Delocalising the parabolic Anderson model through partial duplication of the potential

The parabolic Anderson model on $\mathbb{Z}^d$ with i.i.d. potential is known to completely localise if the distribution of the potential is sufficiently heavy-tailed at infinity. In this paper we investigate a modification of the model in which the potential is partially duplicated in a symmetric way across a plane through the origin. In the case of potential distribution with polynomial tail decay, we exhibit a surprising phase transition in the model as the decay exponent varies. For large values of the exponent the model completely localises as in the i.i.d. case. By contrast, for small values of the exponent we show that the model may delocalise. More precisely, we show that there is an event of non-negligible probability on which the solution has non-negligible mass on two sites.


1.1.
Delocalising the parabolic Anderson model. Given a potential field ξ : Z d → R, the parabolic Anderson model (PAM) is the solution to the Cauchy problem with localised initial condition ∂ t u(t, z) = ∆u(t, z) + ξ(z)u(t, z), (t, z) ∈ (0, ∞) × Z, (1.1) where ∆ is the discrete Laplacian acting on functions f : with | · | the standard ℓ 1 distance. The PAM models the competition between smoothing effects, generated by the Laplacian, and roughening effects, generated by the potential. It is well known that if the potential ξ is sufficiently inhomogeneous, the PAM may undergo a process of localisation in which its solution is eventually concentrated, at typical large times, on a small number of spatially disjoint clusters of sites. Indeed, if ξ is an i.i.d. random field with the law of ξ(·) sufficiently heavytailed at infinity, the solution is known to eventually concentrate on a single site with overwhelming probability, i.e. there exists a Z d -valued process Z t such that, as t → ∞, u(t, Z t ) z∈Z d u(t, z) → 1 in probability.
In this case we say that the PAM completely localises.
While there are many results in the literature establishing localisation in the PAM in various settings (see Section 1.2 for an overview), our understanding of the absence of localisation is much less welldeveloped. In the case that the potential ξ is a random field, there are at least two features of ξ which may prevent complete localisation in the PAM. First, the potential may be too homogeneous on large scales -too close to a constant potential -for sharp peaks in the solution to form. Second, even if the potential is sufficiently inhomogeneous, complete localisation may be prevented by the presence of 'duplicated' regions in which the potential is very similar; in this case, the solution may have no reason to favour one such region over another.
The model we consider is an example of the PAM in a random potential that has spatial correlation.
To the best of our knowledge, the only previous work that has considered the PAM with correlated potential in a discrete setting is [9], in which the motivation was to more accurately model a physical system by introducing long-range correlations. The main result in that paper is an asymptotic formula for moments of the total solution; this shows that the solution is intermittent in a certain weak sense, but is not precise enough to determine the localisation/delocalisation properties of the model.

