Two-site localisation in the Bouchaud trap model with slowly varying traps

We consider the Bouchaud trap model on the integers in the case that the trap distribution has a slowly varying tail at infinity. We prove that the model eventually localises on exactly two sites with overwhelming probability. This is a stronger form of localisation than has previously been established in the literature for the Bouchaud trap model on the integers in the case of regularly varying traps. Underlying this result is the fact that the sum of a sequence of i.i.d. random variables with a slowly varying tail is asymptotically dominated by the maximal term.

where σ := {σ(z)} z∈Z d is a collection of independent identically distributed (i.i.d.) strictly-positive random variables known as the (random) trapping landscape. The process X t describes the position at time t of a particle undertaking a continuoustime random walk on Z based at the origin, where the waiting time at a site z is distributed exponentially with mean σ(z) and the subsequent site is chosen uniformly at random from among the neighbours of z. The BTM has its origins in the statistical physics literature, where it was proposed as a simple way of modelling the long-term dynamics of certain spin-glass models (see, e.g., [3]). For a general overview of the BTM see [2]; for interesting recent work see [6].
Although the BTM may be defined on arbitrary graphs by analogy with the above (see, e.g., [2]), the BTM on Z is of particular interest because it demonstrates intermittency; in other words, its dynamics cannot in general be explained with a simple averaging principle. In the context of the BTM, intermittency manifests in the localisation of the probability mass function u(t, z) := P σ (X t = z) where P σ is a random probability measure depending on the particular realisation of the trapping landscape σ; we reserve P to denote probability taken over σ.
Interestingly, intermittency is not in general observed in the BTM on graphs which are highly connected, for instance Z d , d > 1, or the complete graph (the original variant of the BTM studied in [3]). The BTM on a general tree was the subject of a very recent study in [1]. Henceforth, we shall refer exclusively to the BTM on Z.
1.2. Intermittency and localisation. Broadly speaking, intermittency in the BTM arises because of extremes in the trapping landscape, and so the strength of intermittency naturally depends on the thickness of the upper-tail of σ(·). Indeed, it was proven in [5] that if σ(·) has finite expectation, then the BTM almost surely converges to Brownian motion in the limit. This implies that, for each z uniformly, u(t, z) → 0 almost surely as t → ∞, and so the model does not exhibit intermittency.
On the other hand, it was proven in [4] that, under certain assumptions, if σ(·) has infinite expectation then u(t, z) localises in the sense that sup z∈Z u(t, z) 0 almost surely (2) as t → ∞. In other words, for arbitrarily large times t there is a site where X t can be found with non-negligible probability. This was proven, in particular, where the distribution of σ(·) belongs to the domain of attraction of the totally asymmetric α-stable law with α ∈ (0, 1), i.e. where there exists a slowly-varying function L (i.e. lim x→∞ L(xs)/L(x) = 1 for every s > 0) such that Notably, this class includes the Pareto distributions with parameter α ∈ (0, 1).
What has yet to be established, however, is whether the BTM may exhibit a stronger form of localisation in the case α = 0, i.e. where the distribution σ(·) has a super-heavy tail, such that for L a slowly-varying function. Indeed, to the best of our knowledge, the literature has not yet considered such super-heavy-tailed traps.
1.3. Our results. We consider the BTM where the distribution σ(·) satisfies the following assumption: Assumption 1.1. Characterise the distribution σ(·) by the positive, non-decreasing and unbounded function Then, as x → ∞, the function f satisfies, for some ε > 0: (a) (Super-heavy tail) Eventually

) (Tail regularity) Eventually f is continuous, strictly increasing and, for all
; we separate the assumptions for clarity only. Remark 1.2. An important class of distributions that satisfy Assumption 1.1 are the log-stretched-exponential (or log-Weibull) distributions with parameter γ ∈ (0, 1) (i.e. with f (x) = x γ ). Heavier-tailed distributions, such as the log-Pareto distribution (i.e. with f (x) = log x), are also included. Remark that the distributions considered in [4] and [5] do not satisfy Assumption 1.1 since these have f (x) ∼ αx for some α ∈ (0, 1).
We prove that under Assumption 1.1 the probability mass function u(t, z) eventually localises on exactly two sites with overwhelming probability (with respect to the trapping landscape σ). We further prove that the proportion of probability mass on each of the two localisation sites is uniformly distributed in the limit, and derive a limit formula for the distance of the localisation sites from the origin.
To describe these results explicitly, we first define some notation. Fix a value ε > 0 that satisfies Assumption 1.1. For sufficiently large t, denote by r t the unique positive solution to the equation remarking that r t is well-defined for large enough t since f is eventually strictlyincreasing and continuous. Further, let h t be an auxiliary scaling function satisfying log log t ≪ log h t ≪ (log t) c eventually for any constant c > 0, where f (x) ≪ g(x) denotes lim x→∞ f (x)/g(x) = 0. Finally, define a level l t := th 2 t /r t . Denote by Z t }. Our main results are the following: Theorem 1.3 (Two-site localisation in probability). As t → ∞, Theorem 1.6 (Distance of localisation sites from origin). As t → ∞,

