New York Journal of Mathematics

New York J. Math. 8 (2002) 31–61.

Homogenization of Random Walk in Asymmetric Random Environment

Joseph G. Conlon

Abstract. In this paper, the author investigates the scaling limit of a partial difference equation on the d dimensional integer lattice Z^{d}, corresponding to a translation invariant random walk perturbed by a random vector field. In the case when the translation invariant walk scales to a Cauchy process he proves convergence to an effective equation on R^{d}. The effective equation corresponds to a Cauchy process perturbed by a constant vector field. In the case when the translation invariant walk scales to Brownian motion he shows that the scaling limit, if it exists, depends on dimension. For d = 1, 2 he provides evidence that the scaling limit cannot be diffusion.

Contents

1. Introduction 31 2. Proof of Theorem 1.1 35 3. Proof of Theorem 1.2 46 4. Proof of Theorem 1.3 51 5. Proof of Theorem 1.4 54 References 60

1. Introduction

In this paper we shall be concerned with homogenization of a nondivergence form elliptic equation. Let (Ω, F, μ) be a probability space and b : Ω → R^{d} be a bounded measurable function b(ω) = (b_{1}(ω), …, b_{d}(ω)) with

|b(ω)|^{2} = ∑_{i=1}^{d} |b_{i}(ω)|^{2} ≤ 1, ω ∈ Ω. (1.1)

We assume that Z^{d} acts on Ω by translation operators τ_{x} : Ω → Ω, x ∈ Z^{d}, which are measure preserving ergodic and satisfy the properties τ_{x}τ_{y} = τ_{x+y}, x, y ∈ Z^{d}, τ_{0} = identity. Using these operators we can define a measurable function b : Z^{d} × Ω → R^{d} by b(x, ω) = b(τ_{x}ω), x ∈ Z^{d}, ω ∈ Ω.

Received October 4, 2001. Mathematics Subject Classification. 35R60, 60H30, 60J60. Key words and phrases. random walk, homogenization, random environment.

31^{ISSN 1076-9803/02}

For ε > 0 let Z_{ε}^{d} = εZ^{d} be the ε scaled integer lattice. We define the integral of a function g : Z_{ε}^{d} → R by

∫_{Z_{ε}^{d}} g(x)dx =^{def.} ε^{d} ∑_{x∈Z_{ε}^{d}} g(x).

The space L^{2}(Z_{ε}^{d}) is then the space of square integrable functions u : Z_{ε}^{d} → C with norm, ∥u∥_{ε} satisfying

∥u∥_{ε}^{2} = ∫_{Z_{ε}^{d}} |u(x)|^{2}dx.

For i = 1, …, d, let e_{i} ∈ Z^{d} be the element with entry 1 in the ith position and 0 in the other positions. Suppose γ ∈ C and f : R^{d} → R is a C^{∞} function with compact support. Let p : Z^{d} → R be a probability density, whence

p(x) ≥ 0, x ∈ Z^{d}, ∑_{x∈Z^{d}} p(x) = 1.

With any such p we can associate a translation invariant operator A_{p} on functions u_{ε} : Z_{ε}^{d} → R by

A_{p}u_{ε}(x) = u_{ε}(x) − ∑_{y∈Z^{d}} p(y)u_{ε}(x + εy).

For some probability densities p, the Markov chain with generator A_{p} scales in the large time limit to a Levy process [20]. If A_{p} is such a generator, we denote the index of the corresponding symmetric stable process by |A_{p}|. If p is given by

p(±e_{i}) = 1/2d, i = 1, …, d, (1.2)

then |A_{p}| = 2 since the corresponding Levy process is Brownian motion. We also define a p which scales to a Cauchy process, whence |A_{p}| = 1. To do this consider the standard nearest neighbor random walk on Z^{d+1} started at the point (0, −1), 0 ∈ Z^{d}. Then

(1.3)  p(x) = probability the walk first hits the hyperplane
{(y, 0) : y ∈ Z^{d}} in the point (x, 0), x ∈ Z^{d}.

We shall be interested in solutions to the equation

A_{p}u_{ε}(x, ω) − γ ∑_{i=1}^{d} b_{i} (\frac{x}{ε}, ω) [u_{ε}(x + εe_{i}, ω) − u_{ε}(x − εe_{i}, ω)] + ε^{|A_{p}|}u_{ε}(x, ω) (1.4)
= ε^{|A_{p}|}f(x), x ∈ Z_{ε}^{d} , ω ∈ Ω,

when p is given by either (1.2) or (1.3). For a function g : X → C where X = R^{d} or X = Z_{ε}^{d}, let ĝ be its Fourier transform,

ĝ(ξ) = ∫_{X} g(x)e^{ix·ξ}dx, ξ ∈ R^{d}.

Note that when X = Z_{ε}^{d}, then ĝ(ξ) is periodic in ξ whence we can restrict ξ to the cube [−π/ε, π/ε]^{d}. Suppose now b ≡ 0 in (1.4) and p is given by (1.2). Then it is easy to see that (1.4) has a unique solution u_{ε}(x) in L^{2}(Z_{ε}^{d}). Further, if u(x), x ∈ R^{d}, is the function which satisfies

[1 + |ξ|^{2}/2d]û(ξ) = f̂(ξ), ξ ∈ R^{d},

then lim_{ε→0} ∥u_{ε} − u∥_{ε} = 0. Similarly if b ≡ 0 and p is given by (1.3) then (1.4) has a unique solution u_{ε}(x) in L^{2}(Z_{ε}^{d}). We show in §2 that if u(x), x ∈ R^{d}, is the function which satisfies

[1 + |ξ|]û(ξ) = f̂(ξ), ξ ∈ R^{d}, (1.5)

then

lim_{ε→0} ∥u_{ε} − u∥_{ε} = 0. (1.6)

We can also prove a corresponding theorem when b ≢ 0 in the case p is given by (1.3). In the following the expectation value of a random variable X on Ω is denoted by ⟨X⟩.

Theorem 1.1. For 0 < ε ≤ 1, there exists a constant γ_{d} > 0, depending only on d, such that if γ ∈ C satisfies |γ| < γ_{d} then (1.4) has a unique solution u_{ε}(·, ω) ∈ L^{2}(Z_{ε}^{d}), ω ∈ Ω. There exists a vector q_{γ} ∈ C^{d}, |q_{γ}| < 1, analytic in γ, such that if u(x), x ∈ R^{d}, is the function which satisfies

[1 + |ξ| + iq_{γ} · ξ]û(ξ) = f̂(ξ), ξ ∈ R^{d}, (1.7)

then there is the limit

lim_{ε→0} ⟨∥u_{ε}( , ·) − u∥_{ε}^{2}⟩ = 0. (1.8)

Theorem 1.1 is a homogenization result for p given by (1.3). It says that the scaling limit of the random walk, with transition probability (1.3), in a random environment described by b(x, ω), is a Cauchy process with a constant drift q_{γ}. Corresponding results for uniformly elliptic partial differential equations in divergence form were first proved by Kozlov [11] and later by Papanicolaou and Varadhan [16]. In this case the scaling limit is Brownian motion. One can also prove a Brownian motion scaling limit for parabolic equations in divergence form when the coefficients are random in time as well as space [13]. It is possible to relax somewhat the uniform ellipticity assumption and still prove the scaling limit [5]. For a comprehensive survey of homogenization results in partial differential equations see the book of Oleinik at al [25]. Here we obtain the scaling limit of an asymmetric partial difference equation. The scaling limit of a divergence form partial difference equation has been obtained by Künnemann [12] and of a symmetric partial difference equation in non-divergence form by Lawler [14].

Evidently q_{γ} is in general nonzero even in the case ⟨b(·)⟩ = 0. Nevertheless it is possible to impose some general conditions on b(·) to ensure q_{γ} = 0. We say that the vector field b(·) is reflection invariant if it has the property that

(1.9) ⟨∏_{i=1}^{n} b_{k_{i}}(τ_{x_{i}}·)⟩ = (−1)^{n} ⟨∏_{i=1}^{n} b_{k_{i}}(τ_{−x_{i}}·)⟩ ,

x_{i} ∈ Z^{d}, 1 ≤ k_{i} ≤ d, i = 1, … n, n ≥ 1.

Clearly (1.9) implies that ⟨b(·)⟩ = 0. We show at the end of §2 that if b(·) satisfies (1.9) then q_{γ} = 0 in (1.7).

Suppose now (1.9) holds. Let u_{ε}(x) = ⟨u_{ε}(x, ·)⟩, x ∈ Z_{ε}^{d}, where u_{ε}(x, ω) is the unique solution to (1.4) given by Theorem 1.1. Suppose p̂ : [−π, π]^{d} → R is the Fourier transform of the function p of (1.3) and f̂_{ε} the Fourier transform of the function f restricted to the lattice Z_{ε}^{d}. If û_{ε}(ξ) is the Fourier transform of u_{ε}(x), then one can see that there is a d × d matrix q_{γ,ε}(ζ), ζ ∈ [−π, π]^{d}, continuous in ζ and analytic in γ, such that

û_{ε}(ξ)[1 + ε^{−1}{1 − p̂(εξ)} − ε^{−1}e(εξ)q_{γ,ε}(εξ)e(−εξ)] = f̂_{ε}(ξ) , ξ ∈ [−π/ε, π/ε], (1.10)

where e(ζ), ζ ∈ R^{d}, is the vector e(ζ) = (e_{1}(ζ), ⋯, e_{d}(ζ)), with e_{k}(ζ) = 1 − e^{ie_{k}·ζ}.

Theorem 1.2. Suppose the variables b(τ_{x}·), x ∈ Z^{d}, are independent with ⟨b(·)⟩ = 0 and q_{γ,ε}(ζ) is defined by (1.10) for |γ| < γ_{d}, where γ_{d} is the constant of Theorem 1.1. Then there exists a positive constant γ′_{d} < γ_{d}, depending only on d, such that if |γ| < γ′_{d} the matrix q_{γ,ε}(ζ) converges uniformly in ζ as ε → 0 to a matrix q_{γ}(ζ), ζ ∈ [−π, π]^{d}.

Theorem 1.2 is a considerably deeper theorem than Theorem 1.1. One has to use the Calderon-Zygmund theorem [8, 22] that the Hilbert transform is bounded on L^{p} spaces for p ≠ 2. In contrast, Theorem 1.1 can be proved by just using L^{2} theory.

Next we consider the situation when p is given by (1.2) and b ≢ 0. We then have the following theorem:

Theorem 1.3. For 0 < ε ≤ 1 and γ ∈ C satisfying |γ| < ε/\sqrt{2d}, equation (1.4) has a unique solution u_{ε}(·, ω) ∈ L^{2}(Z_{ε}^{d}), ω ∈ Ω. If (1.9) holds then there is a d × d matrix q_{γ,ε}(ζ), ζ ∈ [−π, π]^{d}, continuous in ζ and analytic in γ, such that the Fourier transform û_{ε}(ξ) of ⟨u_{ε}(x, ·)⟩ satisfies

û_{ε}(ξ)[1 + ε^{−2}{1 − p̂(εξ)} − ε^{−2}e(εξ)q_{γ,ε}(εξ)e(−εξ)] = f̂_{ε}(ξ) , ξ ∈ [−π/ε, π/ε]. (1.11)

Note that (1.11) is simply the analogue for p given by (1.2) of (1.10) for p given by (1.3). Since q_{γ,ε}(ζ) is analytic in γ, we can write it in a series expansion,

q_{γ,ε}(ζ) = ∑_{m=2}^{∞} γ^{m}q_{m,ε}(ζ), (1.12)

which converges for |γ| < ε/\sqrt{2d}. In §5 we investigate the limiting behavior of q_{m,ε} as ε → 0. We restrict ourselves to the situation when the b(τ_{x}·), x ∈ Z^{d}, are given by independent Bernoulli variables. Thus we assume that b_{j}(·) ≡ 0, j > 1, and b_{1}(τ_{x}·) = Y_{x}, x ∈ Z^{d}, where the variables Y_{x}, x ∈ Z^{d}, are assumed to be i.i.d. Bernoulli, Y_{x} = ±1 with equal probability. In that case we have the following:

Theorem 1.4. For m = 2, 3, ⋯ the d × d matrix q_{m,ε}(ζ) has the properties:

(a) q_{m,ε}(ζ) ≡ 0 if m is odd or m = 2.

(b) The entries of the matrix q_{m,ε}(ζ) = [q_{m,ε,k,k′}(ζ)], 1 ≤ k, k′ ≤ d, are zero if k, k′ > 1.

(c) The entries of the matrix q_{m,ε}(0) = [q_{m,ε,k,k′}(0)] are zero if k + k′ > 2.

(d) Assume d = 1 and K ⊂ R is a compact subset. Then ε^{m/2}q_{m,ε}(εξ) converges uniformly for ξ ∈ K to a function q_{m}(ξ), as ε → 0. The function q_{4}(ξ) is given by the formula,

q_{4}(ξ) = \frac{32}{18 + ξ^{2}} . (1.13)

(e) If d = 2 then q_{4,ε,1,1}(0) converges to +∞ as ε → 0.

