Renewal convergence rates and correlation decay for homogeneous pinning models

A class of discrete renewal processes with super-exponentially decaying inter-arrival distributions coincides with the infinite volume limit of general homogeneous pinning models in their localized phase. Pinning models are statistical mechanics systems to which a lot of attention has been devoted both for their relevance for applications and because they are solvable models exhibiting a non-trivial phase transition. The spatial decay of correlations in these systems is directly mapped to the speed of convergence to equilibrium for the associated renewal processes. We show that close to criticality, under general assumptions, the correlation decay rate, or the renewal convergence rate, coincides with the inter-arrival decay rate. We also show that, in general, this is false away from criticality. Under a stronger assumption on the inter-arrival distribution we establish a local limit theorem, capturing thus the sharp asymptotic behavior of correlations.


Introduction and main results
1.1. Set-up and generalities. We consider the probability discrete density K(·) concentrated on N = {1, 2, . . .}. We choose K(·) such that for some α > 0 and some function L(·) which is slowly varying at infinity we have where we are using the notation a N N →∞ ∼ b N when lim N →∞ a N /b N = 1. We assume that K(·) is aperiodic, that is gcd{n : K(n) > 0} = 1. We recall that a function L(·) defined on the positive semi-axis is slowly varying at infinity if it is positive, measurable and if lim t→∞ L(ct)/L(t) = 1 for every c > 0. We refer to [4] for the full theory of slowly varying functions, recalling simply that both L(t) and 1/L(t) are much smaller than t δ (as t → ∞), and this for any δ > 0.
We point out that (1.1) and aperiodicity are implied by Starting from K(·), we introduce a family of discrete probability densities indexed by b ≥ 0: K b (n) := c(b)K(n) exp(−bn), (1.3) and c(b) = 1/ n K(n) exp(−bn) (of course c(0) = 1). Our attention focuses on the renewal process τ (b) := {τ 0 (b), τ 1 (b), τ 2 (b), . . .} with interarrival law K b (·), that is the process defined by τ 0 (b) = 0 and by the requirement that {τ j+1 (b) − τ j (b)} j=0,1,... is a sequence of IID random variables and P(τ 1 (b) = n) = K b (n). Note that τ (b) is an increasing sequence of almost surely finite numbers and it can be looked upon equivalently as a sequence of random variables (in fact, a random walk with positive increments) or as a random subset of N ∪ {0}. With this second interpretation we introduce the so called mass renewal function, that is u b (n) := P (n ∈ τ (b)) , (1.4) so that u b (n) is the probability that the site n is visited by the renewal. Note that u b (0) = 1 and, since K(·) is aperiodic, there exists n 0 > 0 such that u b (n) > 0 for every n ≥ n 0 .
1.2. The Renewal Theorem and refinements to it. We now make an excursus in the general renewal theory on the integer numbers. We consider thus a general renewal process with τ 0 = 0 and with inter-arrival taking values in N. For this we introduce the notation F (n) := P(τ 1 = n), while the mass renewal function is denoted by u(·). The classical Renewal Theorem (see e.g. [1]) says that, if F (·) is aperiodic, we have Much effort has been put into refining such a result. Refinements are of course a very natural question when E[τ 1 ] = +∞ (e.g. [8,10]), as well as if E[τ 1 ] < +∞. In the latter case sharp estimates on u(n) − u(∞) have been obtained for sub-exponential tail decay of the inter-arrival distribution, like for example in the case of F (·) = K(·) and K(·) as in (1.2) (we refer to [13] and references therein).
When instead the inter-arrival distribution decays super-exponentially, like for example if F (·) = K b (·) with b > 0, general sharp results are harder to obtain. What can be proven in general in fact is that, if there exists c 1 > 0 such that lim n→∞ exp(c 1 n)F (n) = 0, then there exists c 2 > 0 such that lim n→∞ exp(c 2 n)|u(n) − u(∞)| = 0. However the precise decay, or even only the exponential asymptotic behavior (that is the supremum of the values of c 2 for which the previous equality holds), in general does not depend only on the tail behavior of the inter-arrival probability. This is definitely a very classical problem [15,14], and a number of results have been proven in specific instances (see e.g [3,17]). We are now going to treat this point in some detail.

