Localization for a Class of Linear Systems

We consider a class of continuous-time stochastic growth models on $d$-dimensional lattice with non-negative real numbers as possible values per site. The class contains examples such as binary contact path process and potlatch process. We show the equivalence between the slow population growth and localization property that the time integral of the replica overlap diverges. We also prove, under reasonable assumptions, a localization property in a stronger form that the spatial distribution of the population does not decay uniformly in space.

Similarly, A = B a.s. mean that P (A\B) = P (B\A) = 0. By a constant, we always means a non-random constant.
We consider a class of continuous-time stochastic growth models on d-dimensional lattice Z d with non-negative real numbers as possible values per site, so that the configuration at time t can be written as η t = (η t,x ) x∈Z d , η t,x ≥ 0. We interpret the coordinate η t,x as the "population" at time-space (t, x), though it need not be an integer. The class of growth models considered here is a reasonably ample subclass of the one considered in [Lig85, Chapter IX] as "linear systems". For example, it contains examples such as binary contact path process and potlatch process. The basic feature of the class is that the configurations are updated by applying the random linear transformation of the following form, when the Poisson clock rings at time-space (t, z): (1.1) where K = (K x ) x∈Z d is a random vector with non-negative entries, and independent copies of K are used for each update (See section 1.1 for more detail). These models are known to exhibit, roughly speaking, the following phase transition [Lig85, Chapter IX, sections 3-5]: i) If the dimension is high d ≥ 3, and if the vector K is not too random, then, with positive probability, the growth of the population is as fast as its expected value as time the t tends to infinity, as such the regular growth phase.
ii) If the dimension is low d = 1, 2, or if the vector K is random enough, then, almost surely, the growth of the population strictly slower than its expected value as the time t tends to infinity, as such the slow growth phase.
We denote the spatial distribution of the population by: In our previous paper [NY09], we investigated the case (i) above and showed under some technical assumptions that the spatial distribution (1.2) obeys the central limit theorem. We also proved the delocalization property which says that the spatial distribution (1.2) decays uniformly in space like t −d/2 as time t tends to infinity. In the present paper, we turn to the case (ii) above. We first prove the equivalence between the slow growth and a certain localization property in terms of the divergence of integrated replica overlap (Theorem 1.3.1 below). We also show that, under reasonable assumptions, the localization occurs in stronger form that the spatial distribution (1.2) does not decay uniformly in space as time t tends to infinity (Theorem 1.3.2 below). These, together with our previous work [NY09], verifies the delocalization/localization transition in correspondence with regular/slow growth transition for the class of model considered here.
It should be mentioned that the delocalization/localization transition in the same spirit has been discussed recently in various context, e.g., [CH02,CH06,CSY03,CY05,HY09,Sh09,Yo08a,Yo08b]. In particular, the last paper [Yo08b] by the second author of the present article can be considered as the discrete-time counterpart of the present paper. Still, we believe it worth while verifying the delocalization/localization transition for the continuoustime growth models discussed here, in view of its classical importance of the model.

The model
We introduce a random vector K = (K x ) x∈Z d which is bounded and of finite range in the sense that random vectors with the same distributions as K, independent of {τ z,i } z∈Z d ,i∈N * . Unless otherwise stated, we suppose for simplicity that the process (η t ) t≥0 starts from a single particle at the origin: (1.5) A formal construction of the process (η t ) t≥0 can be given as a special case of [Lig85, page 427, Theorem 1.14] via Hille-Yosida theory. In section 1.4, we will also give an alternative construction of the process in terms of a stochastic differential equation.
To exclude uninteresting cases from the viewpoint of this article, we also assume that the set {x ∈ Z d ; E[K x ] = 0} contains a linear basis of R d , (1.6) P (|K| = 1) < 1. (1.7) The first assumption (1.6) makes the model "truly d-dimensional". The reason for the second assumption (1.7) is to exclude the case |η t | ≡ 1 a.s.
Here are some typical examples which fall into the above set-up: • The binary contact path process (BCPP): The binary contact path process (BCPP), originally introduced by D. Griffeath [Gri83] is a special case the model, where K = (δ x,0 + δ x,e ) x∈Z d with probability λ 2dλ+1 , for each 2d neighbor e of 0 0 with probability 1 2dλ+1 . (1.8) The process is interpreted as the spread of an infection, with η t,x infected individuals at time t at the site x. The first line of (1.8) says that, with probability λ 2dλ+1 for each |e| = 1, all the infected individuals at site x − e are duplicated and added to those on the site x. On the other hand, the second line of (1.8) says that, all the infected individuals at a site become healthy with probability 1 2dλ+1 . A motivation to study the BCPP comes from the fact that the projected process is the basic contact process [Gri83].
• The potlatch process: The potlatch process discussed in e.g. [HL81] and [Lig85, Chapter IX] is also a special case of the above set-up, in which is a non-random vector and W is a non-negative, bounded, mean-one random variable such that P (W = 1) < 1 (so that the notation k here is consistent with the definition (1.10) below). The potlatch process was first introduced in [Spi81] for the case W ≡ 1 and discussed further in [LS81]. It was in [HL81] where case with W ≡ 1 was introduced and discussed. Note that we do not restrict ourselves to the case |k| = 1 unlike in [HL81] and [Lig85, Chapter IX].

