram-positive bacteria, but also most of fastidious Gram-negative and atypical 438 infections and skin infections due to streptococci or staphylococci. M. genitalium is an 439 important sexually transmitted pathogen responsible for both male and female genital tract 440 disease. Recently, the interest of pristinamycin in the treatment of infections due to this 441 pathogen have been demonstrated and European guidelines recommend to use pristinamycin 442 as a third-line treatment after azithromycin and moxifloxacin treatment failure for M. 443 genitalium infections. Moreover, this antibiotic has been described to be a good alternative to 444 treat infections due to resistant Gram-positive bacteria in patients with cancer in an Australian

	Page 4 of 40

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The spectrum of activity of streptogramins includes a broad range of aerobic and anaerobic 122 Gram-positive bacteria, with a MIC 90 generally ≤1 mg/l (Table

2

)

[START_REF] Bouanchaud | In-vitro and in-vivo antibacterial activity of 554 quinupristin/dalfopristin[END_REF][29][START_REF] Speciale | Antimicrobial activity of 558 quinupristin/dalfopristin, a new injectable streptogramin with a wide Gram-559 positive spectrum[END_REF][START_REF] Pankuch | Activities of a new oral streptogramin, XRP 561 2868, compared to those of other agents against Streptococcus pneumoniae and 562 haemophilus species[END_REF][START_REF] Eliopoulos | In vitro activity of an 564 oral streptogramin antimicrobial, XRP2868, against gram-positive bacteria[END_REF][START_REF] Goldstein | Comparative in vitro activities of XRP 567 2868, pristinamycin, quinupristin-dalfopristin, vancomycin, daptomycin, 568 linezolid, clarithromycin, telithromycin, clindamycin, and ampicillin against 569 anaerobic gram-positive species, actinomycetes, and lactobacilli[END_REF]

. They are active 123 against methicillin-susceptible, MRSA and most of E. faecium isolates (including VREF) 124 (129 -hemolytic streptococci, viridans streptococci, Corynebacterium spp., and Listeria 130 monocytogenes (Table 2) [28,38,39]. In addition, streptogramins exhibit activity against most 131 of Gram-positive anaerobes, such as Actinomyces spp., Clostridium spp., Lactobacillus spp., 132 Peptostreptococcus spp., and Cutibacterium acnes (Table 2) [33,39]. They are also active 133 against Mycoplasma spp., Ureaplasma urealyticum, and Chlamydia spp. Indeed, European 134 guidelines about M. genitalium infections recommend pristinamycin as third-line therapy in 135 patients failing both azithromycin and moxifloxacin therapy [40,41]. Interestingly, they also 136 have a good activity against fastidious Gram-negative bacteria including Moraxella 137 catarrhalis, Neisseria spp., and Legionella pneumophila (Table 2) [28,38,39]. Streptogramins 138 have a variable activity against Bacteroides fragilis group and other Gram-negative 139 anaerobes. Several strains are resistant and some, like Fusobacterium spp. are very 140 susceptible (Table 2) [28,39,40]. Finally, Enterobacterales, Pseudomonas aeruginosa, and 141 Acinetobacter spp. are intrinsically resistant to high levels of streptogramins [28]. 142 Clinical Laboratory Standards Institute breakpoints for quinupristin-dalfopristin against 143 staphylococci, enterococci, and streptococci are as follows: ≤1 mg/l, susceptible; 2 mg/l, 144 intermediate; and ≥4 mg/l, resistant [42]. European Committee of Antimicrobial 145 Susceptibility Testing breakpoints for quinupristin-dalfopristin are available only for URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk 193 cleavage of the ester linkage leading to a linearization of the molecule. A Low-level of 194 resistance to streptogramins B, mediated by ribosomal protection through acquisition of msr-195 like genes, is described in staphylococci, streptococci, or enterococci [57]. Page 8 of 40 URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk Expert Review of Anti-infective Therapy A c c e p t e d M a n u s c r i p t URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk Expert Review of Anti-infective Therapy A c c e p t e d M a n u s c r i p t URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk Expert Review of Anti-infective Therapy A c c e p t e d M a n u s c r i p t URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk Expert Review of Anti-infective Therapy A c c e p t e d M a n u s c r i p t URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk Expert Review of Anti-infective Therapy A c c e p t e d M a n u s c r i p t URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk Expert Review of Anti-infective Therapy A c c e p t e d M a n u s c r i p t URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk Expert Review of Anti-infective Therapy A c c e p t e d M a n u s c r i p t URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk Expert Review of Anti-infective Therapy A c c e p t e d M a n u s c r i p t URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk Expert Review of Anti-infective Therapy A c c e p t e d M a n u s c r i p t bacteria. This synergism also allows overcoming certain mechanisms of resistance to the 405 individual components of the combination and may explain low rates of resistance reported so 406 far. To date, two streptogramins are available in human medicine, oral pristinamycin, and 407 quinupristin-dalfopristin, an injectable formulation approved since 1999 and 2000 in the US 408 and Europe, respectively. In this study we have synthesized all the clinical studies evaluating 409 these antibiotics. Finally, virginiamycin, an oral streptogramin, was therapeutically used in 410 France until late 1990s and as a growth promoter for animals in many European countries 411 until early 2000s and currently in the US. 412 Page 17 of 40 URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk 446 Quinupristin-dalfopristin, an injectable streptogramin, was the first to be developed 447 internationally. Despite its effectiveness in the treatment of severe Gram-positive bacterial 448 infections demonstrated in several studies, and the low rate of reported resistance, the 449 penetration of this molecule into the antibiotic market has remained limited. This can be 450 probably explained in part by the mode of administration, which requires a central venous 451 catheter. The low use of this antibiotic can likely also be explained by the high prevalence of 452 side effects, namely venous intolerance and the risk of thrombophlebitis. Studies to decrease 453 the toxicity of this molecule would be interesting, in order to make it more accessible and 454 easy to use. Currently marketed in North America, its high cost also represents a barrier to its 461 No writing assistance was utilized in the production of this manuscript. 462 Page 19 of 40 URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk  The efficacy of pristinamycin has been largely demonstrated due to its extensive use 473 for 50 years in France and some African countries. 474  The clinical use quinupristin-dalfopristin has remained limited, mainly due to its poor URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk Expert Review of Anti-infective Therapy

