ON THE SPEED OF A PLANAR RANDOM WALK AVOIDING ITS PAST CONVEX HULL. By Martin P.W. Zerner

 Julian Floyd
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1 ON THE SPEED OF A PLANAR RANDOM WALK AVOIDING ITS PAST CONVEX HULL By Martn P.W. Zerner Abstract. We consder a random walk n R 2 whch takes steps unformly dstrbuted on the unt crcle centered around the walker s current poston but avods the convex hull of ts past postons. Ths model has been ntroduced and studed by Angel, Benjamn and Vrág. We show a large devaton estmate for the dstance of the walker from the orgn, whch mples that the walker has postve lm nf speed. 1. Introducton Angel, Benjamn and Vrág ntroduced and studed n [1] the followng model of a random walk (X n ) n 0 n R 2, whch they called the rancher. The walker starts at the orgn X 0 = 0. Suppose t has already taken n steps (n 0) and s currently at X n. Then ts next poston X n+1 s unformly dstrbuted on the unt crcle centered around X n but condtoned so that the straght lne segment X n, X n+1 from X n to X n+1 does not ntersect the convex hull K n of the past postons {X 0, X 1,..., X n }, see Fgure 1. Note that (X n ) n 0 s not Markovan snce n general one needs to know the whole hstory of the process n order to determne the transton probabltes for the next step. Ths makes ths model more dffcult to analyze than a Markovan random walk, a property t shares wth many other selfnteractng processes, see [1] and also [2] for references. We do not clam that ths model s of partcular mportance or that ts defnton s very natural. However, n the class of all selfnteractng processes t seems to be among the few models whch are at least partally analyzable and stll have some strkng propertes. Several such propertes have been conjectured by Angel, Benjamn and Vrág n [1]. Based on smulatons and heurstcs, the authors of [1] beleve 2000 Mathematcs Subject Classfcaton. 60K35. Secondary 60G50, 52A22. Key words: convex hull, large devatons, law of large numbers, random walk, self avodng, speed. Current address: Mathematsches Insttut, Unverstät Tübngen, Auf der Morgenstelle 10, D Tübngen, Germany. Emal: 1
2 2 RANDOM WALK AVOIDING ITS CONVEX HULL X2 X3 X =0 0 K 2 X 1 Fgure 1. Three steps of the walk. X 3 s unformly dstrbuted on the bold arc of the crcle wth radus 1, centered n X 2. e.g. that the drecton of the walk converges,.e. that X n / X n converges P a.s. to a random drecton, see [1, Conjecture 1]. Moreover, f we denote by w n as a measure for transversal fluctuatons the maxmal dstance of a pont n K n from the lne X 0, X n, then Angel, Benjamn and Vrág conjecture that w n grows lke n 3/4, see [1, Conjecture 2]. As far as results are concerned, the only major rgorous result whch has been proved so far for ths model to the best of our knowledge s that the walk has postve lm sup speed,.e. there s a constant c > 0 such that P a.s. lm sup X n /n > c as n, see [1, Theorem 1]. Here (Ω, F, P ) s the underlyng probablty space. The purpose of the present paper s to mprove ths result by showng the followng. Theorem 1. There s a constant c 1 > 0 such that 1 (1) lm sup n n log P [ X n c 1 n] < 0 and consequently, (2) lm nf n X n n c 1 P a.s.. In partcular, (2) proves [1, Conjecture 4]. We expect but were not able to prove that the speed lm X n /n exsts and s P a.s. constant, as conjectured n [1, Conjecture 5]. Let us now descrbe how the present artcle s organzed. The next secton ntroduces general notaton and gves a short overvew of the proof. In Secton 3 we ntroduce some sub and supermartngales, whch enable us n Secton 4 to bound exponental moments of the tme t takes the dameter of the convex hull K n to ncrease. From ths we deduce n Secton 5 estmates