(f) If d ≥ 3 then q_{4,ε}(ζ) converges uniformly for ζ ∈ [−π, π]^{d} to a matrix q_{4}(ζ) = [q_{4,k,k′}(ζ)] satisfying q_{4,1,1}(0) > 0, as ε → 0.

Unlike in the previous theorems, the results of Theorem 1.4 depend on the dimension d. For d = 1 the theorem suggests that if we set γ = γ′\sqrt{ε} in (1.4), fix γ′and let ε → 0, then we should obtain a scaling limit. Note however that one expects the resulting series obtained from (1.12) by letting ε → 0 to be at most asymptotic in γ′, not analytic. The limit here is related to the limits obtained by Sinai [21] and Kesten [9] for one dimensional random walk in random environment. For d = 2 the theorem suggests that if we fix γ in (1.4) and let ε → 0 no scaling limit exists. This appears to contradict a conjecture of Fisher [6] that two dimensional random walk in random environment has a diffusive scaling limit. In contrast Derrida and Luck [4] conjecture a nondiffusive scaling limit in two dimensions. A numerical study of the two dimensional equation (1.4) was made in [2] using a multigrid algorithm. It was observed that the algorithm gave considerable acceleration over pure iterative methods such as Gauss-Seidel. This suggests that there is some stability in the Fourier space behavior of the solution to (1.4) as ε → 0. For d ≥ 3 the theorem suggests a diffusive scaling limit for d dimensional random walk in random environment and that the effective diffusion constant is smaller than for the zero noise case.

There has been some recent work [17, 23, 24] on (1.4) with p given by (1.2) and under the assumption ⟨b(·)⟩ ≠ 0. This situation is very different to the situation studied in Theorem 1.4 since one expects now the drift to dominate diffusion. The methods used in [23, 24] are related to methods used to prove Anderson localisation for the random Schrodinger equation.

In this paper we shall adapt a method developed in [3] to prove the homogenization results for (1.4). The method consists of space translation in Ω followed by Fourier transformation in Z^{d}. The space translation in Ω is similar in spirit to “viewing the medium from an observer sitting on a tagged particle”, as described in [10]. The main advantage here is that one obtains the coefficients of the effective homogenized equation from the zero Fourier mode equation in the new variables. The method also avoids use of Martingales, as for example occurs in the recent paper [15].

2. Proof of Theorem 1.1

We first prove (1.6). Thus we consider (1.4) with b ≡ 0 and p given by (1.3). To obtain an expression for p, let us denote a point in Z^{d+1} by (x, y) with x ∈ Z^{d}, y ∈ Z. For n = 0, 1, 2, … let G(x, y, n) be the probability density of the standard random walk in Z^{d+1}, started at the origin, after n steps. Thus

(2.1)  ∑_{x∈Z^{d}} ∑_{y∈Z} G(x, y, n)e^{i[x·ξ+yζ]} = \{\frac{1}{(d + 1)} [∑_{i=1}^{d} cos(e_{i} · ξ) + cos ζ]\}^{n} .

By the reflection principle, the probability that the d + 1 dimensional walk, started at (0, −1) first hits the hyperplane {(y, 0) : y ∈ Z^{d}} after n + 1 steps at the point (x, 0) is given by

\frac{1}{2(d + 1)} [G(x, 0, n) − G(x, 2, n)] .

Hence we have

p(x) = \frac{1}{2(d + 1)} ∑_{n=0}^{∞} [G(x, 0, n) − G(x, 2, n)] . (2.2)

We can compute the Fourier transform of p by using (2.1). Thus

p̂(ξ) = ∑_{x∈Z^{d}} p(x)e^{ix·ξ} = \frac{1}{4π(d + 1)} ∫_{−π}^{π} dζ \frac{1 − cos 2ζ}{1 − [∑_{i=1}^{d} cos(e_{i} · ξ) + cos ζ] /(d + 1)}. (2.3)

Lemma 2.1. The limit (1.6) holds.

Proof. We rewrite the formula of (2.3) for p̂(ξ) as

(2.4)  p̂(ξ) = 1 − \frac{1}{2π} ∫_{−π}^{π} dζ \frac{(1 + cos ζ) ∑_{i=1}^{d}[1 − cos(e_{i} · ξ)]}{[(1 − cos ζ) + ∑_{i=1}^{d}[1 − cos(e_{i} · ξ)]]} ,

whence, on performing the integration with respect to ζ, we obtain the formula,

(2.5) 1 − p̂(ξ) = \{[1 + \frac{2}{∑_{i=1}^{d}[1 − cos(e_{i} · ξ)]}]^{1/2} − 1\} ∑_{i=1}^{d}[1 − cos(e_{i} · ξ)] .

It follows from (2.3), (2.4) that 0 < p̂(ξ) ≤ 1. We can also easily see from (2.5) that

lim_{ε→0} ε^{−1}[1 − p̂(εξ)] = |ξ|, ξ ∈ R^{d}. (2.6)

Consider now (1.4) with b ≡ 0. Taking the Fourier transform of (1.4), we see that (1.4) is equivalent to the equation,

[1 + ε − p̂(εξ)]û_{ε}(ξ) = εf̂_{ε}(ξ), ξ ∈ [\frac{−π}{ε} , \frac{π}{ε} ]^{d} , (2.7)

where f̂_{ε} is the Fourier transform of f regarded as a function on Z_{ε}^{d}. Since f is C^{∞} of compact support it is easy to see that (2.7) is uniquely solvable in L^{2} ([\frac{−π}{ε} , \frac{π}{ε} ]^{d}) and

û_{ε}(ξ) = f̂_{ε}(ξ)/[1 + ε^{−1} \{1 − p̂(εξ)\}] , ξ ∈ [\frac{−π}{ε} , \frac{π}{ε} ]^{d} . (2.8)

Let û(ξ) be the function given in (1.5). We decompose its Fourier inverse u(x), x ∈ R^{d}, as a sum, u(x) = v_{ε}(x) + w_{ε}(x), where

v_{ε}(x) = \frac{1}{(2π)^{d}} ∫_{[ \frac{−π}{ε} , \frac{π}{ε}]^{d}} û(ξ)e^{−ix·ξ}dξ.

Since f̂(ξ) is rapidly decreasing in ξ it follows that lim_{ε→0} ∥w_{ε}∥_{ε} = 0. We can see this from the identity

∥w_{ε}∥_{ε}^{2} = \frac{1}{(2π)^{d}} ∫_{[ \frac{−π}{ε} , \frac{π}{ε}]^{d}} | ∑_{n∈Z^{d}\\{0\}} û(ξ + \frac{2πn}{ε})|^{2} dξ.

Thus we have

lim_{ε→0} ∥u_{ε} − u∥_{ε} = lim_{ε→0} ∥u_{ε} − v_{ε}∥_{ε},
∥u_{ε} − v_{ε}∥_{ε}^{2} = \frac{1}{(2π)^{d}} ∫_{[ \frac{−π}{ε} , \frac{π}{ε}]^{d}} |û_{ε}(ξ) − û(ξ)|^{2}dξ,

where û_{ε}(ξ) is given by (2.8) and û(ξ) by (1.5). The result follows from (2.6) and the dominated convergence theorem. □

Next we wish to prove existence of a solution to (1.4) provided γ is sufficiently small, depending only on d. To do this we need to define the Green’s function G_{p,η}(x), x ∈ Z^{d}, associated with p. Thus for η > 0, G_{p,η}(x) is the solution to the equation

A_{p} G_{p,η}(x) + η G_{p,η}(x) = δ(x), x ∈ Z^{d}, (2.9)

where δ is the Kronecker δ function, δ(0) = 1, δ(x) = 0, x ≠ 0. The Fourier transform Ĝ_{p,η}(ξ) of G_{p,η}(x) is given by the formula

Ĝ_{p,η}(ξ) = [1 + η − p̂(ξ)]^{−1} , ξ ∈ [−π, π]^{d}. (2.10)

It is easy to see from (2.5) that

0 ≤ Ĝ_{p,η}(ξ) ≤ C/[η + |ξ|] ,

for some universal constant C. We can also obtain a corresponding decay rate on the function G_{p,η}(x).

Lemma 2.2. Let G_{p,η}(x), x ∈ Z^{d}, be the solution to (2.9), and 0 < η ≤ 1. Then there is a constant C_{d}, depending only on d such that if d ≥ 2,

0 ≤ G_{p,η}(x) ≤ C_{d}/ [1 + η^{2}|x|^{2}] [1 + |x|^{d−1}] , x ∈ Z^{d}. (2.11)

Proof. Consider the random walk on Z^{d} with transition probability p given by (2.2). Let p_{n}(x), n = 0, 1, 2, … be the probability that the walk, started at the origin hits x after n steps. Thus p_{0}(x) = δ(x) and p_{1}(x) = p(x), x ∈ Z^{d}. It is easy to see that the solution of (2.9) is given by the formula,

G_{p,η}(x) = ∑_{m=0}^{∞}(1 + η)^{−m−1}p_{m}(x), x ∈ Z^{d}. (2.12)

We can derive a formula for p_{m}(x), m ≥ 2, similar to the formula (2.2) for p(x). In fact we have

p_{m}(x) = \frac{1}{2(d + 1)} ∑_{n=0}^{∞}[G(x, m − 1, n) − G(x, m + 1, n)]. (2.13)

It is easy to see from this and (2.12) that

G_{p,0}(x) = δ(x) + \frac{1}{2(d + 1)} ∑_{n=0}^{∞}[G(x, 0, n) + G(x, 1, n)]. (2.14)

If we use now the bound

0 ≤ G(x, y, n) ≤ \frac{C_{d}}{1 + n^{(d+1)/2}} exp [− min\{|x| + |y|, (|x|^{2} + |y|^{2})/(n + 1)\}/C_{d}] ,

for the standard random walk on Z^{d+1}, then the inequality (2.11) for η = 0 follows from (2.14). To obtain (2.11) for η > 0 we need to use the inequality,

0 ≤ G(x, y − 1, n) − G(x, y + 1, n) ≤

\frac{C_{d}y}{1 + n^{(d+3)/2}} exp [− min\{|x| + |y|, (|x|^{2} + |y|^{2})/(n + 1)\}/C_{d}] ,

x ∈ Z^{d}, y ∈ Z y ≥ 0.

It follows from this and (2.13) that

0 ≤ p_{m}(x) ≤ \frac{C_{d}m}{1 + [|x|^{2} + m^{2}]^{(d+1)/2}} , m ≥ 1.

The inequality (2.11) follows now from this last inequality by estimating the sum (2.12). □

Lemma 2.3. Suppose d ≥ 2 and f : R^{d} → R is a C^{∞} function with compact support. Then there exists a constant γ_{d}, depending only on d such that if |γ| < γ_{d} and 0 < ε ≤ 1, then (1.4) has a unique solution u_{ε}(x, ω) in L^{2}(Z_{ε}^{d}). Further, there is a constant C_{d}, depending only on d, such that ∥u_{ε}(·, ω)∥_{ε} ≤ C_{d}∥f∥_{ε}.

Proof. Suppose u_{ε}(x, ω) is in L^{2}(Z_{ε}^{d}) and satisfies (1.4). Let v_{ε}(x, ω) be defined by

v_{ε}(x, ω) = ε^{−1}[A_{p}u_{ε}(x, ω) + εu_{ε}(x, ω)].

It is easy to see that v_{ε}(x, ω) ∈ L^{2}(Z_{ε}^{d}). In view of (2.9) it follows that

u_{ε}(x, ω) = ε^{1−d} ∫_{Z_{ε}^{d}} dy G_{p,ε} (\frac{x − y}{ε} ) v_{ε}(y, ω). (2.15)

Now for i = 1, …, d let us define operators T_{i} on L^{2}(Z_{ε}^{d}) by

T_{i}w(x) = ε^{−d} ∫_{Z_{ε}^{d}} dy [G_{p,ε} (\frac{x − y}{ε} + e_{i}) − G_{p,ε} (\frac{x − y}{ε} − e_{i})] w(y) .

