Delocalising the parabolic Anderson model through partial duplication of the potential
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Abstract
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.
Keywords
Parabolic Anderson model Localisation IntermittencyMathematics Subject Classification
60H25 (Primary) 82C44 60F10 (Secondary)1 Introduction
1.1 Delocalising the parabolic Anderson model
While there are many results in the literature establishing localisation in the PAM in various settings (see Sect. 1.2 for an overview), our understanding of the absence of localisation is much less well-developed. In the case that the potential \(\xi \) is a random field, there are at least two features of \(\xi \) 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.
Of course, there are trivial ways to prevent complete localisation by introducing duplication. For instance, if the potential is symmetric about some plane through the origin then \(u(t,\cdot )\) is also symmetric about this plane, and so complete localisation cannot occur. This paper considers a model of partial duplication in which we pick a fraction \(p \in (0, 1)\) of the sites to duplicate across the plane of symmetry. It turns out that this model exhibits a rich phenomenon of delocalisation; indeed, if the potential is i.i.d. with Pareto distribution (i.e. with polynomial tail decay), we show that the model exhibits a phase transition in the Pareto parameter.Given a random potential for which the PAM completely localises, what kind of ‘duplication’ of the potential will cause complete localisation to fail?
1.2 Localisation in the PAM
The study of localisation in the PAM has received much attention in recent years. This began with the seminal paper [6] and is by now well-understood, see [7, 10, 13] for surveys. In the i.i.d. case, for a wide class of potentials with unbounded tails it is known that the solution to the PAM is concentrated at typical large times on a small number of spatially disjoint clusters of sites, known as islands. The shape of the potential and the solution \(u(t,\cdot )\) on these islands was first studied in [8] for the case of double-exponential potentials. More recently, it has been shown that for sufficiently heavy-tailed potentials (Pareto [11], exponential [12], Weibull [5, 17]), the solution exhibits the strongest possible form of localisation: complete localisation. In [14] this was shown to also be the case for a model that replaced the Laplacian with the generator of a trapped random walk. By contrast, in very recent work [3] it has been shown that in the double-exponential case the PAM localises on a single connected island, rather than on a single site. This has confirmed the long standing conjecture that, in the i.i.d. case, potentials with double-exponential tail decay form the boundary of the complete localisation universality class.
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 certain additional complications that arise in higher dimensions, while preserving the phenomena that we seek to investigate; we comment on the nature of these complications in Sect. 1.5.
1.4 The phase transition in the model
An example of the event \(\mathfrak {D_t}\) (left) and its complement (right). The filled and empty circles represent the values of \(\Psi _t\) for points in D and and E respectively; we have only plotted the top order statistics of \(\Psi _t\). The dashed lines mark out the sites in \(\Omega _t\)
Our first result is to show that, for all values of the Pareto parameter \(\alpha > 1\), the model always localises on the set \(\Omega _t\). We also show that the event \(\mathfrak {D}_t\) has non-negligible probability. Of course, outside the event \(\mathfrak {D}_t\) this is already enough to conclude that the model completely localises.
Theorem 1.1
Our next two results establish the following phase transition in the model. If \(\alpha \in (1, 2)\), then on the event \(\mathfrak {D}_t\) the two sites in \(\Omega _t\) both have a non-negligible proportion of the solution; in other words the model delocalises. By contrast, if \(\alpha \ge 2\) only one site in \(\Omega _t\) has a non-negligible proportion of the solution; in other words, the model completely localises whether the event \(\mathfrak {D}_t\) holds or not. Surprisingly, the critical value of \(\alpha = 2\) does not depend on the value of p. To state these result, let \(Z^{{\scriptscriptstyle {({1}})}}_t\in \Omega _t\), with \(Z^{{\scriptscriptstyle {({1}})}}_t\) chosen to be positive on the event \(\mathfrak {D}_t\).
Theorem 1.2
Theorem 1.3
Remark 1
At first glance it may seem counter-intuitive that delocalisation occurs for small, rather than large, values of \(\alpha \), 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 \(\alpha \) which results in delocalisation.
