Localization at the boundary for conditioned random walks in random environment in dimensions two and higher *

We introduce the notion of localization at the boundary for conditioned random walks in i.i.d. and uniformly elliptic random environment on Z d , in dimensions two and higher. If d = 2 or 3 , we prove localization for (almost) all walks. In contrast, for d ≥ 4 , there is a phase transition for environments of the form ω ε ( x, e ) = α ( e )(1 + εξ ( x, e )) , where { ξ ( x ) } x ∈ Z d is an i.i.d. sequence of random variables, and ε represents the amount of disorder with respect to a simple random walk.


Introduction
In this paper, we deal with the notion of localization for random walks in random environment (RWRE). Informally, the walk is localized if its asymptotic trajectory is confined to some region with positive probability. Otherwise, it is delocalized. For RWRE, localization has been studied almost entirely in the one-dimensional case (see, for example, the works of Sinai [21] and Golosov [13]). This work aims to start the investigation of this phenomenon in dimensions greater than one, considering the case of the boundary. Let us first define the model on which we will be working.

The model of a random walk in a random environment
Fix d ∈ N, the dimension where the walk moves. For x ∈ R d and p ∈ [1, ∞], the p norm of x is denoted by |x| p . Define V := {x ∈ Z d : |x| 1 = 1} = {±e 1 , · · · , ±e d } the set of allowed jumps of the walk (as usual, e i is the vector (0, · · · , 1 i-th position , · · · 0)). An environment is an element ω in the space Ω := M 1 (V ) Z d (in general, we denote by M 1 (X) to the space of probability measures on X). We usually write ω = {ω(x, e)} x∈Z d ,e∈V . Finally, we can define a random walk in the environment ω ∈ Ω starting at a point x ∈ Z d as the Markov chain X = (X n ) n∈N with law P x,ω that satisfies J k > 0 P − a.s., (1.5) and an RWRE is delocalized at the boundary if lim inf n→∞ 1 n n k=1 J k = 0 P − a.s. (1.6) Note that a priori, the walk can be neither localized nor delocalized. However, in Theorem 2.3 we show that this cannot happen for walks that satisfy Assumption 1.1.

A different formulation
Working on the boundary induces a polymer-like interpretation that makes more transparent the argument we use below. Given ω ∈ Ω, x ∈ Z d , and e ∈ V + , define π(ω, x, e) := ω(x, e) e ∈V + ω(x, e ) , Ψ(ω, x) := log e∈V + ω(x, e) . (1.7) Then ω(x, e) = π(ω, x, e)e Ψ(ω,x) , and π induces an RWRE, with V + as the set of allowed jumps. Its quenched measure (starting at x) is P x,π , and its expectation is E x,π . Therefore, for fixed n ∈ N and A ∈ B((Z d ) N ),  This definition resembles the general framework introduced in [19]. Using the polymer measure, it is direct to verify the identity From now on, we use this scheme (except in Section 4.1), although, of course, both definitions are equivalent.

Main results
This paper's main results are that localization holds for (almost) all uniformly elliptic and i.i.d environments in dimensions two and three and a phase transition in terms of the disorder in dimensions four or higher.
Let c := e∈V + q(e). The following assumption will play a remarkable role throughout the sequel.   A related result in RWRE appears in the article [24] of Yilmaz and Zeitouni. They show that for walks that satisfy a certain ballisticity condition 1 , there is a class of measures P such that the quenched and annealed rate functions differ in a neighborhood of the LLN velocity. In the directed polymer model, Comets and Vargas [10] prove localization in dimension 1 + 1 (one dimension for time, and one for space), while Lacoin [15] proves localization in dimension 1 + 2. Berger and Lacoin improved this result in [5], where they gave the precise asymptotic behavior for the difference between the quenched and annealed free energies, as n → ∞.
This parameter represents how much the distribution of the jumps in an RWRE differs from a simple random walk.
A direct consequence of Theorem 1.5 is the following: Corollary 1.7. Under the same hypotheses and notation of Theorem 1.6, if Assumption 1.3 does not hold, then ε = ε max . Otherwise, and if also d = 2 or 3, then ε = 0.