It is easy to see that T_{i} is a bounded operator on L^{2}(Z_{ε}^{d}). It follows from (2.10) that the norm of T_{i} is bounded by

∥T_{i}∥ ≤ sup_{ξ∈[−π,π]^{d}} [\frac{2| sin(e_{i} · ξ)|}{1 + ε − p̂(ξ)}] . (2.16)

We can see from (2.5) that there is a constant c_{d}, 0 < c_{d} < 1, depending only on d such that

1 − p̂(ξ) ≥ c_{d}|ξ|, ξ ∈ [−π, π]^{d}. (2.17)

It follows from this and (2.16) that ∥T_{i}∥ ≤ 2/c_{d}. Observe now that (1.4) implies that v_{ε}(x, ω) ∈ L^{2}(Z_{ε}^{d}) satisfies the equation

v_{ε}(x, ω) − γ ∑_{i=1}^{d} b_{i}(x/ε, ω)T_{i}v_{ε}(x, ω) = f(x). (2.18)

In view of (1.1), if we take γ_{d} = c_{d}/2 it follows that for |γ| < γ_{d} the function v_{ε}(x, ω) is uniquely determined by (2.18). In view of (2.15) it follows that u_{ε}(x, ω) ∈ L^{2}(Z_{ε}^{d}) is the unique solution to (1.4). To prove existence note that (2.18) has a solution v_{ε}(x, ω) ∈ L^{2}(Z_{ε}^{d}) provided |γ| < γ_{d}. If we then define u_{ε}(x, ω) by (2.15) one can easily see that u_{ε}(x, ω) satisfies (1.4). □

We have proven existence of a unique solution u_{ε}(x, ω) to (1.4) provided |γ| < γ_{d}. Next we follow a development similar to the one in [3]. Thus we put v_{ε}(x, ω) = u_{ε}(x, τ_{−x/ε}ω), x ∈ Z_{ε}^{d}, ω ∈ Ω, whence u_{ε}(x, ω) = v_{ε}(x, τ_{x/ε}ω). Rewriting (1.4) in terms of v_{ε}, we obtain the equation,

v_{ε}(x, τ_{x/ε}ω) − ∑_{y∈Z^{d}} p(y)v_{ε}(x + εy, τ_{y}τ_{x/ε}ω)

− γ ∑_{i=1}^{d} b_{i}(τ_{x/ε}ω)[v_{ε}(x + εe_{i}, τ_{e_{i}}τ_{x/ε}ω)

− v_{ε}(x − εe_{i}, τ_{−e_{i}}τ_{x/ε}ω)] + εv_{ε}(x, τ_{x/ε}ω) = εf(x), x ∈ Z_{ε}^{d}, ω ∈ Ω.

This last equation is evidently the same as

(2.19)  v_{ε}(x, ω) − ∑_{y∈Z^{d}} p(y)v_{ε}(x + εy, τ_{y}ω)

− γ ∑_{i=1}^{d} b_{i}(ω)[v_{ε}(x + εe_{i}, τ_{e_{i}}ω) − v_{ε}(x − εe_{i}, τ_{−e_{i}}ω)]

+ εv_{ε}(x, ω) = εf(x), x ∈ Z_{ε}^{d}, ω ∈ Ω.

We write now v_{ε}(x, ω) = u_{ε}(x) + ψ_{ε}(x, ω), x ∈ Z_{ε}^{d}, ω ∈ Ω, where ⟨ψ_{ε}(x, ·)⟩ = 0, x ∈ Z_{ε}^{d}. It follows that

u_{ε}(x) = ⟨v_{ε}(x, ·)⟩ = ⟨u_{ε}(x, ·)⟩, x ∈ Z_{ε}^{d}.

Substituting into (2.19) we obtain the equation

(2.20)  u_{ε}(x) − ∑_{y∈Z^{d}} p(y)u_{ε}(x + εy)

− γ ∑_{i=1}^{d} b_{i}(ω)[u_{ε}(x + εe_{i}) − u_{ε}(x − εe_{i})] + εu_{ε}(x)

+ ψ_{ε}(x, ω) − ∑_{y∈Z^{d}} p(y)ψ_{ε}(x + εy, τ_{y}ω)

− γ ∑_{i=1}^{d} b_{i}(ω) [ψ_{ε}(x + εe_{i}, τ_{e_{i}}ω) − ψ_{ε}(x − εe_{i}, τ_{−e_{i}}ω)]

+ εψ_{ε}(x, ω) = εf(x), x ∈ Z_{ε}^{d}, ω ∈ Ω.

Taking the expected value of this last equation, we obtain the equation,

(2.21)  A_{p}u_{ε}(x) − γ ∑_{i=1}^{d}⟨b_{i}(·)⟩[u_{ε}(x + εe_{i}) − u_{ε}(x − εe_{i})] + εu_{ε}(x)

− γ ⟨∑_{i=1}^{d} b_{i}(·) [ψ_{ε}(x + εe_{i}, τ_{e_{i}}·) − ψ_{ε}(x − εe_{i}, τ_{−e_{i}}·)]⟩

= εf(x), x ∈ Z_{ε}^{d}.

If we subtract now (2.21) from (2.20) we obtain

ψ_{ε}(x, ω) − ∑_{y∈Z^{d}} p(y)ψ_{ε}(x + εy, τ_{y}ω) (2.22)

− γP ∑_{i+1}^{d} b_{i}(ω)[ψ_{ε}(x + εe_{i}, τ_{e_{i}}ω) − ψ_{ε}(x − εe_{i}, τ_{−e_{i}}ω)] + εψ_{ε}(x, ω)

= γ ∑_{i=1}^{d}[b_{i}(ω) − ⟨b_{i}(·)⟩][u_{ε}(x + εe_{i}) − u_{ε}(x − εe_{i})], x ∈ Z_{ε}^{d}, ω ∈ Ω,

where P : L^{2}(Ω) → L^{2}(Ω) is the projection orthogonal to the constant function. Thus a solution u_{ε}(x, ω) of (1.4) which is in L^{2}(Z_{ε}^{d} × Ω) yields functions u_{ε}(x) in L^{2}(Z_{ε}^{d}) and ψ_{ε}(x, ω) in L^{2}(Z_{ε}^{d} × Ω) which satisfy ⟨ψ_{ε}(x, ·)⟩ = 0, x ∈ Z_{ε}^{d}, and the equations (2.21), (2.22). Conversely, if we can find functions u_{ε}(x) in L^{2}(Z_{ε}^{d}) and ψ_{ε}(x, ω) in L^{2}(Z_{ε}^{d} × Ω) which have the property that ⟨ψ_{ε}(x, ·)⟩ = 0, x ∈ Z_{ε}^{d}, and satisfy (2.21), (2.22) then we can construct from them the solution u_{ε}(x, ω) of (1.4) and u_{ε}(x, ω) is in L^{2}(Z_{ε}^{d} × Ω).

We concentrate now on finding solutions to the system (2.21), (2.22) of equations. To do this we Fourier transform the equations. The Fourier transform of (2.21) is given by

(2.23)  û_{ε}(ξ) \{1 + ε^{−1} [1 − p̂(εξ)] + 2γi ∑_{j=1}^{d}⟨b_{j}(·)⟩\frac{sin(e_{j} · εξ)}{ε}\}

+ γε^{−1} ⟨∑_{j=1}^{d} b_{j}(·) [e^{iεe_{j}·ξ}ψ̂_{ε}(ξ, τ_{−e_{j}}·) − e^{−iεe_{j}·ξ}ψ̂_{ε}(ξ, τ_{e_{j}}·)]⟩

= f̂_{ε}(ξ), ξ ∈ [\frac{−π}{ε} , \frac{π}{ε} ]^{d} .

The Fourier transform of (2.22) is given by

(2.24) ψ̂_{ε}(ξ, ω) − ∑_{y∈Z^{d}} p(y)e^{−iεy·ξ}ψ̂_{ε}(ξ, τ_{y}ω)

+ γP ∑_{j=1}^{d} b_{j}(ω) [e^{iεe_{j}·ξ}ψ̂_{ε}(ξ, τ_{−e_{j}}ω) − e^{−iεe_{j}·ξ}ψ̂_{ε}(ξ, τ_{e_{j}}ω)] + εψ̂_{ε}(ξ, ω)

= γ ∑_{j=1}^{d} [b_{j}(ω) − ⟨b_{j}(·)⟩] [e^{−iεe_{j}·ξ} − e^{iεe_{j}·ξ}] û_{ε}(ξ), ξ ∈ [\frac{−π}{ε} , \frac{π}{ε} ]^{d} , ω ∈ Ω.

Suppose now that Ψ_{ε}^{k}(ζ, ω), 1 ≤ k ≤ d, ζ ∈ [−π, π]^{d}, ω ∈ Ω, are functions in L^{2}([−π, π]^{d} × Ω) which satisfy the equation,

Ψ_{ε}^{k}(ζ, ω) − ∑_{y∈Z^{d}} p(y)e^{−iy·ζ}Ψ_{ε}^{k}(ζ, τ_{y}ω) (2.25)

+ γP ∑_{j=1}^{d} b_{j}(ω) [e^{ie_{j}·ζ}Ψ_{ε}^{k}(ζ, τ_{−e_{j}}ω) − e^{−ie_{j}·ζ}Ψ_{ε}^{k}(ζ, τ_{e_{j}}ω)] + εΨ_{ε}^{k}(ζ, ω)

= b_{k}(ω) − ⟨b_{k}(·)⟩, ζ ∈ [−π, π]^{d}, ω ∈ Ω, 1 ≤ k ≤ d.

Then it is clear that the function

ψ̂_{ε}(ξ, ω) = −2iγû_{ε}(ξ) ∑_{k=1}^{d} sin(εe_{k} · ξ)Ψ_{ε}^{k}(εξ, ω) (2.26)

is in L^{2}([\frac{−π}{ε} , \frac{π}{ε} ]^{d} × Ω) provided û_{ε}(ξ) is bounded. Furthermore ψ̂_{ε}(ξ, ω) given by (2.25), (2.26) satisfies (2.24). If we substitute for ψ̂_{ε} from (2.26) into (2.23) then we obtain the equation,

û_{ε}(ξ)\{1 + ε^{−1} [1 − p̂(εξ)] + 2γi ∑_{j=1}^{d}⟨b_{j}(·)⟩\frac{sin(εe_{j} · ξ)}{ε} (2.27)

− 2iγ^{2}⟨ ∑_{j,k=1}^{d} b_{j}(·)\frac{sin(εe_{k} · ξ)}{ε}[e^{iεe_{j}·ξ}Ψ_{ε}^{k}(εξ, τ_{−e_{j}}·) − e^{−iεe_{j}·ξ}Ψ_{ε}^{k}(εξ, τ_{e_{j}}·)]⟩\}

= f̂_{ε}(ξ), ξ ∈ [\frac{−π}{ε} , \frac{π}{ε} ]^{d} .

Lemma 2.4. Suppose d ≥ 2 and γ_{d} = c_{d}/2, where c_{d} satisfies (2.17). Then if |γ| < γ_{d} and 0 < ε ≤ 1, Equation (2.25) has a unique solution Ψ_{ε}^{k}(ζ, ω) ∈ L^{2}(Ω). Furthermore, Ψ_{ε}^{k}(ζ, ω) viewed as a function from [−π, π]^{d} to L^{2}(Ω) is continuous and satisfies an inequality,

ε^{2} sup_{ζ∈[−π,π]^{d}} ⟨|Ψ_{ε}^{k}(ζ, ·)|^{2}⟩ ≤ C_{d}, (2.28)

where C_{d} depends only on d.

Proof. We follow an argument similar to that in Lemma 2.3. Suppose Ψ_{ε}^{k}(ζ, ω) ∈ L^{2}(Ω) and satisfies (2.25). Let Φ_{ε}^{k}(ζ, ω) be defined by

Φ_{ε}^{k}(ζ, ω) = Ψ_{ε}^{k}(ζ, ω) − ∑_{y∈Z^{d}} p(y)e^{−iy·ζ}Ψ_{ε}^{k}(ζ, τ_{y}ω) + εΨ_{ε}^{k}(ζ, ω). (2.29)

It is easy to see that Φ_{ε}^{k}(ζ, ω) ∈ L^{2}(Ω). Further, it follows from (2.9) that

Ψ_{ε}^{k}(ζ, ω) = ∑_{x∈Z^{d}} G_{p,ε}(x)e^{−ix·ζ}Φ_{ε}^{k}(ζ, τ_{x}ω). (2.30)

Now for η > 0, ζ ∈ [−π, π]^{d}, j = 1, …, d we define operators T_{j,η,ζ} on L^{2}(Ω) by

T_{j,η,ζ}φ(ω) = ∑_{x∈Z^{d}} [G_{p,η}(x − e_{j}) − G_{p,η}(x + e_{j})] e^{−ix·ζ}φ(τ_{x}ω). (2.31)

We can see from Bochner’s theorem [19] that T_{j,η,ζ} is bounded on L^{2}(Ω) and

∥T_{j,η,ζ}∥ ≤ sup_{ζ′∈[−π,π]^{d}} \frac{2| sin(e_{j} · ζ′)|}{1 + η − p̂(ζ′)} ≤ \frac{2}{c_{d}} , (2.32)

where the constant c_{d} is from (2.17). Further, (2.25), (2.29), (2.30) imply that Φ_{ε}^{k}(ζ, ω) satisfies the equation,

Φ_{ε}^{k}(ζ, ω) − γP ∑_{j=1}^{d} b_{j}(ω)T_{j,ε,ζ}Φ_{ε}^{k}(ζ, ω) = b_{k}(ω) − ⟨b_{k}(·)⟩. (2.33)

Hence if we take γ_{d} = c_{d}/2, it follows that for |γ| < γ_{d} the function Φ_{ε}^{k}(ζ, ω) is uniquely determined by Equation (2.33). In view of (2.30) it follows that Ψ_{ε}^{k}(ζ, ω) is the unique solution of (2.25) in L^{2}(Ω). We can similarly prove existence for |γ| < γ_{d}. The inequality of (2.28) follows from (2.30) on using Bochner’s theorem and the fact that Φ_{ε}^{k}(ζ, ·) ∈ L^{2}(Ω) is bounded independently of ζ, ε. □

It is clear now that we can construct the solutions to (2.21), (2.22), by using Lemma 2.4 and Equations (2.26) and (2.27). Next we wish to show that the limit of the coefficient of û_{ε}(ξ) in (2.27) as ε → 0 exists. Evidently

lim_{ε→0} \{1 + ε^{−1}[1 − p̂(εξ)] + 2γi ∑_{j=1}^{d}⟨b_{j}(·)⟩\frac{sin(εe_{j} · ξ)}{ε}\}

= 1 + |ξ| + 2γi ∑_{j=1}^{d}⟨b_{j}(·)⟩(e_{j} · ξ),

so we are left to show existence of the term involving the function Ψ_{ε}^{k}. To prove this first observe from (2.30), (2.31) that

e^{−ie_{j}·ζ}Ψ_{ε}^{k}(ζ, τ_{e_{j}}ω) − e^{ie_{j}·ζ}Ψ_{ε}^{k}(ζ, τ_{−e_{j}}ω) = T_{j,ε,ζ}Φ_{ε}^{k}(ζ, ω).

