Monotonicity and regularity of the speed for excited random walks in higher dimensions

We introduce a method for studying monotonicity of the speed of excited random walks in high dimensions, based on a formula for the speed obtained via cut-times and Girsanov's transform. While the method gives rise to similar results as have been or can be obtained via the expansion method of van der Hofstad and Holmes, it may be more palatable to a general probabilistic audience. We also revisit the law of large numbers for stationary cookie environments. In particular, we introduce a new notion of $e_1-$exchangeable cookie environment and prove the law of large numbers for this case.


Excited random walk
An excited random walk (ERW) with bias parameter β ∈ (0, 1] is a discrete time nearest neighbor random walk (Y n ) n 0 on the lattice Z d obeying the following rule: when at time n the walk is at a site it has already visited before time n, it jumps uniformly at random to one of the 2d neighboring sites. On the other hand, when the walk is at a site it has not visited before time n, it jumps with probability (1 + β)/2d to the right, probability (1 − β)/2d to the left, and probability 1/(2d) to the other nearest neighbor sites. Excited random walk was introduced in 2003 by I. Benjamini and D.B. Wilson [5]. By using Theorem 21 of [3], they obtained a local limit theorem version and proved that for every value of β ∈ (0, 1] and d 2, excited random walks are transient. Furthermore, they proved that for d 4, lim inf n→∞ Y n .e 1 > 0 a.s., where (e i : 1 i d) denotes the canonical generators of the group Z d . This result was extended for dimensions d = 2, 3 by G. Kozma (see [11], [12]). In 2007, J. Bérard and A. Ramírez [6] and in 2012, with the different approaching, M. Menshikov, S. Popov, A. Ramírez and M. Vachkovskaia [13] proved a law of large numbers and a central limit theorem hold for the excited random walk for d 2, namely: • (Law of large numbers). There exists v = v(β, d), 0 < v < +∞ such that a.s.
where (e i : 1 i d) denotes the canonical generators of the group Z d .
In 2008, R. van der Hofstad and M. Holmes [10] proved that the velocity v is strictly increasing in β for d 9 relying on the lace expansion technique. In this paper, we prove that the velocity of an excited random walk is differentiable in β for d 8 and monotone in high dimension using cut times and martingale transforms. This technique is different from [10] and simpler. Moreover, it can be applied to more general models such as excited random walk in random environment. About diffrentiability of the velocity, we are interested in the derivative at the critical point β = 0. When the derivative at 0 is positive and if it is continuous in a neighbourhood of 0 then the velocity is monotone in that neighbourhood. The existence of the derivative at 0 of the velocity of a random process plays an important role in mathematical physic, that is known as name Einstein relation for random process, for instance in work of N. Gantert, P. Mathieu and A. Piatnitski [9]. Our main result for the excited random walk is the following: 1. The velocity is differentiable in β ∈]0; 1[ for d 8.
3. For d 6, the derivative at the critical point 0 exists, is positive and satisfies : where R(0) := lim n→∞ (R n /n), R n being the range at time n of the simple, symmetric random walk on Z d .
Our proof of Theorem 1.1 is based on two ingredients: * Let Z n (resp. X n ) be the vertical (resp. horizontal) component of the excited random walk: Z n := (Y n · e 2 , ..., Y n · e d ) , X n := Y n · e 1 .
(Z n , n 0) is a simple random walk on Z d−1 . For d−1 5 (i.e. d 6), E. Bolthausen, A-S. Sznitman, and O. Zeitouni [7] proved the existence of cut times (T k ), i.e times splitting the trajectory into two non-intersecting paths. Moreover, these cut times are integrable for d − 1 5. Let T := T 1 . Using ergodicity properties, it is possible to express the velocity in the direction e 1 as follows: where E β is the expectation under the law P β of the excited random walk, D is the set of cut times, and X T = Y T · e 1 . * Starting from (1), we use Girsanov transforms to make the dependence of v(β, d) w.r.t to β more explicit. This enables us to compute the derivative of v(β, d) in dimensions d 8 and to prove that this derivative is positive for d high enough.

