Sharp error term in local limit theorems and mixing for Lorentz gases with infinite horizon

We obtain sharp error rates in the local limit theorem for the Sinai billiard map (one and two dimensional) with infinite horizon. This result allows us to further obtain higher order terms and thus, sharp mixing rates in the speed of mixing of dynamically H\"older observables for the planar and tubular infinite horizon Lorentz gases in the map (discrete time) case. In the process, we study families of transfer operators for infinite horizon Sinai billiards perturbed with the free flight function and obtain higher order expansions for the associated families of eigenvalues and eigenprojectors.


Introduction
We recall that discrete time Lorentz gases can be understood as Z d extensions, d = 1, 2 of the Sinai billiard map by the free flight function κ. A precise description of the terminology is provided in Section 1. Limit properties of discrete time Lorentz gases with bounded κ, referred to as finite horizon Lorentz gases, have been obtained in several recent works, among which we mention [24,25,18,19,20,21,22]. Very recent notable progress for continuous time (not considered in the present work) Lorentz gases with finite horizon has been made by Dolgopyat and Nandori [6,7] and Dolgopyat, Nandori, Pène [8].
Local limit theorems (LLT) for the free flight function in the finite horizon case has been obtained by Szász and Varjú in [24]. Sharp error rates in the LLT for the finite horizon case has been obtained by Pène [18] and further refined to Edgeworth expansions of any order by the same author in [20]. As shown in [20, Theorems 1.2 and 1.3], Edgeworth expansions in the LLT lead to similar expansions in mixing for dynamically Hölder observables for the infinite measure preserving, finite horizon Lorentz maps, providing, in particular, sharp estimates for observables with null integral, including coboundaries.
Previous results on LLT and mixing for the infinite measure preserving, infinite horizon Lorentz maps reduce to first order terms. Roughly, infinite horizon means that the free flight function is not L 2 with respect to the Liouville measure of the Sinai billiard. Precise definitions and statements are provided in Section 1. Here we recall that a LLT (without error term) has been established by Szász and Varjú in [25,Theorem 13] via the abstract [2, Theorem 2] of Bálint and Gouëzel, which gives the non standard Gaussian limit law for the stadium billiard. First order mixing for dynamically Hölder observables and infinite horizon Lorentz maps was obtained in [20,Theorems 1.1].
In the present work we obtain higher order terms in both, the LLT [25, Theorem 13] and mixing [20,Theorem 1.1]. LLT for Sinai billiards with infinite horizon goes via establishing the LLT for the associated Young tower as in [25], which is possible due to the work of Young [27] and Chernov [5]. We recall that a classical tool for establishing LLTs for Young towers is the perturbed transfer operator method. As clarified in Section 1, a challenge for obtaining error terms in the LLT in [25,Theorem 13] comes from obtaining 'sufficiently high' expansions (not just continuity) for the families of eigenvalues and eigenprojectors associated with the transfer operator perturbed with non L 2 functions. Propositions 1.5 and 1.7 provide such expansions. The required proof builds on the framework put forward in [2] using several geometrical estimates established in [25]. Under assumptions specific to Young towers for Sinai billiards with infinite horizon, Propositions 1.7 and 1.5 can be viewed as refined version of some technical results of [2]. Another challenge is to use Propositions 1.5 and 1.7 and in obtaining sharp mixing rates for dynamically Hölder observables for the infinite measure preserving, infinite horizon Lorentz maps. Our main results in this sense are Theorems 1.1 and 1.2. Moreover, in Theorem 1.3, we obtain improved error terms for particular zero mean functions, including coboundaries, with leading terms (non null in general) of different orders.
At the end of this introductory section we remark that the mechanism for obtaining mixing and mixing rates for infinite measure preserving Lorenz gases (with finite or infinite horizon) are quite different from the one proper to deal with Markov or non Markov infinite measure preserving intermittent interval maps (and more complicated versions of them such as 2D invertible systems and suspension flows over such maps). Very roughly, periodic Lorentz gases are highly non trivial generalizations of symmetric random walks, while the latter mentioned classes are generalizations of Markov chains. Sharp mixing rates for (not necessarily Markov) infinite measure preserving intermittent interval maps have been obtained by Melbourne and Terhesiu [16], Terhesiu [26]; such results have been generalized to suspension flows in [17,3]. We mention that the results on mixing rates for zero integral functions and coboundaries in the setup of Lorentz maps (in [20] and also in Theorem 1.3 therein) have no existing direct analogue in the set up of infinite measure preserving intermittent interval maps; the previous results [16,26,17,3] for zero integral observables are confined to big O error terms.
Our manuscript is organized as follows. In Section 1, we introduce the Z 2 -periodic billiard model with infinite horizon and state our main results Theorems 1.1, 1.2 and 1.3 together with the key technical results Propositions 1.5, 1.7. In Section 2, we obtain an expansion for the probability of long free flights. In Section 3, we prove our first key result Proposition 1.5, stating an expansion of the dominating eigenprojector. In Section 4, we prove our second key result 1.7, which gives an expansion of the eigenvalue using results contained in the two previous sections. In Section 5, we state an expansion in the LLT and in a general context and use it to prove our main result as well as a general decorrelation result for some Z d -extensions. Some further technical estimates are included in the Appendix.

