Excited deterministic walk in a random environment

Excited deterministic walk in a random environment is a non-Markov integer-valued process ( X n ) ∞ n =0 , whose jump at time n depends on the number of visits to the site X n . The environment can be understood as stacks of cookies on each site of Z . Once all cookies are consumed at a given site, every subsequent visit will result in a walk taking a step according to the direction prescribed by the last consumed cookie. If each site has exactly one cookie, then the walk ends in a loop if it ever visits the same site twice. If the number of cookies per site is increased to two, the walk can visit a site x arbitrarily many times before getting stuck in a loop, which may or may not contain x . Nevertheless, we establish monotonicity results on the environment that imply large deviations.


Introduction
The excited deterministic walk in a random environment (EDWRE) in dimension d ≥ 1 is a discrete time process, (X n ) ∞ . . , b}. We imagine Ω as stacks of M cookies, ω(0, z), . . . , ω(M − 1, z), at each site z ∈ Z d , each with an arrow pointing to an element of the cube [−L, L] d . We assume that Ω is equipped with the product measure P = P L,M such that {ω(j, z) : j ∈ [0, M − 1], z ∈ Z d } are i.i.d. with distribution µ supported on [−L, L] d . Note the abuse of notation here, that ω ∈ Ω is both an element of the set of environments, and a random element (via the identity map) with distribution P. We further assume that µ(k) > 0 for all k ∈ [−L, L] d \ {0}. Excited deterministic walk in a random environment The model of excited random walk was first introduced by Benjamini and Wilson [3] in which the nearest-neighbor random walk was perturbed by adding a cookie to each site of Z. In later studies the random walks in random environments were modified by adding multiple cookies to each of the sites and a number of results were established about recurrence, ballisticity, monotonicity, and return times to zero [2,6,8,9,10,13]. Some of these excited random walks are known to converge to Brownian motion perturbed at extrema [5]. The above results that study the behavior for large n cannot be generalized for our walks if M is kept fixed. One of the properties of EDWRE is almost sure boundedness. However, it is unknown whether different modes of convergence may occur if the assumption M < ∞ is changed to an assumption that each site has a finite (but not uniformly bounded) random number of cookies. For the ERW the finiteness of the expected return time to 0 depends on the average drift per site [10]. The return time to 0 for EDWRE is infinite with positive probability. However, conditioned on the event that the walker returns to 0, it is unknown what the expected lengths of the excursions are.
Large deviations for random walks in random environments were studied in [14]. In the case of random walks in excited random environments very little is known in higher dimensions. The methods are often restricted to nearest-neighbor walks. Our main proof is also restricted to one dimension, however we are allowing our walk to make jumps of sizes bigger than 1.
The case M = 1 corresponds to DWRE and Theorem 1.1 can be obtained in arbitrary dimension d [11]. The main argument of the proof used the fact that once the walk visits a site it has visited before, it will end in a loop. This can be simply stated as the 0 − 1 − ∞−principle, meaning that in DWRE the number of times a given site can be visited by the walk is zero, one, or infinity. However, we will see in Theorem 2.1 that EDWRE is a much richer model, and that a site can be visited arbitrarily many times.
The key ingredient in the proof of Theorem 1.1 is Lemma 4.3 that establishes a monotonicity property among favorable environments. A configuration of cookies on Z is called a favorable environment if it enables the walk starting at 0 to reach λn in fewer than n steps. Lemma 4.3 states that for every favorable environment one can change several cookies in [0, O( √ n)] to make another favorable environment that also allows the walk to avoid any backtrackings over 0. This result was the key to establishing a sub-additivity necessary for proving large deviations.
In the case when the maximal jump size is L = 2 one can replace O( √ n) in Lemma 4.3 with a finite number. It remains unknown whether O( √ n) can be replaced by a finite number when L ≥ 3.
Before delving into properties of the model, it is instructive to consider one concrete example. Example 1.5. Assume that the random environment is created in the following way.