To explain this, consider that if \(\alpha \) is smaller, the advantage of the sites in \(\Omega _t\) relative to other sites is increased. We show that, if \(\alpha \) is small enough, this advantage is so great that the impact of the other potential values (at sites closer to the origin than \(Z^{{\scriptscriptstyle {({1}})}}_t\)) is minimal, and the solution cannot readily distinguish between the sites in \(\Omega _t\). On the other hand, for large values of \(\alpha \) the advantage is less pronounced, and the fluctuations in the other potential values eventually force one of the sites in \(\Omega _t\) to be significantly more beneficial than the other. In the next subsection, we give some heuristics for why the transition occurs at \(\alpha = 2\). \(\square \)
Remark 2
One surprising aspect of the phase transition in the model is that it is not sharp. In particular, the random variable \(\Upsilon \) in Theorem 1.2 does not, as might be expected, degenerate for small \(\alpha \). 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. \(\square \)
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 [11, 16, 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. \(\square \)
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 \(\mathbb {Z}\) with the i.i.d. potential \(\xi _0\), with \(U(t) = \sum _z u(t, z)\) the total mass of the solution. Now consider resampling each potential value independently with probability \(q \in (0,1)\), and let \(\tilde{u}(t, z)\) be the solution of the PAM with this resampled potential, with \(\tilde{U}(t) = \sum _z \tilde{u}(t, z)\) the total mass of the solution. Then our results, suitably translated, demonstrate the following phase transition. If \(\alpha \in (1,2)\), then there exists an event of non-negligible probability on which \(U(t)/\tilde{U}(t)\) converges in distribution to a random variable with positive density on \(\mathbb {R}_+\). By contrast, if \(\alpha \ge 2\), then \(|\log U(t)/\tilde{U}(t)| \rightarrow \infty \) in probability. \(\square \)
1.5 Heuristics for the phase transition
To explain the phase transition in the model at \(\alpha = 2\), we show that the second-order contributions undergo two distinct transitions as \(\alpha \) increases, both of which, seemingly coincidentally, occur at \(\alpha = 2\). The first transition is the negligibility or otherwise of non-direct paths which end at the sites in \(\Omega _t\); 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 (\(\alpha \ge 2\)) to the \(\alpha \)-stable universality class (\(\alpha \in (1, 2)\)), and it is this which turns out to cause the phase transition of the model.
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 \(\alpha > 1\), the only significant contribution to U(t) comes from paths which end in \(\Omega _t\). In Proposition 4.3 we show that, if \(\alpha \in (1,2)\), the only significant contribution to U(t) actually come from the direct paths to \(\Omega _t\); here we give some heuristics for why this should be true. On the other hand, if \(\alpha \ge 2\), 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 \(\alpha \in (1,2)\) and let \(y^{(t)}\) denote the direct path from the origin to \(Z^{{\scriptscriptstyle {({1}})}}_t\). For the purposes of keeping the calculations simple, we will show only that the contribution to U(t) from paths \(\Pi ^{(t, +)}\) from the origin to \(Z^{{\scriptscriptstyle {({1}})}}_t\) 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 \(\Omega _t\).
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 \(\alpha > 1\), it is sufficient to consider only direct paths (even though we strongly believe that this is only true in the case \(\alpha \in (1, 2)\)).
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 \(\alpha = 2\) is the same for all values of \(p \in (0 ,1)\). This leads us to wonder what happens if p is not uniformly bounded away from 1. One way to investigate this is to let \(\xi (z)=\xi (-z)\) with probability \(p=p(|z|)\) that depends on the distance of z from the origin. We can then ask the question: how fast should \(p(n)\rightarrow 1\) so that, for a given value of \(\alpha >2\), complete localisation fails? We conjecture that there is a critical scale for p(n) such that if and only if \(p\rightarrow 1\) slower than this scale then complete localisation holds. We will investigate this model in a future paper.
2 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 \(\alpha > 1\).
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 \(\Omega _t\) or because they make too many jumps. This step is rather standard, and is similar to in [11, 16, 17].