3 RANDOM WALK AVOIDING ITS CONVEX HULL 3 γ := ext tme from +1 after τ σ +1 := ext tme from Kτ +n X τ Z,n,n +n Y D R,n X τ γd τ dτ X k() Fgure 2. General notaton for the dameter of K n smlar to the ones clamed n Theorem 1 for X n and show how ths mples Theorem Notaton and outlne of proof We denote by d n the dameter of K n. Snce (K n ) n s an ncreasng sequence of sets, (d n ) n s nondecreasng. The ladder tmes τ at whch the process (d n ) n 0 strctly ncreases are defned recursvely by τ 0 := 0 and τ +1 := nf{n > τ : d n > d τ } ( 0). It follows from [1] that P a.s. τ < for all 0. In Secton 4 we wll reprove ths fact by showng that the dfferences := τ +1 τ ( 0), between two successve fnte ladder tmes have some fnte exponental moments. Note that τ 1 = 1 snce d 0 = 0 and d 1 = 1 and observe that the τ s are stoppng tmes wth respect to the canoncal fltraton (F n ) n 0 generated by (X n ) n 0. Snce the dameter of a bounded convex set s the dstance between two of ts extremal ponts there s for all 1 wth τ < a (P a.s. unque) 0 k() < τ such that d τ = X τ X k(), see Fgure 2. For x R 2 and r > 0 we denote by B(x, r) the closed dsk wth center x and radus r. If τ < then σ +1 := nf{n 0 X n / B(X τ, d τ ) B(X k(), d τ )}
4 4 RANDOM WALK AVOIDING ITS CONVEX HULL boundary of lens X k() X τ X τ+n Drft Fgure 3. Idealzed pcture, assumng d τ =. Whenever the walker crosses a wedge for the frst tme, whch occurs at a postve rate, t has n the next step a postve drft towards the boundary of the lens. s the ext tme of the walk from the large lens shaped regon shown n Fgure 2, whch we shall refer to as the lens created at tme τ. Observe that K τ s contaned n the lens created at tme τ. Moreover, (3) τ +1 σ +1 snce f σ +1 <, X σ+1 has a dstance from ether X τ or X k() greater than d τ. We have now ntroduced enough notaton to be able to outlne the dea of the proof of Theorem 1. Outlne of Proof. To show that ( X n ) n s growng at a postve rate we shall frst show that (d n ) n s dong so. To ths end we shall bound exponental moments of the s. Ths s the heart of the proof and done as follows. Suppose that d n s already qute large and that we have just reached a new ladder tme τ, at whch the dameter has ncreased. Due to (3) the dameter can stay constant only as long as the walk s hdng n the lens created at tme τ. However, there are two mechansms, correspondng to Lemmas 4 and 5, whch ensure that the walk cannot hde n the lens for too long. The frst one, descrbed n Lemma 4, works whle the walk s stll nsde a small ball around X τ, see Fgure 2. The radus of ths ball s chosen to be proportonal to d τ. Ths stuaton s depcted n an dealzed form n Fgure 3. In ths regme the followng happens: Whenever the walk crosses for the frst tme any of the wedgeshaped lnes shown n Fgure 3, t has a drft away from X k() towards the boundary of the lens. To see ths consder the two boundary lnes of the convex hull whch are emanatng from X τ+n n Fgure 3. Suppose we are to draw n Fgure 3 the lne gong from X τ+n to the left. In the pcture ths lne has slope 0,.e. t s parallel to X τ, X k(), but
5 RANDOM WALK AVOIDING ITS CONVEX HULL 5 of course t could have a dfferent slope. However, ths slope cannot be much smaller than 0 because X τ+n s close to X τ but far away from X k(). On the other hand, the lne emanatng from X τ+n to the rght must have n Fgure 3 a slope of at least 1. Together, these two lnes create a drft downwards and more mportantly to the rght towards the boundary of the lens. (If the left lne has a postve slope then the drft to the rght s even stronger.) So every tme the walker crosses for the frst tme such a helpful wedgeshaped lne t wll be pushed towards the next wedgeshaped lne. If the lttle ball around X τ was nfntely large ths would result n crossngs of the wedgeshaped lnes at a postve rate, whch would propel the walk out of the lens very fast,.e. ths mechansm alone would gve a fnte exponental moment of, see Lemma 4. However, snce the ball s sze s fnte and proportonal to d τ, the walk can escape from ths lttle ball before leavng the lense wth a probablty exponentally small n d τ. (Ths wll happen a fnte number of tmes.) Whenever ths occurs, we rely on the second mechansm, see Lemma 5. The lne X τ, X k(), whch s part of the convex hull, creates