Three favorite sites occurs infinitely often for one-dimensional simple random walk
SSubmitted to the Annals of Probability arXiv:
THREE FAVORITE SITES OCCURS INFINITELY OFTENFOR ONE-DIMENSIONAL SIMPLE RANDOM WALK
By Jian Ding ∗ and Jianfei Shen ∗ University of Chicago
For a one-dimensional simple random walk ( S t ), for each time t we say a site x is a favorite site if it has the maximal local time. Inthis paper, we show that with probability 1 three favorite sites occursinfinitely often. Our work is inspired by T´oth (2001), and disprovesa conjecture of Erd¨os and R´ev´esz (1984) and of T´oth (2001).
1. Introduction.
Let S t , t ∈ N be a one-dimensional simple ran-dom walk with S = 0. We define the local time at x by time t to be L ( t, x ) = { < k ≤ t : S k = x } . At time t , we say x is a favorite site ifit has the maximal local time, i.e., L ( t, x ) = max y L ( t, y ), and we say that three favorite sites occurs if there are exactly three sites which achieve themaximal local time. Our main result states that Theorem . For one-dimensional simple random walk, with probabil-ity 1 three favorite sites occurs infinitely often.
Theorem 1.1 complements the result in [24] which showed that there areno more than three favorite sites eventually, and disproves a conjecture ofErd¨os and R´ev´esz [14, 15, 16] and of [24]. Previous to [24], it was shown in[25] that eventually there are no more than three favorite edges.Besides the number of favorite sites, the asymptotic behavior of favoritesites have been much studied (see [23] for an overview): at time n as n → ∞ ,it was shown in [3, 20] that the distance between the favorite sites and theorigin in the infimum limit sense is about √ n/ poly(log n ) while in the supre-mum limit sense is about √ n log log n ; it was proved in [8] that the distancebetween the edge of the range of random walk and the set of favorites in-creases as fast as √ n/ (log log n ) / ; in [7] the jump size for the position offavorite site was studied and shown to be as large as √ n log log n ; a numberof other papers [12, 2, 21, 17, 13, 18, 6] studied similar questions in broadercontexts including symmetric stable processes, random walks on randomenvironments and so on. ∗ Partially supported by NSF grant DMS-1455049 and an Alfred Sloan fellowship.
MSC 2010 subject classifications:
Keywords and phrases: random walk, favorite sites. a r X i v : . [ m a t h . P R ] O c t J. DING AND J. SHEN
In two dimensions and higher, favorite sites for simple random walks havebeen intensively studied where some intriguing fractal structure arise, see,e.g., [10, 9, 1, 22]. Such fractal structure also plays a central role in thestudy of cover times for random walks, see, e.g., [11, 5, 4]. We refrain froman extensive discussion on the literature on this topic as the mathematicalconnection to the concrete problem considered in the present article is lim-ited. That being said, we remark that analogous questions on the number offavorite sites in two dimensions and higher are of interest for future research,which we expect to be more closely related to the literature mentioned inthis paragraph as well as references therein.Our proof is inspired by [24], which in turn was inspired by [25]. Following[24], we define the number of upcrossings and downcrossings at x by the time t to be U ( t, x ) = { < k ≤ t : S k = x, S k − = x − } ,D ( t, x ) = { < k ≤ t : S k = x, S k − = x + 1 } . It is elementary to check that (see, e.g, [24, Equation (1.6)]) L ( t, x ) = D ( t, x ) + D ( t, x −