Table 1 :

 1 Examples of streptogramins[START_REF] Barrière | Recent 532 developments in streptogramin research[END_REF][START_REF] Canu | Overcoming bacterial resistance by dual target inhibition: the 534 case of streptogramins[END_REF].

	Page 35 of 40
	Streptogramins	Individual components	Producing organism
		A	B
	Natural products
	Madumycin	Madumycin II	Madumycin I	Actinomadura flava
	Mikamycin	Mikamycin A	Mikamycin B	Strepomyces mitakaensis
	Pristinamycin	Pristinamycin II A , II B , II C ,	Pristinamycin I A , I B , I C , I D ,	Streptomyces
		II D , II E , II F , II G	I E , I F , I G , I H , I I	pristinaespiralis
	Synergistin	Synergistin A1, A2	Synergistin B1, B2, B3
	Vernamycin	Vernamycin A	Vernamycin B (, , , )	Streptomyces olivaceus
	Virginiamycin	Virginiamycin M1, M2	Virginiamycin S1, S2, S3,	Streptomyces loidensis
			S4, S5	Streptomyces virginiae
	Semisynthetic derivatives
	Quinupristin-dalfopristin	Dalfopristin	Quinupristin	-
	(RP 59500)	(RP 54476)	(RP 57669)
	Linopristin-flopristin	Flopristin	Linopristin	-
	(XRP 2868)	(RPR 132552)	(RPR 202868)

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Table 2 :

 2 Comparative in vitro activity (MIC in mg/l) of quinupristin-dalfopristin, and pristinamycin against Gram-positive bacteria[START_REF] Pankuch | Activities of a new oral streptogramin, XRP 561 2868, compared to those of other agents against Streptococcus pneumoniae and 562 haemophilus species[END_REF][START_REF] Eliopoulos | In vitro activity of an 564 oral streptogramin antimicrobial, XRP2868, against gram-positive bacteria[END_REF][START_REF] Goldstein | Comparative in vitro activities of XRP 567 2868, pristinamycin, quinupristin-dalfopristin, vancomycin, daptomycin, 568 linezolid, clarithromycin, telithromycin, clindamycin, and ampicillin against 569 anaerobic gram-positive species, actinomycetes, and lactobacilli[END_REF][START_REF] Low | A review of in-vitro antibacterial activity of 572 quinupristin/dalfopristin against methicillin-susceptible and -resistant 573 Staphylococcus aureus[END_REF][START_REF] Singh | An Enterococcus faecalis ABC homologue 575 (Lsa) is required for the resistance of this species to clindamycin and 576 quinupristin-dalfopristin[END_REF][START_REF] Singh | Differences in the Enterococcus faecalis lsa locus that 578 influence susceptibility to quinupristin-dalfopristin and clindamycin[END_REF][START_REF] Dina | Nonsense mutations in the lsa-like gene in 581 Enterococcus faecalis isolates susceptible to lincosamides and Streptogramins A[END_REF][START_REF] Bonfiglio | Novel streptogramin antibiotics[END_REF][39][40].