1.3.
The PAM with partially duplicated potential. In this section we formally introduce the PAM with partially duplicated potential that is the object of our study. For the remainder of the paper we fix d = 1; this avoids the additional complications that arise in higher dimensions, while preserving the phenomena that we seek to investigate.
We henceforth refer to ξ as the potential, and denote its corresponding probability and expectation by Prob and E respectively.
The model that we consider is the parabolic Anderson model on Z -i.e. the solution of equation (1.1) -with the partially duplicated potential ξ. It follows from [6] by the same argument as in the i.i.d. case that the solution exists provided that α > 1, and is given by the Feynman-Kac formula where (X t ) t≥0 is a continuous-time random walk on Z with generator ∆ started at the origin and P and E are its corresponding probability and expectation. We denote by the total mass of the solution.
1.4. The phase transition in the model. We are now ready to introduce our results. Let D = {z ∈ Z : ξ(z) = ξ(−z)} denote the set of integers whose potential values are duplicated, and E = Z\ D the set of positive integers whose potential values are unique (or exclusive) to them. For each t > 0 and z ∈ Z, define the functional Ψ t (z) = ξ(z) − |z| t log ξ(z).
Notice that Ψ t represents a balance between the local potential value and a 'penalty term' which increases in the distance to the origin; it turns out that Ψ t is a good approximation for the asymptotic growth rate of the high peaks solution of the PAM, in the sense that, for a high peak centred at z ∈ Z d , 1 t log u(t, z) ≈ Ψ t (z), see [12] for example. For each t > 0, let Ω t be the set of maximisers of Ψ t ; in Lemma 3.2 we prove that either Ω t = {z} for some z ∈ E, or Ω t = {−z, z} for some z ∈ D. Define D t = {|Ω t | = 2} to be the event that the maximisers of Ψ t are duplicated; an example of this event and its complement are depicted in Figure 1. Our first result is to show that, for all values of the Pareto parameter α > 1, the model always localises on the set Ω t . We also show that the event D t has non-negligible probability. Of course, outside the event D t this is already enough to conclude that the model completely localises.
Theorem 1.1 (Localisation of the model). Let α > 1. As t → ∞, 1 U (t) z∈Ωt u(t, z) → 1 in probability. (1.2) Moreover, as t → ∞, Our next two results establish the following phase transition in the model. If α ∈ (1, 2), then on the event D t the two sites in Ω t both have a non-negligible proportion of the solution; in other words the model delocalises. By contrast, if α ≥ 2 only one site in Ω t has a non-negligible proportion of the solution; in other words, the model completely localises whether the event D t holds or not. Surprisingly, the critical value of α = 2 does not depend on the value of p. To state these result, let Z (1) t ∈ Ω t , with Z (1) t chosen arbitrarily on the event D t . Theorem 1.2 (Delocalisation in the case α ∈ (1, 2)). Let α ∈ (1, 2). As t → ∞, where Υ is a random variable with positive density on R + , L(·) denotes the law of a random variable, and ⇒ denotes weak convergence. In Section 5. 3 we give an explicit construction of the random variable Υ. Theorem 1.3 (Complete localisation in the case α ≥ 2). Let α ≥ 2. As t → ∞, log u(t, Z (1) t ) u(t, −Z (1) t ) → ∞ in probability.
Remark 1. At first glance it may seem counter-intuitive that delocalisation occurs for small, rather than large, values of α, since by analogy with the i.i.d. case we might expect that the heavier the tails of the potential, the stronger the localisation. However, in our model it is precisely the strengthening of the concentration effect for small α which results in delocalisation.
To explain this, consider that if α is smaller, the advantage of the sites in Ω t relative to other sites is increased. We show that, if α is small enough, this advantage is so great that the impact of the other potential values (at sites closer to the origin than Z (1) t ) is minimal, and the solution cannot readily distinguish between the sites in Ω t . On the other hand, for large values of α the advantage is less pronounced, and the fluctuations in the other potential values eventually force one of the sites in Ω t to be significantly more beneficial than the other. In the next subsection, we give some heuristics for why the transition occurs at α = 2. ⋄ Remark 2. One surprising aspect of the phase transition in the model is that it is not sharp. In particular, the random variable Υ in Theorem 1.2 does not, as might be expected, degenerate for small α. As will be further explained in the next subsection, this is ultimately due to two different scales, arising from distinct sources, exactly cancelling each other out. ⋄ Remark 3. The proof of the Theorem 1.1 is relatively straightforward, and is similar to analogous results in the i.i.d. case, see [10,12,17]. The proof of Theorems 1.2 and 1.3 are much more involved, and require us to analyse the model, and indeed the PAM with i.i.d. potential, in much finer detail than has been done in previous work. ⋄ Remark 4. Our main results can be recast as a demonstration of the robustness, or lack thereof, of the total mass of the solution of the PAM with i.i.d. potential under a resampling of some of the potential values. More precisely, suppose u(t, z) denotes the solution of the PAM on Z with the i.i.d. potential ξ 0 , with U (t) = z u(t, z) the total mass of the solution. Now consider resampling each potential value independently with probability q ∈ (0, 1), and letũ(t, z) be the solution of the PAM with this resampled potential, withŨ (t) = zũ (t, z) the total mass of the solution. Then our results, suitably translated, demonstrate the following phase transition. If α ∈ (1, 2), then there exists an event of non-negligible probability on which U (t)/Ũ (t) converges in distribution to a random variable with positive density on R + . By contrast, if α ≥ 2, then | log U (t)/Ũ (t)| → ∞ in probability. ⋄ 1.5. Heuristics for the phase transition. We start by recalling that the high peaks of the solution have the first-order approximation log u(t, z) ≈ tΨ t (z) = tξ(z) − |z| log ξ(z).
The main challenge with analysing our model, compared to i.i.d. case, is that on event D t there are two maximisers of the functional Ψ t . Hence the height of the peak at these sites is identical up to the first-order approximation. As a result, and in contrast to the i.i.d. case, in order to understand the localisation phenomena we must turn to second-order contributions. Notice that the first-order approximation, captured by the functional Ψ t , has the nice feature that it is local, depending on the value of ξ at the site z only. By contrast, the second-order contributions depend on all the potential values along entire paths to Z (1) t and −Z (1) t . This makes them much more challenging to study.
To explain the phase transition in the model at α = 2, we show that the second-order contributions undergo two distinct transitions as α increases, both of which, seemingly coincidentally, occur at α = 2. The first transition is the negligibility or otherwise of non-direct paths which end at the sites in Ω t ; this transition serves mainly as a extra technical difficulty in our proofs, rather than a determining factor in the phase transition of the model. The second transition is a shift in the fluctuations of the second-order contributions from the Gaussian universality class (α ≥ 2) to the α-stable universality class (α ∈ (1, 2)), and it is this which turns out to cause the phase transition of the model.
These transitions are also relevant for the PAM with i.i.d. potential, and give a more nuanced understanding of localisation phenomena in the i.i.d. case than has previously been available. For example, in the case α ∈ (1, 2), our proof of the first transition establishes that the PAM path measure, given by concentrates on a single geometric path (i.e. the direct path to the localisation site), which is much stronger result than the complete localisation of the solution. In the case α ≥ 2, we strongly suspect that the path measure instead concentrates on a class of paths that end at the localisation site but which also contain small loops.
1.5.1. The first transition: Direct/non-direct paths. Recall that the Feynman-Kac formula allows us to consider the contribution to U (t) coming from different geometric paths which start at the origin. Assuming the localisation result in Theorem 1.1, we know that, for all α > 1, the only significant contribution to U (t) comes from paths which end in Ω t . In Proposition 4.2 we show that, if α ∈ (1, 2), the only significant contribution to U (t) actually come from the direct paths to Ω t ; here we give some heuristics for why this should be true. On the other hand, if α ≥ 2, then we strongly believe that certain sets of non-direct paths do make a non-negligible contribution to U (t); since we do not need this for our main results, we do not formally prove this.
Assume that α ∈ (1, 2) and let y (t) denote the direct path from the origin to Z (1) t . For the purposes of keeping the calculations simple, we will show only that the contribution to U (t) from paths Π (t,+) from the origin to Z (1) t obtained by adding a single loop of length two to y (t) , anywhere along the path except at the end, are negligible with respect to the contribution to U (t) from the path y (t) itself. The same argument can be extended, with minor adaption, to cover all non-direct paths to Ω t .
We can assume without loss of generality that Z (1) t ∈ N. For any path y = (y 0 , . . . , y n ) of length n we can write the contribution from y at time t as U (t, y) = e −2t I n (t; ξ(y 0 ), . . . , ξ(y n )), for a function I n (t; a 0 , . . . , a n ) with a rather nice structure; see equations (2.2) and (2.3). In Lemma 3.9 we prove a bound on I which enables us to compare U (t, y) for various paths. This lemma implies that for any path y (t,+) ∈ Π (t,+) we have which reflects the fact that each extra step induces in a 'penalty' of order (ξ(Z (1) t ) − ξ(j)) −1 . In Proposition 5.6 we prove that (up to a small correction) ξ(Z (1) t ) − ξ(j) ≥ (t/ log t) 1/(α−1) for any 0 ≤ j < Z (1) t , and also that |Z (1) t | is asymptotically (t/ log t) α/(α−1) . Since there are no more than 2|Z (1) t | such paths (there are |Z (1) t | places to add the loop and two directions the loop can go in), their total contribution is at most Notice that the exponent is negative if α < 2, which confirms that such paths are negligible with respect to the direct path. As mentioned, we can readily extend this argument to all non-direct paths.
1.5.2. The second transition: The universality class of fluctuations. To keep things simple, and since the intuition is correct, we shall for now assume that, for all α > 1, it is sufficient to consider only direct paths (even though we strongly believe that this is only true in the case α ∈ (1, 2)).
Assume that the event D t holds and denote by y (t,1) the direct path to Z (1) t and y (t,−1) the direct path to −Z (1) t . Abbreviate ι = sgn(Z (1) t ). We derive in Lemma 3.8 that, provided a n = a i for i = n, the function I satisfies I n (t; a 0 , . . . , a n ) = e tan n−1 In Proposition 4.1 we show that the second term in the form of I above can be essentially discarded when considering the direct path, thus giving , and similarly for U (t, y (t,−1) ). Using the assumption that only direct paths are significant, we obtain where we have used a Taylor expansion for the logarithm in the last step (this Taylor expansion does not actually converge if α < 2, but it does give a good insight into the scale of the fluctuations; see Section 6 for precise statements). Note that the summand is zero for each j ∈ D and so in expectation there are q|Z (1) t | non-zero terms, where q = 1 − p. At this point we have reduced the study of the ratio u(t, Z (1) t )/u(t, −Z (1) t ) to the study of fluctuations in the sum of independent (although not identically distributed) random variables, and so we may appeal to the well-developed theory of such fluctuations.
In the case α ∈ (1, 2), these fluctuations belong to the α-stable universality class, and so we obtain where Y is a certain non-degenerate random variable. Since we prove in Proposition 5.6 that the growing scales in (1.3) exactly cancel out. Hence the ratio u(t, Z (1) t )/u(t, Z (2) t ) remains of constant order as t → ∞, and so there is a non-negligible proportion of the solution at both sites in Ω t . In the case α > 2, the fluctuations are instead in the Gaussian universality class, and so we obtain Using (1.4), this gives that The case α = 2 is slightly more delicate, but using the extra logarithmic factor that appears in the fluctuations, we can prove that | log u(t, Z (1) t )/u(t, −Z (1) t )| → ∞ also in this case. 1.6. Future work. Intuitively, the closer p is to 1, the more symmetric the model becomes and the more likely that the model delocalises for a wider class of potentials. Our results show that if p is uniformly bounded away from 1 then this intuition is not realised, since the threshold α = 2 is the same for all values of p ∈ (0, 1). This leads us to wonder what happens if p is not uniformly bounded away from 1. One way to investigate this is to let ξ(z) = ξ(−z) with probability p = p(|z|) that depends on the distance of z from the origin. We can then ask the question: how fast should p(n) → 1 so that, for a given value of α > 2, complete localisation fails? We conjecture that there is a critical scale for p(n) such that if and only if p → 1 slower than this scale then complete localisation holds. We will investigate this model in a future paper.