Step 2: Reduction to subsets of paths that end at \(\Omega \). 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 \(\alpha \in (1, 2)\) and \(\alpha \ge 2\) (see the heuristics in Sect. 1.5).
In Sect. 3.3 we show that I has a rather neat symmetric structure and study its properties. Using this understanding, in Sect. 4 we identify the paths making non-negligible contribution to \(U_0\). For \(\alpha \in (1,2)\) the situation is relatively simple: in Propositions 4.2 and 4.3 we show that only the direct paths to \(\Omega _t\) 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 \(\alpha \ge 2\) 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 \(\xi \) hold, we can limit the significant paths to those that end at \(\Omega _t\) and visit each site in \(\{0\}\cup \mathcal {N}_t\) at most once, where \(\mathcal {N}_t\) 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 \(\mathcal {N}_t\). Since these sites are visited at most once, we can again apply standard fluctuation theory.
This analysis is already enough to finish the proof of Theorem 1.1 assuming the event \(\mathcal {E}_t\) holds; we complete the proof at the end of Sect. 4.
Step 3: Point process techniques. In Sect. 5 we build up a point process approach to study the high exceedences of \(\xi \) and the top order statistics of the penalisation functional \(\Psi _t\). We start by proving that the potential \(\xi \), properly rescaled, converges to a Poisson point process. We then use this convergence to pass certain functionals of \(\xi \), including properties of \(\Psi _t\), 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 \(\mathcal {E}_t\) holds eventually with overwhelming probability. Second, we give an explicit construction for the limit random variable \(\Upsilon \) 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 \(u(t, Z^{{\scriptscriptstyle {({1}})}}_t)/u(t, -Z^{{\scriptscriptstyle {({1}})}}_t)\). At this point we have assembled all the main ingredients, and all that is left is to apply fluctuation theory to analyse the ratio \(u(t, Z^{{\scriptscriptstyle {({1}})}}_t)/u(t, -Z^{{\scriptscriptstyle {({1}})}}_t)\); here we again distinguish between the cases \(\alpha \in (1, 2)\) and \(\alpha \ge 2\) (see the heuristics in Sect. 1.5).
In Sect. 6 we study the case \(\alpha \in (1,2)\) 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 \(u(t,Z^{{\scriptscriptstyle {({1}})}}_t)/u(t,-Z^{{\scriptscriptstyle {({1}})}}_t)\) converges to the limit random variable we identify in Sect. 5.
In Sect. 7 we study the case \(\alpha \ge 2\) and complete the proof of Theorem 1.3. Here we apply a central limit theorem to establish that the fluctuations in \(u(t, Z^{{\scriptscriptstyle {({1}})}}_t)/u(t, -Z^{{\scriptscriptstyle {({1}})}}_t)\) due to the sites \(\mathcal {N}_t\) (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 \(| \log u(t,Z^{{\scriptscriptstyle {({1}})}}_t)/u(t,-Z^{{\scriptscriptstyle {({1}})}}_t)| \rightarrow \infty \), irrespective of the contribution due to the other sites.
3 Preliminaries
In this section we establish some preliminary results. First, we prove asymptotic properties of the potential \(\xi \). 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 \(Z^{{\scriptscriptstyle {({1}})}}_t\) and \(Z^{{\scriptscriptstyle {({2}})}}_t\), and also to establish that \(\mathcal {E}_t^{[2,\infty )}\) holds eventually with overwhelming probability.
Lemma 3.1
Proof
Lemma 3.2
For fixed t, almost surely either \(\Omega _t = \{z\}\) for some \(z \in E\) or \(\Omega _t = \{-z, z\}\) for some \(z \in D\), and the same conclusion holds for the maximisers of \(\Psi _t\) on the set \(\mathbb {Z}\setminus \Omega _t\). Moreover, almost surely \(\Psi _t(Z^{{\scriptscriptstyle {({1}})}}_t)> \Psi _t(Z^{{\scriptscriptstyle {({2}})}}_t)>1\) eventually for all t.