Model and main results
We consider a planar billiard domain Q given by: with J a non empty finite set and with O j, := O j + , where the O j are open convex set with boundary C 3 and with nonzero curvature, such that O j, have pairwise disjoint closures. We assume that the billiard has infinite horizon, i.e. that the interior of Q contains at least one line.

Figure 1. A periodic billiard domain with 4 infinite horizon directions
We denote by (M, T, µ) the original billiard dynamical system map corresponding to collision times. We recall that the configuration space M is the set of position-speed couples (q, v) with q ∈ ∂Q and v a unit reflected vector, i.e. a unit vector v oriented inside Q. The billiard map T maps a configuration (q, v) corresponding to a collision time to the configuration corresponding to the next collision time. The measure µ is the measure on M with density proportional to cos ϕ, where ϕ is the angle of v with the normal vector to ∂Q directed inside Q, normalized so that µ({(q, v) ∈ M : q ∈ j∈J ∂O j }) = 1. The infinite measure preserving dynamical system (M, T, µ) is canonically isomorphic to the Z 2 -extension of (M ,T ,μ) by κ :M → Z 2 , where (M ,T ,μ) is the probability preserving billiard dynamical system in the billiard domain inQ = Q/Z 2 andμ the probability measure with density proportional to cos ϕ.
It is known that (M ,T ,μ) is a factor, under a projection written p 1 :∆ →M , of a Young tower (∆,f ,μ ∆ ) with stable and unstable curves. By factorizing/collapsing the stable curves we can reduce it to a one-dimensional Young tower (∆, f, µ ∆ ) by p 2 :∆ → ∆. Throughout we letκ : ∆ → Z 2 be the version of κ on ∆, that isκ • p 2 = κ • p 1 (the existence of such aκ comes from the fact that κ is constant on the stable curves).
Let P be the transfer operator for (∆, f, µ ∆ ). We consider the family (P t ) t∈R of perturbations of P given by P t := P e it·κ · , where · denotes the standard scalar product on R d . Note that P 0 ≡ P .
Inside the proof of our Proposition 1.7 below, which gives a higher order expansion of λ t at t = 0, we provide a precise summary of the results in [25] needed to obtain (3). The above expansion of λ t implies the LLT forT and thus first order mixing (speed of mixing) for T as in [20,Theorem 1.1]. Here we are interested in higher order terms in both the LLT and mixing for suitable classes of functions.
We state our main result for the two dimensional maps T andT and mention the difference in the one dimensional case (more detailed results in dimension 1 and 2 are given in Section 5.3). To do so we need to introduce some notation. Let R 0 ⊂ M be the set of reflected vectors that are tangent to ∂Q. The billiard map T defines a C 1 -diffeomorphism from M \ (R 0 ∪ T −1 R 0 ) onto M \ (R 0 ∪ T R 0 ). For any integers k ≤ k , we set ξ k k for the partition of M \ k j=k T −j R 0 in connected components and ξ ∞ k := j≥k ξ j k . For any φ : M → R and any η ∈ (0, 1), we set L φ,η := sup By a slight abuse of notations, we define the same notions with same notations with (M ,T ) instead of (M, T ). Set with the notations introduced at the begining of Section 2 and where w ⊗ w represents the matrix w 2 1 w 1 w 2 w 1 w 2 w 2 2 if w = (w 1 , w 2 ). Roughly speaking, C is the set of different "corridors" (distinct modulo Z 2 ) that can be drawn in Q; for each corridor C ∈ C, d C is its length, w C a vector in Z 2 in the direction of the corridor with coprime coordinates and n C is the number of different (distinct modulo Z 2 ) points of tangencies of ∂Q with the corridor C. Theorem 1.1 (Local limit theorem). Let (M, T, µ) be the Lorentz map with infinite horizon in dimension d = 2. Assume that the interior of Q contains at most two lines not parallel to each other.
Let N ∈ Z d andφ,ψ :M → C be two measurable functions such that φ (η) + ψ (η) < ∞, then, uniformly in N ∈ Z d , For any N ∈ Z 2 , we write M N for the set of (q, v) ∈ M such that q ∈ j∈J ∂O j,N .
Theorem 1.2 (Decorrelation in infinite measure). Let η, γ ∈ (0, 1). Let (M, T, µ) be the Lorentz map with infinite horizon in dimension d = 2. Assume that the interior of Q contains at most two lines not parallel to each other. Let φ, ψ : M → R be two measurable observables such that If moreover When φ or ψ has zero integral, then Theorem 1.2 only provides an estimate in O(·). Nevertheless, the spectral method used here enables us to establish of sharp decorrelation rates for a reasonably large class of zero integral observables of the Z d -extension, including smooth coboundaries. Theorem 1.3 (Sharper decorrelation rates for particular functions with zero integral). Let (M, T, µ) be the Lorentz map with infinite horizon in dimension d = 2. Assume that the interior of Q contains at most two lines not parallel to each other.