1.3.
On super-exponentially decaying inter-arrival laws. From the very definition of renewal process one directly derives the equivalent expressions and with the notation f (z) = ∞ n=0 z n f (n) ( f (·) is the z-transform of f (·)) and z is a complex number. Of course f (·) is a power series and |z| a priori has to be chosen smaller than the radius of convergence, which, for the two series appearing in (1.6), is at least 1.
As a matter of fact, we are interested (in particular) in the radius of convergence of the series .
If we assume that lim sup n→∞ exp(cn)F (n) < ∞ for some c > 0, the radius of convergence of F (·) is at least exp(c), however it is not at all clear that the radius of convergence of ∆(·) coincides with the radius of convergence of F (·). In reality the problem does not come from the singularity at z = 1 ( F (1) = 1) since it is easily seen that it is removable. Notice also that, when F (·) is aperiodic, F (z) = 1 on the unit circle only if z = 1. However there may be other solutions z to F (z) = 1 for z within the radius of convergence of F (·).
The main purpose of this note is, however, to point out that, in a suitable class of renewal processes, motivated by statistical mechanics modeling (see Subsection 1.5), the tail decay of u(·) − u(∞) is closely linked with the tail decay of the F (·). We are in fact going to show that if F (·) = K b (·), that is in the set-up of § 1.1, the decay rate of {u b (n) − u b (∞)} n is equal to the decay rate of K b (·), if b is sufficiently small. And under the stronger hypothesis (1.2) we control the sharp asymptotic behavior of u b (n) − u b (∞). (1) For every choice of K(·) satisfying (1.1) we have b 0 ∈ (0, ∞] and for every b (1.9) (2) For every choice of K(·) satisfying (1.2) we have that for every b ∈ (0, b 0 ) (1.11) Remark 1.2. When there exists z 0 , 1 < |z 0 | < exp(b), such that K b (z 0 ) = 1 (therefore b > b 0 ) one can easily write down the sharp asymptotic behavior of {u b (n) − u b (∞)} n in terms of the values of z 0 with minimal |z 0 |. As a matter of fact one has lim sup but the sequence changes sign infinitely often and, in general, the superior limit cannot be replaced by a limit (see Section 4 for details). In Section 4 we also provide explicit examples showing that b 0 can be arbitrarily small by choosing K(·) suitably. In all the examples we have worked out the inequality in (1.9) is strict (for every b > b 0 ), but it is unclear to us whether or not this is a general phenomenon.
The proof of Theorem 1.1(1) can be found in Section 2 which is devoted to the study of which of course is the radius of convergence of ∆ b (·), and to establishing that b 0 is not zero. Theorem 1.1(2) follows instead by a direct application of a well established technique [7]: we detail this application in Section 3. We point out that the validity of the results in [7] go beyond the assumption (1.2), but we do make use of the regularly varying character of K(·) in establishing b 0 > 0. A closer look at the proof of b 0 > 0 however shows that when n nK(n) < ∞ (cf. (2.14)) the regular variation property is used only marginally and in fact Theorem 1.1 holds also for a number of sub-exponential (c.f. [4]) distributions K(·) beyond our assumptions. For example Theorem 1.1 holds also for K(n) = L(n)n q exp(−n γ ), with q ∈ R and γ ∈ (0, 1). .

1.5.