a postve drft out of the lens. Snce the dameter of the lens s of order d τ t wll take the walker a tme of order d τ to reach the boundary of the lens. In fact we shall show, see Lemma 5, that t s exponentally costly for the walk to stay n the lens for a tme longer than the order of d τ. Ths s enough to make sure that the walk s not slowed down too much by occasonally resstng the frst mechansm and gettng lost n the lens. To make ths dea precse we need to ntroduce more notaton. The pont Y := X τ + X k() ( 1) 2 wll serve as the center of K τ and the radus R,n := X τ +n Y ( 1, n 0) s the dstance of X τ +n from ths center. The orthogonal projecton of X τ +n onto the straght lne passng through X τ and X k() wll be called Z,n ( 1, n 0). The dstance of X τ +n from ths lne s denoted by D,n := X τ +n Z,n ( 1, n 0). For the followng defntons we assume, n 1 and τ + n < τ +1. In partcular, due to (3), we assume that at tme τ + n the walk has not yet left the lens created at tme τ. Ths mples that Z,n X τ, X k() and that X τ and X k() are stll boundary ponts of K τ +n, as shown n Fgures 2 and 4. Hence f we start n X τ +n and follow the two boundary lne segments emanatng from X τ +n we wll eventually reach X τ and X k(). The boundary lne segment whose contnuaton leads frst to K τ and then to X k() s called
6 6 RANDOM WALK AVOIDING ITS CONVEX HULL X τ X τ Y ϕ 1 ϕ 2 s 1,,n s X τ +n 2,,n Z,n ψ 1 ψ 2 s 1,,n X τ +n s2,,n X k() X k() Fgure 4. General notaton s 1,,n, whle the other lne segment startng n X τ +n s denoted by s 2,,n, see Fgure 4. The angle between s j,,n and X τ +n, Y s called ϕ j,,n [0, π] (j = 1, 2), see the left part of Fgure 4. Smlarly, the angle between s j,,n and X τ +n, Z,n s denoted by ψ j,,n [0, π] (j = 1, 2), see the rght part of Fgure 4. Occasonally, we wll dropped the subscrpts and n from ϕ and ψ. Snce K τ +n s convex, (4) ϕ 1 + ϕ 2 = ψ 1 + ψ 2 π. Furthermore, ϕ 1 ψ 1 s one of the angles n a rght angled trangle, namely the trangle wth vertces X τ +n, Y and Z,n. Hence, (5) ϕ 1 ψ 1 = ϕ 2 ψ 2 π/2. 3. Some sub and supermartngales The followng result shows that for every 1, both (R,n ) n and (D,n ) n (1 n < τ +1 τ ) are submartngales. Lemma 2. For all, n 1, P a.s. on {τ + n < τ +1 }, (6) (7) and E[R,n+1 R,n F τ +n] sn ϕ 1,,n + sn ϕ 2,,n 2π E[D,n+1 D,n F τ +n] sn ψ 1,,n + sn ψ 2,,n 2π sn ϕ 1,,n 2π sn ψ 1,,n 2π 0, 0 (8) E[D,n+1 D,n + R,n+1 R,n F τ +n] c 2 for some constant c 2 > 0.
7 X τ Xτ RANDOM WALK AVOIDING ITS CONVEX HULL 7 Z,n +n Y X τ +n X τ Z,n Fgure 5. The expected ncrement of R,n s small n the left fgure and large n the rght fgure. For D,n t s the other way round. Fgure 5 shows examples n whch the expected ncrements of R,n and D,n are close to 0, thus explanng, why we are not able to bound n (6) and (7) these expected ncrements ndvdually away from 0. Note however, that n both stuaton depcted n Fgure 5, f the expected ncrement of R,n or of D,n s small then the expected ncrement of the other quantty s large. Ths confrms that the expected ncrements of R,n and D,n cannot both be small at the same tme, see (8). Proof of Lemma 2. We fx, n 1 and drop them as subscrpts of ϕ j,,n and ψ j,,n (j = 1, 2). Then the followng statements hold on the event {τ + n < τ +1 }. Consder the angle between Y, X τ +n and X τ +n, X τ +n+1 whch ncludes s 2,,n. Ths angle s chosen unformly at random from the nterval [ϕ 2, 2π ϕ 1 ]. Hence we get by a change of bass argument usng Y, X τ +n for the frst bass vector, E[R,n+1 F τ +n] = 1 (2π ϕ 1 ) ϕ 2 1 2π ϕ 1 ϕ 2 2π ϕ1 ϕ 2 2π ϕ1 ϕ 2 1 R,n + 2π ϕ 1 ϕ 2 (R,n, 0) (cos ϕ, sn ϕ) dϕ R,n cos ϕ dϕ 2π ϕ1 ϕ 2 cos ϕ dϕ = R,n + sn ϕ 1 + sn ϕ 2 R,n + sn ϕ 1 + sn ϕ 2, 2π ϕ 1 ϕ 2 2π whch shows (6). Smlarly, (7) follows from E[D,n+1 F τ +n] = 1 (2π ψ 1 ) ψ 2 2π ψ1 ψ 2 D,n cos ψ dψ. For the proof of (8) we assume wthout loss of generalty ϕ 1 π/2. Indeed, otherwse ϕ 2 π/2 because of ϕ 1 + ϕ 2 π, see (4), and n the followng proof one only has to replace the subscrpt 1 by the subscrpt 2 and swap... Y......