Theorem 1.4 (Distribution between localisation sites). For
Remark 1.7. Theorems 1.3-1.4 collectively state that, for a large fixed time t, a particle undertaking the BTM is overwhelmingly likely to be located at either of the two (random) sites in Γ t , and that the particle will be uniformly distributed between these two sites. Theorem 1.6 details the location of these sites. Note that the form of localisation in Theorem 1.3 is indeed stronger than that in equation (2). For instance, the results in [4] imply that two-site localisation in probability does not hold when σ(·) belongs to the domain of attraction of the totally asymmetric α-stable law with α ∈ (0, 1).
Finally, although we do not prove it here, we strongly suspect that max z∈Z u(t, z) 1 in probability for arbitrary distributions σ(·), meaning that one-site localisation in probability always fails in the BTM, regardless of the assumptions on σ(·). The reason for our belief is that Assumption 1.1 is specifically chosen to favour localisation, yet one-site localisation fails.
Remark 1.8. Although Γ t is defined in terms of the scaling function h t , with overwhelming probability it is eventually independent of the specific choice of h t .

1.4.
Strategy of the proof. The main idea of the proof is that, for a large fixed t, the set Γ t has been constructed to ensure that: (i) a particle undertaking the BTM random walk is very likely to have hit Γ t before time t; (ii) if the particle hits z ∈ Γ t before time t, it is very likely to still be at z at time t. The underlying fact permitting this construction is that the maximum of a sequence of i.i.d. random variables with a super-heavy tail will, with overwhelming probability, asymptotically dominate the sum of the sequence.
To make this idea precise, fix a trapping landscape σ and a large time t, and consider running the BTM on this landscape. Define the random time τ t 1 := inf{s : X s ∈ Γ t } and consider P σ (τ t 1 > t). Denote by Σ t the sum of the traps lying in By a consideration of the displacement and local time of a simple random walk, it is possible to show that the random time τ t 1 is very likely to be less than Σ t d t h t , where d t := max z∈Γt {|z|}. Hence, on the event (with respect to σ) Further, for i = 1, 2, define the set By a similar consideration, we may show that the random time τ t 2 is very likely to be greater than r t l t /h 2 t = t, and so we find that, as t → ∞, Introduce a new random processX t s on the same probability space as X s satisfying: (i)X t s = X s for all s < τ t denotes the site in I τ t that shares a conjugacy class with z in the quotient space Z/I τ t . Finally, define, for any s > 0, u t (s, z) := P σ (X t s = z, τ t 1 < s) . By construction it is clear that (3) and (4) imply that, on the event E t , u t (t, z) u(t, z) → 1 in probability (5) for each z ∈ I τ t . Note thatû t (s, z) is a probability mass function supported on I τ t which, via Markov chain theory, can be shown to converge monotonically to an equilibrium distribution π τ t as s → ∞. Moreover, we can show that π τ t is proportionate to σ. Hence if, as t → ∞, then π τ t is asymptotically localised at Z τ t . So defining, for each i = 1, 2, the sum on the event (again with respect to σ) Since we show that the events E t and F t hold eventually with overwhelming probability, combining equations (5) and (6) yields Theorem 1.3.
In Section 2 we study preliminary asymptotics for r t and l t . In Section 3 we study properties of the trapping landscape σ, and show that the events E t and F t hold eventually with overwhelming probability. In Section 4, we study the BTM on the events E t and F t , establishing equations (3), (4) and (6), and completing the proof of Theorems 1.3, 1.4 and 1.6.

Preliminary asymptotics
Lemma 2.1 (Asymptotics for r t ). As t → ∞, the following hold: Proof. Note that r t → ∞, since r t is non-decreasing and if log r t < C for all t then which contradicts the fact that f is unbounded. Then by Assumption 1.1 eventually and, since log r t ≪ log t and log h t ≪ (log t) ε , Lemma 2.2 (Asymptotics for l t ). As t → ∞, the following hold: Proof. These follow easily from Assumption 2.1 and the assumptions on h t .

