Large deviations for stable like random walks on $\mathbb Z^d$ with applications to random walks on wreath products

We derive Donsker-Vardhan type results for functionals of the occupation times when the underlying random walk on $\mathbb Z^d$ is in the domain of attraction of an operator-stable law on $\mathbb R^d$. Applications to random walks on wreath products (also known as lamplighter groups) are given.


Introduction
This work addresses two closely related questions of independent interests. From the point of view of random walks on the lattices Z d , we extend the well-known large deviation theorem of Donsker and Varadhan regarding the Laplace transform of the number D n of visited points before time n. The theorem of Donsker and Varadhan, [8], treats random walks driven by measure µ in the domain of normal attraction of a symmetric stable law of index α ∈ (0, 2) (as well as the Gaussian case).
We generalize this result to random walks driven by a measure in the domain of attraction of an operator-stable law. For instance, this includes laws that are "stable" with respect to anisotropic dilations of the type δ t (x 1 , . . . , x d ) = (t 1/α1 x 1 , . . . , t 1/α d x d ) (1.1) with α i ∈ (0, 2), 1 ≤ i ≤ d. In this case, the generalization of the theorem of Donsker and Varadhan reads as follows. Letη denote the Fourier transform of the distribution η on either Z d or R d . for some symmetric finite measure M on S d−1 whose support generates R d . Then lim n→∞ 1 n d/(d+α) log E(e −νDn ) = k(ν, η) ∈ (0, ∞) where η is the probability distribution on R d such thatη = e −Θ and d α = d 1 1 αi . As in the classical Donsker-Varadhan theorem, the constant k(ν, η) is described by a variational formula. In its most natural generality (see Theorem 3.9 with F (s) = 1 (0,∞) (s) = 1 − δ 0 (s)), this theorem involves more general dilation semigroups of the form t E = ∞ 0 where E is an invertible matrix with eigenvalues in [1/2, ∞) (E may not be diagonalizable and, even if E is diagonalizable, it may not be diagonalizable in a basis of Z d vectors). In this case, the associated limit law η is "operator-stable" with respect to the dilation structure t E , t > 0, and the real α ∈ (0, 2) is given by α = tr(E)/d.
In fact, we are also interested in a different generalization of the Donsker-Varadhan Theorem. Given a random walk on Z d driven by a symmetric measure µ, let l(n, x) denotes the number of visits at x up to time n. We are interested in obtaining a large deviation result for the Laplace transform of more general functionals of the occupation time vector (l(n, x)) x∈Z d than the number of visited sites, D n = #{x : l(n, x) = 0}. For instance, we are interested in the asymptotic behavior of − log E e −λ x∈Z d log l(n,x) and, more generally, − log E e −λ x∈Z d F (l(n,x)) when F belongs to some appropriate class of functions. For simplicity, in the next theorem, we consider the case where the function F is simply F (s) = s γ , γ ∈ (0, 1) and the dilation structure is given by (1.1 (1.2). Then, for γ ∈ (0, 1), lim n→∞ n − γ+τ (1−γ) 1+τ (1−γ) log E e −ν x ℓ(n,x) γ = k(ν, η, γ) ∈ (0, ∞) where η is the probability distribution on R d such thatη = e −Θ and τ = d 1 1 αi .
In the case of where µ is symmetric finitely supported and the dilation structure is the isotropic δ t (x) = √ tx (in this case, the limit law η is Gaussian), this result is contained in [3]. See Theorem 1.2 (with p = 1 and H(s) = s γ ) and Section 2.3 in [3]. Indeed, one of the contributions of [3] is to show how to deduce results such as Theorem 1.2 from the Donsker-Varadhan large deviation principle for the scaled version of the occupation measure of the underlying process. In order to treat processes that fall in the operator stable realm, we modify some of the arguments in [3] and rely more on the original techniques of Donsker and Varadhan.
The version of Theorem 1.2 which treats dilations of the form t E and functionals x F (ℓ(n, x)) associated to more general functions F than power functions is given in Theorem 3.9 and in Section 5.2. Except for a few technical adaptations to the operator stable context, the proofs of Theorem 1.1 is a routine generalizations of the proof given by Donsker and Varadhan in the stable context. Similarly, the proofs of Theorems 1.2-Theorem 3.9 involves an adaptation of the techniques of Donsker and Varadhan and [3].
In developing these results in the operator stable context, we are motivated by applications to the study of random walks on a class of groups called wreath products. These groups are also known as lamplighter groups. The wreath product K ≀H, i.e., the lamplighter group with base-group H and lamp-group K, will be defined precisely below. If we think of the elements of K as representing different colors (possibly countably many different colors), then an element of K ≀ H can be viewed as a pair (h, η) where h is an element of H (the position of the lamplighter on the base H) and η = (k h ) h∈H ∈ K H is a finite configuration of colors on H in the sense that only finitely many h ∈ H have k h = e K where e K is the identity element in K (only finitely many lamps are turned on). This description does not explain the group law on K ≀ H but captures the nature of the elements of the wreath product K ≀ H. The identity element in K ≀ H has the lamplighter sitting at e H and all the lamps turned off (k h = e K for all h ∈ H). In one of the simplest instance of this construction, H = Z (a doubly infinite street) and K = Z/2Z (lamps are either off (0) or on (1)).
We are interested in a large collection of random walks on wreath products which can be described collectively as the "switch-walk-switch" walks. See also [16,22]. Namely, we are given two probability measures, one on H, call it µ, and one on K, call it ν. The measure µ drives a random walk on H which describes the moves of the lamplighter (i.e., the first coordinate, h, in the pair (h, η) ∈ K ≀ H). The measure ν drives a random walk on K whose basic step is interpreted as "switching" between lamp colors. Based on this input, we construct a probability measure q = q(µ, ν) on K ≀ H (this measure q is defined precisely later in the paper). The basic step of the walk driven by q can be accurately describes as follows: the lamplighter switches the color of the lamp at its standing position (using ν), takes a step in H (using µ) and switches the color of the lamp at its new position (using ν). These different moves are, in the appropriate sense, made independently of each other hence the name, switch-walk-switch. Let us insist on the fact that we will be interested here in cases when the measures µ and ν are not necessarily finitely supported. Now, an elementary argument shows that the probability of return q (n) (e) of the random walk driven by q on K ≀ H is given by 0 is the random walk on H driven by µ and l * (n, h) is an essentially trivial modification of the number of visits of (X m ) ∞ 0 to h up to time n. The expectation is relative to the random walk (X m ) ∞ 0 on H, started at e H . This observation goes back to [22] and is the basis of the analysis developed in [16]. If we set F (m) = − log ν (2m) (e K ) then it follows under mild assumptions that In words, the log-asymptotic of the probability of return of a switch-walk-switch random walk on the wreath product K ≀ H is given by the appropriate version of the Donsker-Varadhan large deviation theorem for the random walk on the base H driven by µ. The particular functional h F (l(n, h) that needs to be treated depends on the nature of the lamp-group K and the measure ν. Formula (1.3) is particularly interesting because, in the general context of random walks on groups, precise log-asymptotic of the probability of return are hard to obtain. The following result serves to illustrate this point.
Let ν be any finite symmetric measure on Z D with ν(0) > 0 and generating support. Let δ t be the anisotropic dilation on R d defined at (1.1). Let µ be a symmetric measure on Z d as in Theorem 1.1 with δ −1 n (µ (n) ) =⇒ η andη = e −Θ . On the wreath product Z D ≀Z d consider the switch-walk-switch random walks q = ν * µ * ν. Then lim Here, λ 1 (Θ, U ) is the principle eigenvalue of the infinitesimal generator L Θ with Dirichlet boundary condition in U (By definition, L Θ f = Θf ).
Remark 1.4. Assume that the dilations δ t are isotropic with α i = α, i = 1, . . . , d and that there is an Euclidean norm Qx, x such that Θ(ξ) = Qξ, ξ α . Then λ Θ is achieved on an Euclidean ball for the Euclidean structure provided by Q −1 , namely, the Euclidean ball whose volume is one. Note that the volume is computed here with respect to the Lebesgue measure corresponding to the fixed square lattice Z d ⊂ R d . This fact is well-known when α = 2 and follows from [1] when α ∈ (0, 2).
For any finitely generated group G and any α ∈ (0, 2), [2] introduces a non-increasing function which, by definition, provides the best possible lower bound valid for every measure µ on G satisfying the weak-α-moment condition Here |g| is the word-length of G with respect to some fixed finite symmetric generating set and ρ α (g) = (1 + |g|) α . For instance, it is well know and easy to see that Here and throughout this paper, on the relevant real interval or on N. We use ≃ only when at least one of the functions f, g is monotone (or roughly monotone).
The main results of the present work allow us to complement some of the lower bounds proved in [2] for Φ G,ρα with matching upper bounds (note that upper bounds on Φ G,ρα are proved by exhibiting a measure with finite weak-αmoment and the appropriate return probability behavior).

