Complete Localisation and Exponential Shape of the Parabolic Anderson Model with Weibull Potential Field

We consider the parabolic Anderson model with Weibull potential field, for all values of the Weibull parameter. We prove that the solution is eventually localised at a single site with overwhelming probability (complete localisation) and, moreover, that the solution has exponential shape around the localisation site. We determine the localisation site explicitly, and derive limit formulae for its distance, the profile of the nearby potential field and its ageing behaviour. We also prove that the localisation site is determined locally, that is, by maximising a certain time-dependent functional that depends only on: (i) the value of the potential field in a neighbourhood of fixed radius around a site; and (ii) the distance of that site to the origin.

1. Introduction 1.1. The parabolic Anderson model. We consider the Cauchy equation on the lattice (Z d , | · | ℓ 1 ) ∂u(t, z) ∂t = (∆ + ξ)u(t, z) , where ∆ is the discrete Laplacian on Z d defined by (∆f )(z) = y∼z f (y), the set {ξ(z)} z∈Z d is a collection of independent identically distributed (i.i.d.) random variables known as the random potential field, and ½ {0} is the indicator function see [1]). The Cauchy form of the problem in equation (1) arises naturally in a system consisting of a single particle undergoing diffusion while branching at a rate determined by a (random) potential field (see [6][Section 1.2]).
The PAM and its variants are of great interest in the theory of random processes because they exhibit intermittency, that is, unlike other commonly studied random processes such as diffusions, their long-term behaviour cannot be described with an averaging principle. The PAM is said to localise if, as t → ∞, the total mass of the process U (t) := z∈Z d u(t, z) is eventually concentrated on a small number of sites, i.e. if there exists a (random) localisation set Γ t such that z∈Γt u(t, z) U (t) → 1 in probability .
The most extreme form of localisation is complete localisation, which occurs if the total mass is eventually concentrated at just one site, i.e. if Γ t can be chosen in equation (2) such that |Γ t | = 1.
It turns out that complete localisation cannot hold almost surely, since the localisation site will switch infinitely often and so, at certain times, the solution must be concentrated on at least two distinct sites (see, e.g., [11] for an example of almost sure convergence in the PAM on exactly two sites).
Note that elsewhere in the literature (see, e.g., [14]) the convention (∆f )(z) := y∼z (f (y) − f (z)) is used to define the discrete Laplacian in the PAM. This is equivalent to shifting the random potential field by the constant 2d, and makes no qualitative difference to the model.

Localisation classes.
It is known that the strength of intermittency and localisation in the PAM is governed by the thickness of the upper-tail of the potential field distribution ξ(·), and in particular the asymptotic growth rate of g ξ (x) := − log(P(ξ(·) > x)) .
Depending on this growth rate, the PAM can exhibit distinct types of localisation behaviour, which are often categorised along two qualitative dimensions: (1) the number of connected components of Γ t (localisation islands) in the limit (i.e. single, bounded or growing); and (2) the size of each localisation island in the limit (i.e. single, bounded or growing).
Universality classes with respect to the size of each localisation island are wellunderstood (see, e.g., [9] and [5]). It was proven in [5] that the double-exponential distribution forms the critical threshold between these classes. More precisely, if g ξ (x) = O(e x χ ) for some χ < 1 (i.e. tails heavier than double-exponential) then localisation islands consist of a single site. This class includes Weibull-like tails, where g ξ (x) ∼ x γ for γ > 0, and Pareto-like tails, where g ξ (x) ∼ γ log x for γ > d (recall from [6] that if γ < d then the solution to equation (1) is not well-defined; if γ = d then the solution is well-defined only for d > 1). Conversely, if e x χ = O(g ξ (x)) for some χ > 1 (i.e. tails lighter than double-exponential, including bounded tails), then the size of localisation islands grow to infinity.
On the other hand, universality classes with respect to the number of localisation islands are not at all well-understood. In particular, it is not known whether the PAM with g ξ (x) = O(e x χ ) for some χ < 1 always exhibits complete localisation. Indeed, this was conjectured to be false in [11]. Up until now, complete localisation has only been exhibited for the PAM with Pareto potential (in [11]) and Weibull potential with parameter γ < 2 (in [14]), which includes the case of exponential tails.
