EXOTIC LOCAL LIMIT THEOREMS AT THE PHASE TRANSITION IN FREE PRODUCTS

. We construct random walks on free products of the form Z 3 ∗ Z d , with d = 5 or 6 which are divergent and not spectrally positive recurrent. We then derive a local limit theorem for these random walks, proving that µ ∗ n ( e ) ∼ CR − n n − 5 / 3 if d = 5 and µ ∗ n ( e ) ∼ CR − n n − 3 / 2 log( n ) − 1 / 2 if d = 6 , where µ ∗ n is the n th convolution power of µ and R is the inverse of the spectral radius of µ . This disproves a result of Candellero and Gilch [7] and a result of the authors of this paper that was stated in a ﬁrst version of [11]. This also shows that the classiﬁcation of local limit theorems on free products of the form Z d 1 ∗ Z d 2 or more generally on relatively hyperbolic groups with respect to virtually abelian subgroups is incomplete.

Let us consider the random walk (X n ) n driven by µ, defined by X n = g 1 ...g n where g k are independent random variables whose distribution are given by µ.Then, µ * n is the nth step distribution of the random walk, so for all x ∈ Γ, µ * n (x) is the probability that X n = x.We will also always assume that the random walk is admissible, i.e. for every x ∈ Γ, there exists n such that µ * n (x) is positive.In other words, every element of the group can be visited with positive probability, i.e. the support of µ generates Γ as a semi-group (hence as a group, since µ is symmetric).We also say that the measure µ is admissible.
We denote by ρ the spectral radius of the random walk defined by ρ = lim sup µ * n (x) 1/n .
The local limit problem consists in finding the asymptotic behavior of µ * n (x) as n goes to infinity.We assume for simplicity that µ is aperiodic, i.e. there exists n 0 such that for every n ≥ n 0 , µ * n (e) > 0, where e is the identity element of Γ.In many cases, the asymptotics arising in local limit theorems are of the form (1.1) where R is the inverse of the spectral radius.This is for example the case in all abelian groups of rank d, with α = d/2, see [24,Theorem 13.12] and references therein, and more generally in all nilpotent groups of homogeneous dimension D, with α = D/2, see [2,Corollary 1.17].This is also the case in all hyperbolic groups with α = 3/2, see [13], [18] for the case of trees and [14] for the general case.Finally, to our knowledge, this was also the case so far in all known examples of relatively hyperbolic groups.
In the context of free products of the form Γ = Z d1 * Z d2 , Candellero and Gilch [7] gave a complete classification of every possible local limit theorem.In particular, they proved that they always are of the form (1.1), with α = 3/2 or α = d i /2 and the latter case can only happen if d i ≥ 5.Although in this paper we will not work in the general setting of relatively hyperbolic groups, let us mention that free products are the simplest examples of such groups and results of [7] are being generalized to this setting in recent works by the authors, see [8], [9], [11].
Our main goal in this note is to disprove [7,Lemma 4.5] and a similar statement that appeared in a first version of [11].In particular, we prove that the classification obtained in [7] is incomplete: we derive a local limit theorem on Z 3 * Z 5 of the form (1.1) but with unexpected exponent α = 5/3, and a local limit theorem on Z 3 * Z 6 which is not of the form (1.1).Before stating our main results, let us introduce some terminology.
We consider the Green function G(x, y|r) defined by G(x, y|r) = n≥0 µ * n (x −1 y)r n .
If x = y = e, we will often write G(e, e|r) = G(r).Its radius of convergence R is independent of x and y, provided µ is admissible and it is the inverse of the spectral radius of µ.All the groups under consideration in this paper will be non-amenable.Consequently, • by a landmark result of Kesten [17], R > 1 (see also [24,Corollary 12.5]), • by a result of Guivarc'h [16], G(R) is finite (see also [24,Theorem 7.8]).
Following the notations of [8], we define The sums I (k) are related to the kth derivatives of the Green function.Precisely, by [15, Proposition 1.9],I (1) (r) = rG ′ (r) + G(r) and similar formulae hold for higher derivatives.Following [8], we say that the random walk driven by µ is divergent if We say that it is convergent otherwise.
Assume from now on that Γ = Γ 1 * Γ 2 .We define for i = 1, 2 and we set (1.4) 2 (r).Still following [8], we say that the random walk driven by µ is spectrally positive recurrent if it is divergent and if J (2) (R) is finite.
For i = 1, 2, we also consider the first return kernel p Γi,r to Γ i associated with rµ (see (2.2) for a proper definition).Then, p Γi,r defines a transition kernel on Γ i and we denote by R i (r) the inverse of its spectral radius.We say that the random walk driven by µ is spectrally degenerate along Γ i if R i (R) = 1.When both R 1 (R) and R 2 (R) are bigger than 1, we say that the random walk is spectrally non-degenerate.
Roughly speaking, when the random walk is spectrally degenerate along Γ i , the free factor Γ i has strong influence on its asymptotic behavior; we refer to [9] and [10] for further details.This notion should be compared with what is called "typical case" in [24], where another way of measuring influence of a free factor is given.By [9, Proposition 2.9], these two notions coincide, i.e. the "typical case" corresponds to the case of a spectrally non-degenerate random walk.