Assume that K is polycyclic with exponential volume growth. Then
Remark 1.6. The lower bounds are from [2]. The upper bound in the first statement is already in [2] since it is based on the classical large deviation result in [8]. The upper bounds in Statements 2 and 3 make use of the extensions of [8] in the spirit of [3] developed here.

Operator-stable laws
For α ∈ (0, 2), the rotationally symmetric α-stable law with density f α on R d is the probability distribution whose Fourier transform is e −|ξ| α . It is embedded in a convolution semigroup with density f t α which satisfies f t More generally, a probability measure µ on R d is called a (non-degenerate) symmetric α-stable law if its support is R d and it is embedded in a probability semigroup µ t such that δ α t (µ) = µ t . A necessary and sufficient condition for that property is thatμ = e −Θ with where M is a finite Borel measure on S d−1 whose support generates R d (that is, the Lévy measure W of µ satisfies δ α t (W ) = tW and its support generates R d ). In the next section, we briefly review the definition of operator-stable laws. In this definition, the role of the isotropic dilations is played by more general one-parameter groups of transformations t E = ∞ 0 where E is an endomorphism of the underlying vector space. For a detail account of the theory of operator-stable laws, see [13,15]. Given a Borel measure µ, we let t E (µ) be the Borel measure defined by t E (µ)(A) = µ(t −E (A)).

Operator-stable laws
Let V be a finite dimensional vector space equipped with the Euclidean scalar product ·, · . Let M 1 (V) denote the set of probability measures on V. Given µ ∈ M 1 , letμ = e −ψ denotes its Fourier transform. Let ID(V) denotes the set of infinitely divisible laws on V. Throughout this section, we use notation compatible with [13]. Recall that if µ ∈ ID(V) with Fourier transform e −ψ then e −tψ is the Fourier transform of a probability measure µ t and (µ t ) t≥0 is a continuous convolution semigroup of measure (uniquely determined by µ). Of course, for µ ∈ ID(V), the function ψ admits a Levy-Khinchine representation so that xi → ψ(ξ) is the sum of three terms, namely, the drift term −i c, ξ with c ∈ V, the Gaussian term 1 2 Qξ, ξ , where Q ∈ End + (V), and the generalized Poisson term where W is a Levy measure. Following [13], we call the triple (c, Q, W ) the L-K triple of µ (this triple is uniquely determined by µ). We will be interested in the symmetric case where c = 0 and W (x) = W (−x). In this case, the Poisson term of the Levy-Khinchine formula equals In general, we let η Q be the (Gaussian) law associated with the triple (0, Q, 0) and e(W ) the (generalized-Poisson) law associated with (0, 0, W ).
Definition 2.1 (Definition 1.3.11 [13]). A law η ∈ ID(V) is said to be operatorstable if there exist E ∈ End(V) and a mapping a : for all t ∈ R × + . In this case, E is called an exponent of η. Let EXP(η) denote the set of exponents of η. If a ≡ 0, η is said to be strictly operator-stable.
One can always split an operator-stable law into a Gaussian part and a generalized Poisson part that are supported on supplementary linear subspaces of V.
The subspace supporting the Gaussian part is either trivial or associated with the eigenvalues z of E with Re(z) = 1/2 of E. The subspace spanned by the support of W is associated with the eigenvalues z of E with real part strictly larger than 1/2. Both the Gaussian part e(Q) and the Poisson part e(W ) are operator stable with exponent E. Further, T E (W ) = tW . See the splitting theorem, [13, Lemma 1.3.12 and Theorem 1.3.14]. Now we restrict our attention to symmetric operator stable laws (so that c = 0, W (dx) = W (−dx)). Since t E (W ) = tW , the Fourier transform of e(W ) can be written (with S ⊂ V, the unit sphere) where M is a finite measure on S. Compare with the hypothesis in Theorem 1.1 and Theorem 1.2. Choose an orthonormal basis {e i } on V with respect to inner product <, > . The generating functional A of (η t ) t≥0 (see [13, 1.3.16]) is given for f ∈ C 2 (V) by One can also write down the Dirichlet form of the continuous convolution semigroup (η t ) t≥0 as From the splitting theorem [13, Theorem 1.3.14] it follows that (q ij ) is semipositive definite and that the subspace where it is positive definite is the support of the Gaussian part e(Q).
Example 2.1 (Anisotropic radial operator-stable laws). One can construct operator-stable laws with respect to non-isotropic homogeneous norms. On . We may assume that a 1 = min 1≤i≤d a i . Since we can think of t E as dilations scaling differently in different coordinates. The following norm was considered in [14]. Let B = {x : x < 1} be the open Euclidean unit ball, define From Theorem 1 in [14], · * ,E is a sub-additive homogeneous norm. Set Clearly, t E (W ) = tW for all t ∈ R × + . Let η be the generalized Poisson law with L-K triple (0, 0, W ). Then η is operator-stable with exponent E. Note that the assumption a 1 > 1 2 is needed so that W is a Lévy measure. Example 2.2 (Anisotropic axial operator-stable laws). Let E be as in the previous example. For α ∈ (0, 2) let ν α be the one-dimensional symmetric αstable law (so thatν α (y) = e −|y| α ). Let η be the product measure on V = R d given by η = ⊗ d 1 ν 1/ai so thatη(ξ) = e − d 1 |ξi| 1/a i . Clearly, η is operator-stable with exponent E. Note that in this case, the Levy measure is supported on the union of the axes.