Homogeneous pinning models and decay of correlations. What motivated, and what even suggested the validity of the results in this note, is the behavior near criticality of homogeneous pinning models. As it as been pointed out in particular in [9], a large class of physical models boils down to a class of Gibbs measures that, in mathematical terms, are just obtained from discrete renewal processes modified by introducing an exponential weight, or Boltzmann factor, depending on N N (τ ) := |τ ∩ (0, N ]|. More precisely if P is the law of τ and the latter is the renewal sequence with inter-arrival distribution K(·), we consider the family of probability measures {P N,β } N ∈N defined by 13) with Z N,β the normalization constant. Then one can show ( [6], [11,Ch. 2]) that the weak limit P ∞,β of {P N,β } N ∈N exists for every β ∈ R (to be precise, this statement holds for every β assuming (1.2), but it holds also assuming only (1.1) if β > 0). The parameter β actually plays a crucial role. In fact if β < 0 then τ , under P ∞,β , is a transient renewal and it contains therefore only a finite number of points (this is the so-called delocalized phase). If instead β > 0 then τ , again under P ∞,β , is a positive recurrent renewal with inter-arrival distribution given by We point also out that it is not difficult to see that b coincides with the limit as N tends to infinity of (log Z N,β )/N and it is hence the free energy of the system [11,Ch. 1]. In [9] and, more completely in [11,Ch. 2], one can find the analysis of b(β) as β ց 0.
As a consequence τ (b), for b > 0, does describe the localized regime of an infinite volume statistical mechanics system: if b is small, the system is close to criticality. The correlation length is a key quantity in statistical mechanics, see e.g. [9]. Moreover it is expected to scale nicely with β (or, which is equivalent, with b) approaching criticality, typically as β to some (negative) power, possibly times logarithmic corrections. The correlation length may be defined by introducing first the correlation function: (1.14) where we have used the Renewal Theorem. Then the correlation length is just one over the decay rate ξ(b) of c(·): ξ(b) := −1/ lim sup n→∞ n −1 log |c(n)| and therefore which roughly can be rephrased by saying that the correlation length, close to criticality, scales like one over the free energy.
On physical grounds (1.16), or rather the weaker form log ξ(b) ∼ − log b, is certainly expected [9]. A proof of (1.16) has been given in [18] by coupling arguments for the case in which K(·) is given by the return times of a simple random walk (and the proof is given also for disordered models). The result actually holds as an equality for every b (like the case presented in § 4.1 below: we point out that for α = 1/2 the distribution K(·) treated in § 4.1 coincides with the distribution of the returns to zero of a simple random walk in the sense that K(n) is the probability that the first return to zero of a simple random walk happens at time 2n). In general coupling arguments yield sharp results on the rate when suitable monotonicity properties are present (see in particular [16]): the returns of a simple random walk are in this class. In absence of monotonicity properties coupling arguments usually yield only upper bounds on the speed of convergence (and hence lower bounds on the rate, see [1] and references therein): in [19] a coupling argument is given for disordered pinning models and it yields in our homogeneous set-up that lim sup bց0 log ξ(b)/ log(b) ≤ −1, under the stronger hypothesis (1.2).
We conclude this introduction with two important remarks: Some of the papers we have referred to (in particular [3,17]) aim at explicit bounds that hold for every n, possibly at the expense of sharp asymptotic results. Also in our set-up the question of obtaining more quantitative estimates, particularly when b ց 0, is important and relevant for the applications.
Remark 1.4. The class of pinning models we have considered contains the so called (1 + d)-dimensional pinning models. The name comes from the directed viewpoint on Markov chains: if one considers a Markov chain S with state space Z d , the state space of the directed process {(n, S n )} n is Z 1+d . The renewal structure in this case is simply given by the successive returns to 0 ∈ Z d by S or, equivalently, by the intersections of the directed process with the line {(n, 0) ∈ Z 1+d : n = 0, 1, 2, . . .}. This viewpoint is important in order to understand the spectrum of applications of pinning models. We are not going to discuss this further here, and we refer to [11,20], but we do point out that precise estimates catching the order of magnitude of the correlation length in a class of (d+1)-dimensional pinning models, i.e. Gaussian effective surfaces in a (d+1)-dimensional space pinned at an hyper-plane, have been obtained in [5].
2. The radius of convergence of ∆ b (·) In this section we work in the most general set-up, i.e. we assume (1.1). Recall the definition of b 0 from the statement of Theorem 1.1.
and, for every choice of K(·), b 0 > 0 and therefore Note that this result implies (1.8) and (1.9).