The regular and slow growth phases
We now recall the following facts and notion from [Lig85, page 433, Theorems 2.2 and 2.3], although our terminologies are somewhat different from the ones in [Lig85]. Let F t be the σ-field generated by η s , s ≤ t.
is a martingale, and therefore, the following limit exists a.s.
We will refer to the former case of (1.13) as regular growth phase and the latter as slow growth phase.
The regular growth means that, at least with positive probability, the growth of the "total number" |η t | of the particles is of the same order as its expectation e (|k|−1)t |η 0 |. On the other hand, the slow growth means that, almost surely, the growth of |η t | is slower than its expectation.
Since we are mainly interested in the slow growth phase in this paper, we now present sufficient conditions for the slow growth.
2) A sufficient condition for the regular growth phase will be given by (1.25) below.

Results
Recall that we have defined the spatial distribution of the population by (1.2). Interesting objects related to the density would be ρ * t is the density at the most populated site, while R t is the probability that a given pair of particles at time t are at the same site. We call R t the replica overlap, in analogy with the spin glass theory. Clearly, (ρ * t ) 2 ≤ R t ≤ ρ * t . These quantities convey information on localization/delocalization of the particles. Roughly speaking, large values of ρ * t or R t indicate that the most of the particles are concentrated on small number of "favorite sites" (localization), whereas small values of them imply that the particles are spread out over a large number of sites (delocalization).
We first show that the regular and slow growth are characterized, respectively by convergence (delocalization) and divergence (localization) of the integrated replica overlap: To overcome this problem, we will adapt a more general approach introduced in [Yo08b].
Next, we present a result (Theorem 1.3.2 below) which says that, under reasonable assumptions, we can strengthen the localization property where c > 0 is a constant. To state the theorem, we define ( 1.19) We also introduce: where ((S t ) t≥0 , P x S ) is the continuous-time random walk on Z d starting from x ∈ Z d , with the generator , where the special case of the potlatch process is discussed. We consider the dual process ζ t ∈ [0, ∞) Z d , t ≥ 0 which evolves in the same way as (η t ) t≥0 except that (1.1) is replaced by its transpose: Here, q(x, y) is the matrix given by [Lig85, page 445, (3.8)-(3.9)] for the dual process. In our setting, it is computed as: so that (1.27) becomes: Under the assumption (1.25), a choice of such function h is given by h = 1 + cG, where 3) Let π d be the return probability for the simple random walk on Z d . Also, let ·, · and * be the inner product of ℓ 2 (Z d ) and the discrete convolution respectively. We then have that for the potlatch process. (1.28) For BCPP, (1.28) can be seen from that (cf. [NY09, page 965]) β x,y = 1{x = 0} + λ1{|x| = 1} 2dλ + 1 δ x,y , and G(0) = 2dλ + 1 2dλ To see (1.28) for the potlatch process, we note that 1 2 (k +ǩ) * G = |k|G − δ 0 , withǩ x = k −x and that Thus, from which (1.28) for the potlatch process follows.

SDE description of the process
We now give an alternative description of the process in terms of a stochastic differential equation (SDE). We introduce random measures on [0, (1.29) The precise definition of the process (η t ) t≥0 is then given by the following stochastic differential equation: (1.30) By (1.3), it is standard to see that (1.30) defines a unique process η t = (η t,x ), (t ≥ 0) and that (η t ) is Markovian.

Proofs
It is convenient to introduce the following notation: Then, by the Doléans-Dale exponential formula (e.g., [HWY92,page 248,9.39]), where Note also the predictable quadratic variation of M · is given by 1) We start with the "⊃" part of (2.3): Note that (1+u)e −u ≤ e −c 1 u 2 for −1 ≤ u ≤ b K −1, where c 1 > 0 is a constant. We suppose that ∞ 0 R s ds = ∞, or equivalently that, M ∞ = ∞. Then, for large t, This shows that ∞ 0 R s ds = ∞ implies |η ∞ | = 0, together with the bound (1.18). We now turn to the "⊂" part of (2.3): We need to prove that Let p be a transition function of a symmetric discrete-time random walk defined by and p n be the n-step transition function. We set Proof: Since the discrete-time random walk with the transition probability p is the jump chain of the continuous-time random walk ((S t ) t≥0 , P x S ) with the generator (1.21), we have that For d ≥ 3, G(x) < ∞ for any x ∈ Z d and β x,y = 0 only when |x|, |y| ≤ r K , we see from (1) that lim n→∞ x,y g n (x − y)β x,y = (|k| − k 0 ) x,y G(x − y)β x,y .
Thus, (2.4) holds for all large enough n's.

Proof of part (b):
The predictable quadratic variation of the martingale M · can be given by: where F s,z (ξ) = Thus, to prove (2.9), it is enough to show that there is a constant c ∈ (0, ∞) such that

2)
M t ≤ c t 0 R s ds.
We will do so via two different bounds for |F s,z (ξ)|: