Excited against the tide: A random walk with competing drifts

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1 Excite against the tie: A rano walk with copeting rifts arxiv: v1 [ath.pr] 28 Jan 2009 Mark Holes January 28, 2009 Abstract We stuy a rano walk that has a rift β to the right when locate at a previously unvisite vertex an a rift µ to the left otherwise. We prove that in high iensions, for every µ, the rift to the right is a strictly increasing an continuous function of β, an that there is precisely one value β 0 (µ,) for which the resulting spee is zero. 1 Introuction In this paper we stuy what ight be calle an excite rano walk with rift (ERWD), where the rano walker has a rift β in the positive irection of the first coponent each tie the walker visits a new site, an a rift µ in the negative irection of the first coponent on subsequent visits. Moels of this type with rift for a fixe finite nuber of visits to a site have been stuie by Zerner an others, usually in 1 iension, see for exaple [1, 2, 3, 17]. They are generalisations of the excite rano walk introuce in [4]. It is known that excite rano walk has ballistic behaviour when 2 in [4, 13, 14], while there is no ballistic behaviour (when β < 1) in one iension [7]. Laws of large nubers an central liit theores can be obtaine for 2 using renewal techniques (see for exaple [16], [17], [5]). Intuitively, the velocity appearing in the laws of large nubers shoul be increasing in the exciteent paraeter, β. This has been prove in iensions 9 [11] using a perturbative expansion evelope in [10], but the proble reains open for < 9. Using the sae expansion approach it is possible [12] to prove onotonicity for the first coorinate of the spee of rano walk in a partially rano i.i.. environent, in the special case where at each vertex, either the left or the right step is not available. We use the sae arguent in this paper, together with the law of large nubers provie by [6, Theore 1.4] to prove that for 12, for each fixe µ > 0 the spee in the irection of the positive first coorinate is continuous an strictly increasing in β. We give an easy coupling arguent showing that the spee is negative when β is sufficiently sall (epening on µ an ) an show that when 9 the spee is positive when β is sufficiently large. We conclue that when 12, for each µ 0 there is exactly one value β 0 (µ, ) of β for which the spee is zero. Departent of Statistics, The University of Aucklan, Private Bag 92019, Aucklan 1142, New Zealan. E-ail holes@stat.aucklan.ac.nz 1

2 Although we only consier the once-excite rano walk with all subsequent visits having a reverse rift, any of the results of this paper will reain vali uner appropriate relaxations of these conitions, e.g. ulti-excite rano walks with a reverse rift only after the kth visit to a site, or rano walks in site-percolation-like cookie environents. It ay also be possible (using the approach in [11, 12] an this paper) to prove onotonicity in p for the spee of excite rano walk in a site-percolation (paraeter p) cookie environent in high iensions. 