Properties of excited walks
The results in this section serve to outline some of the major differences between excited and non-excited walks. In this section we will restrict ourselves to the case d = 1.  In regular non-excited deterministic walks in random environments, the number of visits to any particular site can be 0, 1, or infinity. The last case corresponds to the situation in which the walk ends in a loop passing through a prescribed number of sites infinitely many times. In an excited environment, the walker may revisit 0, for instance, any number of times 1, 2, . . . , ∞. However, the probability of revisiting 0 a large finite number of times decays exponentially, as the next theorem demonstrates. For convenience, we let Theorem 2.1. Assume that L ≥ 2 and M ≥ 2. Let D 0 be the cardinality of the set {n : X n = 0}. For each k ∈ N the following inequality holds To see this, observe that on the M th visit to 0, all cookies but the last have been consumed. Assume the last cookie points to the right. The site 0 can be visited at most M − 1 additional times without being caught in a loop if the top M − 1 cookies at 1 all point left, and the last cookie points right. It is easy to construct environments that attain each of these values, but it is an interesting problem to compute the distribution of D 0 .
Proof. The lower bound follows from Lemma 2.3 below.

Lemma 2.3.
There exist two functions f, g : Z → {−2, −1, 1, 2} such that the deterministic sequence x n defined by x 0 = 0 and contains exactly k terms equal to 0 and has −2k ≤ x n ≤ 2k − 1 for all n.
Indeed, if we find two such functions, then the event E ⊂ {D 0 = k} can be constructed as follows: For the upper bound, suppose that V j 0 is the time of the jth visit to 0 (so V 1 0 = 0). If V k 0 < ∞ and V k+1 0 = ∞, then the walker cannot get stuck in a loop that includes 0, and the number of visits to 0 must be k. Therefore, between consecutive visits to 0, the walker must see at least one new cookie, otherwise it will be stuck in a loop containing In order for the walker to revisit 0 at time V k 0 , none of the regions [iL, (i + 1)L − 1] that the walker visits before this time can be a trap where the walker gets stuck in a loop. An example of a trapping configuration on the interval [iL, (i + 1)L − 1] has ω(j, iL) = 1 = −ω(j, iL + 1) for j ≥ 0 and ω(0, iL + x) = −x for x = 2, . . . , L − 1. Therefore, the probability that [iL, (i + 1)L − 1] is a trapping region is at least (µ min ) 2M +L−2 . Finally, observe that the set of i ∈ Z such that the walker visits [iL, (i + 1)L − 1] by time V k 0 must be a set of consecutive integers containing 0, since the walker cannot jump over any such region. Therefore, the walker must either visit every such region for 0 ≤ i ≤ k/2LM − 1, or every such region for −k/2LM + 1 ≤ i ≤ 0. The probability that none of these regions is a trap gives the upper bound.
We will now use induction on i to prove that for each i ∈ {0, 1, . . . , k − 1} the following holds:  This is easy to verify for i = 0 and i = 1. Assume that the statement is true for some i and let us prove it for i + 1.
From now on the sequence is periodic and none of the terms will be zero.
This proves that there are exactly k terms equal to 0, and since it is stuck in a loop, no vertices outside [−2k, 2k − 1] are visited.

Laws of large numbers
In this section we assume that the walk is in R d for any d ∈ N. We prove that the walk is almost surely bounded. As a consequence, the law of large numbers holds with the limiting velocity equal to 0. Moreover, all of the moments of the process X n have growth that is slower than any function f (n) that satisfies lim n→∞ f (n) = +∞. This means that the central limit theorem also does not have the classical form for this model.
The following lemma will be essential for the proofs of the boundedness of the walk. This lemma establishes the exponential decay of the probabilities that the walk reaches the annulus A k defined in the following way: This way, A 0 is the hypercube [−L, L] d , while for k ≥ 1, A k is an annulus. For any set A ⊆ R d let us define There exists a positive real number c ∈ (0, 1) and an integer k 0 such that Proof. For each x ∈ Z d , let x + = x be an arbitrarily chosen vertex from the set where x ∞ denotes the largest coordinate of x in absolute value. Observe that if x is outside the hypercube [−kL, kL] d , then so is x + , and that x + can be reached by the walker in one step from x. Denote by G(x) the event that all cookies at x point to x + , and all cookies at x + point to x. That is, On the event G(x), the walk would get stuck in a loop between x and x + if it ever reached the site x.
We obviously have the following relation, The Lemma will be established once we prove that for every k ≥ 0 the following inequality holds: For each x ∈ A k let us introduce the event The event Ω x is in the sigma field generated by the cookies inside the set A 0 ∪ · · · ∪ A k−1 . Therefore, Ω x and G (x) are independent.
We now have This completes the proof of (3.1), and hence the proof of the required inequality.
A consequence of Lemma 3.1 is that the sequence X n is almost surely bounded. We present this result in the following lemma.