	Bacterial species	Quinupristin-Dalfopristin		Pristinamycin
	(no. of tested isolates) a	Range	MIC 50 MIC 90	Range	MIC 50	MIC 90
	MS Staphylococcus aureus (30)
	MR Staphylococcus aureus (45)
	Staphylococcus epidermidis (14)
	VS Enterococcus faecalis (24)
	VanA Enterococcus faecalis (10)
	VanB Enterococcus faecalis (23)
	VS Enterococcus faecium (30)
	VanA Enterococcus faecium (32)
	VanB Enterococcus faecium (31)
	PS Streptococcus pneumoniae (86)
	PI Streptococcus pneumoniae (81)
	PR Streptococcus pneumoniae (94)
	ES Streptococcus pneumoniae (141)
	ER Streptococcus pneumoniae (120)
	Streptococcus pyogenes (27)
	Streptococcus agalactiae (15)
	Viridans streptococci (20)
	Listeria monocytogenes (23)
	Corynebacterium jeikeium (10)
	Actinomyces israelii (13)
	Actinomyces odontolyticus (10)
	Clostridium difficile (14)
	Clostridium perfringens (12)
	Finegoldia magna (11) b

Table 3 :

 3 Mechanisms

		of	resistance	to	streptogramins	in	Gram-positive	bacteria
	(http://faculty.washington.edu/marilynr/) [20, 21, 24, 48-68].
	Resistance	Gene(s)		Resistance	Main bacterial species
	mechanism			phenotype a
	Streptogramins A
	Ribosomal	cfr, cfr(B), cfr(D)	PhLOPS A	Staphylococcus spp., Streptococcus spp,
	methylation					Enterococcus spp.
	Acetylation	vat(A)-(E), vat(G),	S A		Staphylococcus spp., Enterococcus spp.
	Ribosomal	vga(A)-(E),		S A		Staphylococcus spp., Enterococcus spp.
	protection	vga(A) V, vga(A) LC	LS A		Staphylococcus spp.
		lsa(A)-(C) c , lsa(E)	LS A		Staphylococcus spp., Streptococcus spp,
						Enterococcus spp.
		sal (A)		LS A		Staphylococcus sciuri
		eat(A)v		LS A		Enterococcus faecium
	Streptogramins B
	Ribosomal	erm genes b		MLS B	Staphylococcus spp., Streptococcus spp,
	methylation					Enterococcus spp., Cutibacterium acnes
	Ribosomal mutation	rrn (23S rRNA)		MLS B	Streptococcus pneumoniae
		rplD (L4 protein)	M		Streptococcus pneumoniae
		rplV (L22 protein)	MS B		Staphylococcus aureus, Streptococcus
						pneumoniae
	Hydrolysis	vgb(A), vgb(B)		S B		Staphylococcus spp.
	Ribosomal	msr(A)-(D), msr(F)-(H)	MS B
	protection

, Enterococcus faecium Staphylococcus spp., Streptococcus spp., Enterococcus spp., Macrococcus spp. a L: Lincosamides; M: Macrolides; O: Oxazolidinones; P: Pleuromutilins; Ph: Phenicols; S A : Streptogramins A; S B : Streptogramins B. b erm(A)-(C), erm(F), erm(Q), erm(T), erm(X), erm(Y), erm

Table 4 :

 4 Main pharmacokinetic parameters of streptogramins after a single dose[29,[90][91][92].

		Pristinamycin	Quinupristin-Dalfopristin (30:70)
	Parameter a	PI A	(2 g, PO)	PII A	(7.5 mg/kg, IV) Quinupristin Dalfopristin
	C max (mg/L)	0.8			0.6	2.3-2.7	6.1-8.2
	t max (h)	3.2			3.1	0.9	1.1
	AUC (mgh/L)	2.2			1.2	2.7-3.3	6.5-7.7
	t 1/2 (h)	4			2.8	0.9-1.3	0.7-1.1
	CL (L/h/kg)	-			-	0.7-0.9	0.7-0.8
	V d (L/kg)	-			-	0.45	0.24
	PB (%)	40-45	70-80	55-78	11-26
	FE (%)	-			-	75	77
	UE (%)	-			-	15	19
	a AUC: area under the concentration-time curve; CL: Systemic clearance; C max : Maximal plasma concentration; FE:
	Excretion in the faeces; IV: Intravenous; PB: Protein binding; PO: per os; t 1/2 : Plasmatic half-life; UE: Excretion in the
	urine; Vd: Volume of distribution.