(a) Let φ, ψ : M → C be bounded observables such that ∃γ ∈ (0, 1), (Σ −2 N · N ) 1 + (d + 2) log log n 2 log n + O((log n) −1 ) Let us make several observations on this last result. First, whereas in the finite horizon case, we only have leading terms in n −d/2−m in the decorrelation of smooth functions, in the infinite horizon case we can have leading terms in n − d 2 −m log n −d/2 but also in n − d 2 −1 log n −d/2−1 . Other orders are possible. For example, we can easily adapt our proof to obtain sharp decorrelation rate in n − d 2 −m−1 log n −d/2−1 in case (b) with ψ a coboundary of order m.
Observe that, when m = 1, Case (a) of Theorem 1.3 corresponds to the study of Ω φ.ψ • S n dν − Ω φ.ψ • S n−1 dν and the dominating term given by (a) is equivalent to the difference between the two leading terms of Ω φ.ψ • S n dν and of Ω φ.ψ • S n−1 dν obtained in Theorem 1.2. The leading term is of order (n log n) −d −((n+1) log(n+1)) −d ∼ d(n log n) − d 2 −1 log n. Let us observe that the case when φ is a coboundary and the case of two coboundaries is included in item (a) of theorem 1.3. Indeed, by T -invariance of µ, The assumption that the corridors are not parallel ensures that det Σ 2 = 0.
The above results hods true when d = 1 for the billiard in the tubular domain Q/({0} × Z), up to replacing Σ 2 by a 1,1 provided a 1,1 = 0 (i.e. provided the interior of Q contains at least one non vertical line).
We focus on the case d = 2 for all the preliminary results used in the proof of Theorem 1.2 since the similar results in the case d = 1 follow from them. Theorems 1.1 and 1.2 are contained in the more technical Theorems 5.6 and 5.7 (valid for a class of less regular observables) which are consequences of Theorem 5.1 that gives higher order terms in LLT and speed of mixing under abstract assumptions on families of eigenvalues and eigenprojectors. Most of our work consist in proving the two following results enabling the application of Theorem 5.1 to the quotiented tower (∆, f, µ) andκ. These two results are higher order versions of the results in [24] and do not require the assumption that the corridors are not parallel. Proposition 1.5. Let η ∈ (0, 1) and fix p > 2. There exists a functional Banach space B 0 → B → L p (µ ∆ ) such that, for every p ∈ [1, 4 3 ), for every γ ∈ (1, min(2 p−1 p , 4 p − 2)), there exists C > 0 such that for every w ∈ B 0 , there exists Π 0 w belonging in L s (µ ∆ ) for every s ∈ [1, 2) and such that Remark 1.6. A precise formula for Π 0 is given by (13) of Proposition 3.2 together with (16) of Proposition 3.3.
with the notations introduced at the begining of Section 2.
Let us point out the fact that, since the norms of R d are all metrically equivalent, Proposition 1.7 is true for any choice of norm | · | on R d .

Estimate of the probability of long free flights
Our proof of Proposition 1.7 provided in Section 4 is based on estimates established in Section 3 and on an higher order expansion of the tail of the free flight κ. Lemma 2.1 below gives the required expansion. We mention that Lemma 2.1 can be regarded as a refinement of [25,Proposition 6]. First, it provides higher order of the tail of the free flight and second, we compute the dominating term without any restriction on the number of types of scatterer at the boundary of a given corridor.
Note that if (A, w) ∈ A then (A, w/|w|) is in T 1 ∂Q as well as its iterates in the past and in the future under T k . But the set of points of k∈Z T k (T 1 ∂Q) is locally finite. Therefore A is finite.
Given L, w ∈ Z 2 , we write E L,w for the set of (A, B) such that (A, w) ∈ A and B ∈ ∂Q is as described above (on the other line delimiting the corridor).
where c E is the curvature of ∂Q at E for any E ∈ i∈I ∂Q and where B" is the first point of ∂Q met by the half-line B − R + w.
Note that, the set {B ∈ j∈J ∂O j : ∃L ∈ Z 2 , (A, B) ∈ E L,w } is finite and can be ordered (B i ) i∈Z/N A Z in such a way that, for all i,  Proof. Observe first that, for every ε, outside the ε-neighbourhood of k∈Z T k (T 1 ∂Q), κ is uniformly bounded. Moreover, outside the ε-neighbourhood of points (A 0 , w/|w|) with (A 0 , B) ∈ E L,w , for some N 0 large enough, κ ∈ L + (N 0 + N)w. Therefore, it is enough to prove that, for any (A 0 , B) ∈ E L,w and any i 0 , i 1 ∈ J so that A 0 ∈ ∂O i 0 and B ∈ ∂O i 1 ,  We write O for the first obstacle touched by A 0 + R + w (at A 0 ∈ ∂O ) and O" for the first obstacle touched by B 0 − R + w (at B" ∈ ∂O").