Next we prove a version of the von Neumann ergodic theorem [19].

Lemma 2.5. Let φ ∈ L^{2}(Ω) and ξ ∈ R^{d}. Then if ⟨φ⟩ = 0 the function T_{j,ε,εξ}φ converges in L^{2}(Ω) to the function T_{j,0,0}φ.

Proof. First note that we can define the operator T_{j,0,ζ}, ζ ∈ [−π, π]^{d}, by (2.31) and from (2.32) it is a bounded operator on L^{2}(Ω). To be more precise, observe that for η > 0, T_{j,η,ζ}φ = 0 if φ ∈ L^{2}(Ω) satisfies ∂_{j,ζ}φ = 0, where ∂_{j,ζ} is the operator on L^{2}(Ω) given by

∂_{j,ζ}φ(ω) = e^{−ie_{j}·ζ}φ(τ_{e_{j}}ω) − e^{ie_{j}·ζ}φ(τ_{−e_{j}}ω).

Observe that the adjoint ∂_{j,ζ}^{∗} of ∂_{j,ζ} is given by ∂_{j,ζ}^{∗} = −∂_{j,ζ}. Suppose now φ = ∂_{j,ζ}ψ where ψ ∈ L^{2}(Ω). Then from (2.31) we have that

T_{j,η,ζ}φ(ω) = ∑_{x∈Z^{d}} [G_{p,η}(x + 2e_{j}) + G_{p,η}(x − 2e_{j}) − 2G_{p,η}(x)] e^{−ix·ζ}ψ(τ_{x}ω). (2.34)

We can readily see, using the argument of Lemma 2.2 that there is a constant C_{d}, depending only on d such that

|G_{p,η}(x + 2e_{j}) + G_{p,η}(x − 2e_{j}) − 2G_{p,η}(x)|
≤ C_{d}/[1 + η^{2}|x|^{2}][1 + |x|^{d+1}], x ∈ Z^{d}.

Hence if φ = ∂_{j,ζ}ψ, ψ ∈ L^{2}(Ω), we can define T_{j,0,ζ}φ by setting η = 0 in the formula (2.34). If ∂_{j,ζ}φ = 0 we set T_{j,0,ζ}φ = 0. In view of (2.32) this defines T_{j,0,ζ} as a bounded operator on L^{2}(Ω) which has the property that, for any φ ∈ L^{2}(Ω), T_{j,η,ζ} φ converges in L^{2}(Ω) to T_{j,0,ζ}φ as η → 0.

So far we have not used the ergodicity of the translation operators τ_{x}, x ∈ Z^{d}. Suppose now φ ∈ L^{2}(Ω) and ⟨φ⟩ = 0. Then ergodicity of the translation operators implies that for any δ > 0 there exists ψ_{δ} ∈ L^{2}(Ω) such that ∥φ − ∂_{j,0}ψ_{δ}∥ < δ. Now using the fact that

∂_{j,0}ψ_{δ} = ∂_{j,εξ}ψ_{δ} + ψ′_{δ,ε}

where ψ′_{δ,ε} ∈ L^{2}(Ω) satisfies ∥ψ′_{δ,ε}∥ ≤ 2ε|e_{j} · ξ|∥ψ_{δ}∥, it follows from (2.32), (2.34) that T_{j,ε,εξ}[∂_{j,0}ψ_{δ}] converges in L^{2}(Ω) as ε → 0 to the function T_{j,0,0}[∂_{j,0}ψ_{δ}]. The result then follows from (2.32) on letting δ → 0. □

Lemma 2.5 enables us to define the vector q_{γ} ∈ R^{d} of (1.7).

Lemma 2.6. There exists a constant γ_{d} > 0, depending only on d, such that if |γ| < γ_{d} and 0 < ε ≤ 1, Equation (2.27) has a unique solution û_{ε} ∈ L^{2} ([\frac{−π}{ε} , \frac{π}{ε}]^{d}). There is a vector q_{γ} ∈ C^{d}, analytic in γ and satisfying |q_{γ}| < 1, such that if u(x), x ∈ R^{d}, has Fourier transform satisfying (1.7) and u_{ε}(x), x ∈ Z_{ε}^{d}, is the Fourier inverse of û_{ε}, then

lim_{ε→0} ∥u_{ε} − u∥_{ε} = 0. (2.35)

Proof. First observe that we can rewrite (2.27) as

(2.36)  û_{ε}(ξ)\{1 + ε^{−1} [1 − p̂(εξ)] + 2γi ∑_{j=1}^{d}⟨b_{j}(·)⟩\frac{sin(εe_{j} · ξ)}{ε}

+ 2iγ^{2}⟨ ∑_{j,k=1}^{d} b_{j}(·)\frac{sin(εe_{k} · ξ)}{ε}T_{j,ε,εξ}Φ_{ε}^{k}(εξ, ·)⟩\}

= f̂_{ε}(ξ), ξ ∈ [\frac{−π}{ε} , \frac{π}{ε} ]^{d} ,

where Φ_{ε}^{k}(ζ, ω) is the solution of (2.33). Note that from (2.36),(2.17) and (2.32) one has that û_{ε} ∈ L^{2} ([\frac{−π}{ε} , \frac{π}{ε}]^{d}) provided 0 < ε ≤ 1, |γ| < γ_{d} and γ_{d} is sufficiently small, depending only on d. It is also clear from Lemma 2.5 and (2.32) that for any φ ∈ L^{2}(Ω) with ⟨φ⟩ = 0 and nonnegative integer m,

lim_{ε→0} T_{j,ε,εξ}^{m}φ = T_{j,0,0}^{m}φ,

where the convergence is in the L^{2} norm. It follows then from (2.33) that

lim_{ε→0} T_{j,ε,εξ}Φ_{ε}^{k}(εξ, ·) = T_{j,0,0}Φ_{0}^{k}(0, ·).

Hence from (2.36) we define the vector q_{γ} = (q_{1}, …, q_{d}) by

q_{k} = 2γ⟨b_{k}(·)⟩ + 2γ^{2} ∑_{j=1}^{d} ⟨b_{j}(·)T_{j,0,0}Φ_{0}^{k}(0, ·)⟩ . (2.37)

Evidently we may choose γ_{d} sufficiently small, depending only on d, such that |q_{γ}| < 1. The limit (2.35) now follows from the argument of Lemma 2.1. □

Lemma 2.7. Let Ψ_{ε}^{k}(ζ, ω) be the function defined in Lemma 2.4 for |γ| < γ_{d}, 0 < ε ≤ 1. Then for any ξ ∈ R^{d}, there is the limit,

lim_{ε→0} ε^{2} ⟨|Ψ_{ε}^{k}(εξ, ·)|^{2}⟩ = 0. (2.38)

Proof. We first consider the zeroth order contribution in γ to Ψ_{ε}^{k}. Thus in (2.33) we make the approximation Φ_{ε}^{k}(ζ, ω) ≃ b_{k}(ω) − ⟨b_{k}(·)⟩. In this approximation we have from (2.30) and Bochner’s theorem the identity,

ε^{2} ⟨|Ψ_{ε}^{k}(εξ, ·)|^{2}⟩ = ∫_{[−π,π]^{d}} dμ_{φ}(ζ) [\frac{ε}{1 + ε − p̂(ζ − εξ)}]^{2} , (2.39)

where dμ_{φ} is the finite positive Borel measure on [−π, π]^{d}, satisfying

⟨φ(τ_{x}·)\overline{φ(·)}⟩ = ∫_{[−π,π]^{d}} e^{ix·ζ}dμ_{φ}(ζ), x ∈ Z^{d} ,

and φ(ω) = b_{k}(ω) − ⟨b_{k}(·)⟩. Observe now that

(2.40)  [\frac{ε}{1 + ε − p̂(ζ − εξ)}]^{2} ≤ 1, ζ ∈ [−π, π]^{d},

(2.41)  lim_{ε→0} [\frac{ε}{1 + ε − p̂(ζ − εξ)}]^{2} = 0, ζ ∈ [−π, π]^{d}\\{0\}.

Since ⟨φ⟩ = 0, one has by ergodicity of the translation operators τ_{x}, x ∈ Z^{d}, that μ_{φ}({0}) = 0. It follows now by dominated convergence that (2.38) holds in this approximation.

To deal with the general case note that (2.39) holds with φ(ω) = φ_{ε}(ω) = Φ_{ε}^{k}(εξ, ω). The result follows then from the above argument if we use the fact that dμ_{ϕ_{ε}} ≤ 2dμ_{ϕ_{0}} +2dμ_{ϕ_{ε}−ϕ_{0}} and the fact that lim_{ε→0} μ_{ϕ_{ε}−ϕ_{0}}([−π, π]^{d}) = ∥ϕ_{ε} −ϕ_{0}∥^{2} = 0. □

Proof of Theorem 1.1. We just need to prove (1.8). In view of Lemma 2.6 and (2.26) it will be sufficient to show that

lim_{ε→0} ∫_{[ \frac{−π}{ε} , \frac{π}{ε}]^{d}} |f̂_{ε}(ξ)|^{2}|ξ|^{2}ε^{2} ⟨|Ψ_{ε}^{k}(εξ, ·)|^{2}⟩ dξ = 0.

This follows by dominated convergence from (2.28) and Lemma 2.7. □

Finally we show that if (1.9) holds then the vector q_{γ} defined by (2.37) is zero.

Lemma 2.8. Suppose (1.9) holds and the conditions of Theorem 1.1. Then q_{γ} = 0 in (1.7).

Proof. In view of (2.37) we need to show that

⟨b_{j}(·)T_{j,0,0}Φ_{0}^{k}(0, ·)⟩ = 0, 1 ≤ j, k ≤ d, (2.42)

where Φ_{0}^{k} satisfies (2.33). To do this let L_{R}^{2}(Ω) be the subset of functions Φ ∈ L^{2}(Ω) which have the property that

⟨Φ(τ_{x}·) ∏_{i=1}^{n} b_{k_{i}}(τ_{x_{i}}·)⟩ = (−1)^{n+1} ⟨Φ(τ_{−x}·) ∏_{i=1}^{n} b_{k_{i}}(τ_{−x_{i}}·)⟩ ,

x, x_{i} ∈ Z^{d}, 1 ≤ k_{i} ≤ d, i = 1, …, n, n ≥ 0 .

Evidently Φ ∈ L_{R}^{2}(Ω) implies ⟨Φ⟩ = 0. In view of (1.9) b_{i}(·) ∈ L_{R}^{2}(Ω), 1 ≤ i ≤ d. Observe now that for any η > 0 the operator b_{j}(ω)T_{j,η,0} takes L_{R}^{2}(Ω) into L_{R}^{2}(Ω). This follows from the fact that G_{p,η}(x) = G_{p,η}(−x), x ∈ Z^{d}. Now let Φ_{η}^{k}(ω) be the solution in L_{R}^{2}(Ω) to the equation

Φ_{η}^{k}(ω) − γ ∑_{j=1}^{d} b_{j}(ω)T_{j,η,0}Φ_{η}^{k}(ω) = b_{k}(ω). (2.43)

Clearly a unique solution Φ_{η}^{k} exists if |γ| < γ_{d}. Comparing (2.43) and (2.33) one sees that Φ_{ε}^{k}(0, ω) = Φ_{ε}^{k}(ω) ∈ L_{R}^{2}(Ω). Taking the limit as ε → 0 we conclude that Φ_{0}^{k}(0, ω) ∈ L_{R}^{2}(Ω), whence (2.42) follows. □

3. Proof of Theorem 1.2

Since ⟨b(·)⟩ = 0 we have from (2.33), (2.36) that

e(εξ)q_{γ,ε}(εξ)e(−εξ) = −2i ∑_{k=1}^{d} sin(εe_{k} · ξ) ∑_{n=1}^{∞} γ^{n+1} ⟨[∑_{j=1}^{d} b_{j}(·)T_{j,ε,εξ}P ]^{n} b_{k}(·)⟩ .