1.1 Main results an organisation A nearest-neighbour rano walk path η is a sequence {η i } i=0 for which η i Z an η i1 η i is a nearest-neighbour of the origin for all i 0. For a general nearest-neighbour path η with η 0 = 0, we write p η i (x i, x i1 ) for the conitional probability that the walk steps fro η i = x i to x i1, given the history of the path η i = (η 0,...,η i ). We write ω n for the n-step path of excite rano walk with opposing rift (ERWD) an Q for the law of { ω n } n=0, i.e., for every n-step nearest-neighbour path η n, Q( ω n = η n ) = n1 i=0 p η i (η i, η i1 ), (1.1) where, for i = 0, p 0 (0, η 1 ) = (2) 1 (1 βe 1 η 1 ) is the probability that the first step is to η 1, an p η i (η i, η i x) = 1 e 1 x(βi {ηi / η i1 } µi {ηi η i1 }) 2 = 1 e 1 x((β µ)i {ηi / η i1 } µ)). (1.2) 2 Here e 1 = (1, 0,..., 0), an x y is the inner-prouct between x an y. It is always the case that our walks take nearest-neighbour steps, although this is not ae explicit in the notation. We are intereste in the velocity/spee/rift v 1 (, β, µ) of the first coorinate of the rano walk, efine by ω n [1] n v 1 = li n, (1.3) whenever this liit exists. Here x [1] enotes the first coponent of x Z. The ain result of this paper is the following theore, which is prove in Section 5. Theore 1.1 (Monotonicity of the spee). The velocity v 1 (, β, µ) for ERWD in iension is continuous for (β, µ) [0, 1] 2 when 6 an is strictly increasing in β [0, 1] for each µ [0, 1] when 12. In fact we show that the partial erivative of v 1 with respect to β exists an is positive when 12. It is also possible to prove onotonicity in β for β µ sufficiently sall when 8, an for all µ [0, 1] (resp. β [0, 1]) an β (resp. µ) sufficiently sall when 9. In orer to establish Theore 1.1 we first nee to establish that the liiting velocity v 1 actually exists. Although this can be achieve for sufficiently large using the expansion of [10], we can appeal to the ergoic etho of [6], for which it is sufficient that 6. The following result is establishe in [6, Theore 1.4] in the context of rano walk in a rano environent, however the sae arguent works for ERWD, an in fact uch ore generally (e.g. i.i.. rano ulticookie environent). Theore 1.2 (see Theore 1.4 of [6]). For each 6 an (µ, β) [0, 1] 2, there exists v(, µ, β) such that Q(v = li n n 1 ω n ) = 1. 2

3 Sign of velocity of ERW in 2 iensions with copeting rifts beta an u beta u Sign of velocity of ERW in 3 iensions with copeting rifts beta an u beta u Figure 1: Estiates of the sign (,-) of the velocity v 1 of ERWD in 2 an 3 iensions, base on 1000 siulations of 7000-step walks, one in R [15]. In [6] there exist well-behave rano cut ties (inepenent of the environent) with the property that the set of sites visite before an after each cut-tie are isjoint. Thus one can essentially lay own a new i.i.. environent at each of these cut-ties. The equivalent notion in the present context, woul be to replace all of the cookies at these cut ties since none of the sites with issing cookies at a cut tie will be visite again anyway. The following Lea is prove in Section 2 using an eleentary cookie replaceent coupling arguent an iplies that for 6 an each µ > 0 we can choose β sufficiently sall so that the velocity is negative. Lea 1.3. For each 2 an each µ > 0 there exist δ > 0 an β (µ,, δ) > 0 such that for every β < β, Q(li sup n ω [1] n n < δ) = 1. Lea 1.3 iplies transience of ω [1] n for 2 