This has left open the question as to whether the PAM with Weibull potential with parameter γ ≥ 2, which includes the important class of normal tails, also exhibits complete localisation.
1.3. Main results. We consider the PAM with Weibull potential, that is, where ξ(·) satisfies g ξ (x) = x γ , for some γ > 0. We prove that the PAM with Weibull potential is eventually localised at a single site with overwhelming probability (complete localisation) and, moreover, that the renormalised solution has exponential shape around this site. We determine the localisation site explicitly, and derive limit formulae for its distance, the profile of the nearby potential field and its ageing behaviour. We also prove that the localisation site is determined locally, that is, by maximising a certain time-dependent functional that depends only on: (i) the values of ξ(·) in a neighbourhood of fixed radius ρ := ⌊(γ − 1)/2⌋ + around a site, where x + := max{x, 0}; and (ii) the distance of that site to the origin. In particular, if γ < 3 then ρ = 0 and so the localisation site is determined only by maximising a certain time-dependent functional of the pair (ξ(·), | · | ℓ 1 ). We shall refer to ρ as the radius of influence.
In order to state these results explicitly, we introduce some notation. Define a large 'macrobox' γ , identifying its opposite faces so that it is properly considered a d-dimensional torus. Further, for each a ≤ 1, define the associated macrobox level L t,a := ((1 − a) log |V t |) 1 γ and let the subset Π (Lt,a) := {z ∈ V t : ξ(z) > L t,a } consist of sites within the macrobox V t at which ξ-exceedences of the level L t,a occur. Define also, for each z ∈ V t and n ∈ N, the ball B(z, n) := {y ∈ V t : |y − z| ℓ 1 ≤ n}, considered as a subset of V t (i.e. with the metric acting on the torus). Henceforth, for simplicity, we simply write | · | in place of | · | ℓ 1 when denoting distances on Z d or V t .
Fix a constant 0 < θ < 1/2, and abbreviate L t := L t,θ . Letξ := ξ½ Vt\Π (L t ) be the L t -punctured potential field. For each z ∈ V t and n ∈ N, define the L t -punctured HamiltonianH (z) n on B(z, n) with Dirichlet boundary conditions where ∆ Vt denotes ∆ restricted to the torus V t ; and letλ with all distances being on the torus V t . We shall callλ (n) t (z) the n-local principal eigenvalue at z and remark that it is a certain function of the set ξ (n) (z) := {ξ(y)} y∈B(z,n) . Note that the {λ   Remark also that in the special case n = 0,λ (0) t (z) is simply the potential ξ(z).
For any sufficiently large t, define a penalisation functionalΨ Note that, for any t, the site Z (1,n) t is well-defined almost surely, since V t is finite. Moreover, as we shall see, Z (1,n) t will turn out to be independent of the choice of θ. Define a function q : N → [0, 1] by using the convention that 0/0 := 0. Introduce the scales and an auxiliary scaling function κ t → 0 that decays arbitrarily slowly. Finally, let considered as a subset of Z d . Our main results can then be summarised by the following: Theorem 1.1 (Profile of the renormalised solution). As t → ∞, the following hold: is bounded in probability .
Corollary 1.2 (Complete localisation). As t → ∞, Theorem 1.3 (Description of the localisation site). As t → ∞, the following hold: where X is a random vector whose coordinates are independent and Laplace distributed random variables with absolute-moment one; (b) (Local profile of the potential field) For each z ∈ B(Z (1,ρ) t , ρ) uniformly, in probability ; (c) (Ageing of the localisation site) where Θ is a nondegenerate almost surely positive random variable.
Corollary 1.4 (Ageing of the renormalised solution). For any sufficiently small ε > 0, as t → ∞, and Θ is the same almost surely positive random variable as in Theorem 1.3.
is the maximiser of the penalisation func-tionalΨ (ρ) t (z), which balances the magnitude of the ρ-local principal eigenvalue at a site with the distance of that site from the origin. Heuristically, this may be explained as the solution favouring sites with high local principal eigenvalue but being 'penalised' for diffusing too quickly.