All these quantities and definitions can be generalized to the context of relatively hyperbolic groups, replacing free factors with the appropriate notion of maximal parabolic subgroups.The current classification of local limit theorems on relatively hyperbolic groups is as follows.When the random walk is spectrally non-degenerate, the local limit has the form (1.1), with α = 3/2 [9].This was first proved by Woess [23] for random walks on free products in the "typical case" situation.When the random walk is spectrally positive recurrent, we can only prove the rough estimate µ * n (e) ≍ R −n n −3/2 , which means that the ratio of the quantities on the left and right hand-side is bounded away from 0 and infinity [8].When the random walk is convergent and parabolic subgroups are virtually abelian, the local limit theorem has the form (1.1), with α = d/2, where d is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate [11].Moreover, in this situation, one can only have d ≥ 5.
Thus, we recover so far the classification given in [7] and presented above.Furthermore, up to the present paper, for free products of the form Z d1 * Z d2 the case of a divergent and not spectrally positive recurrent random walk was considered as not being able to occur, see [7,Lemma 4.5].As announced, we disprove here this result and we actually construct such a random walk on Γ = Z 3 * Z d , with d = 5 or 6.As a consequence, the classification of possible behaviors of µ * n needs to be completed.We also derive a local limit theorem for the random walk we construct.This is the first step into this program.
When either µ 1 or µ 2 is aperiodic, the same property holds for µ α .From now on, in order to simplify the argument, we assume that both measures µ 1 and µ 2 are aperiodic; this allows us to avoid to consider several sub-cases for the estimation of the Green functions associated with the corresponding random walks on Γ 1 and Γ 2 , see (4.1) and (4.9) below.
Theorem 1.2.Assume that the measures µ 1 and µ 2 are aperiodic.Then, the random walk on Γ driven by µ α * given by Theorem 1.1 satisfies the following local limit theorem: If d = 5, we have and if d = 6, we have where R is the inverse of the spectral radius of µ α * .
Without assuming aperiodicity, the same asymptotics hold for µ * 2n , since µ is symmetric, so its period must be 1 or 2.
Let us state that to our knowledge, the asymptotic for d = 6 in Theorem 1.2 gives the first example of a local limit theorem on a non-amenable group which is not of the form (1.1).For amenable groups, the situation is quite different and there exist many examples where µ * n (e) behaves like exp(−n c ).
Up to a sub-exponential error term, this is the case for all polycyclic groups of exponential growth [21], [1, Theorem 1] and for amenable Baumslag-Solitar groups [20, Theorem 5.2 (5.2)], with c = 1/3.This is also the case for lamplighter groups of the form A ≀ Z d , where A is a finite non-trivial group [20, Theorem 5.2 (5.6)], with c = d/(d + 2).Note that amenable Baumslag-Solitar groups and lamplighter groups are examples of solvable non-polycyclic groups.
For d = 1, a precise local limit theorem for the lamplighter group of the form µ * n (e) ∼ Cn 1/6 exp(−n 1/3 ) was proven by Revelle [19].This was further extended to Diestel-Leader graphs DL(q, r) by Bartholdi, Neuhauser and Woess, see [4, Theorem 5.4] and [3,Corollary 5.26].Diestel-Leader graphs are not amenable when q = r, since the spectral radius of the simple random walk is smaller than 1, see [4, (1.3)].Thus, the examples of [4] and [3] already provide local limit theorems which are not of the form (1.1) but of the form µ * n (e) ∼ CR −n exp(−n c )n α for nonamenable graphs.However, according to [12,Theorem 1.4] when q = r, DL(q, r) is not quasi-isometric to the Cayley graph of a finitely generated group.
We also refer to [6] where many other examples are given, beyond the class of amenable groups.Asymptotics are only given there for − log µ * n (e) though.Thus, for non-amenable groups, these examples only recover the fact that R > 1.
We now briefly outline the content of our paper.In Section 2, we give various characterizations of spectral degeneracy in terms of quantities that are suited to the study of random walks on free products.Along the way, we introduce functions and quantities defined in [24].The conclusion of this section is a useful characterization of spectral degeneracy and divergence in terms of the sign of a single quantity, see precisely Corollary 2.5.
In Section 3, we use Corollary 2.5 to prove Theorem 1.1, that is, we construct a probability measure µ on Z 3 * Z d , d = 5 or 6, which is divergent but not spectrally positive recurrent.We will actually construct a family of probability measure µ α and exhibit a phase transition at some α * .The measure µ α * will have the required properties.
Finally, Section 4 is devoted to derive a local limit theorem for µ α * , thus proving Theorem 1.2.This is done by first finding precise asymptotics of the derivative of the Green function G r (e, e) as r → R and then using Karamata's Tauberian theorem.Most of the intermediate results in this section are of geometric nature and we believe it should be possible to extend them to relatively hyperbolic groups, with (possibly challenging) new arguments replacing those that rely on the combinatorial structure of free products.