an β < β, whence renewal techniques (e.g. [5, 16, 17]) ay be use to prove the existence of the velocity. In high iensions the forula for the velocity in [10], shows that the velocity is positive for β sufficiently large. Lea 1.4. For each 9 an each µ > 0 there exists β (µ, ) > 0 such that for every β > β, Q(li n ω [1] n n > 0) = 1. Leas 1.3 an 1.4 together with Theore 1.1 then ieiately iply the following result. Corollary 1.5. For each 9 an each µ [0, 1] there exists a β 0 (µ, ) [0, 1] for which v = 0, i.e. Q(li n ω n n = 0) = 1. For each 12 an µ [0, 1] there is a unique such β 0 (µ, ). Apart fro Lea 1.3, all of the above results are prove in this paper in high iensions only. We expect the results to hol for all 2, with the exception that Lea 1.4 ay not hol for µ close to 1, when = 2. See Figure 1. 3

4 Conjecture 1.6. For all 2, (µ, β) [0, 1] 2 the velocity v 1 = li n n 1 ω [1] n exists an is onotone increasing in β for fixe µ an ecreasing in µ for fixe β respectively. For each 3 an µ [0, 1] there exists a unique β 0 (µ, ) [0, 1] such that v(, µ, β 0 ) = 0. It woul also be interesting to know whether the first coorinate of the walk is recurrent when β = β 0. Perhaps this can be verifie when 14 using [6, Theore 2.2]. The reainer of this paper is organise as follows. We prove Lea 1.3 in Section 2. In Section 3 we give a forula for the spee of ERWD when 6, an verify Lea 1.4. In Section 4 we consier the erivative of this forula, an estiates of this erivative are use in Section 5 to prove Theore Proof of Lea 1.3 Fix 2 an µ > 0. We prove that for β, δ > 0 sufficiently sall epening on µ an, li sup n n 1 ω [1] 3n < δ alost surely. We o this by coparing the ERWD with a siilar oel where all of the eaten cookies are replace at ties 3, N. Perhaps the results in this section can be iprove by replacing all of the cookies at ties k for k > 3. Let E Z (resp. E x Z \ {x}) enote a cookie configuration, i.e. a set of vertices in Z (resp. Z \{x}) at which cookies are present. Let A x = (1µ(βµ)I x ) an B x = (1µ(βµ)I x ), where I x is the inicator that there is a cookie at x. Then 0 A x, B x 2, A x B x = 2, A x is increasing in I x an B x is ecreasing in I x. Let Q E enote the law of an ERWD η with a given initial cookie configuration E Z, an write x y (resp. x y) if x is a neighbour of y (resp. x is graph istance 2 fro y). Then we have (2) 3 Q E (η [1] 3 = 3) =A 0 A e1 A 2e1 (2.1) (2) 3 Q E (η [1] 3 = 2) =A 0 A e1 (2 2) A 0 A u A u A ue1 (2.2) u e 1 :u e 1 ±e 1 u 0:u ±e 1 (2) 3 Q E (η [1] 3 = 1) =A 0 A e1 B 2e1 A 0 B e1 (1 µ) B 0 A e1 (1 µ) A 0 (2 2) 2 (2.3) A u (2 2) (2 2)(1 µ) A v (2.4) u 0:u ±e 1 v 0:v e 1 =0 (2) 3 Q E (η [1] 3 = 3) =B 0 B e1 B 2e1 (2.5) (2) 3 Q E (η [1] 3 = 2) =B 0 B e1 (2 2) B 0 B u B u B ue1 (2.6) u e 1 :u e 1 ±e 1 u 0:u ±e 1 (2) 3 Q E (η [1] 3 = 1) =B 0 B e1 A 2e1 B 0 A e1 (1 µ) A 0 B e1 (1 µ) B 0 (2 2) 2 (2.7) B u (2 2) (2 2)(1 µ) B v (2.8) u 0:u ±e 1 v 0:v e 1 =0 (2) 3 Q E (η [1] 3 = 0) =1 above ters (2.9) It is easy to see that (2.1) an (2.2) are non-ecreasing in I x for every x, for any fixe cookie environent E x Z \ {x}, an that Q E (η [1] 3 > 0) =(2.1)(2.2)(2.3)(2.4) is nonecreasing in I x for x / {0, e 1 } an any fixe E x. Siple arithetic enables us to write Q E (η [1] 3 > 0) =A 0 K 0 (E 0 ) K 0 (E0 ) = A e1 K 1 (E e 1 ) K 1 (Ee 1 ), (2.10) 4