Proof
For the second statement, let \(z_1,\,z_2\in \mathbb {Z}_+\) be fixed sites satisfying \(\xi (z_1)\wedge \xi (z_2)>1\) (such sites exist almost surely). Then \(\Psi _t(z_1)\wedge \Psi _t(z_2)>1\) for all t sufficiently large and so in particular \(\Psi _t(Z^{{\scriptscriptstyle {({1}})}}_t)\) and \(\Psi _t(Z^{{\scriptscriptstyle {({2}})}}_t)\) are both larger than one eventually. Again since \(\Psi _t(z)\) is a continuous random variable with no point mass, this implies the second statement. \(\square \)
Proposition 3.3
\(\text {Prob}\big (\mathcal {E}_t^{[2,\infty )} \, | \, \mathcal {E}_t \big )\rightarrow 1\) as \(t\rightarrow \infty \).
Proof
3.2 Eliminating the a priori negligible paths
Lemma 3.4
Proof
Lemma 3.5
Proof
Lemma 3.6
Proof
Proposition 3.7
Proof
3.3 Structure of the function I
Lemma 3.8
- (1)If \(a_n \ne a_i\) for \(i \ne n\) thenMoreover, if \(a_0,\dots ,a_n\) are pairwise distinct then$$\begin{aligned} I_n(t; a_0,\dots ,a_n)=e^{ta_n}\prod \limits _{j=0}^{n-1}\frac{1}{a_n-a_j} -\sum _{i=0}^{n-1}I_i(t,a_0,\dots ,a_i)\prod _{j=i}^{n-1} \frac{1}{a_n-a_j} . \end{aligned}$$$$\begin{aligned} I_n(t; a_0,\dots ,a_n)=\sum _{i=0}^{n}e^{ta_i}\prod \limits _{\genfrac{}{}{0.0pt}{}{j=0}{j\ne i}}^{n}\frac{1}{a_i-a_j}; \end{aligned}$$(3.10)
- (2)
\(I_n\) is symmetric with respect to the variables \(a_0,\dots ,a_n\).
Proof
The first statement in (1) follows by induction from (3.9), 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 (3.9). Finally, the symmetry of \(I_n\) for pairwise distinct variables follows from the symmetry of the expression on the right hand side of (3.10). Then it extends by continuity to all variables. \(\square \)
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
Proof
Our ‘negative dependence’ lemma requires the application of a result of [4], which we state below.
Theorem 3.10
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 \((Y_1,\ldots ,Y_n)\) 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. \(\square \)
Lemma 3.11
Proof
Lemma 3.12
Proof
4 Significant paths
The aim of this section is to determine which paths make a non-negligible contribution to \(U_0(t)\). As described in Sect. 1.5, in the case \(\alpha \in (1, 2)\) we prove that only the direct paths to \(\Omega _t\) are significant. In the case \(\alpha \ge 2\), we can only prove the much weaker result that the significant paths are those which end at \(\Omega _t\) and visit the set \(\mathcal {N}_t \cup \{0\}\) at most once, where \(\mathcal {N}_t\) is the set of non-duplicated sites of high potential defined in (2.4) (actually this is true for all \(\alpha > 1\), but is not as strong as what we prove for \(\alpha \in (1,2)\)).
Assuming the event \(\mathcal {E}_t\) holds, this is already enough to prove the localisation statement in Theorem 1.1; we complete this proof at the end of the section.
4.1 The case \(\alpha \in (1, 2)\): direct paths to \(\Omega _t\)
To prove that only direct paths are significant, we first give an approximation for the contribution made by the direct paths, and then use this approximation to show the negligibility of all other paths. Denote by \(y^{{\scriptscriptstyle {({t,1}})}}\in \mathcal {P}_{all}\), \(y^{{\scriptscriptstyle {({t,-1}})}}\in \mathcal {P}_{all}\) the shortest geometric paths from 0 to \(|Z^{{\scriptscriptstyle {({1}})}}_t|\) and to \(-|Z^{{\scriptscriptstyle {({1}})}}_t|\), respectively.