Let us write v α for the unit vector making angle α with w, for α ∈ [0, π/2] and we consider x > 0 small. For a given position where we write β for the angle such that the line A + v β R is tangent to O from above, α N for the angle such that the line q + v α N R is tangent to O (i 1 , 1 +N w) from underneath (at some point B N ) and α N for the angle such that the line q + v α N R is tangent to O" + N w from underneath (at some point B" N ) . Observe first that

For a given position
We have to estimate angles α N and α N . Note that α N is either α N +m 1 for another choice of point B 0 and for some m 1 ∈ Z 2 (depending on B 1 ). Therefore it is enough to estimate α N . The notation O(...) below will be uniform in A. Let B N := B 0 + N w. The tangent line to ∂O (i 1 , 1 +N w) at B N is parallel to w, so that .
where c B 0 denotes the curvature of ∂O (i 1 , 1 ) at B 0 and c B 0 its derivative with respect to arclength (since ∂O (i 1 , 1 ) is C 4 ). Also Since (8) and (9) are equal, we obtain that This leads to (10) Therefore, we also have (11) Hence Moreover (11) combined with (7) implies that β < α N holds true if and only if a condition of the following form is satisfied: and that β < α N if and only AA 0 = O(N −2 ). Note moreover that the set β < α N has also the form and thus theμ-measure of (r, α) such that α N < β < α N and ). Therefore, using the fact that theμ-measure is invariant, To conclude, observe that 3. Regularity of the projector t → Π t at t = 0 In this section we prove Proposition 1.5 and even a more general result Proposition 3.2. We write π 0 : ∆ → Y for the vertical projection from ∆ to its base Y = ∆ 0 given by π 0 (x, ) = (x, 0) and ω : ∆ → N for the level map given by ω((x, )) = . To simplify notations we write µ ∆ (h) for ∆ h dµ ∆ .
3.1. Banach spaces and regularity of the dominating eigenprojector. We start by recalling results from [4,27,5,25]. First recall that the operator P can be written as follows (12) ∀h with α ≡ 0 outside the base of ∆. We write s 0 (·, ·) for the separation time for f on ∆ corresponding to the separation time s(·, ·) in [27]. In particular, if s 0 (x, y) ≥ n then the corresponding elements inM (or in M N ) are in the same atom of ξ n 0 . Recall the class of η ∈ (0, 1)-Hölder functions defined before the statement of Theorem 1.2. Choose β ∈ (η, 1) large enough so that |α(y 1 ) − α(y 2 )| ≤ C α β s 0 (y 1 ,y 2 )+1 if s 0 (y 1 , y 2 ) ≥ 1. The condition β > η will ensure that the Banach spaces B 0 and B described below will be tailored to the approximation of observables considered in Theorem 1.2.
We define the space B 0 -of Lipschitz functions with respect to the metric β s 0 (·,·) , with norm Let p > 2 and choose ε > 0 small enough so that in particular ≥0 e ε θ 1 < ∞. Write ∆ for the -th floor of the tower ∆. We let B be the space of functions f : ∆ → C such that the following quantity is finite The choice of ε so that so that ≥0 e ε θ 1 < ∞ ensures that the Young Banach space B can be continuously injected in L p (µ ∆ ). In what follows we exploit that B is continuously embedded in L p (μ) and satisfies (1) and 2. Using just the information on big tailμ(|κ| > N ) = O(N −2 ), we have with P 0 := P (iκ·).
Proof. Note that q < p. Set r := qp p−q so that 1 q = 1 p + 1 r . With this notation, the upper bound on γ can be rewritten γr < 2 and , where we used the Hölder inequality at the penultimate line and the fact thatκ admits moments of every order smaller than 2 at the last line.
Note that, given q ∈ [1, 2) up to taking p > 2q/(2 − q) large enough (and so up to our choice of B), we can adjust γ so that γ < 2/q is as close to 2/q as we wish (in particular as close to 2 as we wish if we take q = 1). Let Y := ∆ 0 be the base of the tower ∆. Throughout, we let µ Y = µ ∆ (·|Y ). As shown in [5,25], the height of the tower (∆, f, µ ∆ ), which we denote by σ : Y → N, has exponential tail µ Y (σ > n) θ n 1 for some θ 1 ∈ (0, 1). Using Lemma 3.1 and building on the arguments used in [2], we will prove that Proposition 3.2. Let b ∈ (p, +∞]. For every w ∈ B 0 , every v ∈ L b (µ ∆ ) constant on each (a, ) (with a ∈ α and 0 ≤ < σ(a)) such that vw ∈ B, there exists Π 0 (vw) belonging to L η (µ ∆ ) for every η ∈ [1, 2) such that 1 Y Π 0 (vw) ∈ B 0 . Moreover, for every p ∈ (1, 4/3) and every γ ∈ (1, min(2 p−1 p , 4 p − 2)), there exists C > 0 such that, for every (v, w) as above and all t ∈ B δ (0), Proposition 1.5 follows directly from Proposition 3.2 by taking v = 1 ∆ . We postpone the proof of Proposition 3.2 to the end of this section. We remark that Proposition 3.2 cannot be proved merely via the continuity arguments in [15], which is why we resort to building on the arguments in [2].
The key elements used in the proof of Proposition 3.2 are: i) Lemma 3.10, which gives the expansion of t → Y Π t dµ Y ; ii) define 1 Y Π 0 and control 1 Y Π 0 v B 0 for suitable functions v in Proposition 3.3; iii) use i) and ii) together with formula (13) to control Π 0 v in L p for suitable v and p .

3.2.