As claimed,Ψ (ρ) t (z) depends only on the set ξ (ρ) (z) and on the distance |z|. Indeed, in order to determine Z where j := [γ/2] ∈ {ρ, ρ+ 1} and Γ * k (z, ρ) is the set of all length k nearest neighbour paths z =: y 0 → y 1 → . . . → y k := z in B(z, ρ) such that y i = z for all 0 < i < k. This path expansion can be iteratively evaluated to get an expression forλ (ρ) t (z) as an explicit function of ξ (ρ) (z). Note that j is chosen precisely to be the smallest non-negative integer such that a −2j−1 t = o(d t ), which ensures that paths with more than 2j steps contribute at most o(d t ) to the sum. Since we show in Section 4 that the gap between the maximisers ofΨ (ρ) t is on the scale d t , such an expression is sufficient to determine Z (1,ρ) t . Remark 1.6. Our limit theorem for the profile of the renormalised solution holds within a distance r t κ t of the localisation site, where κ t may be chosen to decay arbitrarily slowly. At or beyond this scale, the profile will be interrupted by 'bumps' in the renormalised solution around other high values of the functionalΨ (ρ) t , which occur at distances on the scale r t . In this region, we simply bound the renormalised solution by the height of these bumps, although we also expect a weaker global exponential decay to hold.
Remark 1.7. The ageing of the renormalised solution in Corollary 1.4 is a natural consequence of complete localisation of the renormalised solution (Corollary 1.2) and the ageing of the localisation site (Theorem 1.3). The proof of this result is essentially the same as in [13][Proposition 2.1] for the corresponding result in the case of Pareto potential field; we defer to that paper for the proof. Note also that Corollary 1.4 is a quenched ageing result along the lines of [13], as opposed to the annealed (i.e. averaged over all realisations of the random environment) ageing studied in [8].
Remark 1.8. Recall that it was previously shown in [14] that complete localisation holds in the case γ < 2. The analysis in that paper is broadly similar to ours, but uses the penalisation functional t (z) in the special case ρ = 0. This restricts the validity of the analysis to where there is an exact correspondence between the top order statistics of the fields ξ andλ (ρ) t in V t . Clearly this holds for γ < 3, since then ρ = 0. On the other hand, the exact correspondence has been shown to be false if γ ≥ 3 (in [4]), and so an analysis based on the functional Ψ * t =Ψ (0) t fails in that case.
Remark 1.9. We briefly mention the strong possibility that our results can be extended to the case of fractional-double-exponential potential field, i.e. where g ξ (x) = e x χ for some χ < 1. The main difference in that case is that the radius of influence ρ grows with t, which presents a technical difficulty in extending the results in Proposition 4.2. Nevertheless, we strongly believe such an extension is valid, and since the rest of our proof holds essentially unchanged, we expect complete localisation to also hold in the fractional-double-exponential case.
The paper is organised as follows. In Section 2 we give an outline of the proof, and establish Theorem 1.1 subject to an auxiliary Theorem 2.3. In Section 3 we establish some preliminary results. In Section 4 we use a point process approach to study the random variables Z  ) (and generalisations thereof), and in doing so complete the proof of Theorem 1.3. In Section 5 we collect results from spectral theory that we will apply in Section 6. In Section 6 we complete the proof of the auxiliary Theorem 2.3.

Outline of the Proof
In the literature, the usual approach to study u(t, ·) is with probabilistic methods via the Feynman-Kac representation (for instance, in [5]). Our primary approach is different, applying spectral theory methods to the HamiltonianH (as is done in [2], for instance). We note, however, that these approaches are very similar, and we do at times make use of the Feynman-Kac representation.
2.1. Spectral representation of the solution. The basic idea that underlies our proof is that the solution u(t, ·) is well-approximated by a spectral representation in terms of the eigenfunctions of the HamiltonianH restricted to a suitably chosen domain. It turns out that this spectral representation is asymptotically dominated by just one eigenfunction, which is eventually localised with exponential decay away from the localisation site.