Before we begin, we state a small combinatorial lemma that will be used in Proposition 4.3 below.
Lemma 4.1
Proof
Proposition 4.2
Proof
Proposition 4.3
Proof
4.2 The case \(\alpha \ge 2\): paths to \(\Omega _t\) visiting sites in \(\mathcal {N}_t\) at most once
Our proof proceeds in two stages. First, we analyse the portion of the part up until the first visit to \(\Omega _t\) and after the last visit to \(\Omega _t\), and show that, in this portion of the path, it is never beneficial to visit sites in \(\mathcal {N}_t \cup \{0\}\) more than once. Second, we analyse the portion of the path consisting of the loops that occur between first and last visit to \(\Omega _t\), showing that it is never beneficial for these loops to return to sites in \(\mathcal {N}_t \cup \{0\}\); in fact, we show the stronger result that these loops have length at most \(\lfloor 2\alpha \rfloor \) (although we suspect that the optimal bound is actually \(\lfloor \alpha \rfloor \)).
Denote by \(\mathcal {P}^t\) the set of all geometric paths contributing to \(U_0(t)\), that is, those visiting \(\Omega _t\) and having length at most \(R_t\). Fix \(t>0\) and let \(y\in \mathcal {P}^t\). The skeleton of y, denoted \(\text {skel}(y)\), is the geometric path from the origin to a site in \(\Omega _t\) constructed by chronologically removing all loops in y which start and end at any site belonging to \(\{0\}\cup \mathcal {N}_t\) up until the first visit of \(\Omega _t\) as well as removing any part of the path after the final visit of y to \(\Omega _t\).
We can now partition \(\mathcal {P}^t\) into equivalence classes by saying that paths y and \(\hat{y}\) are in the same class if and only if \(\text {skel}(y)=\text {skel}(\hat{y})\). We write \(\mathfrak {P}^t\) for the set of all such equivalence classes. Note that any such equivalence class \(\mathcal {P}\in \mathfrak {P}^t\) contains the null path, \(y_{\mathrm {null}}^\mathcal {P}\in \mathcal {P}\), defined as \(y_{\mathrm {null}}^\mathcal {P}=\mathrm {skel}(y_{\mathrm {null}}^\mathcal {P})\). Observe that every null path, prior to visiting \(\Omega _t\) for the first time, either (i) visits each site in \(\{0\} \cup (\mathcal {N}_t \cap \mathbb {N})\) exactly once, or (ii) visits each site in \(\{0\} \cup (\mathcal {N}_t \cap -\mathbb {N})\) exactly once. In particular, until the first visit of \(\Omega _t\) 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.
Lemma 4.4
Proof
We now eliminate paths that make loops from \(\Omega _t\) that return to sites in \(\mathcal {N}_t\). Denote by \(\mathrm {Null}^t_1\) the set of all null paths in \(\mathcal {P}^t\) which visit each site in \(\{0\}\cup \mathcal {N}_t\) at most once, \(\mathrm {Null}^t_2\) for all other null paths in \(\mathcal {P}^t\) and \(\mathrm {Null}^t\) for their union.
Lemma 4.5
Proof
Note that by the construction of null paths, the only way for a null path to visit a site in \(\mathcal {N}_t\) more than once is by having a loop from \(\Omega _t\). On the event \(\mathcal {E}^{[2,\infty )}_t\) 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 \(\Omega _t\) of length more than \(k_0\), where \(k_0>2\alpha \), have negligible contribution to the solution compared to the contribution from all other null paths.
To do this we partition \(\mathrm {Null}^t\) 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 \(\Omega _t\) of length at least \(k_0\). For any such equivalence class \(\mathcal {P}\), write \(y^{\mathcal {P}}_\mathrm {min}\) for the path in \(\mathcal {P}\) of minimum length (i.e. the path without any loops from \(\Omega _t\) of length at least \(k_0\)). Further, for any \(k\ge k_0\), write \(\mathcal {P}^k\) for the set of paths in \(\mathcal {P}\) with additional length k compared to \(y^{\mathcal {P}}_\mathrm {min}\). Finally we write \(\mathfrak {N}^t\) for the set of all such equivalence classes.