m-cookies excited random walk in random environment
We now consider a model more general than the excited random walk, the m-cookies excited random walk in random environment (m-ERWRE). A cookie random walk or also called multi-excited random walk was introduced by M. Zerner (see [18], [19]). Monotonicity for multi-excited random walks on intergers was proven (see [2], [4], [14], [18],...). The law of large number for random walks in random environment was studied and holds under some conditions (see also [7], [17]). Let m be a positive integer or m = +∞. We place m cookies on every site of the lattice Z d . Moreover, m random variables (β k (y)) 1 k m with values in [0, 1], are attached to each site y of Z d . The process β := {(β k (y)) 1 k m } y∈Z d serves as a random environment whose law is denoted by Q. Let B := ([0; 1] m ) Z d be the set of random environments.
The m-cookies excited random walk in the random environment β = {(β k (y)) 1 k m } y∈Z d is a discrete time nearest neighbor random walk (Y n ) n 0 on the lattice Z d obeying the following rule: when at time n the walk is at a site that has k cookies where 1 k m, it eats one cookie and jumps with probability (1 + β m+1−k (y))/2d to the right, probability (1 − β m+1−k (y))/2d to the left, and probability 1/(2d) to the other nearest neighbor sites. On the other hand, when the walk is at a site y where there is no more cookie, then it jumps uniformly at random to one of the 2d neighboring sites. When m = 1 and the environment β is non random and constant, we recover the excited random walk.
Hence, when β is fixed, the "quenched" law P β of the m-cookies excited random walk in random environment β, is the probability on the path space (Z d ) N , defined by: , for 1 k m; The "annealed" law P is then defined as the semi-direct product on B × (Z d ) N : P = Q ⊗ P β . We will consider some particular cases of m−cookies excited random walk in random environment.

Excited random walk with m identical cookies (m-ERW.)
This is the case where the m cookies are non random and the same at each site: ∀k such that 1 k m , ∀y ∈ Z d , β k (y) = β , for some real number β ∈ [0; 1]. P m,β denotes the law of m-ERW. When m is large, the m-ERW is more and more like a simple random walk with bias β. Let v(m, β) be the speed of the m-ERW, whose existence is proven for d 2 in [6], [13]. We prove in section 3 the following result: Hence, there exists m 0 such that for m m 0 the speed of the m-ERW is increasing in β on [0; 1].

m-cookie excited random walk in stationary random environment
In this model, the random environment β = {β(y)} y∈Z d is assumed to be: • stationary: β(y + ·) law = β for any y in Z d ; • and/or ∆-exchangeable. To define this notion, we consider a family ∆ = {δ z } z∈Z d−1 of bijective mappings from Z to Z. The mapping σ ∆ : is then a bijection from Z d to Z d , acting on the set B of environment by σ ∆ (β)(y) = β(σ ∆ (y)). The environment is said to be ∆-exchangeable if and only if σ ∆ (β) law = β for any family ∆. Otherwise stated, an environment is ∆-exchangeable if its law does not change when performing permutations of the environment on each horizontal line.
An i.i.d. environment is of course stationary and ∆-exchangeable. Another simple example is provided by a stationary environment not depending on the horizontal component: for To describe our main result about this model, we introduce a partial ordering on the laws of environments. Generally speaking, let Q 1 , Q 2 be two probability measures on a partially ordered set (E, ). We say that a probability measure Q on E × E is a monotone coupling of Q 1 and Q 2 , if when denoting by l 1 and l 2 the coordinate maps from E × E to E: When such a monotone coupling exist, we say that Q 1 ≺ Q 2 .
The set B of environment is provided with the partial ordering: We say that the cookies are identical when the bias does not depend on every cookie: Our main result reads then as follows: Assume that the random environment is stationary and ∆-exchangeable.
1. For d 6, Xn n converges P −a.s. to a non negative random variable V , whose expectation is denoted by v(Q).
2. If the cookies are identical, there exists d 0 ∈ N * such that v(Q) is increasing with Q for d d 0 (w.r.t. the partial ordering ≺).