Regularity of t → 1 Y Π t . Recall that σ corresponds to the first return time of f to the base Y . In what follows, we let F = f σ : Y → Y with F (y) = f σ(y) (y) for all y ∈ Y be the first return map. Recall that (Y, F, µ Y ) is a Gibbs Markov map with respect to a suitable countable partition Y and that σ is constant on each atom of Y (the required definitions are recalled below). Let R : L 1 (µ Y ) → L 1 (µ Y ) be the transfer operator of the Gibbs Markov base map (Y, F = f σ , µ Y ) given by (14) Rv(x) := y : F y=x The transfer operator R of F is related to the transfer operator P of f via the following relation Observe in particular that, on f −1 (Y ), e −α = χ • π 0 . Define d β (y, y ) = β s(y,y ) where the separation time for F , s(y, y ), is the least integer n ≥ 0 such that F n y and F n y lie in distinct partition elements in α. The partition Y separates trajectories with s(y, y ) = ∞ if and only if y = y ; so d β is a metric. The map F is a (full-branch) Gibbs-Markov map, which means that • F | a : a → Y is a measurable bijection for each a ∈ Y, and • | log χ(y) − log χ(y )| ≤ C α d β (y, y ) for all y, y ∈ a, a ∈ Y (since s(·, ·) ≤ s 0 (·, ·)). A consequence of this definition is that there is a constant C > 0 such that and for all a ∈ Y and y, y ∈ a.
Since F is Gibbs Markov, it follows that (see, for instance, [23, Section 5]): • The space (B 1 , · B 0 ) of θ-Hölder continuous functions on Y contains constant functions and B 1 ⊂ L ∞ (µ Y ). Note that B 1 corresponds to functions h ∈ B 0 supported on Y . For this reason and for the reader convenience, we have chosen to write also · B 0 for the norm of B 1 (in order to avoid the introduction of unnecessary notation). • R is quasi-compact on B 1 and 1 is a simple eigenvalue for R, isolated in the spectrum of R. Adapting the argument of [2,Lemma 3.15], in this section we obtain Proposition 3.3. Let b ∈ (p, +∞] and γ ∈ (1, 2 p−1 p ). There exists C > 0 such that, for every w ∈ B 0 and v ∈ L b (µ ∆ ) constant on each (a, ) (with a ∈ A and 0 ≤ < σ(a)) so that vw ∈ B, there exists 1 Y Π 0 (vw) ∈ B 0 such that and with Q 0 (1 Y ) ∈ B 1 given by (26). If moreover vw = v w − (v w ) • f is a coboundary of functions of the same kind, then Recall that σ is the first return time of f to the base Y . Writeκ n := n−1 j=0κ • f j and note thatκ σ := σ(·)−1 j=0κ • f j is the induced (to the base Y ) version ofκ. The main idea to define 1 Y Π 0 v is to take the derivative at t = 0 of left hand side of the following identity with Q t an eigenprojector of a perturbationR t of R. More precisely, while Π t is the eigenprojector of P (λ −1 t e itκ ·) associated to the eigenvalue 1, Q t is the main eigenprojector of R t = R(λ −σ(·) t e itκσ ·) associated to the eigenvalue 1 (see below for the formal definition ofR t ). Our proof of Proposition 3.3 will use several preliminary facts, that are contained in the following lemmas.
Proof. First, due to the Hölder inequality, with p ∈ (1, 2/r) and q = p/(p − 1) so that 1/p + 1/q = 1. Let s = rp/(rp − 1) so that 1/(rp) + 1/s = 1. Using the Hölder inequality for inner products, we have that for any n ≥ 1 We formally defineR t := n≥1 λ −n t R n (e itκn ·), with R n v : ). The next lemma provides some estimates on R n which we will use in the proofs below.
and ε 0 ∈ (0, 1). Then, for every γ ∈ [1, 2/q), there exist C 0 and ρ ∈ (0, 1) so that for all t small enough, all w ∈ B 1 , all n ≥ 1 and all v Y ∈ L b (µ Y ) constant on each atom of the partition α, Note that, in this lemma, 1 < γ < 2(1 − 1 b ) and that up to adapting the value of q, we can take γ as close to 2(1 − 1 b ) as we wish. Proof. By the arguments used in [17, Proof of Proposition 12.1] and exploiting that v Y and κ σ are constant on every a ∈ A, A justification of (20) based on [17, Proof of Proposition 12.1] is provided in Appendix A. We note that since σ has exponential tail, equation (20) and Hölder inequality imply (18) and (19). Next, by Lemma 3.5, for any r ∈ (1, 2),κ σ ∈ L r (µ Y ). Since γq ∈ [1, 2), using the same argument as in the proof of Lemma 3.1 combined with Hölder inequality, we have which leads to (17). Note that γq < 2 ensures thatκ γ σ ∈ L q (µ Y ). The result follows from the previous two displayed inequalities since σ has exponential tail.
Note thatR 0 = R and that (3) implies that λ 0 = d dt λ t | t=0 = 0. The next lemma shows that t →R t ∈ B 1 is differentiable at t = 0 with derivativeR 0 := R(iκ σ ·) = ∞ n=0 R n (iκ n ·), and that this is also true if we replace w ∈ B 1 by v Y w as in Lemma 3.6.