Domain of operator-attraction
For full probability laws, the class of operator-stable laws coincides with limit distributions of normalized sums of i.i.d. random variables and convergence in law of normalized sums can be characterized in terms of convergence of Fourier transforms or convergence of generators as in Trotter's theorem. More precisely, we have the following equivalent characterizations of convergence.

For any
Next, we introduce the definition of strict domain of operator-attraction. Remark 2.4. With this definition, DOA s (η) = ∅ is equivalent to saying η can be obtained as the limiting distribution of convolution powers of some µ after normalization (but without re-centering). The word "strict" refers to the absence of re-centering. When T n can be taken as the isotropic matrix b n Id, b n ∈ R + , this agrees with the definition of the strict domain of attraction.
Example 2.3. Let U (resp. V ) be a random variable on Z in the domain of normal attraction of the α (resp. β) symmetric-stable law ν α (ν β resp.) on R. The measure η on R 2 η(dx, dy) = ν α (dx) ⊗ ν β (dy) is operator stable with exponent and let µ denote the distribution of (X, Y ) T . In order to obtain convergence of µ (n) we need to rotate back by a rotation of angle θ then normalize componentwise. That is, setting we have η = lim n→∞ T n (µ (n) ). So, in this case, µ ∈ DOA s (η) but does not belong to DNOA s (η).
Remark 2.6. Theorem 4.11.5 of [15] gives a practical criterion to show that a given measure ν belongs to the domain of normal attraction of a full operatorstable law η without Gaussian part. More precisely, the following statement is a simple modification of [15,Theorem 4.11.5]. Let η be operator-stable, symmetric, with no Gaussian part and Lévy measure W given by Let E ∈ EXP(η). A necessary and sufficient condition for a probability measure ν to be such that n −E (ν (n) ) =⇒ η is that for any measurable Ω ∈ S d with M (∂Ω) = 0.
The theorem below is a characterization of strictly operator-stable laws as those adapted distributions whose domain of strict operator-attraction is nonempty.
Theorem 2.7 (Theorem 1.6.4 [13] ). For η ∈ M 1 a (V) the following assertions are equivalent: For µ ∈ DOA s (η), the choice of normalization sequence T n is in general not unique. In particular, we can adjust T n using the symmetries of the limiting distribution η and the convergence still holds.
The following technical result is important for our purpose. It says that we can always adjust the normalization sequence by elements in Inv(η), so that the new normalization sequence has nice regular variation properties. Theorem 2.9 (Theorem 1.10.19 [13] ). Suppose µ is in the strict domain of attraction of a full operator stable law η, that is, there exists a sequence of invertible matrices B n ∈ GL(V) such that Then there exists a modified normalization sequence {B ′ n = B n S n }, S n ∈ Inv(η), hence still fulfilling where the convergence is uniform in t on compact subsets of R × + .

Two more examples on Z 2
In this subsection we discuss two examples on Z 2 that are in the strict domain of operator-attraction of some operator-stable laws. For later use, we include the additional requirement that the inverse of the normalization sequence preserve the lattice Z 2 .
Note that a key point in these examples is that they describe probability measures supported on the square lattice Z 2 ⊂ R 2 which implies a certain rigidity in the choice of the Euclidean structure on R 2 .
Example 2.5. Let e 1 , e 2 be the standard basis for R 2 . Consider the two unit vectors u 1 = 1 √ 2 (e 1 + e 2 ) and u 2 = 1 √ 1+π 2 (e 1 + πe 2 ). Let µ 1 , µ 2 be probability measures defined by where c 1 and c 2 are normalizing constants and α, β ∈ (0, 2). Take where λ 1 and λ 2 are positive constants determined by µ. Consider the generalized Poisson law η with Lévy measure W given by Note that W is supported on the union of the one-dimensional subspaces Ru 1 and Ru 2 and η is operator-stable with exponent E. The law η can be viewed as a product of two one-dimensional symmetric stable laws supported on Ru 1 and Ru 2 and of exponents α and β, respectively (the exact scale parameter for each of these stable laws is determined by the constants λ 1 , λ 2 ). From the convergence theorem, Theorem 2.2, µ ∈ DNOA s (η). Set where ⌊·⌋ means take integer parts of each matrix entry. Then n E − B n is a matrix with entries in [0, 1) and it follows that B n · n −E → I. We conclude that Example 2.6. Take u 1 , u 2 , P, α, β and E the same as in the previous example. Write δ r (x 1 , x 2 ) = (r 1/α x 1 , r 1/β x 2 ). Let S denote the Euclidean unit circle, σ the Lebesgue measure on S. Let Γ be the union of the two arcs [0, π 2 ] and [π, 3π 2 ] of S. Define the Lévy measure W as Then W is supported in the cone {x ∈ R 2 : , that is the pushforward of measure W under linear transformation P. Let η be the generalized Poisson law with Lévy measure P W. Take a discrete approximation µ of P W supported on Z 2 by setting

