One-point localization for branching random walk in Pareto environment

We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show a very strong form of intermittency, where with high probability most of the mass of the system is concentrated in a single site with high potential. The analogous one-point localization is already known for the parabolic Anderson model, which describes the expected number of particles in the same system. In our case, we rely on very ﬁne estimates for the behaviour of particles near a good point. This complements our earlier results that in the rescaled picture most of the mass is concentrated on a small island.


Introduction and main result 1.Introduction
We consider a branching process in random environment defined on Z d . We start the system with a single particle at the origin, which can branch and also migrate in space. Given the random potential ξ = {ξ(z) : z ∈ Z d } of non-negative random variables, a particle splits into two particles at rate ξ(z) when at site z. Furthermore, each particle moves independently according to the law of a nearest neighbour simple random walk in continuous time on Z d . This particular model was introduced in [4].
Most of the analysis of this model has concentrated on the expected number of particles. We fix a realization of the environment ξ and denote the expected number of particles by u(z, t) = E ξ [#{particles at site z at time t}], One-point localization for BRW in Pareto environment started with a single particle at site y. Finally, for any measurable set F ⊂ Ω, we define P y (F × ·) = F P ξ y (·) Prob(dξ).
If we start with a single particle at the origin, we omit the subscript y and simply write P ξ and P instead of P ξ 0 and P 0 . We define Y (z, t) to be the set of particles at the point z at time t, and let Y (t) be the set of all particles present at time t. We are interested in the number of particles N (z, t) = #Y (z, t) and N (t) = #Y (t) = z∈Z d N (z, t).
Our main result states that the system is intermittent with one relevant island consisting of a single point. Theorem 1.1. There exists a stochastic process (w t ) t≥0 such that N (w t , t) N (t) → 1 in probability under P as t → ∞.
This theorem says that at any large time, with probability close to 1, the overwhelming majority of the particles are situated at exactly one site. Recall that in [11], we showed that almost all the particles are contained in a "small" ball around one site. The bounds in [11] told us approximately what time each site was hit, and approximately how many particles were at each site, but were not precise enough to prove Theorem 1.1.
In fact, in proving Theorem 1.1, we will use the results in [11] to reduce the number of sites that we need to look at; and since we used a rescaled picture in that article, we will again use a rescaling here, to prove a slightly different statement from Theorem 1.1 that implies the statement above. In particular we will see that w t is the maximizer of a functional m t , one of the quantities that form the lilypad process from [11] that we mentioned above. This will be clarified in Section 4.

Remark 1.2.
Comparison with the PAM. In [7] it is shown that the analogous statement of Theorem 1.1 holds for the PAM, i.e. for the expected number of particles (where the expectation is taken with respect to the measure P ξ and the potential remains random). Moreover, they show that for any time t, the expected number of particles is Prob-almost surely concentrated in at most two sites. For the branching system (without taking expectations), a statement about the almost sure behaviour is beyond the scope of our current techniques. We hope to address this in future work. However, we can compare the maximizing site in our statement with the maximizing site in the PAM. Indeed, we will show that the maximizer is the maximizing site of the lilypad model defined in [11]. Moreover, these two sites (the maximizing site in the lilypad model, and the maximizing site in the PAM) were already compared in [11,Thm. 1.5(ii)], where we showed that at any time, the two sites agree with probability bounded away from 0 and 1.
It is worth noting too that our methods could be used to give a relatively short proof that the solution of the PAM is localized in one point with high probability.
In a companion paper [12], we show that the rescaled branching system converges to a Poissonian system, and as a corollary, we deduce that the maximizing site has the ageing property.

The Many-to-One or Feynman-Kac formula
We suppose that under P ξ y , we have a simple random walk (X(t)) t≥0 , started from y, independent of the environment and of the branching random walk above.
Given a particle v ∈ Y (t), and s ≤ t, we write X v (s) for the position of the unique ancestor of v that was present at time s.
We recall the Feynman-Kac formula or, as we often refer to it, the many-to-one formula. This simple result is key to our analysis, and we will use it regularly. Lemma 1.3 (Many-to-one / Feynman-Kac formula). If f is measurable, then Prob-almost surely, for any s > 0,