Alternative approaches to localization
In the recent article [3], the authors developed two notions related to localization in terms of the endpoint distribution ρ n (·) := P ω 0,n−1 (X n−1 ∈ ·) of a directed polymer in random environment.
We say (ρ i ) i is asymptotically purely atomic (concept introduced by Vargas in [23]) if, for each sequence (ε i ) i converging to zero as i → ∞, it is true that (1.14) For directed polymers, it is shown in the aforementioned article that if β > β c (i.e., the "low temperature" regime), then the endpoint distribution is both asymptotically purely atomic and geometrically localized with positive density. On the other hand, if 0 ≤ β ≤ β c (i.e., the "high temperature" regime), then there is a sequence (ε i ) i that converges to zero as i → ∞ such that Using the framework introduced in Section 1.2.1, the results above should be adaptable to our case for any d ≥ 2 using the parametrization (1.12). On the other hand, in dimensions two and three, the endpoint distribution of an RWRE conditioned at the boundary should be both asymptotically purely atomic and geometrically localized with positive density (as soon as Assumption 1.3 is satisfied).
One of the novelties in [3] is a compactification argument for distributions on Z d , which was inspired by the analog treatment for distributions on R d made in [17]. Further applications of similar ideas can be found in [2], [7], and [1].

An equivalent criterion for localization
In this section, we prove an equivalent criterion of localization/delocalization that will be used throughout the sequel. First, we need to define the following quantities.
The last equality holds since the conditioned walk is directed.
In the directed polymer literature, these limits are known as quenched and annealed free energy, respectively. The existence of p and the fact that it is deterministic hinge upon the following characterization in terms of I q , the quenched rate function for RWRE (which is also deterministic, see Lemma 3.5 in [4]).
The proof of this lemma follows the lines of Lemma 16.12 in [18], so it is omitted. Once the existence of p and λ is established, we proceed to state a criterion of localization/delocalization with regards to whether p = λ or p < λ. Theorem 2.3. Let (X n ) n∈N be an RWRE that satisfies Assumption 1.1.
(i) The RWRE is localized at the boundary if and only if p < λ.
(ii) The RWRE is delocalized at the boundary if and only if p = λ.
In particular, the walk is either localized or delocalized P-a.s.
As a corollary, we obtain a characterization of localization/delocalization in terms of the difference between the infima of the quenched and annealed rate functions.
To prove Theorem 2.3, we need to introduce a couple of definitions. The first is a martingale that relates p and λ, and the second is a random variable linked to J n . Definition 2.5. Given an RWRE (X n ) n∈N that satisfies Assumption 1.1, define the random variable in (Ω, B(Ω), P) The following lemma is straightforward, so its proof is skipped.
The martingale convergence theorem implies that W ∞ := lim n→∞ W n exists and is non-negative P-a.s. Since the event {W ∞ = 0} is invariant under the translations (T e ) e∈V + P-a.s., the ergodicity of P implies that P(W ∞ = 0) ∈ {0, 1}. This consequence will be useful in Proposition 2.7.
Next, we introduce a second random variable, This random variable is F n−1 -measurable. Observe that J 2 n ≤ I n ≤ J n . (2.5) The main ingredient in the proof of Theorem 2.3 is the next proposition, which compares W n and I n . We use the following notation: for sequences (a n ), (b n ) we say that a n = Θ(b n ) if a n = O(b n ) and b n = O(a n ). that is, − log W n = Θ( n k=1 I k ).
Sketch of the proof of Proposition 2.7. The proof of Theorem 2.1 in [9] can be adapted to show Proposition 2.7. It is based on the Doob's decomposition of the submartingale − log W n . More precisely, there exist a martingale {M n } n∈N and an adapted process {A n } n∈N such that for all n ∈ N, − log W n = M n + A n . Indeed, we decompose A n and M n as Exactly as in the aforementioned result, it is enough to prove that there is a constant C > 0 such that for all n ∈ N, (2.8) To check the inequalities above, notice that, by uniform ellipticity, the potential Ψ is bounded P-a.s., so there are constants 0 < C 1 < C 2 such that P-a.s., for all n ∈ N, Wn Wn−1 ∈ (C 1 , C 2 ). Therefore, for some constants C 3 , C 4 > 0. Thus, E[− log(1 + U n )|F n−1 ] is bounded by above by Similarly we get a lower bound and this shows the first inequality in (2.9). Finally, noting that for some constant C 5 > 0, log 2 (1 + U n ) ≤ C 5 U 2 n , repeating the steps from the last display we get the second inequality on (2.9), concluding the proof. The method of fractional moment and change of measure used in the proof was originally introduced by Derrida et al. in [11] for the pinning model. For directed polymers, Lacoin and Moreno used it for the first time in [16] on the hierarchical lattice, and Lacoin in [15] on Z d . Yilmaz and Zeitouni adapted the technique in [24] for random walks in random environment. As the proofs are similar, we only mention the main points of them and refer to the papers above for further details. More precisely, let φ(θ) := e∈V q(e)e θ,z . In [24], the analog of showing that p < λ in the space-time RWRE setting, is to demonstrate that for a sufficiently large set of points θ ∈ R d , lim n→∞ 1 n E log E 0,ω e θ,Xn −n log(φ(θ)) < 0, the main difference between the two models is that the potential Ψ(ω, x) is replaced by a tilt that depends on the steps of the walk, namely, Ψ st (θ, e) := e θ,e . This introduces a correlation that in our case is not present (see the paragraph below (3.9)). Thus, it is natural to apply the methods in [24] to deduce the desired result. We sketch the main ideas and differences in the next pages.
Hence, until the end of the proof we assume that (1.10) holds.
Let {X n } n∈N be a simple random walk with jumps in V + and lawP that satisfieŝ , x ∈ ∂R n , e ∈ V + , and define µ :=Ê(X 1 ). Consider N = nm with n fixed (but large enough) and m → ∞.
. We define, for y ∈ Z d , Now, we estimate each expectation E[W N (ω, Y )] 1/2 , applying the change of measure. The plan is the following (recall that N = mn with fixed n): fix j ∈ {1, · · · , m}, Y ∈ (Z d ) m , and a square integer n. Also, C 1 is a constant to determine, and y 0 := 0. Then define (3.5)