In order to apply this idea, we restrictH to the macrobox V t (i.e. with periodic boundary conditions, recalling that V t is a torus), on which the solution u(t, ·) turns out to be essentially concentrated. So let u Vt (s, z) be the solution to the PAM restricted to V t , that is, defined by the Hamiltonian H := ∆ Vt + ξ, with u Vt (s, z) := 0 outside V t by convention, and let U Vt (t) := z∈Vt u Vt (t, z). Proposition 2.1 (Correspondence between u Vt (t, z) and u(t, z)). As t → ∞ and for any z, where both hold almost surely.
Remark 2.2. Since the error in Proposition 2.1 is of lower order than the bounds in Theorem 1.1, it will be sufficient to prove that Theorem 1.1 holds for u Vt (t, ·). Proposition 2.1 is proved in Section 3.
Denote by λ t,i and ϕ t,i the i'th largest eigenvalue and corresponding eigenvector of H, with each ϕ t,i taken to be ℓ 2 -normalised with ϕ t,i (z) := 0 outside V t by convention. Since V t is bounded, the solution u Vt (t, ·) permits a spectral representation in terms of the eigenfunctions of H: | t and remark that this is chosen so that the magnitude of the i'th term in the sum in equation (4) is e tΨt(i) |ϕ t,i (·)|, using the convention that exp{−∞} := 0.
We refer to {Ψ t (·)} as the penalised spectrum, noting that it represents a tradeoff between the magnitude of the eigenvalue and the (absolute) magnitude of the eigenvector at the origin; the intuition here is the same as in Remark 1.5. We prove that, with overwhelming probability, a gap exists between the largest two values in the penalised spectrum, which implies that the spectral representation in equation (4) is dominated by just one eigenfunction. Moreover, we prove that this eigenfunction is eventually localised at Z To make this precise, let i (1) t for notational convenience. Moreover, introduce auxiliary scaling functions f t , h t , e t → 0 and g t → ∞ as t → ∞ such that Theorem 2.3 (Auxiliary theorem). As t → ∞, the following hold: (a) (Gap in the penalised spectrum) In Section 2.2 immediately below we finish the proof of Theorem 1.1 subject to the auxiliary Theorem 2.3; the other sections of the paper are dedicated to proving Theorems 1.3 and 2.3.
Our proof of Theorem 2.3 is based on the observation that Ψ t (i) is asymptotically approximated byΨ . This is useful, since it is simpler to study the maximisers ofΨ t ) directly. Using a point process approach, we demonstrate a gap between the top two maximisers ofΨ (ρ) t (and generalisations thereof), and also describe the location and the neighbouring potential field of the maximiser Z (1,ρ) t , proving Theorem 1.3. We then establish the validity of the approximation, which requires both a correspondence between eigenvalues and local principal eigenvalues, and an analysis of the decay of eigenfunctions, in particular finding bounds on the value of eigenfunctions at zero; here we draw heavily on the methods in [2] and [3].

2.2.
Proof of Theorem 1.1 subject to the auxiliary Theorem 2.3. Starting from the spectral representation in (4), we pull out the term involving the maximising index i Consider now any z ∈ B t . Combining part (b) of Theorem 2.3 with equations (5) and (6) we have that, with overwhelming probability Remark that the correspondence in Proposition 2.1 implies that, for any z and with overwhelming probability, .
and so, putting these together, we have where o(1) does not depend on z, which proves part (a) of Theorem 1.1.
On the other hand, combining part (b) of Theorem 2.3 with Proposition 2.1 and equation (6), we have that which is bounded in probability. Theorem 1.1 is proved.

Preliminaries
In this section we establish some preliminary results. Denote by ξ t,i the i'th highest value of ξ in V t .
Proof. These follow from the min-max theorem for the principal eigenvalue.
Proof of Proposition 2.1.