Outline of proof
In this section we give an outline of the proof of our main results, and an overview of the rest of the paper. We assume henceforth that α > 1.
Step 1: Trimming the path set. As already remarked, the Feynman-Kac formula allows us to consider contributions to u(t, z) coming from various geometric paths which start at the origin and are at site z at time t. The first step is to eliminate paths that a priori make a negligible contribution to the solution, either because they fail to hit the sites in Ω t or because they make too many jumps. This step is rather standard, and is similar to in [10,12,17].
We now define the a priori negligible paths. Introduce the scales which, as suggested in (1.4), are the asymptotic scales for |Z (1) t | and ξ(Z (1) t ) respectively. For technical reasons, we also introduce some auxiliary positive scaling functions f t → 0 and g t → ∞ which can be thought of as being arbitrarily slowly decaying or growing. We shall need these scales to satisfy (2.1) For any set A ⊆ Z denote by τ A = inf{t > 0 : X t ∈ A} its hitting time by the continuous-time random walk (X s ). Let J t be the number of jumps of (X s ) by time t. We decompose the total mass U (t) into a significant component In Section 3.2 we use standard methods to prove that U 1 is negligible with respect to U as long as certain typical properties of ξ hold. To define these properties, denote, for each n ∈ N 0 , ξ (1) n = max{ξ(z) : |z| ≤ n} and ξ (2) n = max{ξ(z) : |z| ≤ n, ξ(z) < ξ (1) n }.
Let Z (2) t be a maximiser of Ψ t on the set Z \ Ω t ; we prove that Z (2) t exists in Lemma 3.2. The typical properties are contained in the event which in particular guarantees a large gap between the value of Ψ t at sites in Ω t and all other sites. Remark that, by the conclusion of Lemma 3.2, the event E t does not depend on our choice of Z (2) t .
Step 2: Reduction to subsets of paths that end at Ω. At this point understanding U 0 becomes the main goal, and we aim to find out which paths make a non-negligible contribution to it; here we make a distinction between the cases α ∈ (1, 2) and α ≥ 2 (see the heuristics in Section 1.5).
The main input is a careful analysis of the properties of the function I that defines the contribution to U (t) for any path. To define this function precisely, denote by P all = {y = (y 0 , . . . , y ℓ ) ∈ Z ℓ+1 : ℓ ∈ N 0 , |y i − y i−1 | = 1 for all 1 ≤ i ≤ ℓ} the set of all geometric paths on Z. For each path y ∈ P all , denote by ℓ(y) its length (counted as the number of edges). Denote by (τ i ) i∈N0 the sequence of the jump times of the continuous-time random walk (X t ) and by P (t, y) = {X 0 = y 0 , X τ0+···+τi−1 = y i for all 1 ≤ i ≤ ℓ(y), t − τ ℓ(y) ≤ τ 0 + · · · + τ ℓ(y)−1 < t} the event that the random walk has the trajectory y up to time t. Let be the contribution of the event P (t, y) to U (t). By direct computation, we have x i > t dx 0 · · · dx ℓ(y) = e −2t I ℓ(y) (t; ξ(y 0 ), . . . , ξ(y ℓ(y) )). (2.2) where the function I is defined by I n (t; a 0 , . . . , a n ) = e tan R n for each t > 0, n ∈ N, and a 0 , . . . , a n ∈ R. In particular, I 0 (t; a 0 ) = e ta0 .
In Section 3.3 we show that I has a rather neat symmetric structure and study its properties. Using this understanding, in Section 4 we identify the paths making non-negligible contribution to U 0 . For α ∈ (1, 2) the situation is relatively simple: in Propositions 4.1 and 4.2 we show that only the direct paths to Ω t are significant, and approximate their contribution to U 0 (t) by a certain product over the path. This is useful because, since each site is visited at most once, we can invoke standard fluctuation theory to analyse this product.
The situation is more complicated for α ≥ 2 since we strongly suspect that non-direct paths are significant. Instead we show in Proposition 4.6 that, as long as certain additional typical properties of ξ hold, we can limit the significant paths to those that end at Ω t and visit each site in {0} ∪ N t at most once, where N t is a set of non-duplicated sites of high potential. The advantage is that, after careful conditioning, it will be sufficient to study the fluctuations of the contribution from sites in N t . Since these sites are visited at most once, we can again apply standard fluctuation theory.
To define the set N t precisely, we first introduce an additional auxiliary scaling function which is chosen in such a way that, on the one hand, 1/δ t grows slower than (log t) 1 α , but on the other hand, log(1/δ t ) grows faster than any power of g t and 1/f t . For each t > 0, we then let The additional typical properties we need are which guarantees the set N t is large enough and well-separated from Ω t ; in Proposition 3.3 we prove that this event holds eventually with overwhelming probability, assuming the event E t also holds.
This analysis is already enough to finish the proof of Theorem 1.1 assuming the event E t holds; we complete the proof at the end of this section.
Step 3: Point process techniques. In Section 5 we build up a point process approach to study the high exceedences of ξ and the top order statistics of the penalisation functional Ψ t . We start by proving that the potential ξ, properly rescaled, converges to a Poisson point process. We then use this convergence to pass certain functionals of ξ, including properties of Ψ t , to the limit. Since this analysis involves several lengthy computations, some of the proofs are deferred to Appendix A.
To end the section, we draw two main consequences from our point process analysis. First, we establish that the event E t holds eventually with overwhelming probability. Second, we give an explicit construction for the limit random variable Υ appearing in Theorem 1.2; this is done via identifying it as the law of a certain time-inhomogeneous Lévy process stopped at a random time.
Step 4: Fluctuation theory for the ratio u(t, Z (1) t )/u(t, −Z (1) t ). At this point we have assembled all the main ingredients, and all that is left is to apply fluctuation theory to analyse the ratio u(t, Z (1) t )/u(t, −Z (1) t ); here we again distinguish between the cases α ∈ (1, 2) and α ≥ 2 (see the heuristics in Section 1.5). In Section 6 we study the case α ∈ (1, 2) and complete the proof of Theorem 1.2. In particular, since only direct paths contribute significantly to U 0 (t), and since the contribution from these paths can be approximated by a product over the path, we can use standard theory to study these fluctuations. With the aid of our point process analysis, we prove that the ratio u(t, Z (1) t )/u(t, −Z (1) t ) converges to the limit random variable we identify in Section 5. In Section 7 we study the case α ≥ 2 and complete the proof of Theorem 1.3. Here we apply a central limit theorem to establish that the fluctuations in u(t, Z (1) t )/u(t, −Z (1) t ) due to the sites N t (which are visited at most once) are in the Gaussian universality class; the proof of the central limit theorem is deferred to Appendix B. These fluctuations turn out to already be sufficient to prove that irrespective of the contribution due to the other sites.