Characterizations of spectral degeneracy in free products
Let Γ = Γ 1 * Γ 2 be a free product of two groups.Consider finitely supported, symmetric and admissible probability measures µ 1 and µ 2 on Γ 1 and Γ 2 respectively.For α ∈ [0, 1], set In the sequel, we write µ for µ α and we set α 1 = α and α 2 = (1 − α).If α i > 0, the probability measure µ is finitely supported, symmetric and admissible on Γ.Such a probability measure is called adapted to the free product structure.We denote by R the inverse of the spectral radius of µ and by R i the inverse of the spectral radius of µ i .
The Green functions G, G 1 and G 2 of µ, µ 1 and µ 2 respectively are related as follows.For i = 1, 2, for every x, y ∈ Γ i , for every r ≤ R, where ζ i is a continuous function of r, see [24,Proposition 9.18] for an explicit formula.We always have We denote by p Γi,r the first return transition kernel to Γ i associated with rµ, which is defined as We denote by G Γi,r the Green function associated with p Γi,r .By [10, Lemma 4.4], for every x, y ∈ Γ i , it holds which is actually the main reason for introducing p Γi,r .
In fact, because µ is adapted to the free product structure, if the random walk ever leaves Γ i at some point x, it can only come back to Γ i at the same point x.We deduce that the first return kernel p Γi,r can be written in our context as where w i = w i (r) is the weight of the first return to e associated to rµ, starting with a step driven by α j µ j , j = i.Thus, [24,Lemma 9.2] shows that for any x, y In particular, for t = 1, Recall that following [10], we say that the random walk is spectrally degenerate along Γ i if the spectral radius of the first return kernel p Γi,R is 1.In this section, we prove equivalent conditions to spectral degeneracy, using the more standard terminology for free products introduced in [24, Chapter 9].
The following characterization is proved in [9, Proposition 2.9].We detail it here for convenience.
Lemma 2.1.The random walk is spectrally degenerate along Γ i if and only if Proof.By applying (2.4) with t = 1 + ǫ and r = R, we get The condition ǫ > 0 yields αiR(1+ǫ) is finite, which concludes the proof.
In [24], the situation where ζ i (R) < R i for i = 1, 2 is called the "typical case".Thus, Lemma 2.1 shows that this typical case corresponds to being spectrally nondegenerate.
The following statement gives a characterization of spectral degeneracy in terms of θ and θ Lemma 2.3.The random walk is spectrally degenerate along Γ i if and only if and so the random walk is spectrally degenerate along Γ i by Lemma 2.1.
Conversely, if the random walk is spectrally degenerate along Γ i , then Combining this with the inequality θ ≤ θ, we finally obtain θ = θ.
Following [24, Chapter 9], let us introduce two functions Φ and Ψ which are very useful in the context of free products.
On the one hand, he function Φ is defined implicitly by the formula for every r ≤ R.This function is defined in general on an open neighborhood (inside the complex plane) of the interval [0, θ).Since G(R) is finite, it is also defined on [0, θ].

Lemma 2.4. The random walk is spectrally degenerate if and only if
Proof.This statement is a consequence of [24,Theorem 9.22].
Let us conclude this section by summarizing the situation as follows.

A divergent not spectrally positive recurrent random walk
In this section, we construct an adapted random walk on Γ = Z 3 * Z d , d = 5 or 6, which is divergent but not spectrally positive recurrent.Such a random walk is necessarily spectrally degenerate and corresponds to the second case in Corollary 2.5.