Outline of the proof and layout of the paper
We now explain why Theorem 1.1 holds. We already know that almost all the particles are in a small ball around the "good site" w, so really we only need to consider the behaviour of particles within that ball. Imagine, for a moment, that we begin with one particle at w. The site w has very large potential, on a much larger scale than the sites around it, so to simplify the picture we imagine particles breeding only when at w. Then the Feynman-Kac formula tells us that, under this simplification, the expected population ]. Taking z = w we see that the expected population size at w grows at least like e ξ(w)t−2dt , where the e −2dt factor corresponds to the probability that the random walk remains at w up to time t. We state this precisely in Lemma 2.2. In fact, standard methods can be used to show that in fact the population size at w (not just the expected population size) grows at least like e ξ(w)t−2dt , up to some small error. Now let z instead be a neighbour of w. To be at z at time t, the random walk (X(s)) s≤t must spend some time away from w. Spending time s away from w costs e −ξ(w)s , which suggests we should make s small; but jumping within a time interval of length s costs at least s. A simple optimization calculation suggests that we should choose s roughly of the order 1/ξ(w). We therefore guess that the size of the population at any neighbour of w should be at most 1/ξ(w) times as large as the size of the population at w itself. Similarly, at distance 2 from w we expect to see about 1/ξ(w) 2 times as many particles as at w, and so on. Since ξ(w) 1, and there are at most (2d) k sites at distance k from w, the population at w is much larger than the total population elsewhere. This statement will be made precise in Lemma 2.3. This is the basic idea behind the result, but unfortunately there are technical difficulties. The main problem we face is that we do not only have to consider starting with one particle at w. We have particles arriving at w from other sites at unpredictable times: it may be that the first particle to hit w was ahead of its time, and there is a wait before more particles pour in; or it may be that immediately after the first particle hits w, a huge number more arrive hot on its tail. Besides, with huge numbers of particles to consider, some particles will show unusual behaviour. For example, some might visit w only for a very brief period, and end up with more descendants at a neighbouring site than at w.
We take each particle to arrive at w (whose ancestors have not already visited w) in turn. The argument above is enough to show that the expected number -and therefore, by Markov's inequality, the actual number -of descendants at sites other than w is much smaller than the expected number at w. This is Proposition 2.1. The challenge then is to show that the number of particles at w is as large as it should be. We do this by first showing that each particle arriving at w has, with fairly large probability, a reasonable number of descendants at w at time t (not too much smaller than the expected number). This is done in Lemmas 3.2 and 3.3. Say that such particles "behave well".
We then break the time interval [0, t] up into small chunks. Provided that a lot of particles arrive at w within a chunk, the probability that at least half behave well is extremely close to 1. And if at least half behave well, then we have a good number of descendants at w at time t. This is carried out in Lemma 3.4. There are some further technicalities that are taken care of in Lemmas 3.5 and 3.7, but this is essentially enough to show that the population at w is as large as it should be (Proposition 3.1).
Combining Propositions 2.1 and 3.1 does most of the work in proving Theorem 1.1. However, just as in [11], we have to check that various events of small probability do not occur. This is done in Section 4.

Some definitions
In order to apply the heuristic above, we will need to ensure that particles cannot visit points of large potential other than w. This motivates the following definitions. We now work with general points y and z, but it may be helpful to imagine y as the good point w, and z as a point nearby.
Then for θ ≥ 0, define U y,θ = {z ∈ Z d : ξ(z) > ξ(y) − θ} \ {y} and f θ (y, t) = E ξ y [N (y, t; U θ,y )]. One complication is that there are always particles arriving at w from elsewhere, and as soon as a new particle arrives it begins contributing to the population at w.
Controlling this precisely is difficult, and if for example a large number of particles arrive at w just after w is first hit, then the population at later times will be much larger than if just one particle hits w significantly before any others. (Even very small fluctuations can be significant when the potential at w is so large.) Instead of tackling this issue, we instead work on estimating the population at and near w given the information about particles arriving at w for the first time. Again we make our definitions for a general site y. Let That is, L θ (y, t) contains only those particles that hit y before t, but whose ancestors have not hit y or visited U y,θ . Let G L θ (y,t) be the σ-algebra that contains information about which particles are in L θ (y, t) as well as the times that they hit y, If y has much larger potential than any nearby point, then the number of particles that we anticipate seeing at y at time t is (for any θ ξ(y)) and by the heuristic in Section 1.4, we anticipate seeing at most a constant times particles elsewhere. The following two sections make these statements precise, leaving a small amount of room for inaccuracies. The first, Section 2, shows that the population away from y is unlikely to be bigger than ξ(y) −9/10 v∈L θ (y,t) f θ (y, t − τ y (v)), and the second, Section 3, shows that the population at y is unlikely to be smaller than ξ(y) −4/5 v∈L θ (y,t) f θ (y, t − τ y (v)). By the argument above, these are perfectly reasonable statements, and they of course imply that the population at y is bigger than the population away from y.
Throughout this paper, | · | will denote the L 1 -norm on R d and B(x, r) the open ball around x of radius r in L 1 .
We will encounter some rather involved conditions like "ξ(y) ≥ e √ 256+100d ." The reader should view this as "ξ(y) is large", rather than attaching any particular meaning to the specific constant used. However the proofs do rely on certain dependencies between parameters and simply saying "ξ(y) is large" is not enough in several places; we would have to say that ξ(y) is large enough, but small relative to something else, and so on. We feel that-although it does make some results look cluttered-it is better to give precise conditions that work rather than ask the reader to keep track of all the required dependencies.