Note that the weaker statement that |U Vt (t) − U (t)| → 0 is proved in [7, Section 2.5] (although for a slightly different macrobox); we need to control the error more precisely. For . For a fixed t > 0, consider the Feynman-Kac representations of u(t, z) and u Vt (t, z): where {X s } s∈R + denotes the continuous-time random walk on the lattice Z d based at the origin, ½ A denotes the indicator function for the event A, and where the expectation E is taken over the trajectories of the random walk X s . For each n ∈ N, let e n (X) denote the event that max s<t |X s | ℓ ∞ = n. Let u n (t, z) and u n Vt (t, z) denote, respectively, the expectations in (7) and (8) restricted to the event e n (X), and define U n (t) := z∈Z d u n (t, z) and U n Vt (t) := z∈Z d u n Vt (t, z) by analogy with U (t) and U Vt (t) respectively. Then it is clear, for each z, that n<Rt u n (t, z) = n<Rt u n Vt (t, z) .
As n → ∞, we can bound ξ (n) 1 + 2d almost surely with Lemma 3.1: ξ For n ≥ R t and by Stirling's approximation, we can also bound the probability P(e n (X)) by where Pn a (n) denotes the probability mass function for the Poisson distribution with mean a, evaluated at n. Combining these bounds, for n ≥ R t and as t → ∞ eventually max{U n (t), U n Vt (t)} < exp{t(d log n) 1 γ (1 + ε) − n log n + n log t + Cn)} almost surely, for any ε > 0 and for some C > 0. Since n ≥ R t = t(log t) holds almost surely. Combining equations (9) and (10), we get that and, similarly, as required.

A Point Process Approach
In this section, we use point process techniques to study the random variables Z ), and generalisations thereof; the techniques used are similar to those found in [14]. In the process, we complete the proof of Theorem 1.3. 4.1. Point process asymptotics. Fix an 0 < ε < θ and an 0 < η < 2ρ − γ + 3, remarking that the latter is possible by the definition of ρ. Recall also the definition j := [γ/2] ∈ {ρ, ρ + 1}. For each n ∈ N such that n ≤ j, define the annulī B 1 := B(0, min{n, ρ}) \ {0} andB 2 := B(0, j) \ (B 1 ∪ {0}), and the following |B 1 ∪B 2 |-dimensional rectangles: and, after rescaling E (n) 1 in each dimension, where π y is the projection map with respect to y. Finally, for each z ∈ V t , define the event almost surely, where Γ * k (z, n) is the set of all length k nearest neighbour paths z =: y 0 → y 1 → . . . → y k := z in B(z, n) such that y i = z for all 0 < i < k.
Proof. As in [2, Lemma 2], the eigenvalueλ where Γ k (z, n) is the set of all length k nearest neighbour paths i.e. including paths that return to z multiple times; Γ 0 (z, n) is understood to consist of a single degenerate path. Remark that the factor 1/λ (n) t (z) in equation (11) appears sinceξ(y k ) =ξ(z) = 0. Noticing that, by Lemma 3.3,λ By decomposing each path in ∪ k≥0 Γ k,n into a sequence of paths in ∪ k≥0 Γ * k,n , we get that the right hand side of equation (11) is equal to and so equation (11) gives .
Noticing that (L t,ǫ − L t ) −(2j−1) = o(d t e t ), this yields the result by truncating the infinite sum after paths of length 2j, and since, by Lemma 3.2, eventuallyξ = ξ on B(z, n) \ {z} almost surely. t ; see [3, Section 6]). For each n ∈ N such that n ≤ j, there exists a scaling function A (n) t = a t +o(1) such that, as t → ∞ and for each fixed x ∈ R, the following are satisfied: if the sum converges and Q (n) t (C t ; s (n) ) := 0 otherwise, and the functions To motivate these definitions, consider the first statement of Proposition 4.2. Notice that, by Lemma 3.3, as t → ∞, eventuallỹ ε . This means that we can apply the path expansion in Proposition 4.1 toλ with the o(1) bound taking care of the contribution from the s (n) for which Q (n) t (C t ; s (n) ) does not converge, by Lemma 3.2.
To approximate these integrals, we state some properties of the functions P for a positive constant, where 0 here denotes the zero vector. The function R (n) t (C t ; s (n) ) is also strictly increasing in C t , satisfying and, for each y ∈B where 1 here denotes the vector of ones. In particular, this implies that there exists a scaling factor A Remark that if n = 0, then R eventually, for some constant c > 0.