It is easy to see from the argument of Lemma 2.8 that if (1.9) holds, then

⟨[∑_{j=1}^{d} b_{j}(·)T_{j,ε,0}P ]^{n} b_{k}(·)⟩ = 0, 1 ≤ k ≤ d, n ≥ 1. (3.1)

Hence we may write

(3.2) e(εξ)q_{γ,ε}(εξ)e(−εξ) = −2i ∑_{k=1}^{d} sin(εe_{k} · ξ) ∑_{n=0}^{∞} γ^{n+2} ∑_{m=0}^{n}

⟨ [∑_{j=1}^{d} b_{j}(·)T_{j,ε,0}P ]^{m} [∑_{j=1}^{d} b_{j}(·) {T_{j,ε,εξ} − T_{j,ε,0}} P ]

[∑_{j=1}^{d} b_{j}(·)T_{j,ε,εξ}P ]^{n−m} b_{k}(·)⟩.

If x ∈ Z^{d} with x · e_{1} > 0, we denote by L_{1}(x) all the points of Z^{d} which lie on the line segment joining the point x − (x · e_{1})e_{1} and x, excluding the point x. If x · e_{1} = 0, then L_{1}(x) is the empty set. If x · e_{1} < 0, then L_{1}(x) is all the points of Z^{d} which lie on the line segment joining the point x − (x · e_{1})e_{1} and x, excluding the point x − (x · e_{1})e_{1}. Similarly if x · e_{2} > 0 we denote by L_{2}(x) all the points of Z^{d} which lie on the line segment joining the point x − (x · e_{1})e_{1} − (x · e_{2})e_{2} and x − (x · e_{1})e_{1}, excluding the point x − (x · e_{1})e_{1} etc. For η > 0, ζ ∈ [−π, π]^{d}, k = 1, ⋯, d, we define operators S_{k,η,ζ} on L^{2}(Ω) by

(3.3) S_{k,η,ζ}φ(ω) = ∑_{j=1}^{d} b_{j}(ω) ∑_{x∈Z^{d}} [G_{p,η}(x − e_{j}) − G_{p,η}(x + e_{j})] sgn(x · e_{k})

∑_{y∈L_{k}(x)} e^{−iy·ζ}φ(τ_{x}ω).

We can see now from (3.2) and (2.31) that the matrix q_{γ,ε}(ζ) = [q_{γ,ε,k,k′}(ζ)], 1 ≤ k, k′ ≤ d, is given by the formula

q_{γ,ε,k,k′}(ζ) = (3.4)

− ∑_{n=0}^{∞} γ^{n+2} ∑_{m=0}^{n} ⟨ [∑_{j=1}^{d} b_{j}(·)T_{j,ε,0}P ]^{m} S_{k′,ε,ζ}P [∑_{j=1}^{d} b_{j}(·)T_{j,ε,ζ}P ]^{n−m} b_{k}(·)⟩

− ∑_{n=0}^{∞} γ^{n+2} ∑_{m=0}^{n} ⟨ [∑_{j=1}^{d} b_{j}(·)T_{j,ε,0}P ]^{m} e^{−ie_{k}·ζ}S_{k,ε,ζ}P [∑_{j=1}^{d} b_{j}(·)T_{j,ε,ζ}P ]^{n−m}b_{k′}(·)⟩.

We proceed as in [3] to consider the case when the b(τ_{x}·), x ∈ Z^{d}, are given by independent Bernoulli variables. Thus we assume that b_{j}(·) ≡ 0, j > 1, and b_{1}(τ_{x}·) = Y_{x} , x ∈ Z^{d}, where the variables Y_{x}, x ∈ Z^{d}, are assumed to be i.i.d. Bernoulli, Y_{x} = ±1 with equal probability. Using the notation of §4 of [3] we introduce the Fock spaces F^{r}(Z^{d}), 1 < r < ∞, with F^{2}(Z^{d}) isomorphic to L^{2}(Ω). Now we have seen from (2.32) that T_{1,ε,ζ} is a bounded operator on L^{2}(Ω) with norm ∥T_{1,ε,ζ}∥_{2} ≤ 2/c_{d}. We have also seen in Lemma 2.5 that

lim_{ε→0} ∥[T_{1,ε,ζ} − T_{1,0,ζ}]φ∥_{2} = 0, φ ∈ L^{2}(Ω). (3.5)

We note now as in [3] that the Calderon-Zygmund theorem implies that a similar result holds for all Fock spaces F^{r}(Z^{d}), 1 < r < ∞. We shall show that the limit in (3.5) is uniform for ζ ∈ [−π, π]^{d}, provided one projects orthogonal to the constant function.

Lemma 3.1. Suppose 1 < r < ∞ and regard T_{1,ε,ζ} as an operator on F^{r}(Z^{d}). Then there is a constant C_{r,d} depending only on r, d such that ∥T_{1,ε,ζ}∥_{r} ≤ C_{r,d}. There is also the limit,

lim_{ε→0} { sup_{ζ∈[−π,π]^{d}} ∥[T_{1,ε,ζ}P − T_{1,0,ζ}P]φ∥_{r}} = 0, φ ∈ F^{r}(Z^{d}). (3.6)

Proof. Let T_{ε,ζ} be the integral operator on L^{r}(Z^{d}) defined by

T_{ε,ζ}f(x) = ∑_{y∈Z^{d}} K_{ε}(x − y)e^{−i(x−y)·ζ}f(y), x ∈ Z^{d},

where K_{ε}(x) = G_{p,ε}(x − e_{1}) − G_{p,ε}(x + e_{1}). It follows from (2.17) that

|K̂_{ε}(ξ)| ≤ 2/c_{d}, ξ ∈ [−π, π]^{d}. (3.7)

We may also apply the argument of Lemma 2.2 to see that there is a constant C_{d} > 0, depending only on d, such that

(3.8)  |K_{ε}(x)| ≤ C_{d}/[1 + ε^{2}|x|^{2}][1 + |x|^{d}], x ∈ Z^{d},
|K_{ε}(x + e_{i}) − K_{ε}(x)| ≤ C_{d}/[1 + ε^{2}|x|^{2}][1 + |x|^{d+1}], x ∈ Z^{d}.

It follows now from (3.7), (3.8) that T_{ε,ζ} is a bounded operator on L^{r}(Z^{d}) and there is a constant C_{r,d} depending only on r, d such that ∥T_{ε,ζ}∥_{r} ≤ C_{r,d}. This is a discrete version of the Calderon-Zygmund Theorem [22]. Now, arguing as in [3], we see that this also implies ∥T_{1,ε,ζ}∥_{r} ≤ C_{r,d}.

Next we show that

lim_{ε→0} { sup_{ζ∈[−π,π]^{d}} ∥[T_{ε,ζ} − T_{0,ζ}]f∥_{r}} = 0, f ∈ L^{r}(Z^{d}). (3.9)

We first show that (3.9) holds for r = 2. We can do this by going to the Fourier representation of T_{ε,ζ}, which we denote T̂_{ε,ζ}. Thus,

T̂_{ε,ζ} f̂(ξ) = K̂_{ε}(ξ − ζ)f̂(ξ), ξ ∈ [−π, π]^{d}. (3.10)

Observe now that since f̂ ∈ L^{2}([−π, π]^{d}), for any δ > 0 there exists ν > 0 such that

∫_{|ξ−ζ|<ν} |f̂(ξ)|^{2}dξ < δ, ζ ∈ [−π, π]^{d}. (3.11)

It is easy to see that for any ν > 0, K̂_{ε}(ξ) converges uniformly to K̂_{0}(ξ) in the region ξ ∈ [−π, π]^{d}, |ξ| > ν. It follows from this and (3.7), (3.10), (3.11) that (3.9) holds for r = 2.

To prove (3.9) for general r, 1 < r < ∞, we define for n = 1, 2, …, ζ ∈ [−π, π]^{d} an operator A_{n,ζ} on functions f : Z^{d} → C by

A_{n,ζ}f(x) = ∑_{z∈Z^{d}} φ_{n}(z)e^{iz·ζ}f(x + z), x ∈ Z^{d}.

Here φ_{n} is the probability density for the standard random walk in Z^{d} after n steps of the walk. Thus φ_{n}(z) ≥ 0, z ∈ Z^{d}, and

∑_{z∈Z^{d}} φ_{n}(z) = 1.

It follows that A_{n,ζ} is a bounded operator on L^{r}(Z^{d}) with ∥A_{n,ζ}∥_{r} ≤ 1. Observe now that

T_{ε,ζ}[f − A_{n,ζ}f](x) = ∑_{y∈Z^{d}} K_{ε,n}(x − y)e^{−i(x−y)·ζ}f(y), x ∈ Z^{d},

where the function K_{ε,n} is given by

K_{ε,n}(x) = ∑_{z∈Z^{d}}[K_{ε}(x) − K_{ε}(x + z)]φ_{n}(z).

It is easy to see from (3.8) that K_{ε,n} ∈ L^{1}(Z^{d}) with norm ∥K_{ε,n}∥_{1} satisfying ∥K_{ε,n}∥_{1} ≤ C_{d,n}, where the constant C_{d,n} depends only on d and n. Further, for any x ∈ Z^{d}, K_{ε,n}(x) converges to K_{0,n}(x). We conclude that for any function f ∈ L^{r}(Z^{d}) one has

lim_{ε→0} { sup_{ζ∈[−π,π]^{d}} ∥[T_{ε,ζ} − T_{0,ζ}][f − A_{n,ζ}f]∥_{r}} = 0,

for any n = 1, 2, · · ·. The result (3.9) follows then if we can show that for any f ∈ L^{r}(Z^{d}), 1 < r < ∞, and δ > 0, there exists a positive integer N such that

∥A_{n,ζ}f∥_{r} < δ, n ≥ N, ζ ∈ [−π, π]^{d}.

This last inequality follows by approximating f ∈ L^{r}(Z^{d}) with a function of finite support. A similar argument implies (3.6). □

Lemma 3.2. Suppose d ≥ 2, 1 < r < d and q < ∞ satisfies 1/r > 1/d + 1/q. Then the operator S_{k,η,ζ} of (3.3) is a bounded operator from the space F^{r}(Z^{d}) to F^{q}(Z^{d}). There is a constant C_{r,q,d}, depending only on r, q, d, such that the norm ∥S_{k,η,ζ}∥_{r,q} of the operator S_{k,η,ζ} satisfies the inequality ∥S_{k,η,ζ}∥_{r,q} ≤ C_{r,q,d}. There is also the limit

lim_{ε→0} { sup_{ζ∈[−π,π]^{d}} ∥[S_{k,ε,ζ}P − S_{k,0,ζ}P]φ∥_{r,q}} = 0, φ ∈ F^{r}(Z^{d}). (3.12)

Proof. We proceed as in Lemma 3.1. Let S_{ε,ζ} be the operator on functions f ∈ L^{r}(Z^{d}) defined by

S_{ε,ζ}f(x) = ∑_{y∈Z^{d}} K_{ε,ζ}(x − y)f(y), x ∈ Z^{d},

where

K_{ε,ζ}(x) = [G_{p,ε}(x − e_{1}) − G_{p,ε}(x + e_{1})]sgn(x · e_{k}) ∑_{z∈L_{1}(x)} e^{−iz·ζ}.

It follows from (3.8) that there is a constant C_{d} > 0, depending only on d, such that

|K_{ε,ζ}(x)| ≤ C_{d}/[1 + |x|^{d−1}], x ∈ Z^{d}.

Hence K_{ε,ζ} ∈ L^{s}(Z^{d}) for any s > d/(d − 1). It follows now by Young’s inequality that S_{ε,ζ}f ∈ L^{q}(Z^{d}) for any q > 0 satisfying 1/r > 1/d + 1/q. Further, there is a constant C_{r,q,d}, depending only on r, q, d, such that the norm ∥S_{ε,ζ}∥_{r,q} of S_{ε,ζ} as an operator from L^{r}(Z^{d}) to L^{q}(Z^{d}) satisfies ∥S_{ε,ζ}∥_{r,q} ≤ C_{r,q,d}. It is easy to see that for any x ∈ Z^{d},

lim_{ε→0} { sup_{ζ∈[−π,π]^{d}} |K_{ε,ζ}(x) − K_{0,ζ}(x)|} = 0.

Hence, if S_{0,ζ} is the operator with kernel K_{0,ζ}, one has

lim_{ε→0} { sup_{ζ∈[−π,π]^{d}} ∥[S_{ε,ζ} − S_{0,ζ}]f∥_{q}} = 0, f ∈ L^{r}(Z^{d}),

provided q < ∞ satisfies 1/r > 1/d + 1/q. Now, arguing as in [3] we obtain the result. □

Corollary 3.1. Suppose b_{j}(·) ≡ 0, j > 1 and b_{1}(τ_{x}·) = Y_{x}, x ∈ Z^{d}, where the Y_{x}, x ∈ Z^{d}, are independent Bernoulli variables with zero mean. Then Theorem 1.2 holds.