Proof. We are going to show that R b ≤ exp(b) by making use only of K b (exp(b)) < ∞ and of the fact that the radius of convergence of K b (·) is exp(b).
Of course we may assume that ∆ b (·) is analytic in the centered ball of radius exp(b), since otherwise there is nothing to prove. Let us suppose that ∆ b (·) has an analytic extension to the open ball of radius R > exp(b). From (1.7) we immediately derive an expression for K b (z) in terms of ∆ b (z), for |z| < exp(b), and this gives the meromorphic extension of K b (·) to the centered ball of radius R. However we know that the radius of convergence of K b (·) is exp(b) and that | K b (z)| ≤ n K(n) < ∞ if |z| = exp(b). So the singularity of K b (·) cannot be a pole and therefore K b (·) does not have a meromorphic extension. This implies that ∆ b (·) cannot be analytically continued beyond the centered ball of radius exp(b).
The question that we have to address in order to complete the proof of Proposition 2.1, that is proving b 0 > 0, can be rephrased as: do there exist two sequences {b j } j , b j ց 0 and {z j } j , 1 < |z j | ≤ exp(b j ) such that K b (z j ) = 1 for every j? Of course, if this is not the case, K b (z) = 1 if log |z|(> 0) is sufficiently small.
We make some preliminary observations: first, we may assume ℑ(z j ) ≥ 0, since if K b (z) = 1, we have K b (z) = 1 too. Then let us remark that, by writing z j = r j exp(iθ j ), we can pass to the limit in the equation K b j (z j ) = 1: by the Lebesgue Dominated Convergence Theorem we have that every limit point (1, θ) of {(r j , θ j )} j satisfies n K(n) exp(inθ) = 1, (2.1) which gives θ = 0 by aperiodicity. This tells us that, for b small, singularities have necessarily positive real part and small imaginary part (in short, they are close to 1). Moreover, by monotonicity, we see that the imaginary part cannot be zero (and therefore we assume that it is positive, since solutions come in conjugate pairs). Let us now assume by contradiction that there exists a triplet of sequences tending to zero, with the requirements that 0 ≥ δ j < b j , θ j > 0 for every j and such that K b (exp(b j − δ j ) exp(iθ j )) = 1 for every j. Of course the triplet corresponds to the poles of the associated ∆ b j (·) function at z j = exp ((b j − δ j ) + iθ j ). We are going to show that such a triplet does not exist since we are able to extract subsequences such that for every j in the subsequence. Let us consider the auxiliary sequence of non-negative numbers {δ j /θ j } j . By choosing a subsequence we may assume that this sequence converges to a limit point γ ∈ [0, ∞].
If γ < ∞ we have the asymptotic relation that follows from a Riemann sum approximation and the uniform convergence property of slowly varying functions [4, § 1.5] if the sum is restricted to θ j n ∈ (ε, 1/ε). The rest is then controlled for small n's (n ≤ ε/θ j ) by replacing sin(x) with x and using summation by parts which tells us that N n=1 nK(n) is equal to N −1 n=0 K(n) − N K(N ) and the latter behaves for large values of N as N 1−α L(N )/(1 − α) [4, § 1.5]. For large n's the rest is controlled by using | exp(−δ j n) sin(θ j n)| ≤ 1. Overall the absolute value of the rest is bounded by cθ α j L(1/θ j )(ε 1−α + ε α ) for some c > 0, with c not depending on ε, for j sufficiently large (for example, θ j < ε) and (2.4) follows.
Observe that the left-hand side of (2.4) is asymptotically equivalent to the imaginary The integral can be explicitly computed and it is equal to which is positive for every γ ∈ [0, ∞), therefore for j sufficiently large (2.3) holds (the definition of Γ(·) is recalled in Section 4).
If γ = ∞ instead we write n K(n) exp(−δ j n) sin(θ j n) = R < j + R > j , (2.6) with R < j the sum for n ≤ ε/θ j and R > j is the rest (0 < ε ≤ π/2 is a fixed positive constant). Setting s ε := sin(ε)/ε we have To obtain (2.7) we have used summation by parts, namely the identity: (2.8) On the other hand where c, c ′ are positive constants (we have explicitly used the fact that, for every c 1 > 1 and every c 2 > 0 there exists c 3 > 0 such that L(x)/L(y) ≤ c 1 (x/y) c 2 whenever x/y ≥ c 3 [4, Th. 1.5.6]). Therefore |R > j /R < j | → 0 as j → ∞ and for j sufficiently large we have 11) and then also in this regime (2.3) holds.
The marginal case of α = 1 and n nK(n) = +∞ is treated as follows.
If α ∈ [0, ∞) for the step analogous to (2.4) we split the sum according to whether θ j n ≤ ε or θ j n > ε. Summing by parts we obtain is slowly varying and that lim n→∞ L(n)/L(n) = +∞. From this we directly obtain that the first term in the splitting, i.e. the sum over θ j n ≤ ε, is bounded below by a positive constant, depending on ε and γ (this constant can be chosen bounded away from zero for γ in any compact subset of [0, ∞)) times θ j L(1/δ j ). The rest instead is bounded, in absolute value, by a constant (independent of γ) times θ j L(1/θ j ), for j sufficiently large (just use | sin(θ j n) exp(−δ j n)| ≤ 1). Using once again L(n) ≫ L(n) for large n, we obtain that n K(n) exp(−γ j n) sin(θ j n) > 0 for j sufficiently large. If instead γ = +∞ we restart from (2.6) and, by proceeding like in (2.7) and (2.9), we obtain that for j sufficiently large n K(n) exp(−δ j n) sin(θ j n) ≥ 1 2 13) which is positive for j sufficiently large and the case α = 1 and n nK(n) = ∞ is under control.
Let us now consider the case of α > 1, together with the case α = 1 and n nK(n) < ∞ and note that in the latter case L(·) vanishes at infinity.
In these cases for every γ ∈ [0, ∞] we use the splitting in (2.6) and for j sufficiently large we have 14) and the right-hand side is positive (again, for j sufficiently large). This concludes the proof of Proposition 2.1.

Sharp estimates
Throughout this section K(·) satisfies (1.2), we assume b > 0 and we set ∇u b (n) := u b (n) − u b (n − 1) for n = 0, 1, . . . (u b (−1) := 0). We also introduce the discrete probability density µ b on N ∪ {0} defined by with m b := n nK b (n) and K b (n) := j>n K b (j). Let us observe that and that this directly implies the properties We point out also that from (1.6) we get at least for |z| < 1, like for (1.7). Of course the domain of analyticity of φ b (·) is C \ {0} and if we observe that, by direct computation, we have one can then extend the validity of (3.4) to all values of z satisfying |z| ≤ exp(b) and |z| < inf{|ζ| > 1 : Proof of Theorem 1.1 (2). Let us choose b < b 0 . We observe that the two properties in (3.3) are the hypotheses (α) and (β) of [7,Theorem 1]. Hypothesis (γ) of the same theorem, that is that µ b (z) converges at its radius of convergence (exp(b)), is verified too.
the range of the power series µ b (·) is a subset of the analyticity domain of φ b (·). Therefore [7, Theorem 1] yields and by (3.2) we have We conclude by observing that this yields 8) and the proof is complete.
Remark 4.1. It is not difficult to see that one can differentiate, say j times, the expression in (4.1) generating thus sequences which decay like (log n) j /n 1+α and that, for sufficiently large n, do not change sign. This provides examples involving slowly varying functions.
Since we are just developing examples and that generalizations are straightforward, we specialize to the case of −β = α ∈ (0, 1).
4.1. The basic example. In this section we study the case of Note that ∞ n=1 K(n) = 1 follows from (4.1), with β = −α, as well as, with reference to In defining z α for α non integer, we choose the cut line {z ∈ R : z < 0}. With this choice (1 − z exp(−b)) α , and therefore K b (·), has a discontinuity on the line {z ∈ R : z > exp(b)}. We observe that, for every b > 0, K b (z) = 1 for |z| ≤ exp(b) only if z = 1, therefore Theorem 1.1 holds with b 0 = ∞.

