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An extension of the inductive approach to the lace expansion

We extend the inductive approach to the lace expansion, previously developed to study models with critical dimension 4, to be applicable more generally. In particular, the result of this note has recently been used to prove Gaussian asymptotic behaviour for the Fourier transform of the two-point function for sufficiently spread-out lattice trees in dimensions d>8, and it is potentially also applicable to percolation in dimensions d>6.


Motivation
The lace expansion has been used since the mid-1980s to study a wide variety of problems in high-dimensional probability, statistical mechanics, and combinatorics [12]. One of the most flexible approaches to the lace expansion is the inductive method, first developed in [2] in the context of weakly self-avoiding walks in dimensions d > 4, and subsequently extended to a much more general setting in [6]. The inductive approach of [6] was successfully used to prove Gaussian asymptotic behavior for the Fourier transform of the critical two-point function c n (x; z c ) for a sufficiently spread-out model of self-avoiding walk in dimensions d > 4 [8]. Up to a constant, c n (x; z c ) is the probability that a randomly chosen n-step self-avoiding walk ends at x. Other models to which [6] applies include sufficiently spread-out models of oriented

The recursion relation
The lace expansion typically gives rise to a recursion relation for a sequence f n depending on parameters k ∈ [−π, π] d and positive z. We may assume that f 0 = 1. The recursion relation takes the form f n+1 (k; z) = n+1 m=1 g m (k; z)f n+1−m (k; z) + e n+1 (k; z), (n ≥ 0), (1) with given sequences g m (k; z) and e n+1 (k; z). The goal is to understand the behaviour of the solution f n (k; z) of (1). A rough idea of the behaviour we seek to prove can be obtained from the following (nonrigorous) argument. Suppose for simplicity that D(x) is uniformly distributed on a finite box centred at the origin (so that x D(x) = 1), that g 1 (k; 1) = D(k) ≈ 1 − |k| 2 σ 2 /(2d), and that e m , g m+1 ≈ 0 for m ≥ 1. Then we have , and thus The above argument is, however, overly simplistic, and misses important effects on the asymptotic behaviour of the solution to (1) due to the presence of e m (k; z) and g m (k; z). The inductive method of [6] details specific bounds on g m and e n+1 that ensure that there exists a critical value z c and positive constants A, v such that the true asymptotic behaviour is 2d . Verification of these bounds has been carried out for sufficiently spread-out models of self-avoiding walk [8], oriented percolation [7] and the contact process [5], by estimating certain Feynman diagrams in dimensions d > 4. The required bounds are typically of the form |h m (k, z)| ≤ Cm b− d 2 , for some functions h m and exponent b ≥ 0 that varies from bound to bound. What turns out to be important in the analysis is that d 2 = 2 + d−4 2 is greater than 2 when d > 4. In our analysis we introduce two new parameters θ(d), p * and a set B ⊂ [1, p * ]. We will discuss the significance of p * and B following Assumption D in the next section. The most important parameter, θ(d), takes the place of d 2 in exponents appearing in various bounds. As in [6] we require that θ > 2. In [10], the result of this note is applied to lattice trees with the choice θ = 2 + d−8 2 , with d > 8. In general, when the critical dimension is d c , we expect that the correct parameter value is θ = 2 + d−dc 2 , e.g., we expect that θ = 2 + d−6 2 is the appropriate choice for percolation. A detailed proof of the results in this note is available in [4], however, most of the changes to the proof in [6] simply involve replacing d 2 in [6] with θ in [4]. In this note we state the new assumptions and results explicitly, but for the sake of brevity, we present only significant changes in the proof and refer the reader to [6] when the changes are merely cosmetic. The remainder of this note is organised as follows. In Section 3 we state the Assumptions S, D, E θ , and G θ on the quantities appearing in the recursion relation, and the main theorem to be proved. In Section 4, we introduce the induction hypotheses on f n that will be used to prove the main theorem. We then discuss the necessary changes to the advancement of the induction hypotheses of [6]. Once the induction hypotheses have been advanced, the main theorem follows without difficulty.