Consider now the integral in equation (13) restricted to the domain E (n) . As in [3, Theorem 6.3], we may first use equation (14) to integrate out over s (n) 2 , and then apply Laplace's method to approximate the resulting integral over s (n) 1 : with the last line following from an application of Laplace's method to the integral, noticing that the determinant of the Hessian matrix of R Together, these two bounds give Proposition 4.2.

4.2.
Constructing the point processes. We now construct the point processes we shall need to consider. For each n ∈ N such that n ≤ j and each z ∈ V t , denote For each τ ∈ R and q > 0 let   t (0)} has a dependency range 2n, it is sufficient to check the following local dependence condition: as t → ∞, for any x ∈ R. This is satisfied, since by Lemma 3.2 the set Π (Lt) is eventually 2n-separated almost surely, and so eitherλ rt + xd rt eventually, for any x. Observe also that the restriction of N (n) t to H q τ ensures that the intensity measure of the limit process N is such that every relatively compact set has finite measure.
We transform the point process N to a new point process involvingΨ (n) t . For technical reasons, we shall need to consider a certain generalisation of the function-alsΨ (n) t . So for each n ∈ N such that n ≤ j, c ∈ R and sufficiently large t, define the functionalΨ t,c . Note that for any t these are well-defined almost surely, since V t is finite.
Finally, for each τ ∈ R and α > −1 let where M is a point process onĤ α τ with intensity measure ν(dx, dy) = dx⊗e −y−|x| dy. Remark 4.6. Although we prove Proposition 4.5 for each c ∈ R, we shall only apply it to c = 0 and one other value of c that will be determined in Corollary 5.7.
Proof. This follows as in [14, Lemma 3.1] (although note that, due to a different choice of d t , the intensity of the point process in [14, Lemma 3.1] differs by a constant). First choose a pair α ′ and q such that 0 < α ′ + 1 < q < α + 1 and notice that M It was proved in [10, Lemma 2.5] that one can pass to the limit simultaneously in the mapping K t,c and the point process N The density of M is then χ • K −1 = ν, restricted toĤ α τ . We now use the point process M to analyse the joint distribution of the random variables Z n) t,c ). Proposition 4.7. For each n ∈ N such that n ≤ j and each c ∈ R, as t → ∞ converges in law to a random vector with density Proof.  For each c ∈ R, introduce the events which act to collect the relevant information that we shall later need.   (1,j) t and recall that 1/ log log t < e t /g t eventually.
On the event E t,c , the statements and, eventually, all hold, giving a contradiction.
all hold eventually.
Proof. The first two statements follow from the min-max theorem for the principal eigenvalue, since j ≥ ρ. For the third statement, we only need consider the case that j = ρ+1. Then, the event E t,c implies that ξ(y) < a η t for all y such that |y−Z eventually, with the last equality holding since η − 2j < 1 − γ.
Corollary 4.11. For any c ∈ R, on the event E t,c On the event E t,c , Lemma 4.10 implies that all hold eventually. On the other hand, on the event E t,c and by the definition of Z where A(n, ω, t) is the event and D ω (x, y) is the set The random variable Θ can then be defined by
Proposition 5.2 (Correspondence between local and global eigenvalues). The following hold eventually almost surely: Proof. These are proved in [3], as a consequence of the third and first statements of Theorem 4.1 respectively (keeping in mind the definition in (4.3) of that paper).
Remark 5.3. The correspondence in Proposition 5.2 indicates that the i'th highest eigenvalue of H is closely approximated by the principal eigenvalue of the 'single peak' HamiltonianH (zt,i) . In physical terms, this can be interpreted as a lack of 'resonance' between the regions in V t where the potential field ξ is high, i.e. the regions which give rise to a high local principal eigenvalue.