Proof. Observe that b_{k}(·) ∈ F^{r}(Z^{d}) for all r ≥ 1. The result follows now from Lemmas 3.1, 3.2 and the representation (3.4) for q_{γ,ε}(ζ). □

Next we wish to generalize our method to all b(·) such that b(τ_{x}·), x ∈ Z^{d}, are independent random variables. To do this we pursue the method developed in §5 of [3]. Thus we define a spin space S as all s = (m_{1}, …, m_{d}) where the m_{i} are nonnegative integers satisfying |s| = m_{1} + ⋯ + m_{d} ≥ 1. For s ∈ S we define a random variable b_{s} by

b_{s}(·) = ∏_{i=1}^{d} b_{i}(·)^{m_{i}} , s = (m_{1}, …, m_{d}),

and a random variable Y_{0,s} = b_{s} − ⟨b_{s}⟩. Then for s ∈ S, n ∈ Z^{d} we define the variable Y_{n,s} to be the translate of Y_{0,s}, i.e., Y_{n,s}(·) = Y_{0,s}(τ_{n}·). For 1 < r < ∞ we may define Fock spaces F_{S}^{r}(Z^{d}) of many particle wave functions where the particles move in Z^{d} and have spin in S. We can also, for any δ > 0, define a mapping U_{δ} from F_{S}^{2}(Z^{d}) to functions on Ω by

(3.13) U_{δ}ψ = ψ_{0} + ∑_{N=1}^{∞} ∑_{\begin{matrix}\{n_{1},...,n_{N} \}∈Z^{d,N} \\ s_{i}∈S, 1≤i≤N \end{matrix}} ψ_{N}(n_{1}, s_{1}, n_{2}, s_{2}, …, n_{N}, s_{N})

× δ^{|s_{1}|+···+|s_{N} |} Y_{n_{1},s_{1}}Y_{n_{2},s_{2}} … Y_{n_{N} ,s_{N}} .

Following the argument of Lemma 5.1 of [3] we have:

Lemma 3.3. There exists δ > 0, depending only on d, such that the mapping U_{δ} of (3.13) is a bounded operator from F_{S}^{2}(Z^{d}) to L^{2}(Ω) satisfying ∥U_{δ}∥ ≤ 1.

We shall use the operator U_{δ} = U of Lemma 3.3 to transfer the equation of (2.33) from L^{2}(Ω) to F_{S}^{2}(Z^{d}). Thus we shall need to construct operators P_{F}, B_{j,F}, T_{j,ε,ζ,F}, 1 ≤ j ≤ d, with the property that they are bounded operators on F_{S}^{2}(Z^{d}) and satisfy

UP_{F} = P U, UB_{j,F} = b_{j}(·)U, UT_{j,ε,ζ,F} = T_{j,ε,ζ}U. (3.14)

The simplest of these operators to construct is P_{F}. Thus if ψ = {ψ_{N} : N = 0, 1, 2, … } ∈ F_{S}^{2}(Z^{d}) then P_{F}ψ = {(P_{F}ψ)_{N} : N = 0, 1, 2, … }, where (P_{F}ψ)_{0} = 0, (P_{F}ψ)_{N} = ψ_{N}, N ≥ 1. Similarly we have T_{j,ε,ζ,F}ψ = {(T_{j,ε,ζ,F}ψ)_{N} : N = 0, 1, 2, … }, where

(T_{j,ε,ζ,F}ψ)_{0} = [e^{−iζ·e_{j}} − e^{iζ·e_{j}}] Ĝ_{p,ε}(−ζ)ψ_{0},
(T_{j,ε,ζ,F}ψ)_{N}(n_{1}, s_{1}, …, n_{N}, s_{N}) =

∑_{x∈Z^{d}} [G_{p,ε}(x − e_{j}) − G_{p,ε}(x + e_{j})] e^{−ix·ζ}ψ_{N}(n_{1} − x, s_{1}, …, n_{N} − x, s_{N}), N ≥ 1.

The most complicated of the operators to define is B_{j,F}. We have

(B_{j,F}ψ)_{N}(n_{1}, s_{1}, …, n_{N}, s_{N}) = ⟨b_{j}(·)⟩ψ_{N}(n_{1}, s_{1}, …, n_{N}, s_{N})

+ ∑_{s∈S} [⟨b_{j}b_{s}⟩ − ⟨b_{j}⟩⟨b_{s}⟩] δ^{|s|}ψ_{N+1}(0, s, n_{1}, s_{1}, …, n_{N}, s_{N}),

if n_{k} ≠ 0, 1 ≤ k ≤ N, N ≥ 0;

(B_{j,F}ψ)_{N}(0, s_{1}, n_{2}, s_{2}, …, n_{N}, s_{N}) = 0,
if n_{k} ≠ 0, 2 ≤ k ≤ N, s_{1} = (m_{1}, …, m_{d}), m_{j} = 0;
(B_{j,F}ψ)_{N}(0, s_{1}, n_{2}, s_{2}, …, n_{N}, s_{N}) = δ^{−1}ψ_{N}(0, s′_{1}, n_{2}, s_{2}, …, n_{N}, s_{N}),
if n_{k} ≠ 0, 2 ≤ k ≤ N, s_{1} = (m_{1}, …, m_{d}), m_{j} > 0, |s_{1}| > 1,
s′_{1} = (m_{1}, …, m_{j−1}, m_{j} − 1, m_{j+1}, …, m_{d});
(B_{j,F}ψ)_{N}(0, s_{1}, n_{2}, s_{2}, …, n_{N}, s_{N}) = δ^{−1}ψ_{N−1}(n_{2}, s_{2}, …, n_{N}, s_{N})

− ∑_{s∈S}⟨b_{s}⟩δ^{|s|−1}ψ_{N}(0, s, n_{2}, s_{2}, …, n_{N}, s_{N}),

if n_{k} ≠ 0, 2 ≤ k ≤ N and s_{1} = (0, 0, …, 1, 0, …, 0), with 1 in the ith position.

Lemma 3.4. For 1 ≤ k ≤ d let φ_{k} ∈ F_{S}^{2}(Z^{d}) be the wave function φ_{k} = {φ_{k,N} : N = 0, 1, 2, … } where φ_{k,N} ≡ 0 if N ≠ 1 and φ_{k,1} is defined by

φ_{k,1}(n, s) = 1 if n = 0, s = (0, …, 0, 1, 0, …, 0),

with 1 in the kth position, φ_{k,1}(n, s) = 0, otherwise. Then there exists a constant γ_{d} depending only on d such that if |γ| < γ_{d}, the equation

Φ_{ε,F}^{k}(ζ) − γP_{F} ∑_{j=1}^{d} B_{j,F}T_{j,ε,ζ,F}Φ_{ε,F}^{k}(ζ) = φ_{k} , (3.15)

has a unique solution Φ_{ε,F}^{k}(ζ) ∈ F_{S}^{2}(Z^{d}). Further the solution to (2.33) is given by the function Φ_{ε}^{k}(ζ, ω) = UΦ_{ε,F}^{k}(ζ) ∈ L^{2}(Ω).

Proof. Evidently P_{F} is a bounded operator on F_{S}^{2}(Z^{d}) with norm 1. We can also easily see that T_{j,ε,ζ,F} is a bounded operator on F_{S}^{2}(Z^{d}) with norm bounded by the RHS of (2.32). Also B_{j,F} is bounded with norm depending only on d. Hence we may choose γ_{d} small enough, depending only on d, such that (3.15) is uniquely solvable. The fact that UΦ_{ε,F}^{k}(ζ) solves (2.33) comes from (3.14) and Uφ_{k} = b_{k}(·)−⟨b_{k}⟩. □

Proof of Theorem 1.2. Observe from (2.36) that q_{ε,γ}(ζ) is defined in terms of Φ_{ε}^{k}(ζ, ω). In view of Lemma 3.4 we get a representation of q_{ε,γ}(ζ) similar to (3.4) but with the operators T_{j,ε,ζ}, S_{k,ε,ζ}, b_{j}(·) on L^{2}(Ω) replaced by the corresponding operators T_{j,ε,ζ,F}, S_{k,ε,ζ,F}, B_{j,F} on Fock space. Here the operator S_{k,ε,ζ,F} satisfies US_{k,ε,ζ,F} = S_{k,ε,ζ}U. We now argue exactly as in Corollary 3.1. □

4. Proof of Theorem 1.3

We assume in this section that the function p is given by (1.2). Let G_{p,η}(x) be the function defined by (2.9). Then Ĝ_{p,η}(ξ) satisfies the inequality,

0 ≤ Ĝ_{p,η}(ξ) ≤ C/[η + |ξ|^{2}],

for some universal constant C. One can also see that if d ≥ 2 and 0 < η < 1, there is a constant C_{d}, depending only on d, such that

|G_{p,η}(x − e_{k}) − G_{p,η}(x + e_{k})| ≤ C_{d} exp [−\sqrt{η}|x|/C_{d}] /[1 + |x|^{d−1}]. (4.1)

Next we prove the analogue of Lemma 2.3.

Lemma 4.1. Suppose d ≥ 2 and f : R^{d} → R is a C^{∞} function with compact support. Then if γ ∈ C satisfies |γ| < ε/\sqrt{2d} and 0 < ε ≤ 1, Equation (1.4) has a unique solution u_{ε}(x, ω) in L^{2}(Z_{ε}^{d}). Further, there is a constant C_{d}, depending only on d, such that ∥u_{ε}(·, ω)∥_{ε} ≤ C_{d}∥f∥_{ε}.

Proof. Suppose u_{ε}(x, ω) is in L^{2}(Z_{ε}^{d}) and satisfies (1.4). Let v_{ε}(x, ω) be defined by

v_{ε}(x, ω) = ε^{−2}[A_{p}u_{ε}(x, ω) + ε^{2}u_{ε}(x, ω)].

It is easy to see that v_{ε}(x, ω) ∈ L^{2}(Z_{ε}^{d}). In view of (2.9) it follows that

u_{ε}(x, ω) = ε^{2−d} ∫_{Z_{ε}^{d}} dy G_{p,ε^{2}} (\frac{x − y}{ε} ) v_{ε}(y, ω). (4.2)

Now for i = 1, …, d let us define operators T_{i} on L^{2}(Z_{ε}^{d}) by

T_{i}w(x) = ε^{1−d} ∫_{Z_{ε}^{d}} dy [G_{p,ε^{2}} (\frac{x − y}{ε} + e_{i}) − G_{p,ε^{2}} (\frac{x − y}{ε} − e_{i})] w(y).

It is easy to see that T_{i} is a bounded operator on L^{2}(Z_{ε}^{d}). It follows from (2.10) that the norm of T_{i} is bounded by

∥T_{i}∥ ≤ sup_{ξ∈[−π,π]^{d}} [\frac{2ε| sin(e_{i} · ξ)|}{1 + ε^{2} − p̂(ξ)}] . (4.3)

Now (4.3) and (1.1) imply that the operator ∑_{i=1}^{d} b_{i}(·, ω)T_{i} is bounded on L^{2}(Z_{ε}^{d}) and has norm satisfying the inequality

∥∑_{i=1}^{d} b_{i}(·, ω)T_{i}∥ ≤ sup_{ξ∈[−π,π]^{d}} [\frac{2ε \{∑_{i=1}^{d} sin^{2}(e_{i} · ξ)\}^{1/2}}{1 + ε^{2} − p̂(ξ)} ]

≤ 4 sup_{z>0} [\frac{εz}{ε^{2} + 2z^{2}/d}] = \sqrt{2d}.

Observe now that (1.4) implies that v_{ε}(x, ω) ∈ L^{2}(Z_{ε}^{d}) satisfies the equation

v_{ε}(x, ω) − γε^{−1} ∑_{i=1}^{d} b_{i}(x/ε, ω)T_{i}v_{ε}(x, ω) = f(x). (4.4)

Hence if |γ| < ε/\sqrt{2d}, the function v_{ε}(x, ω) is uniquely determined by (4.4). In view of (4.2) it follows that u_{ε}(x, ω) ∈ L^{2}(Z_{ε}^{d}) is the unique solution to (1.4). To prove existence note that (4.4) has a solution v_{ε}(x, ω) ∈ L^{2}(Z_{ε}^{d}) provided |γ| < ε/\sqrt{2d}. If we then define u_{ε}(x, ω) by (4.2) one can easily see that u_{ε}(x, ω) satisfies (1.4).

□

In order to obtain Equation (1.11) we follow a development similar to the one following Lemma 2.3. Thus we put v_{ε}(x, ω) = u_{ε}(x, τ_{−x/ε}ω), x ∈ Z_{ε}^{d}, ω ∈ Ω, and write v_{ε}(x, ω) = u_{ε}(x) + ψ_{ε}(x, ω), x ∈ Z_{ε}^{d}, ω ∈ Ω, where ⟨ψ_{ε}(x, ·)⟩ = 0, x ∈ Z_{ε}^{d}. It follows that

u_{ε}(x) = ⟨v_{ε}(x, ·)⟩ = ⟨u_{ε}(x, ·)⟩, x ∈ Z_{ε}^{d}.