Assumptions and main result
Suppose that for z > 0 and k ∈ [−π, π] d , we have f 0 (k; z) = 1 and that (1) holds for all n ≥ 0, where the functions g m and e m are to be regarded as given. Fix θ > 2. The first assumption, Assumption S, remains unchanged from [6]. It requires that the functions appearing in the recursion relation (1) respect the lattice symmetries of reflection and rotation, and that f n remains bounded in a weak sense.
Assumption S. For every n ∈ N and z > 0, the mapping k → f n (k; z) is symmetric under replacement of any component k i of k by −k i , and under permutations of the components of k. The same holds for e n (·; z) and g n (·; z). In addition, for each n, |f n (k; z)| is bounded uniformly in k ∈ [−π, π] d and z in a neighbourhood of 1 (both the bound and the neighbourhood may depend on n).
The next assumption, Assumption D, is only cosmetically changed from [6]. It introduces a probability mass function D = D L on Z d which defines an underlying random walk model and involves a non-negative parameter L which will typically be large. This serves to spread out the steps of the random walk over a large set. An example of a family of D's obeying the assumption is taking D uniform on a box of side 2L + 1 centred at the origin. In particular, Assumption D implies that D has a finite second moment, and we define We assume that f 1 (k; z) = zD(k) and e 1 (k; z) = 0.
In particular, this implies that g 1 (k; z) = zD(k). In addition, we also assume: (i) D is normalised so thatD(0) = 1, and has 2 + 2ǫ moments for some 0 < ǫ < θ − 2, i.e., (ii) There is a constant C such that, for all L ≥ 1, There exist constants η, c 1 , c 2 > 0 such that Assumptions E and G of [6] are adapted to general θ > 2 as follows. The relevant bounds on f m , which a priori may or may not be satisfied, are that for some p * ≥ 1 and some nonempty B ⊂ [1, p * ], we have for every p ∈ B, for some positive constant K, where the norm is defined by The bounds in (8) are identical to the ones in [6, (1.27)], except the first bound, which only appears in [6] with p = 1 and θ = d 2 . It may be that B = {p * } (i.e. B is a singleton), and then p = p * . This is the case in [10], where the choices p * = 2 and B = {2} are sufficient, as only the p = 2 case in (8) is required to estimate the diagrams arising from the lace expansion and verify the assumptions E θ , G θ which follow below. The set B allows for the possibility that in other applications a larger collection of · p norms may be required to verify the assumptions.
As in the proof of [6, Theorem 1.1], the proof of Theorem 3.1 establishes the bounds (8) for all non-negative integers m, with z in an m-dependent interval containing z c . Consequently, all bounds appearing in Assumptions E θ and G θ follow as a corollary, for z = z c and all m. Also, it follows immediately from Theorem 3.1(d) and the bounds of Assumptions E θ and G θ that Finally, we remark that it is straightforward to extend [6, Theorem 1.2] for the susceptibility to our present setting, with the assumption θ > 2 replacing d > 4. On the other hand, the proof of the local central limit theorem [6, Theorem 1.3] does require θ = d 2 , and does not extend to the more general setting considered in this paper.

Induction hypotheses and their consequences 4.1 Induction hypotheses
Theorem 3.1 is proved via induction on n, as in [6]. The induction hypotheses involve a sequence v n , which is defined exactly as in [6] as follows. We set v 0 = b 0 = 1, and for n ≥ 1 we define b n = − 1 σ 2 n m=1 ∇ 2 g m (0; z), c n = n m=1 (m − 1)g m (0; z), v n = b n 1 + c n .

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The induction hypotheses also involve several constants. Let θ > 2, and recall from (3) that ǫ < θ − 2. We fix γ, δ > 0 and λ > 2 according to Here λ replaces ρ + 2 from [6], which is merely a change of notation. We also introduce constants K 1 , . . . , K 5 , which are independent of β. We define where c is a constant determined in the proof of Lemma 4.6 below. To advance the induction, we need to assume that Here a ≫ b denotes the statement that a/b is sufficiently large. The amount by which, for instance, K 3 must exceed K 1 is independent of β, but may depend on p * , and is determined during the course of the advancement of the induction. Let z 0 = z 1 = 1, and define z n recursively by For n ≥ 1, we define intervals In particular this gives Recall the definition a(k) = 1 −D(k). Our induction hypotheses are that the following four statements hold for all z ∈ I n and all 1 ≤ j ≤ n.
(H3) For k such that a(k) ≤ γj −1 log j, f j (k; z) can be written in the form with r i (k) = r i (k; z) obeying (H4) For k such that a(k) > γj −1 log j, f j (k; z) obeys the bounds Note that these four statements are those of [6] with the replacement in (H4) and the global replacement By global replacement we also mean that d−2 → θ − 2, etc. whenever such quantities appear in exponents.