8 8 RANDOM WALK AVOIDING ITS CONVEX HULL X τ and X k(). By (6) and (7), E[D,n+1 D,n + R,n+1 R,n F τ +n] sn ϕ 1 + sn ψ 1 (9). 2π We wll show that the rght sde of (9) s always greater than c 2 := (4π 2 ) 1. Assume that t s less than c 2. Then (10) sn ϕ 1, sn ψ 1 (2π) 1 and hence ϕ 1 (π/2) sn ϕ 1 1/4 by concavty of sn on [0, π/2]. Smlarly, (10) mples that ether ψ 1 1/4 or π ψ 1 1/4. Due to ϕ 1 ψ 1 π/2, see (5), the latter case s mpossble. Therefore, (11) ϕ 1 ψ 1 max{ ϕ 1, ψ 1 } 1/4. The angle α [0, π/2] between Z,n, X τ +n and X τ +n, X τ s less than or equal to ψ 1. Consequently, (12) sn ψ 1 sn α = X τ Z,n X τ X τ +n X τ Y Y Z,n d τ. However, snce Y, X τ +n K τ +n, (13) R,n = Y X τ +n d τ +n = d τ. Therefore, (12) sn ψ 1 d τ /2 Y Z,n d τ Y X τ +n = 1 2 sn ϕ 1 ψ 1 whch contradcts (10). We fx the constants (14) β := 1 + 4π 8 > 30 and γ := 1 2β < (11) = 1 4, Whenever τ < we denote the frst ext tme after τ from B(X τ, γd τ ) by γ +1 := nf{n > τ : X n X τ > γd τ } ( ), see Fgure 2. If 1 and n 0 then we shall call n good for f n = 0 or f (15) τ + n < τ +1 γ +1 and E[R,n+1 R,n F τ +n] 1 π 8 P a.s.. Ths means, n 1 s good for f at tme τ + n the walker has not yet left the ntersecton of the small ball around X τ and the lens shown n Fgure 2 and, roughly speakng, feels a substantal centrfugal force pushng t away from the center Y. Good tmes help the walker to leave the lens shortly after τ and closely to the pont X τ. Next we ntroduce an Azuma type nequalty for a certan famly of supermartngales, whch wll help make ths dea more precse.
9 RANDOM WALK AVOIDING ITS CONVEX HULL 9 Lemma 3. There are constants c 3 > 0, c 4 > 0 and 1 c 5 < such that P a.s. for all 1, n 0, where M,n := 1{τ + n < τ +1 } exp E [M,n F τ ] c 5 exp( c 4 n), ( c 3 n ) W,m m=1 and W,m := D,m 1 D,m + β(r,m 1 R,m ) + 4 1{m 1 s good for }. Proof. Fx 1. Frstly, we shall prove that for sutable c 4 > 0, (16) E [M,n+1 F τ +n] exp( c 4 )M,n P a.s. for all n 1, thus showng that (M,n ) n 1 s an exponentally fast decreasng submartngale. Snce W,m s F τ +mmeasurable (m 1), we have E [M,n+1 F τ +n] = M,n E[exp(c 3 W,n+1 ), τ + n + 1 < τ +1 F τ +n] M,n E[exp(c 3 W,n+1 ) F τ +n]. Therefore s suffces to show for the proof of (16) that on the event {τ + n < τ +1 }, E[exp(c 3 W,n+1 ) F τ +n] < 1 for some sutable c 3 > 0. However, snce the W,m s (m 1) are unformly bounded by a constant ths follows from the fact that on the event {τ + n < τ +1 }, E [W,n+1 F τ +n] = E[D,n D,n+1 + R,n R,n+1 F τ +n] ( + 4 π ) 8E[R,n R,n+1 F τ +n] + 1{n s good for } c 2 < 0 P a.s. by vrtue of defnton (15) and Lemma 2 (6), (8). Usng (16) for nducton over n we obtan by the tower property, E [M,n F τ ] exp( c 4 (n 1))E [M,1 F τ ] P a.s. for all n 1. Snce M,1 s bounded by a constant ths fnshes the proof. 4. Exponental moments of τ +1 τ. The next two lemmas provde tools needed to bound n Proposton 6 the tmes durng whch the dameter does not ncrease. The frst lemma shows that the walk cannot spend too much tme nsde the lens and the small ball around X τ shown n Fgure 2. Lemma 4. For all 1, n 0, P a.s. P [τ + n < τ +1 γ +1 F τ ] c 5 exp( c 4 n), where c 4 and c 5 are as n Lemma 3. The second lemma gves a frst crude upper bound for.