1) + { 1, let f ( r ) be the (possibly infinite) number of times when thecurrently occupied site is one of the r favorites: f ( r ) = { t ≥ S t ∈ K ( t ) , K ( t ) = r } . We remark that one of the main conceptual contributions in [24, 25] is theintroduction of this function f ( r ). Effectively, f ( r ) counts the clusters ofinstances for r favorite sites; it is plausible that after the random walk leavesone of the favorite sites, within a non-negligible (random) number of stepsthose r favorite sites will remain favorite sites. Therefore, the expectationof f ( r ) is significantly smaller than the expected number of t at which r favorite sites occurs, and in fact it was shown in [24] that E f ( r ) < ∞ forall r ≥ 4. It was then conjectured in [24] that f (3) < ∞ with probability AVORITE SITES OF RANDOM WALKS 1, even though from the computations in [24] it was clear that E f (3) = ∞ .In the current article, we will show, using the idea of counting clusters in[24], that the correlation becomes so small that the first moment dictatesthe behavior. That is to say, we will show that(1.2) f (3) = ∞ with probability 1 , which then yields Theorem 1.1.The rest of the paper is organized as follows: in Section 2 we will set upthe framework of our proof following [24]; in Section 3 we first show that f (3) = ∞ with positive probability and then prove (1.2) by demonstrating a0-1 law. We emphasize that the first moment computation in Subsection 3.1follows from arguments in [24], and the main novelty of our work is on thesecond moment computation in Subsection 3.2. Acknowledgement. We thank Yueyun Hu and Zhan Shi for introducingthe problem on favorite sites and for interesting discussions, and we thankSteve Lalley and B´alint T´oth for many helpful discussions and useful com-ments for an early version of the manuscript. 2. Preliminaries. In this section, we recall the framework of [24] withsuitable adaption to our setup, and collect a number of useful and well-understood facts. We claim no originality in this section, and the existence ofthe current section is mainly for the completeness of notation and definition.2.1. Three consecutive favorite sites. It turns out that in order to show f (3) = ∞ it suffices to consider instances of three favorite sites which areconsecutive. To this end, we define the inverse edge local times by T U ( k, x ) (cid:44) inf { t ≥ U ( t, x ) = k } and T D ( k, x ) (cid:44) inf { t ≥ D ( t, x ) = k } . We consider the events of three consecutive favorite sites, i.e., A ( k ) x,h (cid:44) { K ( T U ( k + 1 , x )) = { x, x + 1 , x + 2 } , L ( T U ( k + 1 , x ) , x ) = h } . We write the events in T U ( k + 1 , x ) rather than T U ( k, x ) as it matches theform of the Ray-Knight representation which we will discuss later. We thenlet I h = ( ( h + √ h ) , ( h + 2 √ h )) and define N H = H (cid:88) h =1 (cid:88) k ∈ I h ∞ (cid:88) x =1 A ( k ) x,h and N = lim H →∞ N H = ∞ (cid:88) h =1 (cid:88) k ∈ I h ∞ (cid:88) x =1 A ( k ) x,h . J. DING AND J. SHEN We observe that for each h , the events A ( k ) x,h are mutually disjoint. In addition,we have that f (3) ≥ u ( x ) where u ( x ) = ∞ (cid:88) t =1 { S ( t − x − , S ( t )= x, x ∈ K ( t ) , K ( t )=3 } = ∞ (cid:88) k =1 { x ∈ K ( T U ( k,x )) , K ( T U ( k,x ))=3 } = ∞ (cid:88) k =0 ∞ (cid:88) h =1 { x ∈ K ( T U ( k +1) ,x ) , K ( T U ( k +1 ,x ))=3 , L ( T U ( k +1 ,x ) ,x )= h } . Therefore, we have that f (3) ≥ N , and thus it suffices to show that N = ∞ .We remark that the preceding discussions are extracted from decompositionsin [24, (2.3), (2.4), (2.5)], and they are the starting point for all computationsin [24] as well as the present article.2.2. Additive processes and the Ray-Knight representation. Throughoutthis paper we denote by Y t a critical Galton-Watson branching process withgeometric