Preliminaries
In this section we establish some preliminary results. First, we prove asymptotic properties of the potential ξ. Second, we establish the negligibility of U 1 (t). Lastly, we study the structure of the function I introduced in (2.3).

3.1.
Asymptotic properties of the potential. To begin, we establish asymptotic properties of the potential. This allows us to deduce properties of the maximisers Z (1) t and Z (2) t , and also to establish that E [2,∞) t holds eventually with overwhelming probability.
Lemma 3.2. Almost surely either Ω t = {z} for some z ∈ E or Ω t = {−z, z} for some z ∈ D, and the same conclusion holds for the maximisers of Ψ t on the set Z \ Ω t . Moreover, Proof. By Lemma 3.1 with 0 < ε < min{1 − 1/α, 1/α}, for all z with |z| sufficiently large which is a bounded function of |z|. Hence Ψ t is bounded for each t > 0. Since Ψ t (z) is a continuous random variable with no point mass, this implies the first statement. For the second statement, let z n ∈ Z be such that ξ(z) = ξ (1) n . For any sufficiently large n we have eventually as t → ∞. Hence there are infinitely many points where Ψ t is larger than one and, in particular, Ψ t (Z (1) t ) and Ψ t (Z (2) t ) are both larger than one eventually.
Proof. Let (1) and observe that we may work on the event

by the law of large numbers.
For each t > 0, denote by G t the σ-algebra generated by D, Z (1) t and ξ(Z (1) t ), and denote the conditional probability with respect to G t by Prob Gt . It is easy to see that, conditionally on G t , the events z ∈ N t z∈E,|z|<|Z (1) t | are independent. Hence we can stochastically dominate the desired properties of N t by equivalent properties of Bernoulli trials, and use standard properties of such trials to complete the proof. For each z ∈ E, |z| < |Z (1) t |, the conditional distribution of ξ(z) with respect to G t is the Pareto distribution with parameter α conditioned on Ψ t (z) uniformly for all z for all sufficiently large t almost surely. Further, Combining two above inequalities with (3.1) we get uniformly for all z for all sufficiently large t almost surely. Using this together with the conditional independence and the properties guaranteed by E ′ t and E ′′ t we infer that eventually where Bin(n, τ ) denotes a binomial random variable with parameters n ∈ N and τ ∈ [0, 1], and ≺ denotes stochastic domination. By looking at the characteristic function of the binomial distribution we see that This condition is clearly satisfied by both binomial random variables in (3.3) by the choice of δ t , f t , and g t . To complete the proof of the inequalities on |N t |, it remains to notice Similarly, the upper bound in (3.2) also implies that, conditionally on G t , where Geo 1 (τ ) and Geo 2 (τ ) denote two independent geometric random variables with parameter τ ∈ [0, 1] supported on N. Observe that