Erroneous Lemma 3.2 (Alternative version of Erroneous Lemma 3.1). Assume the random walk driven by µ is spectrally degenerate along
A more general statement also appeared in a first version of [11], which led the authors to modify their statement.Define for i = 1, 2, By (2.1), the quantity J (2) defined in (1.4) can be written as ) must be finite by (3.4), we deduce that (3.9) Thus, this lemma is a special case of the following wrong statement that appeared in a first version of [11].
Erroneous Lemma 3.3 (Generalized version of Erroneous Lemma 3.1).In the context of relatively hyperbolic groups with respect to virtually abelian subgroups, if G ′ (R) is infinite, then J (2) (R) is finite, i.e. the random walk is spectrally positive recurrent.
We now set Γ = Z d1 * Z d2 and we consider symmetric admissible and finitely supported probability measures µ i on , see for instance [24,Theorem 13.12].Now we choose d 1 and d 2 in such a way that These three conditions, together with the fact that R 1 = R 2 = 1 impose that d 1 = 3 or 4 and d 2 = 5 or 6.From now on, we set d 1 = 3 and we write d = d 2 ∈ {5, 6}.In terms of the functions Φ i and Ψ i , it holds 1).Thus by continuity, there exists α * ∈ (0, α c ) such that Ψ(θ) = 0 when α = α * .This yields for this value α * of the parameter In other words, the random walk driven by µ α * is spectrally degenerate along is finite.The assumptions of Erroneous Lemma 3.1 are hence satisfied, so it would imply that Φ ′′ (θ) is finite, so Φ ′′ 2 (θ 2 ) is finite by (3.6).This is a contradiction, so we disproved Erroneous Lemma 3.1.
Notice that the probability measure µ α * satisfies the following properties.
(1) the random walk is spectrally degenerate along Γ 2 = Z d , (2) the random walk is not spectrally degenerate along Γ 1 = Z 3 , (3) Ψ(θ) = 0, hence the random walk driven by 2) is infinite by (3.9) and the random walk driven by µ α * is not spectrally positive recurrent.If we assume that µ 1 or µ 2 is aperiodic, i.e. µ * n 1 (e) or µ * n 2 (e) is positive for large enough n, then µ α is also aperiodic for every α.This can be obtained for instance assuming that µ 1 (e) and µ 2 (e) are positive, i.e. by considering lazy random walks on the free factors.This ends the proof of Theorem 1.1.
We thus exhibited a phase transition at α = α * , where the sign of Ψ(θ) changes, so does the behavior of the random walk by Corollary 2.5.Moreover, the following holds.
• when Ψ(θ) > 0, the random walk is convergent, hence spectrally degenerate.By [10, Proposition 6.1], it cannot be spectrally degenerate along Z 3 .In this case, it holds by [11,Theorem 1.3] As claimed in the introduction, at the phase transition α = α * , the local limit theorem has an again different form.This is the purpose of Section 4.

3.3.
Identifying the mistakes in Erroneous Lemmas.The mistake in the former version of [11] when proving Erroneous Lemma 3.3 was to assume that the spectral radius ρ H,r of the first return transition kernel p H,r defined in (2.2) were differentiable at r = R.However, this differentiability property is only proved for convergent random walks.The issue in [7] is more subtle.The authors write ζ i (r) = ζ i (R) + X i (r) and first find a linear system of the form , where LP i is a linear polynomial function.Then, they derive a contradiction from this linear system, using the assumptions of Erroneous Lemma 3.1.On Page 19 of [7], they expand (ζ i (R) + X i (r)) n and then switch two sums to identify the coefficients C (i) j , see precisely [7, (4.8)].However, switching sums is not legitimate, because the coefficients in front of X j (r) kj X i (r) ki involve successive derivatives of the Green function G j at ζ j (R) and these successive derivatives can be infinite.This is typically the case when assuming that Φ ′′ (θ) is infinite and In any case, in both [7] and [11], the spotted invalid arguments are only related to the proofs of Erroneous Lemma 3.1 and Erroneous Lemma 3.3 and do not affect the remainder of the papers.