Lemma 3.7. Let b, q, γ as in Lemma 3.6. Then for every γ ∈ [1, 2/q), there exists C > 0 such that for t small enough and all w ∈ B 1 and for any v Y as in Lemma 3.6.
By Lemma 3.8, for t small enough, the eigenvalueλ t ofR t associated with the projection (1 Y Π t )v is so thatλ t = 1; this lemma tells us how the projection acts on B. Let Q t be the eigenprojection forR t associated withλ t = 1.
Since 1 is an isolated eigenvalue in the spectrum ofR 0 = R andR t is a continuous family of operators (by Lemma 3.7), we have thatλ t = 1 is isolated in the spectrum ofR t for every t small enough. Hence, there exists δ 0 > 0 so that for any δ ∈ (0, δ 0 ), , so is the derivative at t = 0 of t → Q t ∈ B 1 and write for δ > 0 small enough. Recall that Q 0 , Q 0 are well defined in B 1 . The next result shows that Q 0 h B 0 is also well defined for h = v Y w with v Y , w as in Lemma 3.6. Lemma 3.9. Let b, q, γ as in Lemma 3.6. Then there exist C 1 , C 2 > 0 such that for t small enough, for every v Y ∈ L b (µ Y ) constant on each a ∈ α and every w ∈ B 1 , Proof. The first estimate will come from our estimates of (Q t − Q 0 − tQ 0 )(v Y w) B 0 and Q 0 (v Y w) B 0 and from (25) ensuring that for some ρ ∈ (0, 1), where we used (21) combined with the spectral properties of R, up to take δ small enough so that (1 − δ) −1 ρ < 1. Second Note that Recall that for all ξ so that |ξ − 1| = δ, (ξI −R t ) −1 B 0 1, for all t small enough. This together with Lemma 3.7 with v Y = 1 Y implies that for all t small enough, Hence, To simplify notations, we write µ Y (·) for Y · dµ Y . We claim that (27) |ξ−1|=δ This implies that Proceeding as in estimating E 2 (t)(v Y w) above and using the first part of the conclusion in Lemma 3.7 and a formula analgous to (27), and thus, the second part of the conclusion follows.

SHARP ERROR TERM IN LLT AND MIXING FOR LORENTZ GASES WITH INFINITE HORIZON 19
Finally, similarly to (27), we claim that The third part of the conclusion follows by putting all the above together.
It remains to prove the claims (27), (28) and (29). Note that for any operatorP bounded in B 1 , we have In the sequel, we will takeP = Id,R 0 −R t ,P =R 0 andP =R 0 −R t − tR 0 , respectively.
SinceR 0 = R has a spectral gap in B 1 with decomposition R j = µ Y (·)1 Y + N j with N j B 0 ≤ Cρ j for some C > 0 and some ρ < 1, for every j ≥ 1, we can write for some C > 0. Thus, there exist C" > 0 and ρ < 1 so that for j ≥ 1, Below we write γ(P ) to indicate that this is a positive number that depend on the operator P ∈ {R 0 −R t ,R 0 ,R 0 −R t − tR 0 }. Using the fact thatR 0 = R and that (ξI − R) −1 = j≥0 ξ −j−1 R j and putting the above together, we obtain due to Lemma 3.7 and to (30), with γ(R 0 −R t ) = 1, γ(R 0 ) = 0, γ(R 0 −R t − tR 0 ) = γ. Therefore The claims follow by choosing δ small enough so that The next estimate, of independent interest, requires a more careful analysis and strongly exploits that the modulus is outside of the integral. Its proof uses arguments somewhat similar to the ones in [15] together with arguments exploiting symmetries on the tower ∆. Recall that p > 2.
We will make use of this choice from equation (51) onwards. Recall that (ξI

SHARP ERROR TERM IN LLT AND MIXING FOR LORENTZ GASES WITH INFINITE HORIZON 21
We set N := (γ + ε) log t/ log θ 0 .
• Let us start with the computation of I 1 (t).

3.5.
Proof of the expansion of Π t : proofs of Propositions 3.3 and 3.2. We first provide the proof of Propositions 3.3 relying on our technical Lemmas 3.9 and 3.10, and then we prove Proposition 3.2 .
Proof of Proposition 3. 3 We proceed as in [2, Proof of Lemma 3.14] using our estimates. Since 1 is a simple eigenvalue ofR t , and since Q t (1 Y ) and 1 Y Π t (vw) are both eigenfunctions belonging to B θ ofR t associated to 1, these two vectors are proportional and so, ) in C , by Lemma 3.10. Moreover, due to Lemma 3.9, Combining the above estimate with (56), we conclude that which leads to (16).

Expansion of λ t : Proof of Proposition 1.7
Let v t be the eigenfunction of P t associated with λ t so that . Using a classical argument (see, for instance, [2]), we write Estimates of V (t) and Ψ(t) are given respectively in Lemmas 4.1 and 4.2. In the above formula, Ψ(t) is the pure scalar part which will be estimated as if dealing with the characteristic function of an i.i.d. process and for this we only need to exploit Lemma 2.1. The function V (t) will be estimated via the estimates used in the proof of Proposition 3.2. We start with the latter.