Furthermore, the Fourier transform û_{ε}(ξ) of u_{ε}(x) satisfies the equation

(4.5)  û_{ε}(ξ) \{1 + ε^{−2} [1 − p̂(εξ)] + 2γi ∑_{j=1}^{d}⟨b_{j}(·)⟩\frac{sin(e_{j} · εξ)}{ε^{2}} \}

+ γε^{−2} ⟨∑_{j=1}^{d} b_{j}(·) [e^{iεe_{j}·ξ}ψ̂_{ε}(ξ, τ_{−e_{j}}·) − e^{−iεe_{j}·ξ}ψ̂_{ε}(ξ, τ_{e_{j}}·)]⟩

= f̂_{ε}(ξ), ξ ∈ [\frac{−π}{ε} , \frac{π}{ε} ]^{d} .

The Fourier transform of ψ_{ε}(x, ω) is given in terms of û_{ε}(ξ) by

ψ̂_{ε}(ξ, ω) = −2iγû_{ε}(ξ) ∑_{k=1}^{d} sin(εe_{k} · ξ)Ψ_{ε}^{k}(εξ, ω) (4.6)

Here Ψ_{ε}^{k}(ζ, ω), 1 ≤ k ≤ d, ζ ∈ [−π, π]^{d}, ω ∈ Ω, are functions in L^{2} ([−π, π]^{d} × Ω) which satisfy the equation

Ψ_{ε}^{k}(ζ, ω) − ∑_{y∈Z^{d}} p(y)e^{−iy·ζ}Ψ_{ε}^{k}(ζ, τ_{y}ω) (4.7)

+ γP ∑_{j=1}^{d} b_{j}(ω) [e^{ie_{j}·ζ}Ψ_{ε}^{k}(ζ, τ_{−e_{j}}ω) − e^{−ie_{j}·ζ}Ψ_{ε}^{k}(ζ, τ_{e_{j}}ω)] + ε^{2}Ψ_{ε}^{k}(ζ, ω)

= b_{k}(ω) − ⟨b_{k}(·)⟩, ζ ∈ [−π, π]^{d}, ω ∈ Ω, 1 ≤ k ≤ d.

If we substitute for ψ̂_{ε} from (4.6) into (4.5) then we obtain the equation,

(4.8) û_{ε}(ξ)\{1 + ε^{−2} [1 − p̂(εξ)] + 2γi ∑_{j=1}^{d}⟨b_{j}(·)⟩\frac{sin(εe_{j} · ξ)}{ε^{2}}

− 2iγ^{2}⟨ ∑_{j,k=1}^{d} b_{j}(·)\frac{sin(εe_{k} · ξ)}{ε^{2}}[e^{iεe_{j}·ξ}Ψ_{ε}^{k}(εξ, τ_{−e_{j}}·)

− e^{−iεe_{j}·ξ}Ψ_{ε}^{k}(εξ, τ_{e_{j}}·)]⟩\} = f̂_{ε}(ξ), ξ ∈ [\frac{−π}{ε} , \frac{π}{ε} ]^{d} .

Let us assume now that (1.9) holds. Then, following the development at the beginning of §3, we have that the matrix q_{γ,ε}(ζ) = [q_{γ,ε,k,k′}(ζ)], 1 ≤ k, k′ ≤ d of (1.11) is given by the formula

(4.9) q_{γ,ε,k,k′}(ζ) =

− ∑_{n=0}^{∞} γ^{n+2} ∑_{m=0}^{n} ⟨[ ∑_{j=1}^{d} b_{j}(·)T_{j,ε^{2},0}P ]^{m}S_{k′,ε^{2},ζ}P [ ∑_{j=1}^{d} b_{j}(·)T_{j,ε^{2},ζ}P ]^{n−m}b_{k}(·)⟩

−∑_{n=0}^{∞} γ^{n+2} ∑_{m=0}^{n} ⟨[ ∑_{j=1}^{d} b_{j}(·)T_{j,ε^{2},0}P ]^{m}e^{−ie_{k}·ζ}S_{k,ε^{2},ζ}P [ ∑_{j=1}^{d} b_{j}(·)T_{j,ε^{2},ζ}P ]^{n−m}b_{k′}(·)⟩,

where the operator T_{j,η,ζ} is defined by (2.31) and S_{k,η,ζ} by (3.3), with p given by (1.2). One can see from the proof of Lemma 4.1 that q_{γ,ε}(ζ) is continuous in ζ and analytic in γ provided |γ| < ε/\sqrt{2d}.

5. Proof of Theorem 1.4

The matrix q_{m,ε}(ζ) is given by the coefficient of γ^{m} in the expansion (4.9). Since the b(τ_{x}·) are given by Bernoulli variables, (a) and (b) of Theorem 1.4 follow easily from (4.9). Next we obtain a formula for q_{4,ε}(ζ). Let K_{ε}(x) be defined by

K_{ε}(x) = G_{p,ε^{2}}(x − e_{1}) − G_{p,ε^{2}}(x + e_{1}), x ∈ Z^{d}. (5.1)

For x ∈ Z^{d}, let x^{R} be the reflection of x in the plane through the origin with normal e_{1}. It is clear that

K_{ε}(x^{R}) = −K_{ε}(x), x ∈ Z^{d}. (5.2)

For 1 ≤ k ≤ d, ζ ∈ [−π, π]^{d}, define a_{k}(ζ), b_{k}(ζ) by

(5.3)  a_{k,ε}(ζ) = ∑_{x∈Z^{d}}[K_{ε}(x)]^{3}sgn(x · e_{k}) ∑_{y∈L_{k}(x)} e^{−iy·ζ},
b_{k,ε}(ζ) = ∑_{x∈Z^{d}}[K_{ε}(x)]^{3}e^{ix·ζ}sgn(x · e_{k}) ∑_{y∈L_{k}(x)} e^{−iy·ζ}.

Note that

a_{k,ε}(0) = b_{k,ε}(0) = ∑_{x∈Z^{d}}[K_{ε}(x)]^{3}(x · e_{k}). (5.4)

Then one can see from (4.9) that

q_{4,ε,1,1}(ζ) = [1 + e^{−ie_{1}·ζ}][2a_{1,ε}(ζ) − b_{1,ε}(ζ)]. (5.5)

One also has for 1 < k ≤ d,

(5.6)  q_{4,ε,1,k}(ζ) = 2a_{k,ε}(ζ) − b_{k,ε}(ζ),
q_{4,ε,1,k}(ζ) = e^{−ie_{k}·ζ}[2a_{k,ε}(ζ) − b_{k,ε}(ζ)].

Since a_{k,ε}(0) = b_{k,ε}(0) it follows that q_{4,ε,1,1}(0) = 2a_{1,ε}(0). From (5.2) and (5.4) we see that a_{k,ε}(0) = 0 if k > 1. Hence q_{4,ε,k,k′}(0) = 0 for k + k′ > 2.

Lemma 5.1. (a) Let d = 1 and K ⊂ R be a compact set. Then ε^{2}a_{1,ε}(εξ) converges uniformly for ξ ∈ K to a function a_{1}(ξ), as ε → 0. Similarly ε^{2}b_{1,ε}(εξ) converges to a function b_{1}(ξ).

(b) If d ≥ 3 and 1 ≤ k ≤ d, then a_{k,ε}(ζ) converges uniformly for ζ ∈ [−π, π] to a function a_{k}(ζ), as ε → 0. Similarly b_{k,ε}(ζ) converges to a function b_{k}(ζ).

Proof. (a) It is easy to see that for d = 1, η > 0, one has

G_{p,η}(x) = \begin{matrix}[1 + η − \sqrt{2η + η^{2}}] & |x| \\ √ & \overline{2η + η^{2}} \end{matrix} , x ∈ Z.

Hence if for ε > 0, we define ν(ε) by

ν(ε) = 1 + ε^{2} − ε\sqrt{2 + ε^{2}}, (5.7) then the function K_{ε}(x) of (5.1) is given by

K_{ε}(x) = 2ν(ε)^{|x|}, x ∈ Z, x > 0. (5.8)

Substituting from (5.8) into (5.3) we obtain the identity

a_{1,ε}(ζ) = \frac{8ν(ε)^{3}}{1 − ν(ε)^{3}} [\frac{1}{1 − ν(ε)^{3}e^{−iζ}} + \frac{e^{iζ}}{1 − ν(ε)^{3}e^{iζ}} ] .

One can see that

lim_{ε→0} \frac{ε}{1 − ν(ε)^{3}e^{−iεξ}} = \frac{1}{3\sqrt{2} + iξ} .

We conclude that

lim_{ε→0} ε^{2}a_{1,ε}(εξ) = \frac{16}{18 + ξ^{2}} .

Since for d = 1 one has a_{1,ε}(ζ) = b_{1,ε}(−ζ), the formula (1.13) follows now from the last equation and (5.5).

(b) This follows from the fact that for d ≥ 3, the limit lim_{ε→0} K_{ε}(x) = K_{0}(x) exists for all x ∈ Z^{d}, and the bound (4.1). □

Lemma 5.2. (a) For d ≥ 3, one has a_{1}(0) > 0.

(b) For d = 2 one has lim_{ε→0} a_{1,ε}(0) = +∞.

Proof. (a) In view of (5.2) and (5.4) it is sufficient to show that K_{0}(x) > 0 if x ∈ Z^{d} satisfies x · e_{1} > 0. To see this observe from (2.9) and (5.2) that K_{ε}(x) satisfies the equation

A_{p}K_{ε}(x) + ε^{2}K_{ε}(x) = g(x), x ∈ Z^{d}, x · e_{1} > 0,

with Dirichlet boundary condition K_{ε}(x) = 0, x · e_{1} = 0. Here g is the function, g(x) = 1, x = e_{1}, g(x) = 0 otherwise. The positivity of K_{0}(x) follows now from the maximum principle.

(b) We shall show that there exists α > 0 and a positive constant c_{α}, depending only on α such that for d = 2,

lim_{ε→0} K_{ε}(ne_{1} + me_{2}) ≥ \frac{c_{α}}{n} , |m| < αn, n = 1, 2, · · · . (5.9)

To prove (5.9) we use the fact that for d = 1 one has

K_{ε}(n) = 2ν(ε)^{n} = \frac{2}{π} ∫_{0}^{π} \frac{sin(nζ) sin(ζ)}{2 sin^{2}(ζ/2) + ε^{2}} dζ n ≥ 1.

Since one also has for d = 2 the representation

K_{ε}(ne_{1} + me_{2}) = \frac{2}{π^{2}} ∫_{0}^{π} ∫_{0}^{π} \frac{cos(mξ) sin(nζ) sin(ζ)}{sin^{2}(ζ/2) + sin^{2}(ξ/2) + ε^{2}} dζdξ,

we obtain for d = 2 the formula

lim_{ε→0} K_{ε}(ne_{1} + me_{2}) = \frac{4}{π} ∫_{0}^{π} ν(\sqrt{2} sin(ξ/2))^{n} cos(mξ)dξ, m, n ∈ Z, n ≥ 1.

Now there exist universal positive constants c, C such that

exp[−Cnξ] ≤ ν(\sqrt{2} sin(ξ/2))^{n} ≤ exp[−cnξ], n ≥ 1, 0 ≤ ξ ≤ π.

We conclude that the LHS of (5.9) is bounded below by

\frac{4}{π} ∫_{0}^{π} exp[−Cnξ] dξ = \frac{4}{Cπn} \{1 − exp[−Cπn]\} ,

if m = 0, and by

\frac{2\sqrt{2}}{π} ∫_{0}^{π/4|m|} exp[−Cnξ] dξ − \frac{4}{π} ∫_{π/4|m|}^{∞} exp[−cnξ] dξ

= \frac{2\sqrt{2}}{Cπn} \{1 − exp [−\frac{Cπn}{4|m|}]\} − \frac{4}{cπn} exp [−\frac{cπn}{4|m|}] ,

if m ≠ 0. The inequality (5.9) clearly follows from the last two identities. The result follows now from (5.4) and (5.9) since the argument of Part (a) implies K_{ε}(ne_{1} + me_{2}) > 0, m, n ∈ Z, n > 0. □

We are left now to prove (c) and (d) of Theorem 1.4 for m > 4. In order to do this we need to obtain a suitable representation of q_{m,ε}(ζ) for general m. We first consider the situation ζ = 0. Recall that we are assuming the b_{j}(·) ≡ 0, j > 1, and b_{1}(τ_{x}·) = Y_{x}, x ∈ Z^{d}, where the variables Y_{x}, x ∈ Z^{d}, are assumed to be i.i.d. Bernoulli, Y_{x} = ±1 with equal probability.

Lemma 5.3. Suppose m ≥ 4 is even. Then q_{m,ε,k,k′}(0) = 0 if k + k′ > 2 and q_{m,ε,1,1}(0) has the representation

(5.10) q_{m,ε,1,1}(0) = −2 ∑_{x_{1},···,x_{m−1}∈Z^{d}} ( ∑_{α=1}^{m−1} x_{α} · e_{1}) K_{ε}(x_{1}) ⋯ K_{ε}(x_{m−1})

⟨Y_{0}Y_{x_{1}}Y_{x_{1}+x_{2}} ⋯ Y_{x_{1}+···+x_{m−1}}⟩ .