Lemma 5.4 (Almost sure asymptotics for eigenvalues). The following hold eventually almost surely: Proof. For part (a), it is sufficient to show that eventually λ t,1 < L t,−ε ′ , which follows by combining Lemma 3.3 with the asymptotics in Lemma 3.1. For parts (b)-(d), by the correspondence in Proposition 5.2 it is sufficient to show that eventually As in Lemma 3.3, by the min-max theorem, for each z ∈ V t , The result then follows from the asymptotics in Lemma 3.1 5.3. Exponential decay of eigenfunctions and upper bound on eigenfunctions at zero. In this subsection, we prove that the eigenfunctions ϕ t,i corresponding to the largest eigenvalues of H eventually localise with exponential decay away from the localisation site z t,i . As a corollary, we bound the value of these eigenfunctions at the origin. Note that these results mimic [3, Theorem 4.1], but with tighter control over the rate of the exponential decay.
Proof. Proposition 5.5 is an application of [3, Theorem B.3] with the following notation: Proof. For all 1 ≤ i ≤ |V t | ε , the asymptotics in part (b) of Lemma 5.4 imply that almost surely. Applying Proposition 5.5 we get that, eventually, Corollary 5.7 (Upper bound on eigenfunctions at zero). There exists a c > 0 such that eventually, for all 1 ≤ i ≤ |V t | ε , Proof. Again, applying Proposition 5.5 and part (b) of Lemma 5.4 we have that almost surely, which yields the result since δ t log log t = o(1), replacing c ′ with some c > c ′ .

5.4.
Lower bound on eigenfunctions at zero. In this subsection, we prove a lower bound on the value of certain eigenfunctions at zero. We only do this for very specific eigenfunctions, which satisfy an assumption defined below. It will turn out that the eigenfunction associated to the localisation site Z Before embarking on the proof of a lower bound, we need to introduce some well-known tools from spectral theory, which are proved, for instance, in [3]. Lemma 5.9 (Path expansion over ∆ Vt ; see [3,Lemma A.2]). Consider the Hamiltonian H ζ := ∆ Vt + ζ on V t , where ζ is any potential field. Denote by G ζ (λ; x, y) the Green's function associated with H ζ . Then for any x, y ∈ V t , provided the series converges. Here the sum Γ(x,y) is taken over all paths such that |v i − v i−1 | = 1 for each 1 ≤ i ≤ m and each m ∈ N (i.e. the nearest neighbour paths in V t starting at x and ending at y); n v (Γ(x, y)) denotes the number of times the path Γ(x, y) visits the site v ∈ V t ; |Γ(x, y)| := v∈Vt n v (Γ(x, y)) ≥ |x − y|. Note that if x = y and Γ(x, x) = 0, then the corresponding summand in equation (17) is equal to (λ − ζ(x)) −1 .
Lemma 5.10 (Cluster expansion; see [3, Lemma A.1]). Fix a non-empty subset Π ⊆ V t . For a positive field ζ and any u ∈ Π let G ζ (λ; x, y),G ζ (λ; x, y) and G :=H ζ + ζ½ {u} respectively. Then for any x, y ∈ V t , G ζ (λ; x, y) =G ζ (λ; x, y) provided that the series converges. Here the sum γ is taken over all ordered sets Below we show that the series in Lemma 5.9 converges in our setting, and results in a lower bound on ϕ t,i (x) for certain indices i and sites x. However, to achieve this lower bound, we need to apply the cluster expansion to the auxiliary set H(λ t,i ) since paths that hit H(λ t,i ) might contribute negative terms in the path expansion with respect to λ t,i . Proposition 5.11. For any λ t,i such that 1 ≤ i ≤ |V t | ε , and for any u / ∈ H(λ) and any x ∈ V t ,G (u) (λ t,i ; x, u) > 0 .
Proof. As in Proposition 5.5, [3, Theorem B.3] is valid for all 1 ≤ i ≤ |V t | ε . Moreover, from the proof of [3, Theorem B.3], we conclude that [3, Theorem B.1] is also valid for the same i's. Hence, by [3, Theorem B.1(ii)], we have that λ t,i / ∈ σ(H (zt,i) ) so we can apply the cluster expansion in Lemma 5.10 with Π ← {u} and ζ ← ξ½ Vt\Π (L t ) ∪{u} to obtain the following resolvent identity: Note that the series in Lemma 5.9 converges forG(λ t,i ; x, u) by Lemma 5.4 and hence the numerator in equation (18) is positive. It then suffices to show that Recall that, for u / ∈ H(λ t,i ), we have that λ t,i >λ t (u). Moreover, by [3, Remark B.5] we know thatλ t (u) is the principal eigenvalue ofH (u) if and only ifλ t (u) is the maximal solution of the equatioñ G(λ; u, u) = 1/ξ(u) .