m-cookies excited random walk in i.i.d. random environment
In section 4.3, we assume that there are m cookies, and that the environment is i.i.d. This is the particular case of the environment that is stationary and ∆-exchageable. In this situation, we still denote by Q the law of β(0) using a slight abuse of notation. In the i.i.d setting, using the contruction an ergodic system, we can prove that the limiting speed is deterministic: 3. If the cookies are identical, there exists σ ∈ [0; 1) such that for any d 10, v(Q) is increasing with Q on the set {Q such that Q(0 β(y) σ, ∀y ∈ Z d ) = 1}.
2 Excited random walk 2.1 An expression for the velocity We begin this section by constructing the excited random walk from some independent sequences of random variables. This plays an important role to prove the monotonicity. First, we consider a simple random walk (SRW) {Z n } n 0 on Z d−1 and three sequences of random variables {η i } i 0 ,{ξ i } i 0 and {ζ i } i 0 independent with each other, independent ofZ and having distribution {Z n } n 0 will give the sequence of vertical moves of the excited random walk; η i = +1 will mean that at time i, the excited random walk performs an horizontal move. The direction of this move is given by ξ i when the ERW is at an already visited sites, and by ζ i otherwise. More precisely, set A k i : We define the vertical component Z of Y by: We now construct the horizontal component X of Y . Set Y 0 := 0 and assume that (Y j , 0 j i) are constructed. Let us define Y i+1 .
• On the event "Y i old" (already visited before time i), set We then set X i+1 := X i + E i , and Y i+1 := (X i+1 , Z i+1 ). With this construction, we obtain: Y is an excited random walk with bias parameter β.
Proof. For the proof of lemma 2.1, we need the following lemma: Lemma 2.2. Let F and G be two sigma-algebras and C ∈ F ∩ G such that F | C := {A ∩ C with A ∈ F } ⊂ G. For any integrable random variable V , we get The proof of Lemma 2.2 is easy. Now, we return to the proof of Lemma 2.1. Set For the case e j = e 1 , on the event "Y k new", The cases e j = −e 1 and Y k old are treated similarly. Lemma 2.1 is now proved.
Next, we give another construction of the ERW, on which we obtain an ergodic dynamical system leading to the formula (1). We begin with Let q be the probability on Z d−1 such that q(e) = 1 2d for all |e| = 1 and q(0) = 1 d . Let p 1 and p 2 be the probabilities on {0, 1} such that p 1 (1) = p 1 (0) = 1 2 and p 2 (1) =(1 + β)/2, p 2 (0)= (1 − β)/2. We define the probability P on Ω by From the sequences (Z k ) k∈Z , (η k ) k∈Z , (ξ k ) k∈Z , (ζ k ) k∈Z , we can construct the ERW (Y n ) n 0 just as in the first construction. We also define the sequence (Z k ) k∈Z as the sequence of "moves" of Z. More precisely, (Z k ) k∈Z is the unique sequence such that: Now, set D := {n ∈ Z such that Z (−∞,n) ∩ Z [n,+∞) = ∅} to be the set of cut times of Z and similarly letD be the set of cut times ofZ. The sequence of cut times of Z is then defined by induction: T 1 := inf{n > 0 such that n ∈ D}, T i+1 := inf{n > T i such that n ∈ D} , for i 1, T i−1 := sup{n < T i such that n ∈ D}, for i 1.
Proof. The idea of the proof comes from the paper of E. Bolthausen, A-S. Sznitman & O. Zeitouni [7]. First, we prove thatP is invariant underθ. Take any set A ⊂ W . Without loss of generality, suppose that A ⊂ (0 ∈ D), then we have: Next, we prove that for any set We will prove that θ 1 θ −1 B =θ −1 B. Using the ergodicity of (Ω, P, θ), it follows that P θ −1 B = 0 or 1, and So, to finish the proof we only need to prove that Firstly, we show that Lemma 2.4. For d 6, there exists v(β) > 0 such that a.s., lim n→∞ n −1 X n = v(β) and we have the following formula: Proof. See in [6], [13] about the law of large number for ERW, then v = lim n→∞ Xn n exists P a.s. This is therefore also trueP-a.s. On the other hand, it is proved in [7] that T 1 iŝ P-integrable for d 6 (withÊ|T 1 | = 1 P(0∈D) ). Using the ergodicity of (W,P,θ),P-a.s., and we also have Then, This finishes the proof of lemma 2.4.
Exactly in the same way, we can prove (see lemma 1.1 of [7] ) that that when f is a P-integrable function, A simple instance of this formula is to take

Girsanov transform
This section is devoted to the Girsanov transformation connecting P β and P 0 . We begin by introducing several σ-algebras.
. We get F n ⊂ G n . Moreover T is not a (F n )-stopping time, but is obviously a (G n )-stopping time, so that we can define the σ-algebra G T of the events prior to T . Remind that E j = (Y j+1 −Y j ).e 1 and define for n 0, and β ∈ [0, 1] Note that the law of Z does not depend on β, so that P β (B) = P 0 (B). Now by definition of the excited random walk, where ε n−1 = (y n − y n−1 ).e 1 . Then we get by induction that for any β ∈ [0, 1], where the last equality comes from the fact that A ∈ F Z −1 . Hence, We have just proved that for all A ∈ F Z −1 , y 1 , · · · , y n ∈ (Z d ) n , and B ∈ σ(Z n+k − Z n , k 0), The result follows since

Differentiability of the speed.
This section is devoted to the proof of point 1. in Theorem 1.1. We begin by giving another expression of the numerator in (4).
Proof. Observe that Hence, where the last equality follows from the integrability of T w.r.tP for d 6. Note that {0 ∈ D} and {T > j} belong to G j . Therefore, Thus, This proves the first equaIity. The second one follows from the fact that N T is G Tmeasurable, {0 ∈ D} ∈ G T , and Lemma 2.5.
We turn now to the derivative of the speed v(β) w.r.t β. We start from (4). Since T and 1 0∈D are σ(Z)-measurable, the denumerator in (4) does not depend on β, and Point 1. is then a consequence of the following lemma: Proof. Set We have Since N T dT and It follows from Lemma 2.9, whose statement and proof are postponed to the end of the section, thatÊ 0 (T 2 ) < +∞ for d 8. Fubini's theorem leads then to 1). To this end, we recall some general result about uniform integrability of positive random variables (see for instance Theorem 5 page 189 in Shiryaev [15]). Observe that For x 0 ∈ [0, 1], we have: It follows then from Lemma 2.8 that the family (T 2 M T (x)1 0∈D , x ∈ [0, 1]) is uniformly integrable. By (12), this is also true for the family {V T (x)} x→x 0 in a neighborhood of x 0 ∈ [0; 1[. Therefore, we obtain, Then, we get Therefore taking the derivative w.r.t β in (6) , we obtain

Monotonicity of the speed.
We focus now on the proof of point 2. in Theorem 1.1, and we use (9) to study the sign of the derivative of v(β). Since T 1, N T N 1 = d1 Z 0 =Z 1 . We remind the reader thatZ is defined as the walk Z when it moves, andD denotes the cut times ofZ. Then, We focus now on the second expectation in (9). It is equal to Note that for j k : (7)). (13) Hence, using the fact that From the computation above, we get It is proved in [7] thatÊT = 1/P(0 ∈ D) < ∞ for d 6, and that ET < +∞ when d 8.
Hence we can take f = T in (5).