10 10 RANDOM WALK AVOIDING ITS CONVEX HULL Lemma 5. There s a fnte constant c 6 such that for all 1, n 0, P a.s. where c 4 and c 5 are as n Lemma 3. P [ > n F τ ] c 5 exp(c 4 (c 6 d τ n)), In the proof of both lemmas we wll use the fact that the sum n the defnton of M,n s n part telescopc,.e. n n (17) W,m = D,n + β(r,0 R,n ) + 4 1{m 1 s good for } m=1 snce D,0 = 0. Proof of Lemma 4. For n = 0 the statement s true snce c 5 1. Fx n 1. The statement follows from Lemma 3 and (17) once we have shown that on the event {τ + n < τ +1 γ +1 }, the rght hand sde of (17) s nonnegatve. Frst we wll show that on {τ + n < τ +1 γ +1 }, m=1 (18) D,n + β(r,n R,0 ) 2(D,n X τ Z,n ). Ths s done by brute force. For abbrevaton we set d := d τ, y := D,n and x := X τ Z,n and note that on {τ + n < τ +1 γ +1 } we have x, y [0, γd]. Observe that x and y play the role of cartesan coordnates of X τ +n, see Fgure 6. Usng R,0 = d/2 and R,n = (d/2 x) 2 + y 2 we see that (18) s equvalent to β (d/2 x) 2 + y 2 y 2x + βd/2. Both sdes of ths nequalty are nonnegatve snce x s less than γd, whch s tny compared to βd. Takng the square and rearrangng shows that (18) s equvalent to (19) x(4x 4y 2βd β 2 x + β 2 d) + y(y + βd β 2 y) 0. Snce x, y [0, γd] and βγ = 1/2, see (14), the terms β 2 d n the frst bracket and βd n the second bracket are the domnant terms, respectvely, whch shows that (19) and thus (18) holds. For the proof that the rght hand sde of (17) s ndeed nonnegatve on {τ + n < τ +1 γ +1 } t therefore suffces to show that, n 1 (20) D,n X τ Z,n 2 1{j s good for }. Both D,j and X τ Z,j can ncrease by at most 1 f j ncreases by 1. Therefore, the left hand sde of (20) s less than or equal to 2(#J,n ) where J,n := {0 j < n 0 m < j : D,m X τ Z,m D,j X τ Z,j }. j=0
11 RANDOM WALK AVOIDING ITS CONVEX HULL X k()... Y K τ K τ +3 ϕ 1 ψ1 Z X τ +3,3 x X τ y γd τ Fgure 6. More realstc and detaled than Fgure 3. The darkly shaded convex hull K τ at tme τ has been enlarged after three steps by the lghtly shaded part. j = 3 s good for snce t satsfes the suffcent crteron ψ 1,,3 π/4, see (22), whch corresponds to the fact that the dashed lne, whch ntersects the horzontal axs at an angle of π/4, does not ntersect Kτ o +3. j = 1 s also good for for the same reason, whle j = 2 mght be good for but fals to satsfy the suffcent condton (22), snce the correspondng dashed lne startng n X τ +2 would have ntersected Kτ o +2. Hence t suffces to show that the elements of J,n are good for. Note that j = 0 J,n s good for by defnton of beng good. So fx 1 j J,n. By Lemma 2 (6) t s enough to show that sn ϕ 1,,j 2 1/2, that s (21) ϕ 1,,j [π/4, 3π/4]. On the one hand, ϕ 1 ψ 1 s close to π/2, as can be seen n Fgure 6. More precsely, sn(ϕ 1 ψ 1 ) = Y Z,j Y X τ +j d τ /2 γd τ d τ /2 + γd τ Y X τ X τ Z,j Y X τ + X τ X τ +j = 1 2γ 1 + 2γ 1 2 = sn π 4. Snce 0 ψ 1 ϕ 1 ths mples ϕ 1 π/4, thus provng the frst part of (21). On the other hand, ϕ 1 ψ 1 π/2, see (5). Hence all that remans to be