The population away from y is not too big
Our main result in this section is the following.
Proposition 2.1. Take y ∈ Z d with ξ(y) ≥ 2 and suppose that θ > 10dξ(y) 19/20 . Then there exists a constant C d depending on d only such that for any t > 0, To prove Proposition 2.1, we will need two fairly straightforward lemmas, which will also be useful later. The first essentially says that the population at y grows at least like e ξ(y)t−2dt .

Lemma 2.2.
For any y, z ∈ Z d , θ > 0, and 0 < s < t, The second lemma says that if we start with one particle at z, the expected population at y decreases as z moves away from y. By reversing time (which is how we will apply this lemma later) we can think of starting with one particle at y, and the expected population decreasing as we look further away from y.
) and θ > 2d + (log 2)/η. Then for any y ∈ Z d , k ≥ 1 and s > η, In particular, These lemmas are not difficult to prove, but we first check that they imply Proposition 2.1.
Proof of Proposition 2.1. By Markov's inequality, almost surely. Also note that for any v ∈ L θ (y, t), We apply Lemma 2.3 with η = 1/(8dξ(y) 19/20 ). (This is valid since by assumption θ > 10dξ(y) 19/20 > 2d + (log 2)/η.) This tells us that for any k ≥ 1, Therefore, summing over the vertices z ∈ D k (y) (noting that |D k (y)| ≤ C d k d−1 ) and then over k, Note that by time reversal and the many-to-one formula, for any z and so we have shown that uniformly in s, as required.
By the Markov property at time t − s, this is at least Reversing time again we get the result.
Proof of Lemma 2.3. Let J be the time of the first jump of our random walk, J = inf{u > 0 : Therefore, by the Markov property, Rearranging, we obtain Note that if η < 1/(8d) and θ > 2d + (log 2)/η, then 2dηe 2dη 1−e (2d−θ)η < 8dη < 1. Therefore either the left-hand side above is zero -in which case the result trivially holds -or the supremum on the right-hand side must be attained at some point in B k−1 \ B k = D k−1 (y). The first statement in the lemma follows by induction on k. The second statement follows from the first together with, for the last equality, the many-to-one formula.

The population at y is big enough
Our main aim in this section is to prove the following result.
The idea is simple: each particle that hits y for the first time gives rise to at least cf θ (y, t − τ y (v)) particles at y at time t, for some small constant c, with high probability. The conditions on ξ(y) can be read as "ξ(y) is large" (which will be true in the case we are interested in, since we will apply this result to the optimal point in the whole of Z d ).
Unfortunately the details are quite intricate, since there could be large fluctuations in how many particles are hitting y at different times. We proceed via a series of lemmas.