Proposition 4.6
Proof
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 \(\mathcal {E}_t\) holds; the fact that \(\mathbb {P}(\mathcal {E}_t) \rightarrow 1\) as \(t \rightarrow \infty \) will be proven in Proposition 5.6. The second statement of Theorem 1.1, that \(\mathbb {P}(\mathfrak {D}_t) \rightarrow p/(2-p)\), will be proven in Proposition 5.7.
By Proposition 3.3 we may work on the event \(\mathcal {E}_t\cap \mathcal {E}^{[2,\infty )}_t\). 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 \(\Omega _t\) is negligible. For \(\alpha \in (1,2)\) this follows from Propositions 4.3; for \(\alpha \ge 2\) this follows from Propositions 4.6. In fact, the latter argument works for all \(\alpha >1\) but we prefer to use the much simpler argument for \(\alpha \in (1,2)\).
5 Point process analysis
In this section we develop a point processes approach to analyse the high exceedences of \(\xi \) and top order statistics of the penalisation functional \(\Psi _t\). We use this analysis to prove that the \(\mathcal {E}_t\) holds eventually with overwhelming probability. We also use it to give an explicit construction for the limiting random variable \(\Upsilon \) from Theorem 1.2. Since the proofs in this section are quite technical, we defer some of them to “Appendix A”.
Recall that \(E = \mathbb {Z}{\setminus }D\) 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.
Lemma 5.1
As \(s\rightarrow \infty \), \((\Pi ^{{\scriptscriptstyle {({d,+}})}}_s, \Pi ^{{\scriptscriptstyle {({d,-}})}}_s, \Pi ^{{\scriptscriptstyle {({e}})}}_s)\) converges in law to \((\Pi ^{{\scriptscriptstyle {({d, +}})}}, \Pi ^{{\scriptscriptstyle {({d, -}})}},\Pi ^{{\scriptscriptstyle {({e}})}})\), and in particular, \(\Pi _s\) converges in law to \(\Pi \).
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 \(\Psi _t\). We first introduce the limiting versions of \(Z^{{\scriptscriptstyle {({1}})}}_t\), \(Z^{{\scriptscriptstyle {({2}})}}_t\) and \(\mathfrak {D}_t\) and study their properties, before arguing that we may successfully pass to the limit.
At the end of this section we shall identify \((X^{{\scriptscriptstyle {({i}})}}, Y^{{\scriptscriptstyle {({i}})}}), i = 1,2\), and \(\mathfrak {D}\) as the limiting versions of \((|Z^{{\scriptscriptstyle {({i}})}}_t|, \xi (Z^{{\scriptscriptstyle {({i}})}}_t)), i = 1,2\), and \(\mathfrak {D}_t\) respectively. For now, we establish some properties of these objects.
Lemma 5.2
Almost surely, the random variables \(X^{{\scriptscriptstyle {({1}})}}, X^{{\scriptscriptstyle {({2}})}}, Y^{{\scriptscriptstyle {({1}})}}\) and \(Y^{{\scriptscriptstyle {({2}})}}\) are well-defined and satisfy \(Y^{{\scriptscriptstyle {({1}})}}-\rho X^{{\scriptscriptstyle {({1}})}}>Y^{{\scriptscriptstyle {({2}})}}-\rho X^{{\scriptscriptstyle {({2}})}} >0\).
Proof
Lemma 5.3
Proof
Lemma 5.4
\({\mathrm {Prob}}_*(\mathfrak {D})= p / (2-p)\).
Proof
We now argue that we can successfully pass to the limit. As a consequence, we prove that the event \(\mathcal {E}_t\) holds eventually with overwhelming probability. Since the proof of these results are rather technical, we defer them to “Appendix A”.