Functionals of the occupation time vector
Given a probability measure µ on the lattice Z d , let (X i ) ∞ 0 be the associated random walk. Let (l(n, x)) x∈Z d be the occupation time vector at time n where l(n, x) = #{k ∈ {0, . . . n} : In this section we introduce basic natural hypotheses on µ and F under which we can derive the log-asymptotic behavior of E e − x∈Z d F (l(n,x)) . Definition 3.1 (Convergence assumption). We say that µ satisfies the convergence assumption (C-B n ) if there exists a sequence of invertible matrices B n ∈ Z d×d and a probability distribution η such that Note that (C-B n ) requires the matrices B n to have integer entries so that B n Z d ⊂ Z d . Note also that the distribution η is strictly operator-stable. Under the convergence assumption (C-B n ), [12] provides a local limit theorem that plays an important role in the proof of the uniform large deviation principle. Theorem 3.3 (Theorem 6.4 [12]). Suppose µ is in the domain of attraction of a symmetric, adapted strictly operator-stable law η on R d with density g, that is, there exists a sequence of invertible matrices B n such that Remark 3.4. Note that the density g of an operator-stable law is always smooth.
In [8] it is essentially proved, although not stated explicitly, that given the local limit theorem, the scaled occupation time measures satisfy a uniform large deviation principle in L 1 . We will state and outline the proof of the large deviation principles later in this paper.
Remark 3.5. It is somewhat surprising that, in this case, the "weak limit assumption" always implies the local limit theorem. The proof in [12] relies on the Fourier transform. On the Heisenberg group, there are measures that converges to a (Heisenberg group) Gaussian law, but do not satisfy the local limit theorem.
Example 3.1. Fix α, β ∈ (0, 2) and consider the probability measure µ on Z 2 ⊂ R 2 given by one dimensional symmetric stable laws with parameters α, β, respectively. Then provides a local limit theorem for µ (n) ((x, y)) in the form where f t α is the density of the symmetric stable semigroup, i.e., has Fourier transform e −t|ξ| α and c 1 , c 2 are appropriate constants.
Next we introduce a scaling assumption regarding the function F . It is the operator-stable analog of the scaling assumption in [3]. Definition 3.6 (Scaling assumption). Let B n be as in condition (C-B n ). We say that a function F : [0, ∞) → [0, ∞) satisfies the scaling assumption (S-B n -a n ) if F is concave, sub-additive, increasing with F (0) = 0 and there exist a nondecreasing sequence n → a n ∈ N and a limiting function uniformly over compact sets in (0, ∞).
The following technical proposition is crucial. It is analogous to [3, Proposition 1.1]. The proof is given in the Appendix.
Proposition 3.7. Assume the convergence assumption (C-B n ) and the scaling assumption (S-B n -a n ) as above. Then there exists γ ∈ [0, 1] such that Moreover, there exists κ > 0 such that lim n→∞ a ⌊λn⌋ a n = λ κ for all λ ∈ R + and lim n→∞ log a n log n = κ.
Definition 3.8. Following [3], given µ satisfying (C-B n ) and a function F : , we say that the pair (F, (B n )) is in the γ-class, if there is a sequence a n such that the scaling assumption (S-B n -a n ) is satisfied, and the limiting function F is homogeneous with exponent γ.
The following statement is the main result of this paper. The proof is given in Section 5.2.
Further, for any ǫ > 0 small enough there is R > 1 such that Example 3.2. Theorems 1.1 and 1.2 are special cases of Theorem 3.9. In both cases, let E be the diagonal matrix with i-th diagonal entry 1/α i ∈ (2, ∞). Let η be an operator-stable law with exponent E and Fourier transformη = e −Θ . Let µ be a measure such that It is easy to see that condition (S-B n -a n ) with F = 1 (0,∞) .
In this case the theorem yields the existence of a constant k ∈ (0, ∞) such that .
The previous examples treat cases where µ belongs to the domain of normal attraction of η. It is worth pointing out that Theorem 3.9 does not require normal attraction. For example, consider the case where µ is supported on Z and is of the form is continuous and regularly varying of index 1 + α, α ∈ (0, 2). By a classical result (see [10]), µ is in the domain of attraction of an α-stable law η on R. The nor- Note that ψ is regularly varying of index −1/α. Suppose that F (s) = s γ , γ ∈ (0, 1). The sequence a n in Theorem 3.9 is then given by equation (S-B n -a n ), that is, n −1 a n b an (n/b an ) γ = 1, equivalently, a n ψ(1/a n ) 1−γ = κ γ−1 n 1−γ . It follows that a n varies regularly of index α(1 − γ)/(1 + α − γ). Of course, a n can be computed more explicitly in terms of φ.

Applications to random walks on groups
This section applies the large deviation asymptotics of Theorem 3.9 to obtain precise information about the decay of the return probability of random walks on wreath products with base Z d . We treat certain classes of random walks with unbounded support on the base and we allow a large class of lamp groups.

Random walks on wreath products
First we briefly review definition of wreath products and a special type of random walks on them. Our notation follows [16]. Let H, K be two finitely generated groups. Denote the identity element of K by e K and identity element of H by e H Let K H denote the direct sum: The elements of K H are functions f : H → K, h → f (h) = k h , which have finite support in the sense that {h ∈ H : f (h) = k h = e K } is finite. Multiplication on K H is simply coordinate-wise multiplication. The identity element of K H is the constant function e K : h → e K which, abusing notation, we denote by e K . The group H acts on K H by translation: The wreath product K ≀ H is defined to be semidirect product In the lamplighter interpretation of wreath products, H corresponds to the base on which the lamplighter lives and K corresponds to the lamp. We embed K and H naturally in K ≀ H via the injective homomorphisms Let µ and ν be probability measures on H and K respectively. Through the embedding, µ and ν can be viewed as probability measures on K ≀ H. Consider the measure q = ν * µ * ν on K ≀ H. This is the switch-walk-switch measure on K ≀ H with switch-measure ν and walk-measure µ.
Let (X i ) be the random walk on H driven by µ, and let l(n, h) denote the number of visits to h in the first n steps: l(n, h) = #{i : 0 ≤ i ≤ n, X i = h}.