Lemma 3.2.
For any y ∈ Z d with ξ(y) ≥ 8d, any θ ≥ 2d + (log 2)/16d, and any t ≥ 1/ξ(y), Proof. This is a relatively straightforward second moment calculation. Fix t and y. We use the many-to-two formula [6,Section 4.2]. This says that where for each s ≥ 0, (X 1,s (u)) u≥0 and (X 2,s (u)) u≥0 satisfy • (X i,s (u)) u≥0 is a simple symmetric continuous-time random walk on Z d , jumping to each neighbouring vertex at rate 1, for each i = 1, 2; • X 1,s (u) = X 2,s (u) for all u ≤ s; • (X 1,s (s + u) − X 1,s (s)) u≥0 and (X 2,s (s + u) − X 2,s (s)) u≥0 are independent, and for each i = 1, 2, A i,s is the event Note that on A 1,s , we have ξ(X 1,s (u)) ≤ ξ(y) for all u ≤ s ≤ t, so Also, since (X 1,s (s + u) − X 1,s (s)) u≥0 and (X 2,s (s + u) − X 2,s (s)) u≥0 are independent, by the Markov property the right-hand side above is at most ξ(y)e ξ(y)s sup   Then (Υ s ) s≥0 is a birth-death process with birth rate ξ(y) and death rate 2d. Let D 1 be the time of the first death (i.e. the first time a particle leaves y), and let T n = inf{s ≥ 0 : Υ s = n}. If D 1 ≥ T n then T n is simply the time of the (n − 1)th birth. Therefore, for any u ≥ 0, where under P , the random variables (V j ) j≥1 are independent, and V j is exponentially distributed with parameter ξ(y)j for each j. Thus so fixing n = exp(ξ(y)u/4) we get P ξ y (u ≤ T n ≤ D 1 ) ≤ 2e −ξ(y)u/4 .

One-point localization for BRW in Pareto environment
We now concentrate on the event {T n ≤ D 1 ∧ u}. On the event {T n ≤ D 1 ∧ u}, at time T n we have n particles at y (that have never left y). Each of these has an independent descendance, and therefore (almost surely on the event {T n ≤ D 1 ∧ u}) Clearly f θ (y, t) ≤ e ξ(y)s f θ (y, t − s) for any s ∈ [0, t], and thus by our choice of u, on the event {T n ≤ D 1 ∧ u}, Applying Lemma 3.2 tells us that this is at most (15/16) n , which is smaller than ξ(y) −1/16 , as required.
Recall the notation from Lemma 3.3: when starting with one particle at y, Υ s is the number of particles that have stayed at y up to time s, D 1 is the first time a particle leaves y, and T n is the first time there are n particles at y who have never left. Let u = 2 ξ(y) + log ξ(y) 4ξ(y) and n = ξ(y) 1/2 . Then P ξ N v (y, t; U y,θ ) < 1 G L θ (y,t) ≤ P ξ y (Υ u = 0).
Applying the Chernoff bound ( [11,Lemma 2.6]) in the same way as in the proof of Lemma 3.4, we get we can combine Lemmas 3.4 and 3.5 to get the following corollary.
We now look after those j for which the number of particles absorbed in I j (y) is small.
We are now in a position to prove Proposition 3.1. The strategy is the following. We have shown in Corollary 3.6 that the number of particles "coming from J(y)" cannot be much smaller than the expected number of particles "coming from J(y)". We would also like to say that the number of particles "not from J(y)" can't be much less than the expected number of particles "not from J(y)". But this is difficult, because in a "non-J (y)" time interval we might-for example-have only one particle that just happens to hit y very briefly and then run off before it has time to breed.
We instead show that the number of particles descended from v 0 , defined to be the first particle to hit y, can't be much less than the expected number of particles "not from J(y)". To this end, we have shown that the expected number of particles "not from J(y)" is not much bigger than the expected number of particles descended from v 0 (Lemma 3.7); and that the number of particles descended from v 0 is not much smaller than the expected number of particles descended from v 0 (Lemma 3.3). Putting these elements together gives us a proof.