5 where K 0, K 0 0 (resp. K 1, K 1 0) are constants that epen on E0 but not I 0 (resp. E e 1 but not I e1 ). This verifies that (2.1)(2.2)(2.3)(2.4) is nonecreasing in I x for each x, an each fixe E x. Siilarly, (2.5), (2.6) an (2.5)(2.6)(2.7)(2.8) are nonincreasing in I x for each x, E x. It follows that we can efine rano walks ω an ν on the sae probability space so that ω has the law of ERWD an where ν is the su of inepenent rano vectors X i, each with the istribution of ω 3 (corresponing to the situation above with I x = 1 for all x), an such that the following hol ω [1] 3n ω [1] 3(n1) = 3 ν[1] 3n ν [1] 3(n1) = 3, ω [1] 3n ω [1] 3(n1) = 2 ν[1] 3n ν [1] 3(n1) 2, ω [1] 3n ω [1] 3(n1) = 1 ν[1] 3n ν [1] 3(n1) 1, ν[1] 3n ν [1] 3(n1) = 3 ω[1] 3n ω [1] 3(n1) = 3, ν[1] 3n ν [1] 3(n1) = 2 ω[1] 3n ω [1] 3(n1) 2, ν[1] 3n ν [1] 3(n1) = 1 ω[1] 3n ω [1] 3(n1) 1. In particular, uner this coupling, ν [1] 3n ω [1] 3n is non-negative an non-ecreasing in n, alost surely. Since n 1 n i=1 X i E[ω 3 ] alost surely, an E[ω [1] 3 ] is continuous in β, it is enough to prove that E[ω [1] 3 ] < 0 when β = 0. In this case A x = B x = 1 for each x so we can use (2.1)-(2.8) to get (2) 3 E[ω [1] 3 ] = 4µ < 0 when β = 0 as require. Note that by keeping track of the β epenence in (2.1)-(2.8), one can get an explicit boun on how sall β shoul be as a function of µ an. 3 The forula for the spee We recall soe notation an results fro [10] an [11] giving an expression for the velocity v of the ERWD. If η an ω are two paths of length at least j an respectively an such that η j = ω 0, then the concatenation η j ω is efine by { ηi when 0 i j, ( η j ω ) i = (3.1) ω ij when j i j. Given η such that Q( ω = η ) > 0, we efine a conitional probability easure Q η on walks starting fro η by Q η ( ω n = x n ) Set j 0 = 0, an for N 1 let n1 i=0 For ERWD, N is non-zero precisely when ω (N) j N ω (N) j N 1, so that N = p η x i (x i, x i1 ) = Q( ω n = η x n ω = η ). (3.2) N = ( p ω(n1) j N1 1 ω(n) j N ω p (N) ) j N (ω (N) j N, ω (N) j N 1). (3.3) has alreay been visite by ω (N1) j N1 1 but not by (β µ)e 1 (ω (N) j N 1 ω (N) [ ] j N ) I (N) 2 I {ω j / ω (N1) N j ω (N) N1 j N 1 } {ω (N) j / ω (N) N j N 1 }, an N β µ 2 I I. (3.4) {ω (N) j N 1 =ω(n) j ±e 1 } {ω (N) N j ω (N1) N j } N1 5

6 Define A,N = {(j 1,...,j N ) Z N : N l=1 j l = N 1}, A N = A,N an π (N) (x, y) := j A,N ω (0) 1 ω (1) j 1 1 ω (N) j N 1 I {ω (N) j N =x,ω (N) j N 1 =y}p0 (0, ω (0) N 1 ) n=1 Note that this quantity epens on β, µ, an, as o the following j n1 n i n=0 ( p ω(n1) j n1 1 ω(n) in ω (n) i n, ω (n) i n1). (3.5) π (x, y) := π (N) (x, y), π(n) (x, y) := π (N) (x, y), an π (y) := π (N) (x, y). (3.6) x In [10], it was shown that if li n n=2 x xπ (x) exists an n 1 ω n v in probability, then v(β, µ, ) = xp 0 (0, x) xπ (x) x =2 x (3.7) = βe 1 (y x)π (N) (x, y). =2 x,y (3.8) Let P enote the law of siple syetric rano walk in iensions, beginning at the origin, an let D (x) = (2) 1 I { x =1} be the siple rano walk step istribution. For absolutely suable functions f, g on Z, efine the convolution of f an g by (f g)(x) = y f(y)g(x y). (3.9) Let f k (x) enote the k-fol convolution of f with itself, an let G (x) = k=0 D k Green