Set also
From [16], probability that the random walk on K ≀ H driven by q is at (h, g) ∈ K ≀ H at time n is given by Note that E stands for expectation with respect to the random walk (X i ) ∞ 0 on H started at e H .
From now on we assume that ν satisfies ν(e K ) = ǫ > 0 so that Under these circumstances, we have so that we can essentially ignore the difference between l and l * .
so that, for any g ∈ H, Definition 4.1 (weak scaling assumption). We say that ν satisfies the upper weak scaling assumption (US-B n -a n ) if there exist a constant c 0 > 0 and a function F : [0, ∞) → [0, ∞) satisfying (S-B n -a n ) and such that (US-B n -a n )) We say that ν satisfies the lower weak scaling assumption (LS-B n -a n ) if there exist a constant C 0 < ∞ and a function F : If F K satisfies both the upper and lower conditions, then we say it satisfies the weak scaling assumption (WS-B n -a n ).
We can now use the large deviation asymptotics to estimate the return probability on wreath product K ≀ Z d .
Theorem 4.2. Let µ be a symmetric probability measure on Z d which satisfies the convergence assumption (C-B n ). Let ν be a symmetric probability measure on K with ν(e K ) > 0.
• Assume that ν satisfies (US-B n -a n ). Then the switch-walk-switch measure q = ν * µ * ν on K ≀ Z d satisfies lim sup n→∞ a n n log q (n) (e) ≤ −k η, c 0 F .
• Assume instead that ν satisfies (LS-B n -a n ). Then we have lim inf n→∞ a 2n 2n log q (2n) (e) ≥ −k η, C 0 F . Remark 4.3. Roughly speaking, this theorem says the following: Assume we know how to normalize µ (n) on the base Z d via a transformation B n so that it converges to a limiting distribution η. Assume we know the behavior of the probability of return of the random walk on K driven by ν in the sense that log(ν (2n) (e K )) ≃ −F (n). Then q (n) (e) ≃ exp(− n an ), where a n can be computed from the scaling relation a n det(B an ) n F n det(B an ) ≃ 1.
Proof. The first statement follows immediately from (3.1) in Theorem 3.9. The second statement is deduced from (3.2) as follows. Since q is symmetric, we have q (2n) (e H ) ≥ q (2n) (g) for any g ∈ K ≀ Z d . In particular, if B(r) = B Z d (r) is the ball of radius r in the lattice Z d then, by (4.1), Picking r = Ra 2n and using the fact that a n has regular variation of order κ > 0 (see Proposition 3.7), one easily deduces from (3.2) that lim inf n→∞ a 2n 2n log q (2n) (e) ≥ −k η, C 0 F . as desired. Assume that F is of the form F (y) = y γ ℓ(y) where γ ∈ [0, 1] and ℓ is a slow varying function (at infinity) such that ℓ(t a ℓ(t) b ) ∼ c(a)ℓ(t) for any a > 0 and b ∈ R. (e.g., ℓ(t) = (log t) β , β ∈ R). Let q be the switch-walk-switch measure on K ≀ Z d associated with µ and ν. Then .
The next theorem captures the fact that a better understanding of the return probability on the lamp-group K leads to a more precise asymptotics for q (n) (e).
Theorem 4.4. Let µ be a symmetric probability measure on Z d which satisfies the convergence assumption (C-B n ). Let ν be a symmetric probability measure on K with ν(e K ) > 0. Assume that the function F K (n) = − log ν (2n) (e K ) satisfies the scaling assumption (S-B n -a n ). Then the measure q = ν * µ * ν on

Assorted examples
In this section we describe a number of explicit applications of Theorems 4.2 and 4.4.
This measure is quite obviously in the domain of normal operator attraction of η a = ⊗ d 1 η αi where η αi is a measure on R which is symmetric and α i -stable. In particular, the diagonal d × d matrix E a with i-th diagonal entry 1/α i is in EXP(η a ). The Dirichlet form E ηa associated to the limit law η a is best described via Fourier transform as . . d (the scale parameters c i are related but not equal to c(α i )).  For S and a as defined above, let µ S,a the the probability measure on G = K ≀ H defined by In words, this walk takes steps along the (discrete) one parameter groups s i = {s n i , n ∈ Z} ⊂ G and the steps along s i are distributed according to a symmetric stable-like power law with exponent α i . These measures µ S,a are very natural from an algebraic point of view and one expects that the properties of the associated random walks depend in interesting way on the structure of the group G, the generating k-tuple S and the choice of the k-dimensional parameter a.
The Dirichlet forms E µS,a and E q associated with the measures µ S,a and q on G satisfy E µS,a ≃ E q .
Hence it follows from [17] that log µ (n) S,a (e) ≃ −n d/(d+β) (log n) β/(d+β) where β is as above. Note that β depends only on the first d coordinates of the parameter a = (α i ) d+D 1 . In this sense, the random walks associated with the collection of the measures µ S,a when a varies can distinguish among the d + D generators s i , 1 ≤ i ≤ d + D of K ≀ H between those which come from H and those which come from K.
Note that ≀ is not associative so that this iterated wreath product is different from the iterated wreath product Z 2 ≀ (· · · ≀ (Z ≀ Z) . . . ) considered in [9]. Here we are iterating the lamps while in [9] the base is iterated. Set For each i = 1, . . . , k, fix α i ∈ (0, 2) and a probability measure µ i on Z di which is symmetric, satisfies µ i (0) > 0 and is in the domain of normal attraction of the rotationally α i -stable law η i on R di . Let q 0 be the uniform measure on Z 2 = {0, 1}. Iteratively, define the switch-walk-switch probability measure Applying Corollary 4.4 iteratively, we obtain where the constant c k can be obtained as follows. The constant c 1 is given by [8] whereas, for 2 ≤ i ≤ k and referring to (3.1)- Similarly, we can consider the iterated wreath product starting with lamp group Z d0 instead of Z 2 and q 0 = µ 0 in the domain of normal attraction of the rotationally symmetric α 0 -stable distribution on R d0 . In this case, we obtain The constant c k can be obtained iteratively with c 1 = k η, d0 α0 1 {y>0} and c i = k υ i , F i , with F i as above for 1 < i ≤ k.