Notice that
by choosing b large enough. Combined with (69), this ends the proof of (68). Second, due to Proposition 3.3 and finally, using the fact that 1 > |λ t | > e − t 2 log(1/|t|) 2 if t is small enough, by taking p < 2 close to 2 and using the fact that for |t| small enough e To complete the proof of Proposition 1.7, we still need to estimate Ψ(t) in (63). The next lemma can be viewed as an extension of the asymptotic of the pure scalar quantity in [1, Proof of Theorem 3.1] under the conclusion of Lemma 2.1 via the arguments used in [16,26]. Then, as t → 0, Ψ(t) = (L,w)∈S (1 + it · (L + nw) − e it·(L+nw) ) c L,w n 3 + O(t 2 ) , using the properties of M 0 , S. Setting a L,w,n := c L,w n 3 , A L,w,n := k≥n a L,w,k and b L,w,n := 1 + it · (L + nw) − e it·(L+nw) , using the Abel transform, we obtain and thus Proof of Proposition 1.7. We first observe that Lemma 2.1 ensures that the general conditions of Lemma 4.2 are satisfied with c L,w := and with S is the set of (w, L) ∈ (Z 2 ) 2 , with w prime and L ∈ E w (where E w is the set of L ∈ Z 2 such that L · w ≥ 0 > (L − w) · w) for which there exist (A, B) ∈ E L,w . The finiteness of S comes then from the finiteness of A and the finiteness of the possibilities for B once (A, w) is fixed, the finitess of L once A, w, B are fixed comes from our constraint on L. The disjointness assumption of Lemma 4.2 comes from our first conditions on w and L. Since 1 − λ t = Ψ(t) + V (t), due to Lemmas 4.1 and 4.2, we know that Moreover for any prime w ∈ Z 2 , due Lemma 2.1 combined with (6) ensures that Now, for a corridor C, the corresponding (A, w) ∈ A are given by (A, ±w C ) with A taken among the n C points E ∈ j∈J ∂O j such that E + Z 2 intersects the boundary of the corridor C. Therefore 5. Expansions in the local limit theorem and of decorrelation rate 5.1. Expansion in the local limit theorem in a general context. Let (∆, f µ ∆ ) be a probability preserving dynamical system with transfer operator P . Setκ n := n−1 k=0κ • f k , withκ : ∆ → Z d integrable with zero mean. Assume that, for every t ∈ [−π, π] d , the operator P t : f → P (e it·κ f ) acts on a complex Banach space B of functions f : ∆ → C and satisfies the following properties. Assume that B → L p 1 (µ ∆ ), for p 1 ∈ [1, +∞] and that there exists β ∈ (0, π) such that for every t ∈ [−β, β] d , Assume moreover that there exists an invertible positive symmetric matrix A such that .
or there exist B and p 0 ≥ 1 and γ ∈ (0, 1] such that Notice that (74) can be rewritten We set a n := √ n log n. Let q 0 ∈ [1, +∞] such that 1 q 0 + 1 p 0 ≤ 1 and q 1 ∈ [1, +∞] such that Under these assumptions, we state the following general local limit theorem with expansion, that can be read first considering k = 0. The generalization to k = O(n/ log n) will be useful in the proof of our main results (Theorems 1.1 and 1.2) due to approximations of observables φ, ψ using functions that are constant on every stable curve.
for k = 0 (without additional assumption on p 1 ) and uniformly in k ≤ Cn/ log n.

Decorrelation expansion for Z d -extensions.
We are now interested in decorrelation expansion for Z d -extensions satisfying the set up of Proposition5.1. In this subsection, we consider the Z d -extension (Ω, ν, S) of (∆, f, µ ∆ ) with step functionκ : ∆ → Z d , where Ω = ∆ × Z d , the transformation S is defined by S(x, L) = (f x, L +κ(x)) and preserves the infinite measure ν = µ ∆ ⊗ m d with m d being the counting measure on Z d . Then Suppose that (76)(instead of (75)) holds with γ > 0 and assume that there exists γ 0 > 0 such that Then Proof. Applying Proposition 5.1 to the couples (φ(·, N 1 ), ψ(·, N 2 )) with k = 0 leads to where we used the fact that for every γ ∈ (0, 2], there exists C γ such that, for every X ∈ R 2 , where we used the same argument as before with γ ∈ (1, 2] combined with the fact that I 1 and I 3 are uniformly γ 0 -Hölder with γ 0 ∈ (0, 1) and that I 1 (0) = I 3 (0) = 0.
Observe that when φ or ψ has null integral, then Corollary 5.3 only provides an upper bound (given by the term in O(·)). Nevertheless, the method we used to establish Proposition 5.1 and Corollary 5.3 enables the establishment of explicit decorrelation rates for some specific but natural null integral observables of the Z d -extension, including coboundaries. Before stating these decorrelation results, let us introduce the following notations. Set ∃δ ∈ (0, 1), then, for all integer m ≥ 1, Let us observe that item (B) of Proposition 5.4 (with h = g = 1 ∆ ) implies in particular that µ ∆ (κ n = 1) + µ ∆ (κ n = −1) − 2µ ∆ (κ n = 0) ≈ a −d−2 n , as n → +∞.