s function for this rano walk. Then G k For i 0 an > 2(i 1) 1 efine For > 5 efine (x) < when > 2k. Define G k (x) enote the = G k (0). E i () = ( ) i1g (i1) 1 1. (3.10) 1 a = ( 1) 2G 2 1. (3.11) The following proposition is prove exactly as in the proof of [11, Proposition 3.2] with β replace by β µ in each of the bouns. Proposition 3.1 (Bouns on the expansion coefficients). For all N 1, π (N) x,y (x, y) (β µ) 1 E 0 () N = 1, (β µ) N 1 ( 1) 1 G 1 E 1 () (a ) N2 N > 1. (3.12) Since the spee is known to exist when 6 by the ergoic arguent of [6] (see Theore 1.2), the following corollary is an easy consequence of [10, Propositions 3.1 an 6.1] together with Proposition 3.1, an the fact that 2a 6 < 1 since G 2 5 < 25/12 [9]. 6

7 Corollary 3.2. For all 6 an β, µ [0, 1], v(β, µ, ) = βe 1 =2 x xπ (x) = βe 1 Proof of Lea 1.4. Corollary 3.2 iplies that for 6, v 1 β x,y =2 x,y (y x)π (N) (x, y). (3.13) π (N) (x, y). (3.14) Thus to prove the lea, it is sufficient to show that there exists δ > 0 inepenent of β such that x,y π (N) (x, y) < 1 δ. Fro Proposition 3.1 we have x,y (x, y) 2E G 1 E 1 () 0() 4 ( 1)(1 2a ), (3.15) π (N) an since the right han sie is inepenent of µ an β an ecreasing in, it is enough to prove that it is less than 1 when = 9. This result can easily be checke using the rigorous upper bouns G an G [8, 9]. 4 The ifferentiation step We follow the analysis in [11]. We fix µ [0, 1], ifferentiate the right han sie of (3.8) with respect to β, an prove that this erivative is positive for all β [0, 1], when 12. Letting ϕ (N) (x, y) = β π(n) (x, y) an assuing that the liit can be taken through the infinite sus, we then have Since ϕ (N) v 1 β (β, µ, ) = 1 (y 1 x 1 )ϕ (N) (x, y). (4.1) =2 x,y (x, y) 0 unless x y = 1, we have that v 1 β (β, ) 1 =2 x,y ϕ (N) (x, y). (4.2) We conclue that v 1 is positive when =2 β x,y ϕ (N) (x, y) < 1. To verify the exchange of liits in (4.1), it is sufficient to prove that x,y(y x)π (N) (x, y) is absolutely suable in an N an that =2 sup β,µ [0,1] x,y(y x)ϕ (N) (x, y) <. By Proposition 3.1 an the fact that y x = 1 for x, y nearest neighbours, the first conition hols provie that (β µ)a < 1. (4.3) In fact we shall see that this inequality for β = µ = 1 is also sufficient to also establish the secon conition. We now ientify ϕ (N) (x, y). 7

8 It follows fro (3.5) that ϕ (N) (x, y) = ϕ(n,1) (x, y) ϕ(n,2) (x, y) ϕ(n,3) (x, y), (4.4) where (by Leibniz rule), ϕ (N,1) (x, y), ϕ (N,2) (x, y) an ϕ (N,3) (x, y) arise fro ifferentiating p 0 (0, ω (0) 1 ), Nn=1 ( ) jn1 i n=0 p ω(n1) j n1 1 ω(n) in ω (n) i n, ω (n) i n1 an Nn=1 jn1 i n=0 N, respectively. Observe that if η = x l then β p η β (x l, x) = I ( ) {x l / η 1 } I{xxl =e 2 1 } I {xxl =e 1 }, (4.5) an hence, using I A I A C = I A C c we have ( p η β β (x l, x) p ωn η β (x l, x) ) = 1 2 I ( ) {x l / η 1,x l ω n1 } I{xxl =e 1 } I {xxl =e 1 }. (4.6) Clearly then ( p η β β (x l, x) p ωn η β (x l, x) ) 1 2 I ( ) {x l ω n1 \ η 1 } I{xxl =e 1 } I {xxl =e 1 }. (4.7) Let ρ (N) be obtaine by replacing p 0 (0, ω (0) 1 ) in (3.5) with (2) 1 I (0) {ω erivative) an by bouning N by N for all n = 1,..., N. For k = 1,...,N, let γ (N) k an by replacing j k 1 i k =0 p ω(k1) j k1 1 ω(k) i k 1 =±e 1} (a boun on its be obtaine fro (3.5) by bouning N