Application to fastest decay under moment conditions
This section describes applications of Theorem 4.2 to the computation of the group invariants Φ G,ρ introduced in [2]. Recall that [17] introduce a group invariant Φ G which is a decreasing function of n (defined up to the equivalence relation ≃) such that for all finitely supported symmetric probability measure φ with generating support. Let ρ be a function ρ : G → [1, ∞).
The weak ρ-moment of the probability measure µ is defined as Definition 4.6 (Definition 2.1 [2]: Fastest decay under weak ρ-moment). Let G be a locally compact unimodular group. Fix a compact symmetric neighborhood Ω of e. Let S Ω,K G,ρ be the set of all symmetric continuous probability densities φ on G with the properties that φ ∞ ≤ K and W (ρ, φdλ) ≤ K sup Ω 2 {ρ}. Set Here we will only consider the case when G is finitely generated and ρ is one of the power function ρ α (x) = (1+|x|) α , α ∈ (0, 2) where |·| is the word distance on a fixed Cayley graph of G. We are concerned with the decay of Φ Ω,K G,ρα when n is large. By Proposition 1.2 [2], we can drop the reference to Ω and K. Lower bounds on Φ G,ρ follow from general comparison and subordination results, see [2]. Here, we are interested in obtaining upper bounds on Φ G,ρ .
By definition, for any probability measure φ on G which satisfies the weak ρ-moment condition, n → φ (2n) (e) provides an upper bound for Φ G,ρ . When G is a wreath product G = K ≀Z d , we can use measures of the form φ = ν * µ * ν and apply Theorem 4.2 to estimate φ (2n) (e). Also because of the natural embedding of K and Z d in the wreath product K ≀ Z d , it's not hard to estimate the needed weak ρ-moment of φ. We shall see that, in certain cases, the measures φ of this type actually achieve the fastest decay rate given by Φ G,ρ , up to the equivalence relation ≃. This technique was already used in [2, Theorem 5.1] to determine Φ Z2≀Z d ,ρα . In this case, the classical result of Donsker and Varadhan [8] is all one needs. In the examples below, we use Theorem 4.2 to obtain precise upper bounds on Φ K≀Z d in some other cases.
Example 4.6. In this example we consider G = K ≀ Z d when K is either finite or has polynomial growth or has exponential volume growth and Φ K (n) ≃ exp(−n 1/3 ). The first case is already treated in [2]. We note that these three cases exhaust all possibilities when K is a polycyclic group. The third case also covers the situations when K is the Baumslag-Solitar group or the lamplighter group Z 2 ≀ Z.

Assume that K has polynomial volume growth. Then
3. Assume that K has exponential growth and satisfies Φ K (n) ≃ exp(−n 1/3 ). Then Proof. The lower bounds can be obtained by applying [2,Theorem 3.3]. For this purpose, one needs to compute the function Φ G . For q = ν * µ * ν on G = K ≀ Z d , where µ and ν are associated with simple random walk (with holding) on Z d and K respectively, we can apply Theorem 4.2 to obtain q (2n) (e) ≃ Φ G (n). The case when K is finite is already treated in [16,2]. When K has polynomial volume growth, then ν (2n) (o) ≍ n − D 2 and, as in Example 4.2, These estimates on Φ G allow us to appeal to [2,Theorem 3.3] to obtain the stated lower bounds for Φ G,ρα .
To prove the stated upper bounds, it suffices to exhibit a probability measure measure in S G,ρα that has the proper decay. On Z d , set Then µ α is in the domain of normal attraction of the rotationally symmetric α-stable distribution on R d and it has a finite weak α-moment.
In the case when K is of polynomial volume growth, take q α = ν * µ α * ν, where ν is simple random walk on K. Then ν * µ α * ν has weak α-moment and, by This matches the previously proved lower bound.
In the second case, when K has exponential growth, let U be a symmetric generating set of K. As in [2,Theorem 4.10], pick p i = c α 4 −iα with ∞ 1 p i = 1 and set Then ν α has weak α-moment on K and, by [2, Theorem 4.1] Then ν α * µ α * ν α has weak α-moment on G. Applying Theorem 4.2 and the computations of Example 4.1 to q α = ν α * µ α * ν α , we obtain Fix α ∈ (0, 2). If K is finite then we we have If K has polynomial volume growth, then These are the results stated as Theorem 1.7 in the introduction.

Donsker and Varadhan type large deviations
The goal of this section is to outline the proof of Theorem 3.9, the key result of this article. The proof follows [8] closely. Several other classical sources are also needed to put together the necessary details.

Statement of the large deviation principle in L 1
On Z d , we fix a symmetric probability measure µ and an operator-stable law η such that the convergence assumption (C-B n ) is satisfied. We need to introduce some notation from [8] in order to state the results. Let π be the projection map π : R d → R d /Z d , and let T denote the d-dimensional torus which we also identify with the fundamental domain That is, we take the image of the original lattice Z d under the transformation B −1 ⌊λan⌋ , and project it to the torus T. Then L (n) λ is a cocompact lattice on T and the volume of the fundamental domain T/L (n) ⌊λan⌋ . This is the case because we assume that the matrices B m , m = 1, 2 . . . , have integer entries so In what follows, symbols decorated with are always used to describe quantities associated with the projected random walk on the torus. Note that the construction depends on the choice of sequence a n and parameter λ, for simplicity we will drop reference to a n and λ when no confusion arises.
Under the projection map π, we can push forward the measure B −1 ⌊λan⌋ µ on B −1 ⌊λan⌋ (Z d ) to a measure µ n,λ on L Let P n k be the distribution of L n k in M 1 (T). With this mollification, L n k is absolutely continuous with respect to Lebesgue measure on T. Let f (n) k,λ denote the density of P n k with respect to Lebesgue measure. Let Q (n) k,λ be the distribution of f (n) k,λ in L 1 (T). We will use the following function spaces (this notation is consistent with [7,8]): where Ω is an open subset of R d .
Theorem 5.1 (Large deviation principle in L 1 (T)). Assume that the convergence assumption (C-B n ) is satisfied. Let a n to be any sequence of positive integers increasing to infinity and satisfying a n |det B an | ≤ n. Let Q (n) n,λ be the distribution of f (n) n,λ on L 1 (T). Then we have the large deviation principle in the strong L 1 (T) topology. Namely, for any Borel set D in L 1 (T), and the rate function is given by This result will be useful in the upper bound direction. To obtain a lower bound, we need to have a version with Dirichlet boundary condition.
Let L (n) k be the occupation time measure of the random walk S (n) k = B −1 an (S k ). Perform the same mollification as above but on R d , setting Then L n k is absolutely continuous with respect to Lebesgue measure. Let f (n) k denotes the corresponding density. Let G be the collection of all bounded domain Ω in R d such that 0 ∈ Ω and ∂Ω has Lebesgue measure 0. For any Borel set That is, Q (n) k,Ω is the distribution of the occupation time measure of S (n) j at time k with Dirichlet boundary on ∂Ω. As in the case of the projected version, we have the following large deviation principle.
Theorem 5.2 (Large deviation principle in L 1 (Ω)). Under the convergence assumption (C-B n ), let a n be any sequence of positive integers increasing to infinity satisfying a n |det B an | ≤ n. Let Q (n) n,Ω be the distribution of f (n) n in L 1 (Ω). Then we have large deviation principle in the strong L 1 (Ω) topology. Namely, for any Borel set A ⊂ L 1 (Ω), and The outline of the proof of these results is given in Section 5.3. It follows [8] closely.