Proof. Proposition 5.12 follows from Lemma 5.9 since the sum in equation (17) is convergent by the asymptotics in Lemma 5.4 and the definition ofH. Moreover, every sum along any path is positive, so we may bound the sum in equation (17) by the contribution from the shortest path x → · · · → z t,i .
Lemma 5.14. For all 1 ≤ i ≤ |V t | ε satisfying Assumption 5.8, and any Proof. From the proof of [3, Theorem 4.1] it follows that the series in the cluster expansion in Lemma 5.10 is convergent with Π ← Π (Lt) \ {z t,i } and ζ ← ξ½ Vt\{zt,i} . Denote by u 1 → · · · → u k an ordered set consisting of points from a set Π (Lt) \{z t,i } G (zt,i) (λ t,i ; x, z t,i ) =G(λ t,i ; x, z t,i ) The first summation on the right hand side is positive by Proposition 5.11. It remains to show that the second summation is negligible, i.e. that First apply Proposition 5.13, which gives that this summation is bounded above by B(λ t,i , u l )e −A(λt,i)|u l−1 −u l | e −A(λt,i)|u k −zt,i| . Now use the bound A(λ t,i )|u j−1 −u j | > (1−δ t )A(λ t,i )|u j−1 −u j |+δ t r(Π (Lt) ) for each j in the product, and let u l ∈ H(λ t,i ) for the ordered set γ in the above summation. Applying |u 1 −x|+|u 1 −u 2 |+· · ·+|u l−1 −u l | > |u l −x|, |u l+1 −u l+2 |+· · ·+|u k −z t,i | > 0, and the fact that the number of ordered sets is bounded above by |Π (Lt) | k , we have that the summand is bounded above by Note that the last step is justified by [3, B.6], which is valid since [3, Theorem B.1] is applicable, as in Proposition 5.11. There are now two cases to consider: (i) x = 0; and (ii) x ∈ B(z t,i , h t |z t,i |/3). Case (i). Assumption 5.8 implies that |u l | > |z t,i |(1 + h t ), and so by Proposition 5.12 we get that since h t log log t → ∞ and δ t = o(h t ). Note that in the last step we also bounded λ t,i − L t from below using the asymptotics in part (b) of Lemma 5.4. Case (ii). Apply the bound |u l − x| ≥ |u l − z t,i | − |z t,i − x| ≥ h t |z t,i | − |z t,i − x| then, similarly to as in Case (i), by Proposition 5.12 we get that   (1 + f t ) log log t .
The result then follows from part (a) of Lemma 5.4, since 1/ log log t = o(f t ).
Proof. This follows from Proposition 5.16 since r t g t f t log log t = o(td t e t ).

Completing the Proof
In this section, we complete the proof of the auxiliary Theorem 2.3 using the results obtained in Sections 4 and 5. Throughout this section we fix the constant c from Corollary 5.7, and also fix constants 0 < ε ′′ < ε < θ as in Sections 4 and 5. For notational convenience, abbreviate z (j) t := z t,i (j) t for j = 1, 2. Recall also the event E t,c from equation (16), whose probability goes to 1 by Corollary 4.8. Proof. As in Proposition 4.1, there is a path expansion forλ t (z) with z ∈ Π (Lt,ε) : where Γ * k (z) is the set of all length k nearest neighbour paths z =: y 0 → y 1 → . . . → y k := z in V t such that y i = z for all 1 < i < k. Since paths in Γ * Assumption 5.8. Then, make the correspondence in Lemma 6.5, and remark that κ t r t = o(h t |Z (1,ρ) t |). The third statement follows by applying the upperbound on eigenfunctions in Corollary 5.6 to the index i (1) t , and summing over all z ∈ V t \ B t .