Lemma 3.2.
Denote by B the event that X n is a bounded sequence. More precisely, B = {∃M 0 such that X n ∞ ≤ M 0 for all n ≥ 0}, where x ∞ denotes the biggest coordinate of the d-dimensional vector x in absolute value. Then P (B) = 1.

Large deviations
In this section we prove Theorem 1.1, and henceforth take d = 1. For λ ∈ [0, L] we want to show the existence of the limit lim n→∞ 1 n log P (X n ≥ λn). As stated earlier, we will prove this under the assumption that there are at least 3 cookies on each site, i.e. M ≥ 3. Before we can prove the theorem we need to introduce the following notation. For k ∈ N and x ∈ Z let us denote by V k x the time of the kth visit to the site x. The hitting time V k x can be inductively defined as: Instead of V 1 x we will often write V x . As introduced earlier, for any set A ⊆ R we will denote its hitting time by T A = inf{n : X n ∈ A}. If x > 0 we will write T x instead of T [x,+∞) . The following two inequalities are easy to establish: for some constant C independent of n. Therefore, it is sufficient to prove that lim denote the event that the walk reaches λn by time n before backtracking to the left of 0. It is trivially true that A n ⊂ {T λn ≤ n} so P (A n ) ≤ P (T λn ≤ n).

Definitions
If a = (a ) K =1 ∈ Z K where K ∈ N ∪ {∞} and B ⊂ Z, then the restriction of a to B is denoted a B , and is the sequence of terms in a that belong to B with their order intact.
denote the sequence of locations of the walker from steps t 1 through t 2 .
In other words, we will write ω ≺ ,m ω if (1) the two environments are identical to the right of m, (2) the walkers on both environments visit the same sites in the same order to the right of m and until exceeding , but (3) the walker on ω may avoid some parts of the path followed by the walker on ω to the left of m. Observe that ≺ ,m gives a partial ordering of the environments in {T < ∞}.

Monotonicity results
The next theorem provides the asymptotic equivalence of probabilities P (T λn ≤ n) and P (A n ) on the logarithmic scale.   Proof. We will use the following result whose proof will be presented later. Sinceω andω coincide on sites in [0, +∞) and X(ω) does not visit negative sites, we conclude that X(ω) does not visit negative sites. Therefore, for each ω ∈ {T λn ≤ n} there existsω ∈ A n such that ω andω coincide on all sites except possibly for the sites in [0, 2L √ n]. We can now define a function f : {T λn ≤ n} → A n in the following way. For each ω ∈ {T λn ≤ n} we pick oneω with the properties established in the previous paragraph and define f (ω) =ω.
Let us fix n. We can now define P n on the restriction Ω n of Ω that corresponds to the portion of the integer axis between the numbers −Ln and Ln. The purpose of this restriction is so that P n (ω) > 0 for each ω ∈ Ω n . Formally, and P n is defined to be the restriction of P. Then we have P n (T λn ≤ n) = P (T λn ≤ n) and P n (A n ) = P (A n ), where each ω ∈ Ω is identified with an element of Ω n by truncation, which will also be denoted ω. It suffices to prove that there is C ∈ R + (independent of n) such that P n (T λn ≤ n) ≤ C √ n P n (A n ). (4.4) Observe that if environment ω ∈ Ω n differs from environment ω ∈ Ω n at exactly one site, z ∈ [−Ln, Ln], then P n (ω) ≤ P n (ω )/(µ min ) M . Let C 1 = 1 µmin M and C 2 = (2L + 1) M .
We will prove inequality (4.4) for C = ( This completes the proof of inequality (4.4) which implies (4.3).
In order to prove Lemma 4.3 we first need to establish the following result.