Asymptotics of functional expressions
Throughout this short section, we fix a symmetric probability measure µ on Z d and an operator-stable law η on R d such that the convergence assumption (C-B n ) and scaling assumption (S-B n -a n ) of Definitions 3.1-3.6 are satisfied. In particular, in what follows, (a n ) is the non-decreasing and regularly varying sequence of integers provided by Definition 3.6 (see also Proposition 3.7). The functions F and F are as in Definition 3.6. Let (l(n, x)) x∈Z d be the occupation time vector up to time n for the random walk driven µ. The goal of this subsection is to use the large deviation principles in L 1 to prove Theorem 3.9.

Proposition 5.3. Under the above hypotheses, we have the lower bound
Proof. The proof is essentially the same as for [ Next, follow the line of reasoning used to prove the Corollary of Theorem 6 in [8], using the large deviation upper bound in L 1 (T) and Varadhan's lemma. From the (lower bound part of) the scaling assumption and the regular variation property of det B an , we have for any parameter λ > 0, The last step comes from Varadhan's lemma. Since the choice of parameter λ is arbitrary, we can optimize over all λ > 0.
The following lemma is proved in the appendix. It shows that the constants appearing in the upper and lower bounds actually match up. In particular, since this constant appears as both a sup and an inf of some nonnegative quantities, it follows clearly that the constant k(η, F ) defined below takes value in (0, ∞).

Proof of the large deviation principle in L 1
In this section we indicate how to adapt [8] to prove the large deviation principles as stated in Theorems 5.1 and 5.2. For this purpose we first develop a large deviation principle in the weak topology following Lemma 3.1 and Appendix A in [11].
Throughout this subsection we assume First we establish asymptotics for exponential moment generating functions. Compare with [11,Lemma A.1] which treats simple random walk on Z d .
Proposition 5.6. For the projected occupation measure, for any f ∈ C(T) and any sequence (a n ) satisfying a n → ∞ and a n = o(n) as n → ∞, lim n→∞ 1 n/a n log E exp n a n < f, For the occupation measure with Dirichlet boundary condition, for any function f ∈ C 0 (Ω), lim n→∞ 1 n/a n log E exp n a n < f, L (n) n > 1 {supp(L (n) Proof. In the lower bound direction, we have the following Feynman-Kac estimates as consequences of functional limit theorem. For the proof, adapt the arguments given in [4,Theorem 7.1]. For any sequence (a n ) satisfying a n → ∞ and a n = o(n) as n → ∞, and any f ∈ C(T), Similarly, for f ∈ C 0 (Ω), In the upper bound direction, as a consequence of the convergence assumption, we can adapt the proof of [8,Theorem 3] to have the following large deviation upper bound. Let C be a closed of M 1 (T). Then lim sup n→∞ 1 n/a n log P (n) where Similarly, if C is compact in M 1 (Ω), Either on T or in Ω, apply Varadhan's lemma ([21, Theorem 2.2]) to the large deviation upper bound to obtain the upper bounds needed for Proposition 5.6.
By the Gartner-Ellis theorem (e.g., [5,Theorem 4.5.20]), we obtain the large deviation principle in the weak topology stated in the following Theorem. Compare with [11, Lemma 3.1].
Theorem 5.7. For any Borel set B in M 1 (T) and any sequence (a n ) satisfying a n → ∞ and a n = o(n) as n → ∞,

Similarly, for any Borel set
Next, following [8], we use the local limit theorem to upgrade the large deviation principle in the weak topology to a result in the strong L 1 -topology. This is a rather technical task. As shown in [8,Theorem 6], the key point is to obtain a super-exponential estimate on the L 1 -distance of the density function to its smooth mollification. This, in turn, requires uniform properties of the transition probabilities that are provided by the local limit theorem.
Let {ψ ǫ }, ǫ → 0, be an approximation of the identity on R d with ψ ǫ is smooth, symmetric, compactly supported inside (− ǫ 2 , ǫ 2 ) d . Thinking of ψ ǫ also as a function on T, set K ǫ : L 1 (T) → L 1 (T) as Theorem 5.8. For every δ > 0, λ > 0 and sequence a n tending to infinity such that a n |det B an | ≤ n, we have lim sup ǫ→0 lim sup n→∞ 1 n/a n log Q (n) n,λ f : We follow step by step the proof of [8,Theorem 5]. To this end, we adapt to our situation the sequence of lemmas in [8] that are used to prove this theorem. The first lemma provides an elementary way to select a δ-net of functions. Recall that For any δ > 0, there exist functions V 1 , ..., V J such that for any V ∈ M n,ǫ , Proof. The proof is identical to [8, Lemma 4.1 ].
The second lemma is similar to [8,Lemma 4.2] and concerns the uniform control of the transition probabilities. Such uniform control appears as Assumption (U) in [6, Section 4.1] and [5, Section 6.3] to obtain the large deviation principle in L 1 .
Lemma 5.10. There exists n 0 ∈ N and constant c < ∞ such that for all n ≥ n 0 , Proof. Recall the local limit theorem in [12], Applying this local limit result along the sequence {⌊λa n ⌋} and projecting onto T, we have lim sup n→∞ sup y∈L (n) ||det B an | µ * an n (y) − g λ −1 (y)| = 0.
Since the density g is continuous, the desired result follows.
We have the following uniform estimate with respect to the starting point. Lemma 5.11. Let x and y be any two points in L (n) λ . There exists an integer n 0 such that for any n ≥ n 0 and any θ > 0, where C n,θ = c exp(4θa n ) and c and n 0 are as in Lemma 4.2.
Proof. This follows from Lemma 5.10 as in the proof of [8,Lemma 4.3].
Lemma 5.12. There exists an integer n 0 such that for any n ≥ n 0 and any θ > 0, where C n,θ is as in Lemma 5.11 and I * n is Legendre transform of that is, Proof. Follow [8,Lemma 4.4].
Finally, we need the following technical lemma that controls error terms as n → ∞.