Geometric(ε) random variables. We let
We have similarly . Therefore, in order to prove that sup d 8 ET < +∞, it is enough to prove that sup d 8 E(T ε ) < +∞ for some fixed ε. We call {T ε n } n∈Z the cut times of Z ε , T ε := T ε 1 and D ε is the set of cut times. Then P(0 ∈ D ε ) = εP(0 ∈D) converges to ε when d → ∞ and P(0 ∈ D ε ) is bounded by ε.
On the other hand, repeating the proof of (1.12) in [7], we obtain for k j = 1 + Lj (j 0, L 1, J 1 two fixed integers), Using the fact that P (Z ε n = 0) decreases with d 8 (we delay the proof of this fact at the end), we get Choosing a large enough γ depending on ε, and setting J = [γ log n], L = [ n 3J ] then where c depend only on ε. This implies that sup d 8 ET ε < ∞. Now, in order to finish the proof of Lemma 2.9, we have to prove that P[Z ε n = 0] decreases with d 2. Remark that for n odd P[Z ε n = 0] = 0, so we consider n even. Using characteristic functions, we obtain where we consider a sequence {Θ i } d−1 i=1 of i.i.d. random variables having uniform distribution U[−π, π]. Now, we consider the function f (x) = x n . n being even, f is a convex function on R and For a 1 , a 2 , ..., a d ∈ R, choose then we get Now, take a i = ε cos Θ i + 1 − ε for i = 1, · · · , d and take the expectation. It comes (23) It means that P[Z ε n = 0] decreases with d 2.
2.5 Differentiability of the speed at 0.
We are now interested in proving point 3 in Theorem 1.1, that is to compute the derivative at the critical point 0. By Lemma 2.6 we get Note that By lemma 2.8, {T 1 0∈D M T (β)} β is uniformly integrable in a neighborhood of 0. This is also true for {N T 1 0∈D M T (β)} β→0 since N T T . Therefore, we get On the other hand, with R n is the range of the simple symmetric random walk on Z d then Therefore This finishes the proof of Theorem 1.1.

ERW with several identical cookies
In this section, we consider a multi excited random walk m cookies (m-ERW) that is a particular case of m-ERWRE when the random environment β k (y) is deterministic and with constant value β ∈ [0; 1] for all 1 k m and y ∈ Z d . We denote with P m,β the law of m-ERW defined by: • If Y n has been visited less than m − 1 times before time n, then • If Y n has been visited more than m times before time n then for 1 i d .
We use the notation Y n / ∈ m or Y n / ∈ m {Y 0 ; Y 1 ; ...; Y n−1 } to mean that Y n has not been visited more than m times before time n. Set Then, as in case m = 1, we get and the following formulas for the speed and its derivative when d 8: where In order to prove the uniform convergence of (∂v/∂β)(m, β) as m goes to +∞, we use the following lemma, whose proof is given below: 3. X n (β) converges in probability to X(β), uniformly in β: for any ε > 0, lim n→+∞ sup β∈J P(|X n (β) − X(β)| > ε) = 0 .

Excited random walk in random environment.
Let β be a environment (β ∈ B). We denote by {Y n ∈ k } the event {Y n ∈ k } = {Y n has been visited k − 1 times before time n} .
We recall from the introduction that the "quenched" law P β of the ERW in the environment β, is defined by the following conditions: 3. on the event {Y n ∈ k } where k > m, then The "annealed" law is defined by P = Q ⊗ P β . Observe that the cut-times are still well defined. In section 4.1 and 4.2 we consider the case when there is only one cookie (m = 1) (1-ERWRE) and the environment is then denoted by β = {β(y)} y∈Z d .
In this case, we see that the "annealed" law of ERWRE is the law of an ERW. Proof. We have to prove that Then, we get the law of large numbers and the fact that the speed is increasing in β 0 = E Q [β(0)] from the results on the excited random walk.