by N for all n = 1,...,N ( ω (k) i k, ω (k) i k 1) with the following boun on its erivative j k 1 l=0 I {ω (k) l k 1 ω(k) l k =±e 1 } 2 j k 1 i k = 0 i k l ( p ω(k1) j k1 1 ω(k) i k ω (k) i k, ω (k) i k 1). (4.8) Siilarly, let χ (N) k be obtaine by replacing (k) j k 1 in (3.5) by (2) 1 I {ω (k) j k ω (k1) j k1 1 \ ω(k) j k 1 }I {ω (k) j k 1 ω(k) j k =±e 1 } (a boun on its erivative) an by bouning n for n k by n. Letting γ (N) = N k=1 γ (N) k an χ (N) = N k=1 χ (N) k, we obtain that ϕ (N,1) (x, y) ρ (N), ϕ (N,2) (x, y) γ (N), an x,y x,y x,y ϕ (N,3) (x, y) χ (N). (4.9) Define 2 ǫ() = ( 1) 4G 1G 3 1 E 1() ( 1) 2G 2 1, (4.10) The following proposition is prove exactly as in Corollary 4.5 of [11] except that we replace β with β µ in all of the bouns. Proposition 4.1 (Suary of bouns). For all β, µ [0, 1], an such that 2a < 1, ρ (N) 2E 0() 4G 1 E 1 () ( 1) ( 1 2a ) (4.11) χ (N) E 0 () 2G 1E 1 ()(2 2a ) ( 1) ( 1 2a ) 2 (4.12) γ (N) 2G 2 1 ( 1) ( 4ǫ() ) 16E 1()G 1 G ( ) 2. (4.13) 1 2a ( 1) 4 1 2a 8

9 5 Proof of Theore 1.1 Continuity of v 1 as a function of µ an β follows fro the fact that the forula for the spee is a su of functions that are continuous in µ an β an absolutely suable, uniforally in (µ, β) [0, 1] 2 by Proposition 3.1 an the fact that 2a < 1 for 6. For 6, the bouns of Proposition 4.1 hol. Fro (4.9) we have the require absolute suability conitions in the iscussion after (4.1), an in particular (4.1) hols for all β. To coplete the proof of the theore, it reains to show that the right han sie of (4.2) is no ore than 1. By (4.9) an Proposition 4.1, we have boune ties the right han sie of (4.2) by the su of the right han sies of the bouns in Proposition 4.1. Since these ters all involve siple rano walk Green s functions quantities, we will nee to use estiates of these quantities. By [9, Lea C.1], G n is onotone ecreasing in for each n 1, so that it suffices to show that the su of ters on the right han sies of (4.11), (4.12) an (4.13) is boune by 1 for = 12. For this we use the following rigorous Green s functions upper bouns [8, 9]: G 11 (0) , G 2 11 (0) , G 3(0) (5.1) 11 Putting in these values we get that the su of the right han sies of the bouns in Proposition 4.1 is at ost 0.847, for 12. Acknowlegeents. This work was supporte in part by a FRDF grant fro the University of Aucklan. We woul like to thank Takashi Hara for proviing the SRW Green s functions upper bouns. References [1] T. Antal an S. Rener. The excite rano walk in one iension. J. Phys. A: Math. Gen., 38: , [2] A.-L. Basevant an A. Singh. On the spee of a cookie rano walk. Probab. Theory Relat. Fiels., 141(3-4): , [3] A.-L. Basevant an A. Singh. Rate of growth of a transient cookie rano walk. Electr. Journ. Probab., 13:no. 26, , [4] I. Benjaini an D. B. Wilson. Excite rano walk. Electron. Co. Probab., 8:86 92 (electronic), [5] J. Bérar an A. Raírez. Central liit theore for excite rano walk in iension 2. Electr. Co. Probab., 12: , [6] E. Bolthausen, A.-S. Sznitan, an O. Zeitouni. Cut points an iffusive rano walks in rano environent. Ann. Inst. H. Poincaré Probab. Statist., 39(3): , [7] B. Davis. Brownian otion an rano walk perturbe at extrea. Probab. Theory Relat. Fiels., 113: ,

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