3.2.
Eliminating the a priori negligible paths. We begin by decomposing U 1 (t) into We first find a lower bound for U in Lemma 3.4 and upper bounds for U ′ 1 and U ′′ 1 in Lemmas 3.5 and 3.6 respectively, before combining these to prove the negligibility of U 1 . This approach is standard and similar to [10,12,17].
Proof. The idea of the proof as the same as of [12,Prop. 4.2]. Let ρ ∈ (0, 1] and z ∈ Z, z = 0. Following the lines of [12,Prop. 4.2], we obtain . Observe that on the event E t this ρ eventually belongs to (0, 1] since as required.
Lemma 3.5. Almost surely, on the event E t as t → ∞.
Proof. Observe that the number of jumps J t of the continuous-time random walk by the time t has Poisson distribution with parameter 2t. Estimating the integral in (3.4) by tξ (1) n on the event {J t = n} and using Lemma 3.1 we obtain Denote by n t > R t the maximiser of the expression on the right-hand side, and by If |z t | > R t then by monotonicity n t = |z t | and Lemma 3.6. Almost surely, Proof. For any n ∈ N 0 , let Denote by n t ≤ R t the maximiser of the expression on the right-hand side, and by z t ∈ Z a point such that ξ(z t ) = ζ nt .
If |z t | < r t (log t) −2 then by monotonicity and Lemma 3.1 with small ε > 0 Proposition 3.7. Almost surely, Second, On the event E t , as t → ∞, we have for the first term and for the second term By (2.1) the first term is negligible with respect to the second term and so which proves the claim.
It remains to show that U ′′ 1 (t)/U (t) → 0 on the event E t . Combining Lemmas 3.4 and 3.6 we have on the event E t 3.3. Structure of the function I. In this section we study the structure of the function I introduced in (2.3). Our point of departure is the recursion I n (t; a 0 , . . . , a n ) = 1 a n − a n−1 I n−1 (t; a 0 , . . . , a n−2 , a n ) − I n−1 (t; a 0 , . . . , a n−2 , a n−1 ) (3.8) whenever a n = a n−1 , obtained by integrating the expression in (2.3) over x n−1 . By iterating this recursion we establish the following.
Lemma 3.8. The following hold: (1) If a n = a i for i = n then I n (t; a 0 , . . . , a n ) = e tan Moreover, if a 0 , . . . , a n are pairwise distinct then (2) I n is symmetric with respect to the variables a 0 , . . . , a n .
Proof. The first statement in (1) follows by induction from (3.8), where we apply induction to the first term in the recursion and keep the second term. The second statement in (1) also follows by induction once we notice that it is true for n = 0 and the expression on the right hand side satisfies the recursion (3.8). Finally, the symmetry of I n for pairwise distinct variables follows from the symmetry of the expression on the right hand side of (3.9). Then it extends by continuity to all variables.
We now establish two upper bounds on the function I. The first bounds the effect of adding additional steps onto a base path. The second bounds the effect of changing the largest value of a i along a path; for this we shall need an additional lemma that establishes 'negative dependence' in the effect on I due to changes in the a i . Lemma 3.9. Let m, n ∈ N 0 and suppose a j < a n for all 0 ≤ j < n and a j = a n for all n ≤ j ≤ n+m. x s (a s − a n ) 1 x s (a s − a n ) 1 Further, it follows from (3.8) and symmetry of I proved in Lemma 3.8 that I n+m−i (t; a 0 , . . . , a n+m−i ) ≤ I n+m−i−1 (t; a 1 , . . . , a n+m−i ) 1 a n − a 0 ≤ · · · ≤ I n+m−i−k (t; a k , . . . , a n+m−i ) k j=1 1 a n − a j−1 .
Proof. First we remark that a density being log-concave is equivalent to the density being a Polya frequency function of order 2 (or PF 2 , using the terminology from [4]). Then [4,Theorem 4.1] implies that (Y 1 , . . . , Y n ) is reverse regular of order 2 in pairs (again using the terminology of [4]). Then the discussion following Definition 2.2 in [4] demonstrates that this implies the result.
Lemma 3.11. Let n ≥ 2. For any k = j ∂ 2 ∂a k ∂a j log I n (t; a 0 , . . . , a n ) ≤ 0. (3.10) Proof. By symmetry of I proved in Lemma 3.8 it suffices to prove the statement for j, k = n. Denote a = (a 0 , . . . , a n ). It is easy to see that (3.10) is equivalent to showing that Fix t > 0. Let W i , 0 ≤ i ≤ n, be independent random variables with density c i e (ai−an)x on [0, t] and zero otherwise, where c i is a normalising constant, and let W n+1 be uniform on [0, t]. We remark that each of W i , 0 ≤ i ≤ n + 1, has a log-concave density. Further, letŴ i , 0 ≤ i ≤ n, be defined by Since the densities of W i , 0 ≤ i ≤ n + 1, are log-concave, by Theorem 3.10 we have To prove (3.11), it suffices now to show that For this note that x i ≤ t dx 0 · · · dx n = ce −tan ∂ ∂a k I n (t; a), where c is a normalising constant and thus satisfies x i ≤ t dx 0 · · · dx n = ce −tan I n (t; a).
Plugging this value of c into the above equations gives the required identities.
Proof. Since the function s → log I n (t; a, s) is continuous we can write I n (t; a, x) I n (t; a, y) = exp − y x ∂ ∂s log I n (t; a, s)ds . Since all a i ≤ y we can use monotonicity proved in Lemma 3.11 to obtain ∂ ∂s log I n (t; a − y, s − y) ≥ ∂ ∂s log I n (t; 0, s − y), (3.13) where 0 = (0, . . . , 0). This implies It is easy to see that I n (t; 0, 0) = t n n! . (3.14) Using y > x and the substitution u i = x i , 0 ≤ i ≤ n − 2 and u n−1 = x 0 + · · · + x n−1 in the definition (2.3) of I n , we also obtain integrating over u n−1 that Combining (3.12), (3.13), (3.14) and (3.15) gives the stated result.

Significant paths
The aim of this section is to determine which paths make a non-negligible contribution to U 0 (t). As described in Section 1.5, in the case α ∈ (1, 2) we prove that only the direct paths to Ω t are significant.
In the case α ≥ 2, we can only prove the much weaker result that the significant paths are those which end at Ω t and visit the set N t ∪ {0} at most once, where N t is the set of non-duplicated sites of high potential defined in (2.4) (actually this is true for all α > 1, but is not as strong as what we prove for α ∈ (1, 2)).
Assuming the event E t holds, this is already enough to prove the localisation statement in Theorem 1.1; we complete this proof at the end of the section.
Proof. Fix t > 0, ι ∈ {−1, 1}, and assume the corresponding event E t or E t ∩ D t holds. Denote According to (2.2) and Lemma 3.8 we have U (t, y (t,ι) ) = e −2t I n (t; a 0 , . . . , a n ) Observe that a n − a j > 1 for all 1 ≤ j < n. Further, again by (2.2) we have for all 0 ≤ i < n, where w (i) is the shortest path to iι. Since n−1 i=0 U (t, w (i) ) ≤ U ′′ 1 (t), by Lemma 3.6 we have n−1 i=0 e −2t I i (t; a 0 , . . . , a i ) Combining this with the lower bound for U (t) from Lemma 3.4 and also taking into account that which completes the proof.  ∈ (1, 2). Almost surely, on the event E t , as t → ∞.
Proof. For a path y ∈ P all that hits Ω t , let z t (y) ∈ Ω t be the first point where y hits Ω t , i.e., Denote by m t (y) the number of times Ω t is visited minus one, i.e., Denote by 2w t (y) the difference between the hitting time of z t (y) and |Z (1) t |, i.e., Finally, denote by s t (y) the number of points on the path after the first visit to Ω t that do not belong to Ω t , i.e., Observe that s t (y) ≥ m t (y).

Using Lemma 4.3 below we have
For any y ∈ P t m,w,s we use (2.2) and Lemma 3.9 with n + m being the length of y, i = m, k = n, a 0 , . . . , a n−1 being the values of ξ along y except when it visits Ω t , and a n , . . . , a n+m = ξ(Z (1) t ) and obtain , on the event E t . We will keep |Z (1) t | terms in the product corresponding to one visit to each of the points iι, 0 ≤ i ≤ |Z (1) t | − 1, where ι = sgn(z t (y)), and estimate the rest by Rt > a t f t . This implies By Proposition 4.1 we obtain on E t Let us show that the total mass corresponding to all paths from P t m,w,s except those corresponding to (m, w, s) = (0, 0, 0) is negligible. Indeed, (1), and 2t atft = o(1) as α ∈ (1, 2). Combining this with (4.1) we obtain on the event E t which gives the required result by Proposition 3.7.