5.3.
Decorrelation for the Z d -periodic billiard map. We recall from Section 1 that the billiard map T can be represented as the Z 2 -extension of (M ,T ,μ) with step function κ :M → Z 2 . Analogously the billiard map in the domain Q/({0} × Z) can be represented as the Z-extension of (M ,T ,μ) with step function κ 1 :M → Z the first coordinate of κ. Therefore these two billiard maps can be represented as the Z d -extension of (M ,μ,T ) by a step function κ : We treat together these two models in the following. We consider the Banach spaces B 0 and B defined in Section 3.1. As recalled in Section 3.1, with these choices, (1) and (2) hold. Since B 0 is continuously embedded in B, (72) and (73) holds true with the Banach space B 0 and for any p 1 ∈ [1, +∞). Moreover (74) has been proved in Proposition 1.7 and, due to Proposition 3.2 (the fact that the symmetric matrix is invertible follows from the assumption of total dimension of the horizon), (76) holds true for B 0 for any p 0 ∈ (1, 2). Thus (75) holds also true with for any p 1 ∈ (1, 2) and any γ ∈ (0, 1]. Therefore Proposition 5.1 can be applied in this context. A first consequence to billiards of Proposition 5.1 with k = 0 is the following result. Corollary 5.5. Let q 0 > 2. If φ, ψ :M → C are observables constant on all the stable curves and such that φ (η) + ψ L q 0 (µ) < ∞, then, uniformly in N ∈ Z d , In particular, due to Remark 3.4, Analogously, Theorem 1.2 for η-Hölder observables φ, ψ that are constant along stable curves (e.g. observables that are constant on every obstacle) is a direct consequence of Corollary 5.3 which used Proposition 5.1 just with k = 0. Indeed such observables φ, ψ can be represented by observables belonging to B 0 .
Our goal here is to prove Theorems 1.1 and 1.2 for general η-Hölder observables. To this end, we will use approximations by functions on ∆ and Proposition 5.1 with k = k(n) → +∞.
Recall that we have defined ξ k k , ξ ∞ k := j≥k ξ j k and · (η) before Theorem 1.2. For any φ : M → C or φ :M → C and −∞ < k ≤ k ≤ ∞, we define the following local variation: where we write ξ k k (x) for the element of ξ k k containing x. Before stating the decorrelation result (Theorem 1.2), we start with a local limit theorem (generalizing Theorem 1.1). Set κ n := n−1 k=0 κ •T k and (k(n)) n be a sequence of integers diverging to +∞ such that k(n) = O(a n /(log n) q 1 ), for some q 1 > 2.
• Proof of (124). First, for every integers j, k j ≥ 0, we have Now using the fact that P 2k jφ (k j ) − µ ∆ (φ (k j ) ) B 0 φ ∞ (which comes from (105)) and using (2) for t = 0, we conclude that Due to (99), k ω k −k (φ, ·) L p 2 (μ) < ∞ and thus, taking k j = j/2, we obtain This ends the proof of (124). • Proof of (123). Recall that Eμ[ψ (k) − ψ] ≤ ω +∞ −k (ψ, ·) L 1 (μ) = O((log n) −1 ). Note that to prove (123) it suffices to show that Observe that, due to (125) with k j := j/2 and k = k(n), where in the last equality we have used (99). This ends the proof of (123). ). Since I 0 is bounded and due to the first item of (129) and to (132), the first error term in (136) is in O(a −d n / log n). This completes the proof of (133). Finally, we assume (134) and (135) and prove the last point of the lemma. We start from (111) for (φ N 1 ,ψ N 2 , N 2 − N 1 ) instead of (φ, ψ, N ). The fact that |I 2j (x) − I 2j (0)| x 1+γ combined with (134) ensures that we can replace I 0 (N/a n ) and I 2 (N/a n ) in J 1 up to an error (after summation over N 1 , N 2 ∈ Z 2 ) in O(a −d−1−γ n ). Moreover the first condition in (129) and (135) also ensures that we can replace φ (k) and ψ (k) by respectively φ and ψ in J 1 up to a total error (after summation) in O(a −d−1 n / log n). The fact that |I 2j+1 (x)| x γ , combined with the first condition in (129), also implies that the contribution (after summation) of J 2 is in O(a −d−1 n / log n). We prove as in the previous result that the contribution (after summation) of J 3 is in O(a −d−1 n / log n).
In view of Theorem 1.3, we focus now on observables with null expectation and state a result analogous to Proposition 5.4. We keep the notation I 1 , 2 ,N introduced just before Proposition 5.4 with A = Σ 2 , so that Φ(x) = e − 1 2 x·x (2π) d/2 and where ∇ and ∆ are the usual differential gradient and Laplacian operators.
• (one observable is a coboundary of order m) If φ, ψ : M → C are bounded observables such that (138) N : M → C given by Due to our choice of k and to our assumptions on φ, ψ, Observe that the error term in the previous formula is in O(a −d−2m n ). So we focus on the following integral M φ (k) .ψ (k) • T n dµ = Recall, from the proof of Theorem 5.6 that, due the definition of h (k) as seen at the begining of the proof of Proposition 5.1.