Local limit theorems
We consider from now on the adapted probability measure µ α * on Z 3 * Z d , with d = 5 or 6.The random walk driven by µ α * is spectrally degenerate along Z d , divergent, but not spectrally positive recurrent.Now that α is fixed, we write µ = µ α * for simplicity.
For simplicity, we assume that µ 1 and µ 2 are aperiodic, i.e. µ * n 1 (e) and µ * n 2 (e) are positive for large enough n, so that µ is also aperiodic.Our goal is to prove Theorem 1.2.4.1.Asymptotic differential equations.By (3.1) and (3.3), the two quantities I (1) (r) and I (2) (r) are related to the first and second derivatives of the Green function G. Similarly, by (3.7) and (3.8), the quantity J (2) is related to the second derivatives of the Green functions G i , i = 1, 2. One of the main results in [8] in the context of relatively hyperbolic group is the following rough formula that links the quantities I (2) , I (1) and J (2) : which means that the ratio of these two quantities is bounded from above and below.
In the context of adapted measures on free products, the above rough estimates ≍ can be improved to the more accurate asymptotics ∼ as follows.
Proposition 4.1.Consider an adapted probability measure µ α on Γ = Γ 1 * Γ 2 , with 0 < α < 1 and assume that G ′ (R) = ∞.Then, there exist constants C, c 1 , c 2 and C ′ such that the following holds.As r → R, we have In particular, if Proof.On the one hand, by (3.5) it holds The term (rG ′ (r)+G(r)) 3 converges to 0 as r tends to R and G(r) converges to G(R) which is finite.Thus, On the other hand, by (2.7) and Lemma 2.2 ).Therefore, (3.5) applied this time to the Green functions G i yields This concludes the proof.
Proof.We write r 2 I i (r).
By combining Proposition 4.1 and Lemma 4.2, we get the following statement.
We will also use the following result later on.Lemma 4.4.We have Everything is now settled to prove Theorem 1.2.We treat separately the odd and even cases.4.2.The case d = 5.We consider the adapted probability measure µ α constructed in Section 3 and we set α = α * and write µ = µ α .Recall that the measures µ 1 and µ 2 are assumed to be symmetric, admissible, aperiodic and finitely supported on Γ 1 = Z 3 and on Γ 2 = Z 5 respectively.In particular, R 1 = R 2 = 1 by [24, Corollary 8.15] and • the random walk is not spectrally degenerate along Γ 1 , so Moreover, the function G ′′ 2 (t) has the following asymptotic expansion at 1 : By applying Corollary 4.3, there exists C 2 > 0 such that as r → R, 2 (r) .
Integrating this asymptotic differential equation between r and R and using the fact that G Indeed, For every ǫ > 0, there exists r 0 such that if r ≥ r 0 , we have 1 This follows from the classical local limit theorem µ * n 2 (e) ∼ Cn −5/2 given for instance by [24, Theorem 13.12] and from Karamata's Tauberian theorem [5,Corollary 1.7.3].See also [24,Proposition 17.16] where the singular expansion at 1 of the Green function is given for simple random walks on Z d .

and so
.

4.3.
The case d = 6.The proof for d = 6 is very similar.We still have that G ′′ 2 (ρ)dρ.
This time, using (4.9) yields ).We deduce from (4.3) that I  Integrating between r and R, we have Re-injecting this in (4.10), we deduce that (4.12) .
Thus, by integration between r and R,  It remains to derive from (4.14) the asymptotic behavior of µ * n (e).In the previous case when d = 5, the estimation (4.8) allowed us to apply directly [15,Theorem 9.1], whose proof is based on a version of Karamata's Tauberian Theorem given in [5].Due to the presence of the factor log(R − r), which does not appear in [15], we need to detail the proof.
We introduce the power series This concludes the proof of the case d = 6 in Theorem 1.2.
e|r) = n≥0 P((first return time of the µ-random walk to e) = n)r n and U i (r) = U i (e, e|r) = n≥0 P((first return time of the µ i -random walk to e) = n)r n .By [24, Lemma 1.13 (a)], G(r)(1 − U (r)) = G i (r)(1 − U i (r)) = 1.Following [24, Proposition 9.18 (b)], the weight w i may be written as w i = U (r) − H i (r), where H i satisfies the equation

3. 1 .
Several erroneous lemmas.We first restate [7, Lemma 4.5] (switching the indices 1 and 2) and then explains how it leads to a contradiction.This contradiction is what alerted us in the first place.The flaw in the argument is quite subtle and we will come back to it in Section 3.3.Erroneous Lemma 3.1.[7, Lemma 4.5] Assume that θ = θ = θ 2 /α 2 and that , y|r)G(y, x|r)G(x, e|r) + r x∈Γi y∈Γ G(e, x|r)G(x, y|r)G(y, e|r).