12 12 RANDOM WALK AVOIDING ITS CONVEX HULL shown for the completon of the proof of (21) s that (22) ψ 1,,j π/4. Consder the half lne (dashed n Fgure 6) startng at X τ +j whch ncludes an angle of π/4 wth X τ +j, Z,j that contans s 1,,j. We clam that ths lne does not ntersect Kτ o +j. Ths would mply (22). To prove ths clam observe that for any c > 0 the set of possble values for X τ +m wth D,m X τ Z,m = c s a lne parallel to the half lne just descrbed. Snce j J,n the walker dd not cross between tme τ and tme τ + j 1 the dashed lne passng through X τ +j. Consequently, t suffces to show that the dashed lne does not ntersect Kτ o. If t dd ntersect Kτ o then ths would force the walker on ts way from X τ to X τ +j to cross the dashed lne strctly before tme τ + j, whch s mpossble as we just saw. Proof of Lemma 5. Agan, the statement s true for n = 0 snce c 5 1. For n 1, by the Pythagorean theorem and (13), D,n R,n d τ on the event {τ + n < τ +1 }. Thus on ths event the rght hand sde of (17) s greater than or equal to d τ + β(0 d τ ) + 0. Hence due to Lemma 3 and (17), P a.s. for all n 1, exp( c 3 (1 + β)d τ )P [τ + n < τ +1 F τ ] c 5 exp( c 4 n), whch s equvalent to the clam of the lemma wth c 6 := c 3 (1 + β)/c 4. The followng result s stronger than Lemma 5. Proposton 6. τ < P a.s. for all 0. Moreover, there are postve constants c 7, c 8 and fnte constants c 9 and c 10 1 such that for all 0 and n 0, P a.s., (23) (24) P [ n F τ ] c 9 exp( c 7 n) and E [exp(c 8 ) F τ ] c 10. Proof. (24) s an mmedate consequence of (23). For the proof of (23) choose (25) c 7 := γc 4 c 6 + γ and c 9 := c 5 e c 4 wth c 4, c 5 and c 6 accordng to Lemma 5. We only need to show that (23) holds for all 1 wth τ <. Indeed, the case = 0 s trval snce τ 0 = 0, τ 1 = 1 and hence 0 = 1. Moreover, snce (23) mples < P a.s., we then have τ = < as well. Fx 1 and n 0. We dstngush three cases: {n < γd τ }, {γd τ n < (c 6 + γ)d τ } and {(c 6 + γ)d τ n}.
13 RANDOM WALK AVOIDING ITS CONVEX HULL 13 Note that these events are elements of F τ and partton Ω. By defnton of γ +1, (26) γ +1 τ + γd τ snce the walker takes steps of length one. Therefore, on {n < γd τ }, P [ > n F τ ] On {γd τ n < (c 6 + γ)d τ }, P [ > n F τ ] P [τ + γd τ < τ +1 F τ ] Lemma 4 c 5 exp( c 4 (γ d τ 1)) (25) Fnally, on {(c 6 + γ)d τ n}, P [ > n F τ ] (26) = P [τ + n < τ +1 γ +1 F τ ] Lemma 4 c 5 exp( c 4 n) (25) c 9 exp( c 7 n). (26) P [τ + γd τ 1 < τ +1 γ +1 F τ ] c 9 exp( c 7 n). Lemma 5 c 5 exp(c 4 (c 6 d τ n)) (25) where the last nequalty can easly be checked. c 9 exp( c 7 n), 5. Lnear growth of the dameter and proof of Theorem 1 The followng result (wth m = 0) mples that (d n ) n has a postve lm nf speed. Lemma 7. There are constants c 11 > 0 and c 12 < such that for all 0 m n, E[exp(d m d n )] c 12 (n + 1) exp(c 11 (m n)). For the proof of ths lemma and of Theorem 1 we need the followng defnton: Gven n 0 let n := sup{ 0 τ n}. Note that (27) d τn = d n and n τ n n < τ n+1. Proof of Lemma 7. The case n = m s trval. So let 0 m < n and set (28) f(m, n) := n m > 0 and g(m, n) := c 8f(m, n) > 0, 2n 2 ln c 10 where c 8 and c 10 are accordng to Proposton 6. A smple unon bound yelds E[e dm dn ] I + II + III, where I := P [τ m+1 m f(m, n)n], II := P [τ m+1 m < f(m, n)n, n < m + g(m, n)n ] and III := E [exp (d m d n ), n m + g(m, n)n ],