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Table 5 :

 5 Main clinical trials assessing the efficacy of pristinamycin[START_REF] Pessey | Pristinamycin use in the treatment of acute maxillary sinusitis in 757 adults][END_REF][START_REF] Gehanno | Efficiency of a four-day course of 759 pristinamycin compared to a five-day course of cefuroxime axetil for acute 760 bacterial maxillary sinusitis in adult outpatients[END_REF][START_REF] Trémolières | Efficacy and 766 safety of pristinamycin vs amoxicillin in community acquired pneumonia in 767 adults[END_REF][96][START_REF] Léophonte | Treatment of exacerbations of chronic 772 obstructive pulmonary disease with pristinamycin[END_REF][98][START_REF] Bernard | Oral 778 pristinamycin versus standard penicillin regimen to treat erysipelas in adults: 779 randomised, non-inferiority, open trial[END_REF][START_REF] Bernard | Pristinamycin in the treatment of acute 781 bacterial dermohypodermitis in adults. An open study of 42 patients[END_REF][START_REF] Chosidow | Cloxacillin versus pristinamycin for 784 superficial pyodermas: a randomized, open-label, non-inferiority study[END_REF] and quinupristin-dalfopristin[START_REF] Drew | Treatment 805 of methicillin-resistant Staphylococcus aureus infections with quinupristin-806 dalfopristin in patients intolerant of or failing prior therapy. For the Synercid 807 Emergency-Use Study Group[END_REF][START_REF] Raad | Prospective, randomized dose-ranging open phase 809 II pilot study of quinupristin/dalfopristin versus vancomycin in the treatment of 810 catheter-related staphylococcal bacteremia[END_REF][START_REF] Linden | Treatment of vancomycin-resistant 813 Enterococcus faecium infections with quinupristin/dalfopristin[END_REF][START_REF] Raad | Prospective, randomized study comparing 816 quinupristin-dalfopristin with linezolid in the treatment of vancomycin-resistant 817 Enterococcus faecium infections[END_REF][START_REF] Moellering | The 819 efficacy and safety of quinupristin/dalfopristin for the treatment of infections 820 caused by vancomycin-resistant Enterococcus faecium. Synercid Emergency-Use 821 Study Group[END_REF][START_REF] Nichols | Treatment of hospitalized patients 823 with complicated gram-positive skin and skin structure infections: two 824 randomized, multicentre studies of quinupristin/dalfopristin versus cefazolin, 825 oxacillin or vancomycin. Synercid Skin and Skin Structure Infection Group[END_REF][START_REF] Fagon | Treatment of gram-positive nosocomial 828 pneumonia. Prospective randomized comparison of quinupristin/dalfopristin 829 versus vancomycin. Nosocomial Pneumonia Group[END_REF].

	Results (%) c

  AMC: Amoxicillin-clavulanate; AMX: Amoxicillin; ATM: Aztreonam; CFZ: cefazolin; CLO: Cloxacillin; CXM: Cefuroxime axetil; LIN: Linezolid; OXA: Oxacillin; PG: Penicillin G; PV: Penicillin V; PRI: Pristinamycin; QD: Quinupristin-dalfopristin; VAN: Vancomycin c Success rates observed in per-protocol and/or intention-to-treat populations (end of treatment and/or follow-up). NA: not available.

	Page 39 of 40		Expert Review of Anti-infective Therapy			Page 40 of 40
	Not published	180		NXL103 q12h	7-14	NA	NA
	(NCT00949130)	180	NXL103 q12h LIN q12h	7-14
			M a n u s c r i p t
	A c c e p t e d	1 g q12h, or CFZ 1 g q8h	7 8	68 71	46-66 48-73
	Nosocomial pneumonia
	Fagon et al. [115]	150	QD 7.5 mg/kg q8h + ATM 2 g	10	43-56	53-59
		148		q8h	9	45-58	55-64
			VAN 1 g q12h + ATM 2 g q8h
	Community acquired pneumonia
	Not published	302	NXL103 500 mg q12h	NA	NA	NA
	(NCT00653172)		NXL103 600 mg q12h
			Comparator q12h
	Acute bacterial skin infections

URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk a COPD: Chronic obstructive pulmonary disease. b URL: https://mc.manuscriptcentral.com/eri Email: IERZ-peerreview@journals.tandf.co.uk Expert Review of Anti-infective Therapy