4.2.
The case α ≥ 2: Paths to Ω t visiting sites in N t at most once. Our proof proceeds in two stages. First, we analyse the portion of the part up until the first visit to Ω t and after the last visit to Ω t , and show that, in this portion of the path, it is never beneficial to visit sites in N t ∪ {0} more than once. Second, we analyse the portion of the path consisting of the loops that occur between first and last visit to Ω t , showing that it is never beneficial for these loops to return to sites in N t ∪ {0}; in fact, we show the stronger result that these loops have length at most ⌊2α⌋ (although we suspect that the optimal bound is actually ⌊α⌋).
Denote by P t the set of all geometric paths contributing to U 0 (t), that is, those visiting Ω t and having length at most R t . Fix t > 0 and let y ∈ P t . The skeleton of y, denoted skel(y), is the geometric path from the origin to a site in Ω t constructed by chronologically removing all loops in y which start and end at any site belonging to {0} ∪ N t up until the first visit of Ω t as well as removing any part of the path after the final visit of y to Ω t .
We can now partition P t into equivalence classes by saying that paths y andŷ are in the same class if and only if skel(y) = skel(ŷ). We write P t for the set of all such equivalence classes. Note that any such equivalence class P ∈ P t contains the null path, y P null ∈ P, defined as y P null = skel(y P null ). Observe that every null path, prior to visiting Ω t for the first time, either (i) visits each site in {0} ∪ (N t ∩ N) exactly once, or (ii) visits each site in {0} ∪ (N t ∩ −N) exactly once. In particular, until the first visit of Ω t each null path visits either only positive integers, or only negative integers.
The importance of the null path is through the following lemma, which states that the contribution to the solution coming from an equivalence class is dominated by that coming from the null path. uniformly for all P ∈ P t on the event E t ∩ E Proof. For k ∈ N, write P k for the subset of P consisting of the paths with additional length k compared to y P null . We have on E t since each of the additional k pieces will be added to a loop at a site in {0} ∪ N t or at the end in at most two ways. Note that no null path can visit both sites in Ω t since each null path is in P t and has length at most R t < 2|Z (1) t |. Using (2.2) and Lemma 3.9 with m + 1 being the number of visits of y to Ω t , n + m the length of y, i = 0, a 0 , . . . , a k−1 the values of ξ at the additional points of y, a k , . . . , a n−1 the values of ξ along y P null except when it visits Ω t , and a n = · · · = a n+m the value of ξ on Ω t , we obtain on E t . Since none of the additional sites visited by any path in P are in Ω t , we have on E t as a t f t δ 2α t → ∞. We now eliminate paths that make loops from Ω t that return to sites in N t . Denote by Null t 1 the set of all null paths in P t which visit each site in {0} ∪ N t at most once, Null t 2 for all other null paths in P t and Null t for their union.    Proof. Note that by the construction of null paths, the only way for a null path to visit a site in N t more than once is by having a loop from Ω t . On the event E [2,∞) t this loop must have length at least g t . We shall show a stronger result than is needed: that all null paths with loops from Ω t of length more than k 0 , where k 0 > 2α, have negligible contribution to the solution compared to the contribution from all other null paths.
To do this we partition Null t into equivalence classes by saying two null paths are in the same class if and only if they are identical after removing all loops from Ω t of length at least k 0 . For any such equivalence class P, write y P min for the path in P of minimum length (i.e. the path without any loops from Ω t of length at least k 0 ). Further, for any k ≥ k 0 , write P k for the set of paths in P with additional length k compared to y P min . Finally we write N t for the set of all such equivalence classes. Observe that for all k ≥ k 0 and P ∈ N t , any path y ∈ P k can make no more than ⌊k/k 0 ⌋ extra visits to Ω t compared to y P min . Using (2.2) and Lemma 3.9 with m + 1 being the number of visits of y to Ω t , n + m the length of y, i the number of additional visits to Ω t compared to y P min , a 0 , . . . , a k−1−i the values of ξ at the additional points of y except when it visits Ω t , a k−i = · · · = a k−1 the value of ξ on Ω t , a k , . . . , a n−1 the values of ξ along y P min except when it visits Ω t , and a n = · · · = a n+m the value of ξ on Ω t , we obtain since there are at most ⌊k/k 0 ⌋ additional loops, at most (R t − |Z (1) t | + 1)/2 points where such a loop can be created, and at most 2 k shapes of the loops.
Hence, for any P ∈ N t , on the event E t y∈P Since k 0 > 2α this implies as t → ∞ uniformly over the equivalence classes. To conclude the proof, note that Proposition 4.6. Almost surely, Proof. This is a direct consequence of Lemmas 4.4 and 4.5. Indeed, , as t → ∞.

4.3.
Completion of the proof of Theorem 1.1. We are now in a position to prove the localisation statement in Theorem 1.1 on the event that E t holds; the fact that P(E t ) → 1 as t → ∞ will be proven in Proposition 5.6. The second statement of Theorem 1.1, that P(D t ) → p/(2 − p), will be proven in Proposition 5.7.

By Proposition 3.3 we may work on the event
Since U 1 is negligible with respect to U by Proposition 3.7, it remains to show that the contribution to U 0 from the paths not ending in Ω t is negligible. For α ∈ (1, 2) this follows from Propositions 4.2; for α ≥ 2 this follows from Propositions 4.6 and 3.3. In fact, the latter argument works for all α > 1 but we prefer to use the much simpler argument for α ∈ (1, 2).

Point process analysis
In this section we develop a point processes approach to analyse the high exceedences of ξ and top order statistics of the penalisation functional Ψ t . We use this analysis to prove that the E t holds eventually with overwhelming probability. We also use it to give an explicit construction for the limiting random variable Υ from Theorem 1.2. Since the proofs in this section are quite technical, we defer some of them to Appendix A.
Recall that E = Z \ D denotes the set of positive integers whose potential values are exclusive, and abbreviate q = 1 − p.

5.1.
Point process convergence for the rescaled potential. The first step is to establish that the potential, properly rescaled, converges to a Poisson point process. The limiting point process will arise as a superposition of two distinct independent Poisson point processes that are, respectively, the limit of the potential restricted to the duplicated and the exclusive sites. Let us begin by defining the limiting point process. Consider the measure