14 14 RANDOM WALK AVOIDING ITS CONVEX HULL m n τ m m τ +1 m τ n n τ n +1 Fgure 7. see also Fgure 7. Here term I corresponds to the stuaton n whch after tme m the dameter does not ncrease for an untypcal long whle. Term II handles the case n whch the dameter does ncrease shortly after tme m, as t should, but not often enough n the remanng tme untl n. The thrd term III consders the orgnal random varable on the typcal event that the number of tmes at whch the dameter ncreases s at least proportonal to n wth a constant of proportonalty not too small. It suffces to show that each of these three terms decays as n n the way clamed n Lemma 7 wth constants c 11 and c 12 ndependent of n and m. As for the frst term, I P [ m f(m, n)n ] (23) c 9 (m + 1)e c 7 f(m,n)n (27) = m P [ m =, f(m, n)n ] =0 c 9 (n + 1)e c 7(m n)/2, whch s an upper bound lke the one requested n Lemma 7. The second term s estmated as follows. II (27) = P [ ] τ m+1 < m + f(m, n)n, n < τ n+1 τ m+ g(m,n)n (28) P [ τ m+ g(m,n)n τ m+1 n m f(m, n)n = f(m, n)n ] E [ exp ( ( c 8 τm+ g(m,n)n τ m+1 f(m, n)n ))] (29) = e c 8f(m,n)n k 1 = e c 8f(m,n)n E k 1 E [ exp ( c 8 ( τk+ g(m,n)n 1 τ k )), m + 1 = k ] g(m,n)n 2 =0 exp (c 8 k+ ), m + 1 = k. Note that { m + 1 = k} s the event that τ k s the frst tme after tme m at whch the dameter ncreases. Therefore, (30) { m + 1 = k} F τk. Moreover, the ncrements k+ are measurable wth respect to F τk++1. Consequently, by condtonng n (29) on F τk+ g(m,n)n 2 and applyng Proposton
15 RANDOM WALK AVOIDING ITS CONVEX HULL 15 1/2 X k() 1 Xτ X τ+u =X τ,0 τ +1 K τ Fgure 8. The event A occurs f the frst tral pont sampled les on the bold arc. 6 (24) wth = k + g(m, n)n 2 we conclude g(m,n)n 3 II e c8f(m,n)n c 10 E exp (c 8 ( k+ )), m + 1 = k. k 1 =0 Contnung n ths way we obtan by nducton after g(m, n)n 1 steps, II e c8f(m,n)n c g(m,n)n 1 10 e c8f(m,n)n c g(m,n)n 10 (28) = e (c 8/4)(m n), whch s agan of the form requred n Lemma 7. In order demonstrate that also the thrd term III behaves properly we wll show that the ncrements d τ+1 d τ, 1, have a unformly postve chance of beng larger than a fxed constant, say 1/2, ndependently of the past. To ths end, we may assume that the process (X n ) n s generated n the followng way: There are..d. random varables U n,k, n 0, k 0, unformly dstrbuted on the unt crcle centered n 0 such that X n+1 = X n + U n,k, where k s the smallest nteger such that X n, X n + U n,k does not ntersect K n. Then for any 1, by defnton of τ, (31) {d τ+1 d τ + 1/2} {d τ +1 d τ + 1/2} { U τ,0 Xτ X k() 1 } =: A. d τ 2 The reason for ncluson (31) s llustrated n Fgure 8: If A occurs then K τ and X τ + U τ,0 are separated by a slab of wdth 1/2. In partcular, the lne connectng X τ and X τ + U τ,0 does not ntersect K τ. Hence τ +1 = τ + 1 and X τ+1 = X τ +1 = X τ + U τ,0. Therefore, d τ +1 X τ +1 X k() X τ X k() + 1/2 = d τ + 1/2.