Initialisation of the induction
The verification that the induction hypotheses hold for n = 0 remains unchanged from the p = 1 case, up to the replacements (13-14).

Consequences of induction hypotheses
The key result of this section is that the induction hypotheses imply (8) for all 1 ≤ m ≤ n, from which the bounds of Assumptions E θ and G θ then follow, for 2 ≤ m ≤ n + 1. Throughout this note: • C denotes a strictly positive constant that may depend on d, γ, δ, λ, but not on the K i , k, n, and not on β (which must, however, be chosen sufficiently small, possibly depending on the K i ). The value of C may change from one occurrence to the next.
the only difference being that the constant 2 appears here instead of a constant 1 in [6, (2.19)]. This does not affect the proof in [6]. To verify (15), we use the fact that 1 1−x ≤ 1 + 2x for 0 ≤ x ≤ 1 2 and note that for small enough β it follows from [6, (2.20)] that Here we have used the bounds of (H2-H3) as well as the fact that θ − 1 > δ.
Lemma 4.3. Let z ∈ I n and assume (H2-H3) for 1 ≤ j ≤ n. Then for k with a(k) ≤ γj −1 log j, The middle bound of (8) follows, for 1 ≤ m ≤ n and z ∈ I m , directly from Lemma 4.3. We next state two lemmas which provide the other two bounds of (8). The first concerns the · p norms and contains the most significant changes to [6]. As such we present the full proof of this lemma.
Lemma 4.4. Let z ∈ I n and assume (H2), (H3) and (H4). Then for all 1 ≤ j ≤ n, and p ≥ 1, where the constant C may depend on p, d.
Proof. We show that For j = 1 the result holds since |f 1 (k)| = |z D(k)| ≤ z ≤ 2, and, since p ≥ 1, it therefore follows from (4) and the Parseval relation that D 2 f 1 (·; z) p p ≤ 2 p D 2p 1 ≤ 2 p D 2 1 = 2 p D 2 2 ≤ 2 p CL −d . We may therefore assume that j ≥ 2 where needed in what follows, so that in particular log j ≥ log 2. Fix z ∈ I n and 1 ≤ j ≤ n, and define The set R 2 is empty if j is sufficiently large. Then We will treat each of the four terms on the right side separately.
On R 1 , we use (5) in conjunction with Lemma 4.3 and the fact thatD(k) 2 ≤ 1, to obtain for all p > 0, Here we have used the substitution k ′ i = Lk i √ pj. On R 2 , we use Lemma 4.3 and (6) to conclude that for all p > 0, there is an α(p) > 1 such that where |R 2 | denotes the volume of R 2 . For j ≥ 2, j −1 log j takes its largest value when j = 3, so |R 2 | is maximal when j = 3 and On R 3 and R 4 , we use (H4). As a result, the contribution from these two regions is bounded above by We first consider R 3 , where we applyD(k) 2 ≤ 1. Recall that we can restrict our attention to j ≥ 2. From (5), k ∈ R 3 implies that L 2 |k| 2 > Cj −1 log j, and we have the upper bound For d > 2λp, we have an upper bound on (16) of For d = 2λp, (16) is and θp = θd 2λ > d 2 since λ < θ. This gives an upper bound in this case of CK p 4 j − d 2 L −d . Lastly, for d < 2λp, since λ < θ, (16) is bounded, as required, by
This corresponds to [6, (3.62)], and is used to advance the first and second bounds of (H4). Once the induction has been advanced, the proof of Theorem 3.1 is then completed exactly as in [6], with the global replacement (14). Full details can be found in [4].