Proof in the case d = 2
The idea is to define a function that depends on the different blocks B j . Let  By Cauchy-Schwarz inequality, One can show that for K large enough, we can the estimates in Pages 251-252 from [24] to deduce that The bound (3.4) tell us that p − λ < 0 once we are able to prove the following: Lemma 3.1. For n, K, and C 1 large enough, i=0 Ψ(ω,Xi)+f K (δnD(B1))−n log(c) ; X n − nµ ∈ J y < 1/2.
The proof of the lemma above is followed almost exactly from Section 2.5 in [24]. The main difference rests in display (2.22) in the aforementioned paper. In our case, we need to check that for some α > 0, We can decompose it as The first term is nEE 0,π e Ψ(ω,X1)−log(c) (ω(X 1 ) − α) 2 . As c n := then by independence, the second term in (3.8) is zero. By comparison, the analog of α (called µ in [24]) is greater than zero due to a positive correlation that in our case is not needed.
Combining the previous results, such election of constants help us to deduce that Lemma 3.1 is true, and therefore p − λ < 0.

Proof in case d = 3
In this case, the proof in spirit is essentially the same, but some technical details need to be adapted to this situation. In particular, we need to redefine δ n and D(B j ). First, for a constant C 2 > 0 to determine, let Also, recall thatω is defined as in (3.6). Then we redefine δ n := n −1 (log n) −1/2 , D(B j ) := y,z (y,i),(z,j)∈Bj V (y, z)ω(y)ω(z).
The proof of Theorem 1.6 in [24] can be followed almost word by word, and our situation is a little bit simplified since the correlation issue is not present as in the d = 2 case.
Details are omitted.
The rest of this section is devoted to prove the second part of Theorem 1.6. The main ingredient to show that ε > 0 is the next lemma, a particular case of Lemma 3.1 with θ = 0 in [4].
It only remains to show an example in dimension greater or equal than 4 where 0 < ε < ε max .

An example on which ε < ε max
For simplicity, we consider d = 4, and i.i.d random variables (ξ(x)) x∈Z d ∈ Γ α such that ξ(x, e) = ξ(x, e ) for all e, e ∈ V + , and ξ(x, −e) = −ξ(x, e). If y = (y 1 , · · · , y d ) ∈ ∂D + is a point to determine, for i = 1, · · · , d, define α(e i ) = α(−e i ) := Moreover, assume that the distribution of ξ(0) under Q is the Rademacher distribution, namely, Q(ξ(0) = 1) = Q(ξ(0) = −1) = 1 2 . By Corollary 2.4, localization occurs as soon as inf x∈∂D + I a (x) < inf x∈∂D + I q (x). However, in this case, the infimum on the left is exactly I a (y), and it is achieved only at this point (see Theorem 2.3 in [4]). On the other hand, by the continuity of I q , the infimum on the right is also achieved at some point x ∈ ∂D + . If x = y, then I a (y) < I a (x) ≤ I q (x), so we are done. Thus, assume that x = y. Denote by (y n ) n∈N any sequence as in Lemma 3.5 in [4] for the point y. Then we decompose −I q (y)  Also, the sum and maximum above are over all directed paths 0 = x 0 , x 1 , · · · , x n such that x n = y n . Let C(y n ) be the number of such paths. It's easy to check that there exists some constant C > 0 such that, for all n ∈ N, C(y n ) ≤ Ce nf (y)− d−1 2 log n , where f (y) = − d i=1 y i log(y i ). To estimate the maximum above, we can use Hoeffding inequality (see Theorem 2.8 in [6]) to obtain, for a > 0,  = −I a (y) + log 1 + ε 1 − ε f (y)/2 + 1 2 (log(1 + ε) + log (1 − ε)) .