Lemma 4.4.
Fix ω ∈ Ω. Suppose a, b ∈ Z with |a − b| ≤ L, and 0 ≤ t a < t b are such that X ta (ω) = a, X t b (ω) = b, and one of the following two conditions is satisfied: Then there exists ω ∈ Ω such that (ω) and all j ≥ 0.
Furthermore, if ω contains no self-loop cookies (ω(j, x) = 0) and a = b, then ω also contains no self-loop cookies.
Proof. Let C x = L t b (ω, x) − L ta (ω, x) be the number of times the site x is visited by the sequence X [ta,t b −1] (ω). Under the assumption (a) we obtain the environment ω from ω by removing the cookies visited by the walker X(ω) in the time interval [t a + 1, t b ] and rewiring the top cookie at a at time t a to point at b. That is, From the definition of ω , it is clear that (iii) is satisfied, and (i) is satisfied because, from the perspective of the walker, ω and ω are identical up until time t a . Finally, (ii) is satisfied because the remaining environments at time t b in X(ω) and at time t a + 1 in X(ω ) are identical. The assumption (a) guarantees that rewiring the top cookie at a is allowed. (Without this assumption we would be rewiring the cookie labeled (M − 1), which would modify all cookies j ≥ M − 1, thus affecting the future path of the walk.) If (b) is assumed instead of (a), then no rewiring is necessary since the cookie at a at the time t a points to b in both ω and ω , i.e. ω (L ta + C a , a) = b − a and we can keep the same definition for ω as when working under the assumption (a).
Let us now define Note that we must have a 1 ≤ x + L < L √ n. We can be certain that α 2 is well defined because X [V 1 Having defined a 1 < · · · < a i and assuming that a i < L √ n we inductively define the times α i+1 and β i+1 in the following way: Clearly, a i+1 ∈ (a i , a i + L], b i+1 > a i and |a i+1 − b i+1 | ≤ L. As above, we are certain that  We can continue the induction until we have x < a 1 < · · · < a I where I is the smallest index such that a I ≥ L √ n. Since a i+1 − a i ≤ L for each i ≤ I − 1, we must have EJP 20 (2015), paper 44.

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ejp.ejpecp.org Figure 3: The walker must travel a long distance (at least L √ n) between the second and third visits to a site marked a i ≤ L √ n. Each of these excursions takes at least √ n steps.
I ≥ √ n. From (4.6), we have that for each i ≤ I − 1, before the second visit to the site a i , the walk X(σ) visits the site a i+1 at least M times. Furthermore, (4.7) implies that V s+1 ai (σ) − V s ai (σ) ≥ √ n for each i ≤ I and s ≤ M − 1, that is, the walk spends at least √ n steps between consecutive visits to each site a i . Now we will use our assumption that M ≥ 3. We know that a 1 is visited at least three times before x is visited for the first time. Between the second and third visit to a 1 the walk spent at least √ n steps, as depicted in Figure 3.
The second visit to a 1 has occurred after the site a 2 is visited at least M times, hence the second visit to a 1 occurred after the third visit to a 2 . Therefore n. Since the second visit to a 2 occurred after M visits to a 3 we know that the second visit to a 2 occurred after the third visit to a 3 .
a I (σ) + n ≥ n, which contradicts the assumption that V x (σ) < T λn (σ) ≤ n. This completes the proof of Lemma 4.3.

Large deviations
In this subsection we provide the proof to Theorem 1.1.
Proof of Theorem 1.1. It suffices to prove that P (A m+n ) ≥ P (A n ) · P (A m ). We notice the following inclusion: Let us define the walkX(ω) for ω ∈ A n ∩ {T λn ≤ n, inf{X k : 0 ≤ k ≤ T λn } ≥ 0} in the following way:X k (ω) = X k+T λn (ω) − X T λn (ω). The walkX starts at 0. In analogy to the stopping time T x for the walk X we defineT x for the walkX. The precise definition is: In analogy to A n we define the eventÂ m for the walkX: On the event A n ∩Â m , by time T λn +T λm the walk X reaches the site X T λn +XT λm ≥ λ (n + m). Therefore A n ∩Â m ⊆ A n+m . We will now prove that P A n ∩Â m = P (A n ) · P Â m . For each x ∈ [λn, λn + L], conditioned on X T λn = x, the events A n andÂ m are EJP 20 (2015), paper 44.
which implies the inequality for all n, m > 0. Fekete's subadditive lemma (see [18]) implies the existence of the limit The proof is completed using the inequalities (4.1) and (4.2) and Theorem 4.2.
5 Case L = 2 or M = 1 In the case when L = 2 or the number of cookies per site is 0 we can obtain the exponential decay of probabilities P (X n ≥ λξ(n)) for every positive function ξ that satisfies ξ(n) + ξ(m) ≥ ξ(n + m). In particular this holds for ξ(x) = x θ for θ ∈ (0, 1]. Theorem 5.1. Let ξ : R + → R + be a positive super-additive function and assume that either L = 2 or M = 1. Then there is a function ϕ : R + → R such that for every λ > 0 the following holds: lim n→∞ 1 n log P (X n ≥ λξ(n)) = ϕ(λ).
Proof. We will prove the theorem for the case L = 2. The proof when M = 1 is a simple generalization of the proof from the case of deterministic walks in random environments in [11]. First of all, the following inequalities are obtained in the same way as in the proof of Theorem 1.1: lim sup 1 n log P (X n ≥ λξ(n)) ≤ lim sup 1 n log P T λξ(n) ≤ n .
In the same way as in the proof of inequality (4.3) we now establish P T λξ(n) ≤ n ≤ CP (A n ) .
An argument analogous to the one presented in the proof of Theorem 1.1 allows us to prove the existence of the function ϕ such that lim n→∞ 1 n log P (A n ) = ϕ(λ).