Proposition 5.5
- (i)
\( \Big (\frac{|Z^{{\scriptscriptstyle {({1}})}}_t|}{r_t},\frac{|Z^{{\scriptscriptstyle {({2}})}}_t|}{r_t},\frac{\xi (Z^{{\scriptscriptstyle {({1}})}}_t)}{a_t},\frac{\xi (Z^{{\scriptscriptstyle {({2}})}}_t)}{a_t}\Big ) \Rightarrow (X^{{\scriptscriptstyle {({1}})}},X^{{\scriptscriptstyle {({2}})}},Y^{{\scriptscriptstyle {({1}})}},Y^{{\scriptscriptstyle {({2}})}}),\)
- (ii)
\(\Big (\frac{\Psi _t(Z^{{\scriptscriptstyle {({1}})}}_t)}{a_t}, \frac{\Psi _t(Z^{{\scriptscriptstyle {({2}})}}_t)}{a_t}\Big ) \Rightarrow (Y^{{\scriptscriptstyle {({1}})}}-\rho X^{{\scriptscriptstyle {({1}})}},Y^{{\scriptscriptstyle {({2}})}}-\rho X^{{\scriptscriptstyle {({2}})}}).\)
Proposition 5.6
\(\text {Prob}(\mathcal {E}_t)\rightarrow 1\) as \(t\rightarrow \infty \).
Proposition 5.7
\({\mathrm {Prob}}(\mathfrak {D}_t)\rightarrow \frac{p}{2-p}\) as \(t\rightarrow \infty \).
5.3 An explicit construction of the limiting random variable
Lemma 5.8
Proof
6 Fluctuation theory in the case \(\alpha \in (1,2)\)
In this section we study the fluctuations in the ratio \(u(t, Z^{{\scriptscriptstyle {({1}})}}_t)/u(t, Z^{{\scriptscriptstyle {({2}})}}_t)\) in the case \(\alpha \in (1, 2)\), building on our analysis in Sect. 4.1, and hence complete the proof of Theorem 1.2.
Lemma 6.1
Proof
We next show that the truncated sum \(S_t^{{\scriptscriptstyle {({\delta }})}}\) converges to the variable \(S^{{\scriptscriptstyle {({\delta }})}}\) 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
We are now ready to put everything together to complete the proof of Theorem 1.2, in particular showing that the ratio \(u(t, Z^{{\scriptscriptstyle {({1}})}}_t)/u(t, -Z^{{\scriptscriptstyle {({1}})}}_t)\) converges in distribution to \(\Upsilon \), where \(\Upsilon \) is the random variable defined in Lemma 5.8.
6.1 Completion of the proof of Theorem 1.2
7 Fluctuation theory in the case \(\alpha \ge 2\)
In this section we study the fluctuations in the ratio \(u(t, Z^{{\scriptscriptstyle {({1}})}}_t)/u(t, Z^{{\scriptscriptstyle {({2}})}}_t)\) in the case \(\alpha \ge 2\), and hence complete the proof of Theorem 1.3. Due to our analysis in Sect. 4, we know that it is sufficient to study only the contribution from paths which visit the sites in \(\mathcal {N}_t\) at most once.
The first step is to show that, by conditioning on the information not contained in the sites in \(\mathcal {N}_t\), 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 \(\mathcal {N}_t\) are already enough to imply that \(|\log u(t, Z^{{\scriptscriptstyle {({1}})}}_t)/u(t, Z^{{\scriptscriptstyle {({2}})}}_t) | \rightarrow \infty \), regardless of the contributions from all other sites; we achieve this by invoking a central limit argument.
The next lemma establishes that, after conditioning on \(\mathcal {F}_t\), the contribution to the ratio \(u(t, Z^{{\scriptscriptstyle {({1}})}}_t)/u(t, Z^{{\scriptscriptstyle {({2}})}}_t)\) due to the sites in \(\mathcal {N}_t\) is well-approximated by a product over these sites.