16 16 RANDOM WALK AVOIDING ITS CONVEX HULL It follows from (31) that for all 1 j 1 j 2, (32) j 2 1 j 2 1 d τj2 d τj1 1 1{d τ+1 d τ + 1/2} 1 1{A }. 2 2 =j 1 =j 1 Ths estmate wll be useful snce the random varables (33) 1{A } ( 1) are..d. wth P [A ] > 0. Indeed, let F n (n 0) be the σfeld generated by U m,k, 0 m < n, 0 k. Because of F n F n we have A j F τ for all 1 j <. Moreover, snce the unform dstrbuton on the unt crcle s nvarant under rotatons, (34) A s ndependent of F τ ( 1) and P [A Fτ ] = P [A ] s just the length of the bold crcle segment shown n Fgure 8 dvded by 2π. Ths mples (33). Now we estmate III by (27) III = E [ exp ( ) d τm d τn, n m + g(m, n)n ] E [ exp ( ) d τm+1 d τn, n ( m + 1) + g(m, n)n 1 ] [ ) ] E exp (d τk d τk+ g(m,n)n 1, m + 1 = k k 1 (35) (32) k 1 E exp 1 2 k+ g(m,n)n 2 =k 1{A }, m + 1 = k. As seen n (30), { m + 1 = k} F τk F τk. Therefore, after condtonng n (35) on F τk, we see wth the help of (34) for k and (33) that the rght hand sde of (35) equals [ ( E exp 1 g(m,n)n 1 1)] 2 A, whch decays as requred n Lemma 7, see (28). Lemma 7 drectly mples a weaker verson of Theorem 1 n whch X n s replaced by d n. For the full statement we need the followng addtonal argument. Proof of Theorem 1. (2) follows from (1) by the BorelCantell lemma. For the proof of (1) pck c 11 and c 12 accordng to Lemma 7 and choose c 13 > 0 and c 1 > 0 small enough such that (36) 2c 13 c 11 < 0 and 2c 1 c 11 (c 13 c 1 ) < 0.
17 RANDOM WALK AVOIDING ITS CONVEX HULL 17 We denote by M n := max{ X m m n} the walker s maxmal dstance from the orgn by tme n. Note that M n and d n are related va (37) M n d n 2M n for all n 0 because of X 0 = 0. By a unon bound for any n 0, (38) P [ X n c 1 n] P [ n c 1 n] + P [M n c 13 n] + P [B n ], where B n := { n < c 1 n, M n > c 13 n, X n c 1 n}. It suffces to show that each one of the three terms on the rght hand sde of (38) decays exponentally fast n n. As for the frst term, n P [ n c 1 n] (27) = P [ n =, c 1 n ] (23) c 9 (n + 1)e c 7c 1 n, =0 whch decays exponentally fast n n ndeed. So does the second term n (38) snce by Chebyshev s nequalty, P [M n c 13 n] (37) P [d n 2c 13 n] e 2c 13n E[e dn ] Lemma 7 c 12 (n + 1)e (2c 13 c 11 )n, whch decays exponentally fast due to the choce of c 13 n (36). Fnally, we are gong to bound the thrd term n (38), P [B n ]. Defne the ladder tmes (µ j ) j of the process (M n ) n 0 recursvely by µ 0 := 0 and µ j+1 := nf{n > µ j M n > M µj }. In analogy to ( n ) n for (τ ) we defne for (µ j ) j the ncreasng sequence (j n ) n by j n := sup{j 0 µ j n} and note that µ jn n < µ jn+1 and M n = X µjn. Hence on the event B n, X µjn X n X µjn X n = M n X n (c 13 c 1 )n. Snce the walker takes steps of length one, ths mples n µ jn (c 13 c 1 )n and therefore, on the event B n, (39) µ jn (1 c 13 + c 1 )n. On the other hand, on B n, (40) d n (27) = d τn = X τn X k(n) X n + X n X τn + X k(n) c 1 n + n + M n, where we used agan the fact that the steps have length one to estmate the second term and k( n ) n n to bound the thrd term. Therefore (40) can be estmated on B n from above by (37) (39) c 1 n + c 1 n + M µjn 2c 1 n + d µjn 2c 1 n + d (1 c13 +c 1 )n.
18 18 RANDOM WALK AVOIDING ITS CONVEX HULL Consequently, by Chebyshev s nequalty and Lemma 7, P [B n ] P [d n d (1 c13 +c 1 )n 2c 1 n] c 12 (n + 1)e (2c 1 c 11 (c 13 c 1 ))n, whch decays exponentally n n due to the choce of c 13 and c 1 n (36). References [1] O. Angel, I. Benjamn and B. Vrág. Random walks that avod ther past convex hull. Elect. Comm. n Probab. 8 (2003) [2] I. Benjamn and D.B. Wlson. Excted random walk. Elect. Comm. n Probab. 8 (2003) Department of Mathematcs Stanford Unversty Stanford, CA 94305, U.S.A.
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