The trapping landscape
In this section, we study properties of the trapping landscape σ, and in particular we prove that the events We begin by stating three general propositions on sequences of i.i.d random variables with common distribution σ(·); let X := {X n } n∈N be such a sequence. For a level l, let n l := min{n : X n > l} and m l := n<n l X n be, respectively, the index of the first exceedence of the level l and the sum of all previous terms in the sequence. Proof. Construct the sequence Y := {Y n } n∈N with Y n := f (log X n ) and remark that Y is a sequence of i.i.d. standard exponential distributions. It is well-known (see, e.g. [8, Lemma 4.1]) that, as n → ∞, eventually | max i≤n Y i − log n| < (log log n) δ (7) almost surely, for some δ ∈ (0, 1). Since n l = min{n : Y n > f (log l)}, this yields the result.
Proof. As in the proof of Proposition 3.1, construct the sequence Y with Y n := f (log X n ). Fix a 0 < δ < ε/2 and define and so log m l ≤ log l + log K + max . Hence it is sufficient to prove that, for any C, Consider first each N i l for 2 ≤ i ≤ K. Since, for n < n l , the random variable Y n is distributed as a standard exponential distribution conditioned on not exceeding l, we have Consider now N 1 l := |{n < n l : Y n > l 1 }|. Denote by E := {E n } n∈N the sequence of progressive maxima of Y . By the memoryless property of the exponential distribution, for each i we have that and so the sequence E can be considered as the arrival times of a Poisson process on R + with unit intensity. By the time-reversibility of a Poisson process, f (log l) − max{Y n : n < n l } d = Y 1 and so P(max{Y n : n < n l } < f (log l) − (log l) −δ/2 ) → 1 since (log l) −δ/2 ≪ 1. On the other hand, by Assumption 1.1, eventually and so P(max{X n : n < n l } < l 1 ) → 1 .

This implies that
Combining equations (9) and (10) establishes equation (8) as required.  Bound on partial sum). For any C, as t → ∞, Proof. First remark that if r t /h t = O(1) then the statement is immediate. So assume r t /h t → ∞. As in the proof of Proposition 3.1, construct the sequence Y with Y n := f (log X n ). Applying equation (7), eventually eventually almost surely. Since eventually f is invertable, eventually almost surely i<rt/ht σ(X i ) ≺ m lt and so, applying Proposition 3.2 where we have replaced h lt with h t by Lemma 2.2.
We are now in a position to prove that the events E t and F t hold eventually with overwhelming probability.
Proof. Applying Proposition 3.2 to the sequences {σ(z)} z∈N + and {σ(z)} z∈N − ∪{0} and setting l = l t , we have that where we have again replaced h lt with h t by Lemma 2.2. Similarly, applying Proposition 3.1 to the same sequences, P (log d t < f (log l t ) + log log l t ) = 1 eventually. Recalling that log log l t ≪ log h t and f (log l t ) = log r t + o(1), we have that P (log d t < log r t + log h t ) = 1 (12) eventually. Combining equations (11) and (12) yields Proposition 3.4.
Proof. By symmetry, it is sufficient to prove that  Proof. These bounds can be derived by combining the law of the iterated logarithm with well-known almost sure bounds on the local time of the simple discrete-time random walk in term of the number of steps (see, e.g., [7,Theorems 11.1,11.3]). For any x ∈ Z + and y ∈ Z − , Proof. This is a well-known property of the simple discrete-time random walk, following easily from the optional stopping theorem.
Proof. This is a well-known result from Markov chain theory. It is proved by considering the spectral representation of P(M t = 0) in terms of the eigenvalues λ i and eigenfunctions ϕ i of the generator of M t : recalling that the detailed balance condition ensures that λ i and ϕ i are real. Since P(M t = 0) stays bounded in time, each λ i ≤), resulting in monotonic convergence to a certain constant, which must be the equilibrium distribution at 0 if it exists.
We are now in a position to establish equations (3), (4) and (6): Let Q z denote the local time at z of the geometric path induced by {X s : s ≤ τ t 1 }, which follows the simple discrete-time random walk. By Proposition 4.1, there exists a constant C > 0 such that eventually max z Q z ≺ max z L dt z < Cd t log log d t almost surely. Consider τ t 1 as the sum of the jump times along the geometric path. Since on the event E t , where each Gam(n, µ) is an independent gamma distribution with mean nµ and variance nµ 2 . By Chebyshev's inequality, Since Σ t d t h t < t on E t , combining equations (14) and (15) yields the result. Proposition 4.5. As t → ∞, First note that if r t /h t = O(1) then the result follows easily, since then t ≪ l t and P σ (τ t 2 < t) ≤ P σ (τ t 2 − τ t 1 < t) ≤ P(Exp(l t ) < t) → 0 where Exp(l t ) denotes the exponential distribution with mean l t . So assume r t /h t → ∞. Let Q denote the local time at Z τ t of the geometric path induced by {X s : τ t 1 ≤ s ≤ τ t 2 }. By Proposition 4.1, there exists a constant C > 0 such that eventually Q d = L ⌊rt/ht⌋ 0 > r t h t (log r t ) C almost surely, and so eventually On the other hand, by Lemma 2.1 and the conditions on h t , r t l t h t (log r t ) C = th t (log r t ) C ≫ t and so Chebyshev's inequality gives P Gam r t h t (log r t ) C , l t < t → 0 .