5.2.
Asymptotic properties of the top order statistics of the penalisation functional. We now show how to use the convergence of the potential to extract asymptotic properties of the top order statistics of the penalisation functional Ψ t . We first introduce the limiting versions of Z (1) t , Z (2) t and D t and study their properties, before arguing that we may successfully pass these properties to the limit. for any Borel set A, whereÂ denotes the reflection of A with respect to the y-axis. Remark that the three components ofΠ are independent Poisson point processes with the intensity measures qµ, qµ and pµ, respectively, and soΠ is itself a Poisson point process with intensity measure (2q + p)µ = (2 − p)µ. Abbreviate and let the positive random variables X (1) , X (2) , Y (1) and Y (1) be defined by the properties that (X (1) , Y (1) ) ∈Π, and if (x, y) ∈Π and (x, y) = (X (1) , Y (1) ) then y − ρx < Y (1) − ρX (1) , (X (2) , Y (2) ) ∈Π, and if (x, y) ∈ Π \ (X (1) , Y (1) ) then y − ρ|x| < Y (2) − ρX (2) .
At the end of this section we shall identify (X (i) , Y (i) ), i = 1, 2, and D as the limiting versions of (|Z (i) t |, ξ(Z (i) t )), i = 1, 2, and D t respectively. For now, we establish some properties of these objects. Lemma 5.2. Almost surely, the random variables X (1) , X (2) , Y (1) and Y (2) are well-defined and satisfy Proof. For any a > 0 compute Since this is finite, almost surely there are finitely many points of (x, y) ∈Π satisfying y − ρx > a.
On the other hand, since (5.2) tends to infinity as a ↓ 0, almost surely there exist points (x, y) ∈Π satisfying y − ρx > 0. This implies the result.
Proof. We have To complete the result, compute Proof. Since the three components Π (e) , Π (e) and Π (d,+) that make upΠ are independent Poisson point processes with the intensity measures qµ, qµ and pµ respectively, we have We now argue that we can pass the above properties to the limit. As a consequence, we prove that the event E t holds eventually with overwhelming probability. Since the proof of these results are rather technical, we defer them to Appendix A.
3. An explicit construction of the limiting random variable. We complete this section by giving an explicit construction of the limiting random variable Υ in Theorem 1.2. For any δ > 0, let (1) and on the event D, and zero otherwise, and let Observe that these variables are well-defined since, for every (x, y) ∈ Π (e) such that |x| < X (1) , we have y < Y (1) − ρX (1) + ρ|x| < Y (1) .
In the next lemma we show that, as δ → 0, for a certain random variable Υ with positive density on R + . In Section 6 we identify Υ with the random variable appearing in Theorem 1.2.
Lemma 5.8. As δ ↓ 0, where Υ is a random variable with positive density on R + . The variable log Υ can be realised as the value at time X of a time-inhomogeneous Lévy process with zero drift, no Brownian component and the Lévy measure otherwise, where (X, Y ) ∈ R 2 is a random variable with density given by (5.3).
Proof. Denote for brevity X = X (1) and Y = Y (1) . Conditionally on Π (d) , D and (X, Y ), the point process Π (e) is Poissonian with the intensity measure where we consider δ < Y − ρX. Further, conditionally on Π (d) , D and (X, Y ), the variable S (δ,−) is independent and identically distributed with S (δ,+) . Due to symmetry, S (δ) is therefore the value of a time X of a time-inhomogeneous Lévy process with zero drift, no Brownian component and the Lévy measure As δ ↓ 0, S (δ,−) converges weakly to the value at time X of a time-inhomogeneous Lévy process with zero drift, no Brownian component and the Lévy measure given by (5.5), where the limiting Lévy measure is valid because Since a Lévy process has positive density on R at positive times, and since the law of log Υ is obtained by averaging over the law of Lévy processes at positive times, Υ also has positive density on R + .
6. Fluctuation theory in the case α ∈ (1, 2) In this section we study the fluctuations in the ratio u(t, Z (1) t )/u(t, Z (2) t ) in the case α ∈ (1, 2), building on our analysis in Section 4.1, and hence complete the proof of Theorem 1.2.
Recall that E denotes the set Z \ D. For any t > 0, let on the event E t and zero otherwise. In Section 4 we showed that the ratio u(t, Z (1) t )/u(t, −Z (1) t ) was well-approximated by exp{S t }, so it remains to study the convergence of S t . To do this, we first truncate the sum at potential values above a certain threshold and show that this is a good approximation of the full sum; we then study the convergence of the truncated sums.
For any δ > 0, define on the event E t and zero otherwise, and letŜ (δ) t = S t −S (δ) t . Denote by Prob (e) and E (e) the conditional probability and expectation given D and {ξ(z) : z ∈ D}. The next lemma shows that the truncated sum S (δ) t is a good approximation for the full sum S t .
eventually for all t.
Proof. LetÊ , where c 1 , c 2 > 0 are chosen according to Proposition 5.5 so that for all t ≥ t 1 . By Proposition 5.6, let t 2 be such that for all t ≥ t 2 For each δ > 0, let t 3 be such that δc 1 a t3 > 1. Let t 0 = max{t 1 , t 2 , t 3 }. Consider the event D t ∩E t ∩Ê t , δ > 0 and t ≥ t 0 .
By Chebychev's inequality we have Since the summands are independent and consist of the differences of two independent identically distributed terms under Prob (e) we obtain For each 0 < z < |Z (1) t |, z ∈ E, the conditional distribution of ξ(z) is the Pareto distribution with parameter α conditioned on Ψ t (z) < Ψ t (Z (1) t ), that is, on Observe that for all δ ≤ c 1 This implies Observe that δξ(Z (1) t ) > δc 1 a t ≥ 1. Using the change of variables y = uξ(Z (1) t ), we obtain We next show that the truncated sum S (δ) t converges to the variable S (δ) introduced in (5.4); since the proof is similar to those appearing in Appendix A, we also defer it to the appendix. Proposition 6.2. As t → ∞, L(S (δ) t |D t ) ⇒ L(S (δ) |D). We are now ready to put everything together to complete the proof of Theorem 1.2, in particular showing that the ratio u(t, Z (1) t )/u(t, −Z (1) t ) converges in distribution to Υ, where Υ is the random variable defined in Lemma 5.8. 6.1. Completion of the proof of Theorem 1.2. By Propositions 3.7 and 4.2 on the event D t only the shortest paths to Z (1) t and −Z (1) t are non-negligible and hence we have by Proposition 4.1 It suffices to show that where Υ is the random variable defined in Lemma 5.8. Let x ∈ R, ε > 0 and chooseε > 0 so that Choose δ 0 according to Lemma 6.1 with ε 1 =ε and ε 2 < εProb(D t )/4 for all t, which is possible by Proposition 5.7 since Prob(D t ) converges to a positive limit. By Lemma 5.8, choose δ ≤ δ 0 so that and choose t 0 according to Lemma 6.1. We have Hence by Lemma 6.1 for all t > t 0 , Combining this with (6.3) and (6.4) we obtain Together with (6.1) and (6.2) this gives for all t ≥ t 1 .
7. Fluctuation theory in the case α ≥ 2 In this section we study the fluctuations in the ratio u(t, Z (1) t )/u(t, Z (2) t ) in the case α ≥ 2, and hence complete the proof of Theorem 1.3. Due to our analysis in Section 4, we know that it is sufficient to study only the contribution from paths which visit the sites in N t at most once.
The first step is to show that, by conditioning on the information not contained in the sites in N t , we are left with an expression that is amenable to applying standard fluctuation theory; here we use our analysis of the function I. The final step is to show that the fluctuations due to N t are already enough to imply that | log u(t, Z (1) t )/u(t, Z (2) t )| → ∞, regardless of the contributions from all other sites; we achieve this by invoking a central limit argument.
Observe that Q t (z), z ∈ N t , are conditionally independent with respect to F t , which implies that Further, it is easy to see that for each z ∈ N t , the conditional distribution of ξ(z) is the Pareto distribution with parameter α conditioned on Ψ t (z) < Ψ t (Z (1) t ) and ξ(z) > δ t ξ(Z (1) t ). The next lemma establishes that, after conditioning on F t , the contribution to the ratio u(t, Z (1) t )/u(t, Z (2) t ) due to the sites in N t is well-approximated by a product over these sites.
Proposition 7.1. There exists an F t -measurable random variable P t such that almost surely, as t → ∞.
, as t → ∞. Combining (7.4) and (7.5) we obtain the desired result with which is obviously F t -measurable.
We now study the scale of the fluctuations due to the sites in N t , showing in particular that these fluctuations are unbounded.
The final step is to apply a central limit theorem to show that the fluctuations due to the sites in N t are in the Gaussian universality class. For each z ∈ N t , denote (7.14) and denote by where Φ denotes the distribution function of a standard normal random variable.
Proof. This result follows from an application of the central limit theorem that we state and prove in Appendix B. It remains to verify that the conditions of the theorem are satisfied.
First note that, conditionally on F t , the random variables V t (z), z ∈ N t , are independent. Moreover, by construction, for each t > 0 and z ∈ N t , E Ft V t (z) = 0 and z∈Nt E Ft V 2 t (z) = 1 almost surely.
Hence it remains to verify that, for each ε > 0, t → 0 almost surely. (7.15) To see this, assume the event E t ∩ E [2,∞) t holds, and remark that, according to (7.14) Since Q t (z) and E Ft Q t (z) are either both non-negative or non-positive almost surely, we obtain, using that Var Ft Q t diverges and that E Ft Q t (z) tends to zero by Proposition 7.2, Combining this with (7.1) we observe that in order to prove (7.15) it suffices to show that Var Ft Q t (z) → 0 (7.17) uniformly in z almost surely. Denote Similarly to the proof of Proposition 7.2 we use the change of variables y = uξ(Z (1) t ) to compute, for k ∈ {0, 1, 2}, (1) t ) α with some c k > 0. By the second part of Proposition 7.2 and using (7.6) we obtain uniformly in z almost surely. Combining this with (7.13) we arrive at (7.17).
We are now ready to complete the proof of Theorem 1.3. The point is that, since we have shown that the fluctuations due to N t are unbounded and in the Gaussian universality class, they place negligible probability mass on any bounded scale. Hence we have the result.
7.1. Completion of the proof of Theorem 1.3. Let c > 0. As a direct corollary of Theorem 1.1, so it remains to show the convergence on the event D t .
Let c > 0. Since Prob(D t ) → 0 by Theorem 1.1 and Prob(E t ∩ E By Proposition 7.1 it is then enough to proof that, as t → ∞, for which, in turn, it suffices to show that Observe that even though the event E t does not belong to F t we can take it out of Prob Ft by Proposition 5.6 and since the function under E is bounded. Now, by the dominated convergence theorem, it remains to prove that holds and observe that Hence (7.18) is equivalent to showing that, almost surely, Since P t , E Ft Q t , and Var Ft Q t are F t -measurable, and the length of the interval on the right-hand side of ∈ tends to zero by Proposition 7.2, (7.19) now follows from Proposition 7.3.