Applying bounds at w T (t)
We essentially have everything we need to prove one-point localization, by combining the results from the previous two sections with our previous work from [11]. We therefore need to recall some of the notation from [11]. We introduce a rescaling of time by a parameter T > 0. We also rescale space and the potential. If q = d α−d , the right scaling factors for the potential, respectively space, turn out to be We now define the rescaled lattice as is the open ball of radius R about z in R d . For z ∈ L T , the rescaled potential is given by and we set ξ T (z) = 0 for z ∈ R d \ L T . For any site z ∈ L T , we set we showed in [11] that these two quantities are close in a suitable sense. We also set, for z ∈ L T and t ≥ 0, Again we showed that these two quantities are close. In order to apply our results from earlier sections, we need to ensure that several irritating events do not occur. We check, via a sequence of lemmas, that these events have small probability. All these lemmas are either easy to prove, or are restatements of results from [11]. First we fix some parameters: Then P(A T ) → 0 as T → ∞.
Proof. Since there are at most C 2 d r(T ) 2d ρ 2d T pairs of points in L T (0, ρ T ), for any pair z 1 = z 2 we have Now, ξ T (z 1 ) and ξ T (z 2 ) are independent, and P(ξ T (z 2 ) ∈ [x, x + y]) is decreasing in x for fixed y, so which tends to 0 as T → ∞.
Proof. By the many-to-one lemma, which tends to 0 as T → ∞ since ν T → 0 and d − α < 0. Therefore, with high probability, there exists some point z 0 ∈ L T (0, ν T ) with ξ T (z 0 ) > ν T . But by [11,Lemma 3.4], which is less than t/10 for large T . Since m T (z 0 , t) ≥ ξ T (z 0 )(t − h T (z 0 )), the first part of the result follows.
By [11,Proposition 5.7], with high probability we have M T (z 0 , t) ≥ 9 10 tν T − log −1/4 T , and since ν T = log −d/16α T ≥ log −1/16 T , this is at least 4 5 tν T when T is large. Therefore N (r(T )z 0 , tT ) ≥ e 4 5 a(T )T ν T t with high probability, and the second part of the lemma follows.
The third part is a consequence of the first part, since if w maximizes m T (z, t), then m T (w, t) = ξ T (w)(t − h T (w)) (so if ξ T (w) ≤ ν T /2 then m T (w, t) ≤ ν T t/2 < 9 10 tν T ).
Proof. The first assertion holds by [11,Proposition 4.9 and (7)]. The second and third hold by [11,Lemma 2.7(ii)]. Lemma 4.5. Let w = w T (t) be any point in Z d such that m T (w/r(T ), t) ≥ m T (z, t) for all z ∈ L T . Then, for any ε > 0 Proof. The first statement is simply a rewording of [11,Theorem 1.3]. By Lemma 4.3 we may assume that ξ(w) > ν T a(T )/2, and by the first part of Lemma 4.4 we may assume that w ∈ B(0, r(T )ρ T ). The second statement then follows from the fact that P(∃y ∈ B(0, r(T )ρ T ) : ξ(y) > ν T a(T )/2, ∃z ∈ B(y, ε T ) with ξ(z) > ν T a(T )/2) Recall that κ T = {y ∈ B(0, ρ T r(T )) : ξ(y) ≥ ν T a(T )/2} as the points in κ T that get hit fairly late for the first time and the first particle is not followed by many other particles immediately afterwards. Contributions from these points will be negligible and we first show how to control the points in the complement. Lemma 4.6. For any t ≥ 0, P(∃y ∈ κ T ∩L θ T (tT ) c : N (y, tT ; U y,θ T ) < ξ(y) 1/10 z =y N (z, tT ; U y,θ T , y)) → 0 as T → ∞.
By the second part of Lemma 4.4, we may assume that there are at most K T points in κ T , and a union bound gives the result.
Finally, we can control the points inL θ T (tT ) that only get hit by a few particles that do not have much time to grow. P ∃y ∈L θ T (tT ) : N (y, tT, U y,θ T ) ≥ ν(T ) −1 a(T ) → 0.
Proof. Let y ∈L θ T (tT ). We recall that then ξ(y) ≥ ν T a(T )/2, t − H(y) ∈ [0, . By Lemma 4.4 we can assume that ξ(y) ≤ ν −1 T a(T ) and that there at most K T points in κ T , so that a union bound gives the result.
(viii) For all y ∈L θ T (tT ), we have N (y, tT ; U y,θ T ) ≤ ν −1 T a(T ). For y ∈ Z d , letŨ y = {z ∈ B(0, r(T )ρ T ) : ξ(z) ≤ ξ(y)}. Note that by (iii), any particle that is present at time tT must either have remained within Q or must have travelled via y without exitingŨ y for some y ∈ B(0, ρ T r(T )) with ξ(y) ≥ ν T a(T )/2. By (i) and (iii), such a particle must in fact not have hit U y,θ . Thus for any z ∈ Z d , N (z, tT ) ≤ N (z, tT ; Q c ) + y∈κ T N (z, tT ; U y,θ , y). Therefore we have shown that for any large T , with high probability, the site w T (t) satisfies N (w T (t), tT )/N (tT ) > 1 − ε.
In particular taking t = 1 completes the proof of Theorem 1.1.