Proof. First note that by the argument of Lemma 2.8 the representation (4.9) for q_{γ,ε}(0) continues to hold when we delete the projection operator P from the RHS of the equation. It is easy to see from this that (5.10) holds. If k > 1 then one has a similar representation to (5.10) for q_{m,ε,1,k}(0) and q_{m,ε,k,1}(0) but with e_{1} in (5.10) replaced by e_{k}. In that case (5.2) implies q_{m,ε,k,1}(0) = q_{m,ε,1,k}(0) = 0. □

In order to prove (d) of Theorem 1.4 for general m we use (5.10) to obtain a representation of q_{m,ε,1,1}(0) as a sum indexed by certain types of graph. We shall use the terminology of [1]. For q = 2, 3, ··· let F_{q} be the set of unlabeled, connected, directed multigraphs on q vertices with the properties: (A) The graph has no loops. (B) Each vertex has equal indegree and outdegree. (C) The degree of each vertex is a multiple of 4. (D) The number of edges in the graph is 4q. For a graph G ∈ F_{q} let V [G] denote the vertex set of G and E[G] be the set

of directed edges of G. Each directed edge e has two vertices e_{+} and e_{−}, with the direction of the edge being from e_{−} to e_{+}. We associate with G and a directed edge e ∈ E[G], a number K_{ε}(G, e) defined by

K_{ε}(G, e) = ∑_{{y_{v}∈Z^{d}:v∈V [G]}}(y_{e_{−}} · e_{1})δ(y_{e_{+}}) ∏_{e′∈E[G], e′≠e} K_{ε}(y_{e′_{+}} − y_{e′_{−}}). (5.11)

In (5.11) δ(x), x ∈ Z^{d} is the Kronecker delta, δ(0) = 1, δ(x) = 0, x ≠ 0.

Lemma 5.4. Let m ≥ 4 be an even integer and q = m/2. Then

q_{m,ε,1,1}(0) = ∑_{G∈F_{q}, e∈E[G]} c(G, e)K_{ε}(G, e), (5.12)

for some integers c(G, e) depending only on the graph G and the directed edge e ∈ E[G].

Proof. Observe that

K_{ε}(x_{1}) … K_{ε}(x_{m−1}) ⟨Y_{0}Y_{x_{1}}Y_{x_{1}+x_{2}} … Y_{x_{1}+···+x_{m−1}}⟩ ≠ 0,

only if the sequence 0, x_{1}, x_{1} + x_{2}, …, x_{1} + ⋯ + x_{m−1} of points in Z^{d} has the property that it consists of q pairs of points with the property that no two adjacent points of the sequence are the same. The q pairs of points do not have to be distinct. Hence we may write

q_{m,ε,1,1}(0) = −2 ∑_{i=1}^{q} a_{i}, (5.13)

where a_{i} corresponds to the term on the RHS of (5.10) where i of the q pairs are distinct. For a graph G ∈ F_{q} let K′_{ε}(G, e) be defined in the same way as K_{ε}(G, e) except that the summation on the RHS of (5.11) is over all y_{v} ∈ Z^{d} : v ∈ V [G] such that the q points y_{v} ∈ Z^{d} are distinct. Define F′_{q} as the subset of F_{q} consisting of regular graphs of degree 4 i.e., the degree of each vertex is exactly 4. Now by (B) of the definition of F_{q} and Euler’s theorem [1] a graph G ∈ F_{q} has an Eulerian path. Let c(G) be the number of Eulerian paths. For a graph G ∈ F_{q}, one can group directed edges into equivalence classes where two directed edges e, e′ are equivalent if there is a graph isomorphism of G taking e to e′. Let E′[G] be this set of equivalence classes. Then we may define K′_{ε}(G, ê) for ê ∈ E′[G] by K′_{ε}(G, ê) = K′_{ε}(G, e) for any e ∈ ê. Then one can see that a_{q} of (5.13) is given by

a_{q} = ∑_{G∈F′_{q}, ê∈E′[G]} c(G)K′_{ε}(G, ê).

Since we can obtain a similar representation for the a_{i}, 1 ≤ i < q, we have the formula (5.12) up to replacing K′_{ε}(G, e) by K_{ε}(G, e). The result therefore holds if we can show that for any graph G ∈ F_{q} and e ∈ E[G] there exist integers c_{G,e}(G′, e′), G′ ∈ F_{q}, e′ ∈ E[G′] such that

K′_{ε}(G, e) = ∑_{G′∈F_{q}, e′∈E[G′]} c_{G,e}(G′, e′)K_{ε}(G′, e′). (5.14)

The previous identity follows from Mayer’s trick to obtain an expansion for an non-ideal gas [7]. That is for a function φ : Z^{d} → R, one writes

exp [− ∑_{1≤i<j≤N} φ(y_{i} − y_{j})] = ∏_{1≤i<j≤N}[1 + f_{i,j}], (5.15)

where f_{i,j} = exp[−φ(y_{i} − y_{j})] − 1, and expands the RHS of (5.15) out. Now (5.14) follows from (5.15) on taking φ to be the function, φ(y) = 0, y ≠ 0, φ(0) = +∞. □

Lemma 5.5. Suppose d = 1, q ≥ 2 and G ∈ F_{q}, e ∈ G. If |G| is the number of vertices of G, then the limit

lim_{ε→0} ε^{|G|}K_{ε}(G, e) = K(G, e) (5.16)

exists.

Proof. Evidently it is sufficient to show that (5.16) holds with K_{ε}(G, e) replaced by K′_{ε}(G, e). Suppose |G| = k ≤ q and label the vertices of G as 1, 2, …, k with e_{+} = 1, e_{−} = k. Then

K′_{ε}(G, e) = ∑_{π∈S_{k}} K_{ε,π}(G, e),

where S_{k} is the group of permutations on 1, 2, …, k and

K_{ε,π}(G, e) = ∑_{y_{π1}<y_{π2}<···<y_{πk}} δ(y_{1})y_{k} ∏_{e′∈E[G], e′≠e} K_{ε}(y_{e′_{+}} − y_{e′_{−}}).

In view of (5.8) there are positive integers N_{j}, 1 ≤ j ≤ k − 1 such that

K_{ε,π}(G, e) = ± ∑_{y_{π1}<y_{π2}<···<y_{πk}} δ(y_{1})y_{k} ∏_{j=1}^{k−1} 2^{N_{j}}ν(ε)^{N_{j}[y_{π(j+1)}−y_{π(j)}]}.

Suppose now that πk_{1} = 1, πk_{2} = k and k_{1} < k_{2}. Then

K_{ε,π}(G, e) = ∑_{i=k_{1}}^{k_{2}−1} K_{ε,π,i}(G, e), (5.17)

where

K_{ε,π,i}(G, e) = ± ∑_{y_{π1}<y_{π2}<···<y_{πk}} δ(y_{1})[y_{π(i+1)} − y_{π(i)}] ∏_{j=1}^{k−1} 2^{N_{j}}ν(ε)^{N_{j}[y_{π(j+1)}−y_{π(j)}]}.

It is easy to see now just as in Lemma 5.1 that the limit

lim_{ε→0} ε^{k}K_{ε,π,i}(G, e)

exists. Hence from (5.17) the limit,

lim_{ε→0} ε^{k}K_{ε,π}(G, e) (5.18)

exists if k_{1} < k_{2}. A similar argument shows that the limit (5.18) also exists if k_{1} > k_{2}. It follows then from the previous identities that the limit (5.16) exists. □

Remark 1. Lemma 5.5 proves that (d) of Theorem 1.4 holds with ξ = 0. Observe that the number of graphs in F_{q} is of order q!. One can see this from an asymptotic formula for the number A_{q} of labeled 4-regular simple graphs on q vertices [18],

A_{q} ∼ [(4q)!e^{−15/4}]/[(2q)!96^{q}].

Next we wish to obtain the analogue of Lemma 5.4 for q_{m,ε,k,k′}(ζ) with ζ ∈ [−π, π]^{d} and at least one of k, k′ being 1. To do this we define for any G ∈ F_{q}, e ∈ E[G], a function K_{ε}(G, e, ζ) by

K_{ε}(G, e, ζ) = ∑_{{y_{v}∈Z^{d}:v∈V [G]}} exp[−iy_{e_{−}} · ζ]δ(y_{e_{+}}) ∏_{e′∈E[G], e′≠e} K_{ε}(y_{e′_{+}} − y_{e′_{−}}). (5.19)

Note that K_{ε}(G, e, ζ) is defined similarly to K_{ε}(G, e) of (5.11) and has the property K_{ε}(G, e, 0) = 0. For 1 ≤ k ≤ d we define K_{ε,k}(G, e, ζ) by

(5.20) K_{ε,k}(G, e, ζ) = ∑_{{y_{v}∈Z^{d}:v∈V [G]}} δ(y_{e_{+}}) [ ∏_{e′∈E[G], e′≠e} K_{ε}(y_{e′_{+}} − y_{e′_{−}})]

[ ∑_{e′∈E[G], e′≠e} exp[i(y_{e′_{+}} − y_{e_{−}}) · ζ] sgn[(y_{e′_{+}} − y_{e′_{−}}) · e_{k}] ∑_{y∈L_{k}(y_{e′_{+}} −y_{e′_{−}})} e^{−iy·ζ} ] .

Note that K_{ε}(G, e) of (5.11) is given by K_{ε}(G, e) = K_{ε,1}(G, e, 0). In the following lemma we need to define F_{q} for q = 0, 1. The set F_{1} is the single point vertex graph G with K_{ε}(G, e, ζ) = 0. The set F_{0} is also the single point vertex graph G with K_{ε}(G, e, ζ) = 1.

Lemma 5.6. Let m ≥ 4 be an even integer and q = m/2. Define A_{m,ε,k}(ζ) by

(5.21) A_{m,ε,k}(ζ) = ∑_{q′=2}^{q} ∑_{G∈F_{q′} , G′∈F_{q−q′}}

∑_{e∈E[G], e′∈E[G′]} c(G, e, G′, e′)K_{ε,k}(G, e, ζ)K_{ε}(G′, e′, ζ),

for suitable integers c(G, e, G′, e′) depending only on the graphs G, G′ and the directed edges e ∈ E[G], e′ ∈ E[G′]. Then

q_{m,ε,1,1}(ζ) = [1 + e^{−ie_{1}·ζ}] A_{m,ε,1}(ζ),
q_{m,ε,1,k}(ζ) =  A_{m,ε,k}(ζ),  1 < k ≤ d,
q_{m,ε,k,1}(ζ) =  e^{−ie_{k}·ζ}A_{m,ε,k}(ζ),  1 < k ≤ d.

Proof. We define A_{m,ε,k′}(ζ) as the coefficient of γ^{m} in the first term on the RHS of (4.9). Evidently the formulas for q_{m,ε,k,k′}(ζ) in the statement of the lemma follow from this. We need then to establish (5.21). This follows by the same argument as in Lemma 5.4. Note now that, unlike in Lemmas 5.3 and 5.4, the projection operators P in (4.9) make a contribution. Thus we obtain the factorisations K_{ε,k}(G, e, ζ)K_{ε}(G′, e′, ζ) in (5.21) with G ∈ F_{q′}, G′ ∈ F_{q′′}, where q′ + q′′ = q. □

Lemma 5.7. Suppose d = 1 and K ⊂ R is a compact set. Let q ≥ 2 and G ∈ F_{q}, e ∈ G. If |G| is the number of vertices of G, then the limits

(5.22)  lim_{ε→0} ε^{|G|−1}K_{ε}(G, e, εξ) = K(G, e, ξ)
(5.23)  lim_{ε→0} ε^{|G|}K_{ε,1}(G, e, εξ) = K_{1}(G, e, ξ)

exist, uniformly for ξ ∈ K.

Proof. The proof of (5.22) follows exactly the proof of Lemma 5.5. To prove (5.23) we proceed similarly. Thus we focus on one term in the summation over e′ ∈ E[G], e′ ≠ e in (5.20). As in Lemma 5.5 we fix a permutation π and consider the sum over y_{π1} < y_{π2} < ⋯ < y_{πk}. Suppose that πk_{1} = e′_{−}, πk_{2} = e′_{+} and 60 Joseph G. Conlon

k_{1} < k_{2}. Then

sgn(y_{e′_{+}} − y_{e′_{−}}) ∑_{y∈L_{1}(y_{e′_{+}} −y_{e′_{−}})} e^{−iyζ} =

∑_{i=k_{1}}^{k_{2}−1} (exp[−i(y_{π(i)} − y_{π(k_{1})})ζ] ∑_{z=0}^{y_{π(i+1)}−y_{π(i)}−1} e^{−izζ} ) .

If we consider a fixed i on the RHS of the previous equation and sum over y_{π1} < y_{π2} < ⋯ < y_{πk}, then we see as in Lemma 5.5 that the contribution of this summation to (5.23) converges uniformly for ξ ∈ K as ε → 0. Since we can similarly argue for all the other contributions we conclude that the uniform limit (5.23) exists. □

It is clear that (d) of Theorem 1.4 follows from Lemmas 5.6 and 5.7.

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University of Michigan, Department of Mathematics, Ann Arbor, MI 48109-1109 conlon@math.lsa.umich.edu http://www.math.lsa.umich.edu/˜ conlon/

This paper is available via http://nyjm.albany.edu:8000/j/2002/8-3.html.