Appendix A. Point process arguments
In this appendix we give the details of our point process arguments in Section 5, and in particular provide the proofs of Lemma 5.1 and Propositions 5.5-5.7 and 6.2. All notation in this appendix is carried over from the main part of the paper.
In Lemma A.1 we establish convergence for the maximisers ofΨ t ; in Lemma A.2 we prove that the maximisers of Ψ t andΨ t are the same with overwhelming probability.
For all the arguments in this section, we shall make use of the point processΠ s defined bȳ for any Borel set A, whereÂ denotes the reflection of A with respect to the y-axis. This is the prelimit version of the Poisson point processΠ.
To do so, it suffices to see that Proof. Fix ε > 0 and let a > 0 be sufficiently small that A = (x 1 , x 2 , y 1 , y 2 ) : is such that which is possible according to Lemma 5.2. Observe that where It is easy to see that F t (x, y) → y −ρx as t → ∞.
Let b > 0 and denote K b = [0, b] × (0, ∞). Observe that since ξ(z) > 1 for all z the point processΠ t has no points below the level 1/a t .
First, by examining the graphs of F t one can see that the first set in (A.7) is close to the third and the second is close to the fourth if we restrict them to K b , that is, and as t → ∞ uniformly on A. Moreover, we have and eventually for all t uniformly on A, and the set on the right hand side of (A.8) and (A.9) has finite measure µ according to (5.2).
We are now ready to give the proof of Propositions 5.5 and 5.6.
Proof of Proposition 5.7. Let ε > 0 and let c > 0 be such that In this appendix we state and prove the central limit theorem that we apply in Section 7. This theorem is similar in spirit to the Lindeberg-Feller central limit theorem for triangular arrays, albeit in a slightly non-classical set-up. The notation used in this appendix is independent of the rest of the paper.
For each t > 0, let F t be a σ-algebra, and N t be an F t -measurable N-valued random variable. Denote by E Ft conditional expectation with respect to F t . Let {V t,i : 1 ≤ i ≤ N t }, t > 0, be a triangular array of random variables. For each t > 0, denote and let F Vt (x) = E Ft 1{V t ≤ x} be the conditional distribution function of V t . |F Vt (x) − Φ(x)|1 Et → 0 almost surely, and moreover, by the continuity of Φ, the pointwise convergence of the cumulative distribution functions necessarily takes place uniformly. To proceed, abbreviate σ 2 t,i = E Ft V 2 t,i , and use conditions (1) and (2) to write where in the last step we used the fact that, for complex numbers {z i } 1≤i≤n and {z ′ i } 1≤i≤n of modulus at most 1, Hence, applying the triangle inequality, exp −u 2 σ 2 t,i /2 − 1 − u 2 σ 2 t,i /2 1 Et .
It remains to show that each of A t and B t converge to zero almost surely.
Turning then to B t , we use the fact that, for x ≥ 0, The sum equals one according to (2). Fixing ε > 0, we also have that Applying condition (3) we have the result, since ε > 0 was arbitrary.