Properties of the rate function
The next theorem states that the rate function φ from (1.1) is concave in λ.
Since {X n ≥ λn} ⊆ {T A k < +∞} we use Lemma 3.1 to conclude that P (X n ≥ λn) ≤ c k for some constant c ∈ (0, 1) and all sufficiently large n. For sufficiently large n we have n . This implies that The finiteness of φ(λ) follows from the fact that {T λn ≤ n} contains the event which is the event that the top cookies at each of the sites 0, L, 2L, . . . , nL point to the location that is L units to its right. The probability of the last event is at least µ n min hence φ(λ) ≥ log µ min .
Also, denote byT x the hitting time of the walkX, i.e.T x = T X T αn λn +x − T αnλn . We now obtain T (αλ+βγ)n ≤ n, inf From our choice of sequences (α n ) ∞ n=1 and (β n ) ∞ n=1 we derive the following two inequalities Observe that (1 − α n − β n ) n ∈ {0, 1, 2}. In each of the three cases we have P T (αλ+βγ−αnλ−βnγ)n ≤ (1 − α n − β n )n, inf 0≤k≤T (αλ+βγ−αn λ−βn γ)nX Since the walks X,X andX occupy disjoint parts of the environment (on the events that there are no backtrackings to the left of 0), by independence we obtain P T (αλ+βγ)n ≤ n, inf Taking logarithms of both sides of the last inequality, dividing by n, and taking the limit as n → ∞ we conclude The first limit on the right-hand side of the last inequality is equal to 0. For the second EJP 20 (2015), paper 44.

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ejp.ejpecp.org Excited deterministic walk in a random environment limit we use that α n n is a positive integer, hence lim n→∞ 1 n log P T αnλn ≤ α n n, inf 0≤k≤T αn λn Similarly we obtain that the last term on the right-hand side of (6.3) is equal to βφ(γ) which completes the proof of the concavity.
In a similar way we can prove that the function ϕ from (5.1) is concave in λ.
It suffices to prove that the second limit from the right-hand side is greater than or equal to αϕ(λ). The number α n n is a positive integer and since α n < 1 the following inequality holds α n ξ(n) ≤ ξ(α n n). Therefore lim n→∞ 1 n log P T αnλξ(n) ≤ α n n, inf 0≤k≤T αn λξ(n) X k ≥ 0 ≥ lim n→∞ 1 n log P T λξ(αnn) ≤ α n n, inf 0≤k≤T λξ(αnn) X k ≥ 0 = lim n→∞ α n α n n log P T λξ(αnn) ≤ α n n, inf  Similarly we obtain that the last term on the right-hand side of (6.4) is greater than or equal to βϕ(γ) which completes the proof of the concavity.
For functions ξ(n) that are not the identity map we cannot guarantee that ϕ(λ) < 0.
We believe this not to be true, but we do not have a proof of this.

Open problems
We believe that the main result, Theorem 1.1, holds in higher dimensions. Our proof for EDWRE differs from the proof of the analogous result for DWRE in that it requires Lemma 4.3, and our proof for this lemma relied heavily on the dimension being d = 1. Our proof of Lemma 4.3 required that M ≥ 3 to guarantee the existence of the increasing sequence (a k ) I k=1 in Z that the walk cannot cover in time n. We believe that this technical condition can be removed, but we were only able to do so in the case L = 2. Our next question is related to the excursions that are well understood in the case of ERW [10]. Recall that V 1 0 is the time of the first visit to 0. Since the walk starts at 0 we have that V 1 0 = 0. We can define the time of the entrance to the loop in the following way: Z(ω) = inf {n : ∃k ≤ n, X k = X n , ∀j ∈ {k, k + 1, . . . , n} L j (ω, X j ) ≥ M − 1} .