m 1 and stationary random environment.
We focus now on the case of a stationary and ∆-exchangeable environment, and on the proof of Theorem 1.3.
The random environment being ∆-exchangeable, δ(β) has the same law has β. Hence, Using the stationarity of the environment, we get then Now, set U n = X Tn − X T n−1 for n 1. We have just seen that the sequence {U n } n 1 is stationary underP . Furthermore,Ê|U n | Ê T < ∞ for d 6. By Birkhoff's and Khinchin's theorem,P − as where F U is the σ-algebra generated by the invariant sets of the sequence {U n }. Therefore lim n→∞

Monotonicity of the speed.
Now, we prove that the expectation [1], that if there exists a monotone coupling of Q 1 and Q 2 , then there exists also a stationary monotone coupling of Q 1 and Q 2 , as soon as Q 1 and Q 2 are stationary.
Therefore we can suppose that {(β 1 , β 2 )(y)} y∈Z d is stationary. Set β t (y) = (1 − t)β 1 (y) + tβ 2 (y) for t ∈ [0; 1]. β t = {β t (y)} y∈Z d is a stationary environment . Consider Note that β t is not necessarily exchangeable, so that we can not assert that v(t) is the mean of the speed of the ERW in the random environment β t . Nevertheless, β 1 and β 2 being exchangeable, we get v(0) = v(Q 1 ), v(1) = v(Q 2 ), so that it is enough to prove that v(t) is increasing in t. First of all, we need the Girsanov's transform. Define where Y j / ∈ m denotes the event that Y j has not been visited more than m − 1 times before time j. As in section 2.2, and using the same notations, Remark that as in section 2.3, Hence, we get For d 8,Ê(T 2 ) < ∞. We can take the derivative in t, to get that We study now the sign of the derivative on the set of bounded environment, and from now on we assume that for i = 1, 2, β i (y) σ < 1 a.s. for any y of Z d . As in section 2.4, the first term is bounded from below by its first item corresponding to j = 0.
Now, we focus on the second term. Since using the notations of section 2.3, Therefore, we get It is similar to Lemma 2.9 to prove that Then, for d σd 0 , we have (∂/∂t)v(t) 0, wich implies that v(0) v(1) so that v(Q β 1 ) v(Q β 2 ) on the set of probability measures on bounded environment.
Choose σ = 10 d 0 , then we have the monotonicity for environments bounded by σ for any d 10.

m 1 and i.i.d random environment.
We consider now the case of an i.i.d environment with m cookies. In this situation, we can prove that the derivative is deterministic. To this end, we construct an ergodic dynamical system on which the m−ERWRE is defined. Let µ be the law of β = (β 1 , β 2 , ..., β m )(0) ∈ [0; 1] m .
We decompose the first product according to the value of the first visit to y i . 1 + 1 yn 1 / ∈ β k (y n 1 ) ε j 1 y j =yn 1 1 y j ∈ k .
The last equation comes from the independence of the random variables β k (y i ) for y i / ∈ . On the other hand, using the construction above, 1 + 1 y i / ∈ γ k (n 1 ) ε i 1 y j =y i 1 y i ∈ k .
Proof. The idea of proof comes from [7]. Firstly, we prove that θ is a measure-preserving transformation. Consider a measurable set A × B of W , where A ⊂ Γ, and B ⊂ (Z d−1 ) Z × {0, 1} Z × ({0, 1} m ) Z . We have that Now, we prove that θ is ergodic. Let A be a measurable subset of W, invariant under θ and ε > 0. There exists an integer m ε > 0 and a measurable subset A ε depending only on (w m ) |m| mε such that Then, for L 0, with |c ε | 2ε. For L > 2m ε , Because that p kn (γ) depend only on γ n , we prove that the sequence (γ n , I n , ζ n , ξ n ) n∈Z is the sequence of independent variables under P. Indeed, let i < j, i, j ∈ Z, we take two measurable sets Letting ε tend to 0, we have that P (A) = 0 or 1. Proof. The existence of the limit, the fact that it is deterministic and the expression of v(Q) for d 6 follow from the ergodicity of (Ŵ ,θ,P ), and the integrability of T w.r.tP for d 6.