Remark 4.2.
In the special case under consideration, but also in all the other cases considered in this section, one can obtain and go beyond Theorem 1.1 by direct computations. In fact if we set q(z) := (1 − z exp(−b)) α we have for |q(z)| < |q(1)| Now we set and we note that (q(z)) j = (1 − z exp(−b)) jα and therefore once again (4.1) provides the expansion for (q(z)) j if jα / ∈ N and the n-th term in the power series (of (q(z)) j ) behaves, as n → ∞, like c exp(−nb)n −1−jα , c = 0. Note that if jα ∈ N the arising expression is just a polynomial and hence does not contribute to the asymptotic behavior of the series expansion.
Finally, the series expansion n r (m) (n)z n of R m (·) can be controlled by observing that this function is analytic in the centered ball of radius exp(b) and by using the formula (4.8) where the contour in the middle term is (say) |z| = r, for r ∈ (0, exp(β)), and the last term is obtained by letting r ր exp(b), using the fact that R m (exp(b + iθ)) is bounded. In fact, from the explicit expression and by construction, one readily sees that R m (exp(b + iθ)) is smooth except at θ = 2πk, k ∈ Z, where it is C ⌊(m+1)α⌋ . By using the fact that n-th Fourier coefficient of a C k function is o(n −k ), we see that r (m) (n) = exp(−bn)o(1/n ⌊(m+1)α⌋ ). The chain of considerations we have just made leads to an explicit expansion to all orders for exp(bn)(u b (n) − u b (∞)) as a sum of terms of the form c j 1 ,j 2 n −j 1 −αj 2 , for suitable (explicit) real coefficients c j 1 ,j 2 (j 1 , j 2 ∈ N).

4.2.
Singularities and slower decay of correlations. From the basic example one can actually build a large number of exactly solvable cases that display the more general phenomenology hinted by Theorem 1.1: in particular that, in general, b 0 < ∞.
For example, fix m ∈ N and define Remark 4.5. In general, once the solutions to K b (·) = 1 of minimal absolute value (in the annulus {z : 1 < |z| < exp(b)) are known, it is straightforward to write the sharp asymptotic behavior of u b (n) − u b (∞). For example if z 0 is a complex solution, then also its conjugate is a solution. If these have minimal absolute value among the solutions and if they are simple solutions, for a suitable (and computable) real constants c 1 and c 2 (|c 1 | + |c 2 | > 0) we have n→∞ ∼ |z 0 | −n (c 1 cos (n arg (z 0 )) + c 2 sin (n arg (z 0 ))) . An analogous formula is easily written in the general case.
Proof of Proposition 4.4. In reality, we are going to do something rather cheap, but we are actually proving more than what is stated: we are going to show that for every b > 0 and every r ∈ (0, exp(b)) we can find an m such that there are m zeros of K b (·) − 1 in the annulus {z : 1 < |z| < r}. Given b > 0, since the only solution z to 1 − (1 − z exp(−b)) α ) = 0 is z = 0, then for every r ∈ (1, exp(b)) we have Therefore (recall (4.10)) | K b (z)| ≥ r m x r , if |z| = r. Therefore for m sufficiently large we have | K b (z)| > 1 for |z| = r: let us fix such a couple (m, r). Rouché's Theorem (e.g. [2, p. 153]) guarantees that if f and g are analytic in a simply connected domain containing the simple closed curve γ and if |f (z) − g(z)| < |f (z)| for z ∈ γ, then f and g have the same number of zeros enclosed by γ. Let us apply Rouché's Theorem with f (z) := K b (z) and g(z) := 1 − K b (z) and γ := {z : |z| = r}, so that |f (z) − g(z)| = 1 < |f (z)| for z ∈ γ, by the choice of m. But K b (·) has precisely m + 1 zeros (they are all in 0) and therefore also 1 − K b (·) has m + 1 zeros enclosed by γ. Of course 1 − K b (·) has a zero in 1 and all the other zeros have absolute value in (1, r).