Proposition 7.1
Proof
We write \(\mathrm {Null}^t_{1+}\), \(\mathrm {Null}^t_{1-}\) for the set of null paths in \(\mathrm {Null}^t_1\) ending in \(Z^{{\scriptscriptstyle {({1}})}}_t\) and \(-Z^{{\scriptscriptstyle {({1}})}}_t\), respectively. Further, we denote by \(\mathcal {N}_t^+\) and \(\mathcal {N}_t^{-}\) the subsets of \(\mathcal {N}_t\) consisting of the points lying between 0 and \(Z^{{\scriptscriptstyle {({1}})}}_t\), and 0 and \(-Z^{{\scriptscriptstyle {({1}})}}_t\), respectively. Finally, we denote by \(N_t^{+}\) and \(N_t^{-}\) the respective cardinalities of \(\mathcal {N}_t^+\) and \(\mathcal {N}_t^{-}\).
We now study the scale of the fluctuations due to the sites in \(\mathcal {N}_t\), showing in particular that these fluctuations are unbounded.
Proposition 7.2
Proof
Proposition 7.3
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.
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 \(\mathcal {N}_t\) 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
References
- 1.Billingsley, P.: Convergence of Probability Measures, p. xii+253. Wiley, New York (1968)zbMATHGoogle Scholar
- 2.Billingsley, P.: Probability and Measure. Wiley Series in Probability and Mathematical Statistics, 3rd edn, p. xiv+593. A Wiley-Interscience Publication. Wiley, New York (1995)zbMATHGoogle Scholar
- 3.Biskup, M., Konig, W., dos Santos, R.S.: Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails. Probab. Theory Relat. Fields (2017). doi: 10.1007/s00440-017-0777-x
- 4.Block, H.W., Savits, T.H., Shaked, M.: Some concepts of negative dependence. Ann. Probab. 10(3), 765–772 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Fiodorov, A., Muirhead, S.: Complete localisation and exponential shape of the parabolic Anderson model with Weibull potential field. Electron. J. Probab 19(58), 27 (2014)MathSciNetzbMATHGoogle Scholar
- 6.Gartner, J., Molchanov, S.A.: Parabolic problems for the Anderson model. I. Intermittency and related topics. Commun. Math. Phys. 132(3), 613–655 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Gartner, J., Konig, W.: The Parabolic Anderson Model, Interacting Stochastic Systems, pp. 153–179. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
- 8.Gartner, J., Konig, W., Molchanov, S.: Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35(2), 439–499 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Gartner, J., Molchanov, S.A.: Moment asymptotics and Lifshitz tails for the parabolic Anderson model. In: Stochastic Models (Ottawa, ON, 1998), CMS Conf. Proc. Amer. Math. Soc., Providence, RI, vol. 26, pp. 141–157 (2000)Google Scholar
- 10.Konig, W.: The Parabolic Anderson Model Pathways in Mathematics, p. xi+192. Birkhauser, Basel (2016)zbMATHGoogle Scholar
- 11.Konig, W., et al.: A two cities theorem for the parabolic Anderson model. Ann. Probab. 37(1), 347–392 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Lacoin, H., Mörters, P.: A scaling limit theorem for the parabolic Anderson model with exponential potential. In: Deuschel, J.D., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds.) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol. 11, pp. 247–272. Springer, Berlin (2012)Google Scholar
- 13.Morters, P.: The parabolic Anderson model with heavy-tailed potential. In: Blath, J., Imkeller, P., Roelly. S. (eds.) Surveys in Stochastic Processes. Proceedings of the 33rd SPA Conference in Berlin, 2009, pp. 67–85. EMS Series of Congress Reports (2011)Google Scholar
- 14.Muirhead, S., Pymar, R.: Localisation in the Bouchaud-Anderson model. Stoch. Proc. Appl. 126, 3402–3462 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
- 15.Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust, vol. 4, p. xii+320. Springer, New York (1987)CrossRefzbMATHGoogle Scholar
- 16.Sidorova, N., Twarowski, A.: Localisation and ageing in the parabolic Anderson model with Weibull potential. Ann. Probab. 42(4), 1666–1698 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
- 17.van der Hofstad, R., Morters, P., Sidorova, N.: Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials. Ann. Appl. Probab. 18(6), 2450–2494 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
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