Proof of regular variation properties
In this subsection we deduce from the regular variation of the sequence (B n ) the properties of F stated in the technical proposition 3.7. We follow [3] closely.
Proof of Proposition 3.7. By Theorem 2.9 (i.e., [13,Theorem 1.10.19 ]), under the convergence assumption B −1 n µ (n) =⇒ η, there exists a modified normalization sequence (B ′ n ) with B ′ n = B n S n , S n ∈ Ivn(η), such that (B ′ n ) has the regular variation property where the convergence is uniform in t on compact subsets of R × + . Since S n ∈ Ivn(η) we have (B ′ n ) −1 µ (n) =⇒ η. Further, since Inv(η) is a compact group, we must have det S n = 1, det B ′ n = det B n . Hence we can replace B n in the scaling assumption by B ′ n and we have lim n→∞ a n det(B ′ an ) n F n det(B ′ an ) y = F (y), uniformly over compact sets in (0, ∞). Set F n (y) := a n det(B ′ an ) n F n det(B ′ an ) y .
The scaling assumption now reads lim n→∞ F n (y) = F (y). Note that by assumption, F : [0, ∞) → [0, ∞) is a concave, increasing function with F (0) = 0, therefore both F n and F are concave, non-decreasing, not identically zero with value 0 at 0. Hence F n and F are continuous and strictly positive in (0, ∞), and by concavity, y → Fn(y) y and y → F (y) y are both non-increasing functions. Now we show that for any λ ∈ (0, 1), a ⌊λn⌋ an tends to a finite non-zero limit as n → ∞ . Fix a y > 0 and write y .
As (a n ) is an increasing sequence and det(B n ) is non-decreasing with respect to n, we have det(B ′ a ⌊λn⌋ )/ det(B ′ an ) ≤ 1 Since y → Fn(y) y is non-increasing, we have Letting n → ∞ on both sides, the scaling assumption yields F (y) ≤ lim inf n→∞ a ⌊λn⌋ a n 1 λ F (λy).
Since (a n ) is an increasing sequence by assumption, we have λ F (y) F (λy) ≤ lim inf n→∞ a ⌊λn⌋ a n ≤ lim sup n→∞ a ⌊λn⌋ a n ≤ 1.
Replacing λ by 1 λ , we conclude that for all λ ∈ (0, ∞), a ⌊λn⌋ an is uniformly bounded away from 0 and uniformly bounded from above.
Let φ(λ) be defined for each λ ∈ (0, ∞) as a sub-sequential limit of a ⌊λn⌋ an . Namely, choose some (λ-dependent) sub-sequence t n → ∞ and set φ(λ) = lim n→∞ a ⌊λtn ⌋ at n . From the above reasoning we know that φ(λ) ∈ (0, ∞). Consider the equation F ⌊λn⌋ (y) = a ⌊λn⌋ a n det(B ′ a ⌊λn⌋ )/ det(B ′ an ) ⌊λn⌋ /n F n ⌊λn⌋ /n det(B ′ a ⌊λn⌋ )/ det(B ′ an ) y , and take the limit along the sub-sequence (t n ). We can indeed take the limit on the right hand side of the equation because F is continuous, and the convergences F n (y) → F (y) and B ′ n (B ′ ⌊nt⌋ ) −1 → t −E are uniform over compact sets. This yields Note that the function y is non-increasing. As z → F (y) z is strictly decreasing, the solution z 0 = z(λ, y) to Observe that Therefore φ is multiplicative and φ(λ) = λ κ with κ = log 2 φ(2). Plugging this back in Setting y = 1 gives The fact that lim n→∞ log a n log n = κ, follows exactly from the reasoning in [3].
6.2 Discussion of the constant k(η, F ) of Lemma 5.5 In this subsection, we follow the truncation argument in [7] to prove Lemma 5.5. With the notation of Section 5, let be the upper bound appearing in Proposition 5.4. Let ǫ > 0 be an arbitrary small number. To prove Lemma 5.5, it suffices to find Ω ∈ G and g ∈ F Ω such that E η ( √ g, √ g) + Ω F (g(x))dx ≤ J + ǫ.
For any λ (we will choose λ large enough later on), by the definition of J, there exists f ∈ F T such that We can think of functions on T also as functions on the fundamental domain [0, 1) d . Following [7,Lemma 3.4], let Note that there exists a ∈ T such that the translated function f a (x) = f (x − a) satisfies Because of translation invariance of the expression on the torus, we can replace f by f a in the expression without changing the value. Therefore we may assume that f ∈ F T satisfies and Consider a smooth bump function φ 0 on R such that φ 0 = 1 on [ 1 Now, set Ω = λ E (0, 1) d where λ is sufficiently large and Then, we claim that To see this, first note that g is supported on T λ and that, by the scaling properties t E (W ) = t · W of the Lévy measure, we have Also, since F (y) = F (1)y γ , we have Ω F (g(x))dx = λ (1−γ) tr E T F (g 1/λ (x))dx.

Explicit computation of constants
In this section, we illustrate Theorem 4.4 concerning switch-walk-switch random walks on certain wreath products K ≀ H and give some indications concerning exact the computation of the constant k(η, F K ). Let ν denote a symmetric probability measure on the lamp-group K and let F : (0, ∞) → (0, ∞) be such that F (n) = − log ν (2n) (e K ). Assume that µ is a symmetric measure on the base-group H = Z d satisfying the convergence assumption that is, µ is in the domain of normal attraction of η where η is an operator-stable law with exponent E. Setη = e −Θ and L Θ f = Θf .
We will treat cases where F satisfies the scaling assumption (S-n E -a n ) for some sequence a n . Let F be the corresponding limit function and γ be the associated scaling exponent. See Definition 3.6 and Proposition 3.7.
Let q be the switch-walk-switch measure on G = K ≀Z d given by q = ν * µ * ν.

(6.2)
Here where the infimum is taken over all bounded open sets B ⊂ R d such that the Lebesgue measure |B| = 1, |∂B| = 0, and λ 1 (Θ, B) is the principle eigenvalue for L Θ with Dirichlet boundary on B.
We can recover this result using Theorem 4.4 (actually, a rather trivial special case). We have F (y) = (log ρ 2 )y + o(y) and F = (log ρ 2 )y.
When H = Z d , E η (f, f ) = a R d |∇f | 2 dx, and γ is 1/2 -for example, this is achieved by the switch-walk-switch random walk on the wreath product Z 2 ≀ Z 2 -the minimizer is given by a Bessel function and the constant k(η, s → cs 1/2 ) is explicitly computable, see [18,Proposition 5.2].