offspring distribution and by Z t , R t critical geometric branchingprocesses with one immigrant in each generation (in different ways). Moreprecisely, we let X t,i ’s be i.i.d. geometric variables with mean 1 and recur-sively define(2.1) Z t +1 = (cid:80) Z t +1 i =1 X t,i and R t +1 = 1 + (cid:80) R t i =1 X t,i . One can verify that Y t , Z t and R t are Markov chains with state space Z + and transition probabilities: P ( Y t +1 = j | Y t = i ) = π ( i, j ) (cid:44) (cid:40) δ ( j ) , if i = 0 , − i − j ( i + j − i − j ! , if i > , (2.2) P ( Z t +1 = j | Z t = i ) = ρ ( i, j ) (cid:44) π ( i + 1 , j )and P ( R t +1 = j | R t = i ) = ρ ∗ ( i, j ) (cid:44) π ( i, j − . Let k ≥ x be fixed integers. When x ≥ 1, define the following threeprocesses:1. ( Z ( k ) t ) t ≥ , is a Markov chain with transition probability ρ ( i, j ) andinitial state Z = k .2. ( Y ( k ) t ) t ≥− , is a Markov chain with transition probabilities π ( i, j ) andinitial state Y − = k . AVORITE SITES OF RANDOM WALKS 3. ( Y (cid:48) ( k ) t ) t ≥ , is a Markov chain with transition probabilities π ( i, j ) andinitial state Y (cid:48) ( k )0 = Z ( k ) x − .The three processes are independent, except for the fact that Y (cid:48) ( k ) t startsfrom the terminal state of Z ( k ) t . We patch the three processes together to asingle process: ∆ ( k ) x ( y ) (cid:44) Z ( k ) x − − y , if 0 ≤ y ≤ x − ,Y ( k ) y − x , if x − ≤ y ≤ ∞ ,Y (cid:48) ( k ) − y , if − ∞ < y ≤ . We also define Λ ( k ) x ( y ) (cid:44) ∆ ( k ) x ( y ) + ∆ ( k ) x ( y − 1) + { 0, we define the processes1. ( R ( k ) t ) t ≥ , is a Markov chain with transition probability ρ ∗ ( i, j ) andinitial state R − = k .2. ( Y ( k ) t ) t ≥ , is a Markov chain with transition probabilities π ( i, j ) andinitial state Y = k .3. ( Y (cid:48) ( k ) t ) t ≥− , is a Markov chain with transition probabilities π ( i, j ) andinitial state Y (cid:48) ( k ) − = R ( k ) − − x .In this case, we patch the three processes together by∆ ( k ) x ( y ) (cid:44) Y (cid:48) ( k ) y , if − ≤ y < ∞ ,R y − x , if x − ≤ y ≤ − ,Y ( k ) x − − y , if − ∞ < y ≤ x − . The corresponding Λ ( k ) x is defined byΛ ( k ) x ( y ) (cid:44) ∆ ( k ) x ( y ) + ∆ ( k ) x ( y − − { x By classical Ray-Knight Theorems, we get the couplings for the case k ≥ x ≤ 0: ( D ( T U ( k + 1 , x ) , y ) , y ∈ Z ) law = (∆ ( k ) x ( y ) , y ∈ Z ) , (2.6) ( L ( T U ( k + 1 , x ) , y ) , y ∈ Z ) law = (Λ ( k ) x ( y ) , y ∈ Z ) . (2.7)In this paper, we will mainly use the Ray-Knight representation (2.4) and(2.5), while (2.6) and (2.7) will be used in the calculation of E N H . In thefollowing, we default x > Three favorite sites under Ray-Knight representation. To utilize (2 . Y ( k ) t , Z ( k ) t and Y (cid:48) ( k ) t , we define˜ Z ( k ) t (cid:44) Z ( k ) t + Z ( k ) t − + 1 , ˜ Y ( k ) t (cid:44) Y ( k ) t + Y ( k ) t − , ˜ Y (cid:48) ( k ) t (cid:44) Y (cid:48) ( k ) t + Y (cid:48) ( k ) t − . For h ∈ Z + , define the first hitting time of [ h, ∞ ) for Y ( k ) t and Z ( k ) t to be σ ( k ) h and τ ( k ) h respectively and the extinction time of Y ( k ) t to be ω ( k ) . Thatis, σ ( k ) h (cid:44) inf { t ≥ Y ( k ) t ≥ h } , τ ( k ) h (cid:44) inf { t ≥ Z ( k ) t ≥ h } , and ω ( k ) = inf { t ≥ Y ( k ) t = 0 } . (2.8)Correspondingly, we define the first hitting time of [ h, ∞ ) for the process˜ Y ( k ) t and ˜ Z ( k ) t to be ˜ σ ( k ) h and ˜ τ ( k ) h respectively. Namely,˜ σ ( k ) h (cid:44) inf { t ≥ Y ( k ) t ≥ h } , ˜ τ ( k ) h (cid:44) inf { t ≥ Z ( k ) t ≥ h } . Using the notation above, we can write P ( A ( k ) h,x ) in its Ray-Knight represen-tation form. That is, P ( A ( k ) h,x ) is equal to P (cid:0) Y ( k )0 = h − k − , Y ( k )1 = k + 1 , Y ( k )2 = h − k − , { ˜ Y ( k ) t < h, for t ≥ } , { ˜ Z ( k ) t < h, for 1 ≤ t ≤ x − } , { ˜ Y (cid:48) ( k ) t < h, for t ≥ } (cid:1) . For all the notations above, when the initial state of a process is obvious,we omit the superscript “( k )” to avoid cumbersome notations. We will alsouse conditional probability P ( · | Y = k ) to indicate the initial state. AVORITE SITES OF RANDOM WALKS Standard lemmas. In this subsection we record a few well-understoodlemmas that will be useful later. Lemma . [24, (6.14) – (6.15)] For any ≤ k ≤ h ≤ u the followingovershoot bounds hold: P (cid:0) Y σ h ≥ u (cid:12)(cid:12) Y = k, σ h < ∞ (cid:1) ≤ P ( Y ≥ u | Y = h, Y ≥ h ) , P (cid:0) Z τ h ≥ u (cid:12)(cid:12) Z = k (cid:1) ≤ P ( Z ≥ u | Z = h, Z ≥ h ) . Lemma . We have that(i) For i, j ∈ (cid:16) ( h − √ h ) , ( h + 10 √ h ) (cid:17) , there exist positive constants c and C such that c h − ≤ π ( i, j ) ≤ C h − for all h ≥ .(ii) For i + j = h , π ( i, j ) ≤ O (1) h − .(iii) For j < i < i , π ( i , j ) > π ( i , j ) . Proof. Properties (i) and (ii) follow from straightforward computationusing Stirling’s formula and (2.2). For Property (iii), we see that π ( i +1 ,j ) π ( i,j ) = i + j i < j < i , and (iii) follows from induction. Lemma . We have that E τ h = E Z τ h − Z . In particular, we have that E [ τ h | Z = k ] ≥ h − k . Proof. Applying the Optional Stopping Theorem to the martingale Z t − t at time τ h , we get E τ h = E Z τ h − Z ≥ h − k , as desired. 3. Proof of Theorem 1.1. The current section contains three parts:in Subsection 3.1 we adapt the arguments in [24] and provide a lower boundon the first moment for the number of instances for the consecutive threefavorite sites; in Subsection 3.2 (which contains the main novelty of thepresent paper), we show that the second moment is of the same order as thesquare of the first moment, thereby proving that three favorite sites occurswith non-vanishing probability; in Subsection 3.3 we prove a 0-1 law forthree favorite sites and thus complete the proof of Theorem 1.1.3.1. Lower bound on the first moment. For x > h ∈ N , in orderto bound the probability for three consecutive favorite sites with local time h at vertices x , x + 1 and x + 2, the main part is to control the probabilityfor the local times below h everywhere except at x , x + 1 and x + 2. To thisend, it suffices to consider the edge local times (i.e., number of downcross-ings) in the Ray-Knight representation with appropriate conditioning in the J. DING AND J. SHEN region of ( x, x + 2). Then in the region outside of (0 , x + 2), these edge localtimes evolve as martingales (when looking forward spatially in ( x + 2 , ∞ )and backward spatially in ( −∞ , h ; in the region (0 , x ), the edge localtimes are not exactly a martingale (when looking backward spatially; see(2.1)) and the analysis is slightly more complicated. In the next lemma,we prove a lower bound on the first moment of (cid:80) τ h t =1 h − Z t h . Combined withstandard martingale analysis in the region outside of (0 , x + 2) and a changeof summation when summing over x (see (3.5)), this will then give a lowerbound on the first moment of N H (see Proposition 3.2). Lemma . Suppose that Z = k ∈ [ h − √ h, h − √ h ] . Then there existsa constant c > such that E ( (cid:80) τ h t =1 h − Z t h ) ≥ c √ h . Proof. Let M t = (cid:80) ts =1 ( Z s − s ) − t ( Z t − t ), and let F t = σ ( Z , Z , . . . , Z t ).We see that E ( M t +1 | F t ) = (cid:2)(cid:80) ts =1 ( Z s − s ) + ( Z t − t ) (cid:3) − ( t + 1)( Z t − t ) = M t . Thus ( M t ) is a martingale. By the Optional Stopping Theorem, we see that E ( (cid:80) τ h t =1 ( Z t − t )) = E τ h ( Z τ h − τ h ) and hence E (cid:0)(cid:80) τ h t =1 h − Z t h (cid:1) = (1 + h ) E τ h − h E [ τ h Z τ h − τ h ] . (3.1)Now consider the process M (cid:48) t = − Z t + tZ t − t + t . By (2.1), we see that E ( M (cid:48) t +1 | F t ) = − ( Z t +4 Z t +3)+( tZ t + Z t + t +1) − ( t +2 t +1)+ ( t +1) , where equal to M (cid:48) t . So ( M (cid:48) t ) is a martingale. Using the Optional StoppingTheorem to ( M (cid:48) t ) at τ h , we have E (cid:2) τ h Z τ h − τ h (cid:3) = E (cid:2) Z τ h − τ h (cid:3) − Z = E ( Z τ h − Z ) − E τ h . (3.2)Combining (3.1), (3.2) and Lemma 2.3, we get E (cid:2)(cid:80) τ h t =1 h − Z t h (cid:3) =(1 + 14 h ) E τ h − h E (cid:2) Z τ h − Z (cid:3) =(1 + 14 h ) E ( Z τ h − Z ) − h E [( Z τ h − Z )( Z τ h + Z )] ≥ h E [( Z τ h − Z )(4 h − ( Z τ h + Z ))] . Obviously Z τ h − Z ≥ h − k ≥ √ h and by Lemma 2.1 we have that E ( Z τ h − Z )( Z τ h + Z − h ) = O ( h ). Therefore there is a constant c suchthat E (cid:2)(cid:80) τ h t =1 h − Z t h (cid:3) ≥ c √ h for sufficiently large h . AVORITE SITES OF RANDOM WALKS Proposition . For a constant c > we have E N H ≥ c log H . Proof. In what follows, c i for i ≥ c are all constants. By theRay-Knight representation, E N H is equal to the following product: H (cid:88) h =1 (cid:88) k ∈ I h P (cid:16) Y ( k )0 = h − k − , Y ( k )1 = k + 1 , Y ( k )2 = h − k − , { ˜ Y ( k ) t < h, for t ≥ } (cid:17) × ∞ (cid:88) x =1 P (cid:0) { ˜ Z ( k ) t < h, ≤ t ≤ x − } , { ˜ Y (cid:48) ( k ) t < h, for t ≥ } (cid:1) . Thus, we get that E N H ≥ H (cid:88) h =1 (cid:88) k ∈ I h π ( l, h − k − π ( h − k − , k + 1) π ( k + 1 , h − k − · P ( Y ( h − k − t < h for t ≥ · ∞ (cid:88) x =1 P (˜ τ h ≥ x, { ˜ Y (cid:48) ( k ) t < h, for t ≥ } ) . By Lemma 2.2 (i), all π ( · , · ) in the above equation are at the scale h − .Since Y t is a martingale, by using the Optional Stopping Theorem at σ h ∧ ω where σ h and ω are defined in (2.8), we have P ( Y ( h − k − t < h for t ≥ 0) = P ( Y ( h − k − t hits 0 before h ) ≥ h/ − ( h − k − h/ ≥ c h − . So we get E N H ≥ c H (cid:88) h =1 (cid:88) k ∈ I h ∞ (cid:88) x =1 h − P (cid:16) ˜ τ h ≥ x, { ˜ Y (cid:48) ( k ) t < h, t ≥ } (cid:17) . (3.3) J. DING AND J. SHEN Let k = ( h − √ h ). By independence in the Ray-Knight representation, ∞ (cid:88) x =1 P (˜ τ h ≥ x, { ˜ Y (cid:48) ( k ) t < h, for t ≥ } ) ≥ ∞ (cid:88) x =1 P ( Z ( k )1 ≤ k , Z ( k ) t < h ≤ t ≤ x − , { Y (cid:48) ( k ) t < h , for t ≥ } ) ≥ ∞ (cid:88) x =2 [ h − (cid:88) l =0 (cid:16) P ( Z ( k )1 ≤ k ) · P ( Z ( k ) t < h for 1 ≤ t ≤ x − , Z ( k ) x − = l ) × P ( Y ( l ) t hits 0 before h ) (cid:17) . By Lemma 2.2 (i), P ( Z ( k )1 ≤ k ) ≥ c . Using the Optional Stopping Theoremagain, we have P (cid:16) Y ( l ) t hits 0 before h (cid:17) ≥ h/ − lh/ . So ∞ (cid:88) x =1 P (cid:16) ˜ τ h ≥ x, { ˜ Y (cid:48) ( k ) t < h, t ≥ } (cid:17) ≥ c · ∞ (cid:88) x =1 [ h − (cid:88) l =0 P (cid:16) τ ( k ) h/ ≥ x, Z ( k ) x − = l (cid:17) · h/ − lh/ . (3.4)By interchange of the summation and the expectation (which is valid by theMonotone Convergence Theorem) and Lemma 3.1, we have that the righthand side of (3.4) is equal to c · E (cid:104) [ h − (cid:88) l =0 τ ( k h/ (cid:88) x =1 h/ − lh/ · (cid:8) Z ( k x − = l (cid:9)(cid:105) = c E (cid:16) τ ( k h/ − (cid:88) t =0 h/ − Z ( k ) t h/ (cid:17) ≥ c √ h , (3.5)where in the second inequality we did change of variable t = x − 1. Thus by(3.3) and (3.5), E N H ≥ H (cid:88) h =1 (cid:88) k ∈ I h c h − ≥ c · H (cid:88) h =1 h ≥ c log H , completing the proof of the proposition. AVORITE SITES OF RANDOM WALKS Upper bound on the second moment. The calculation of second mo-ment involves the two three favorite sites that happen in chronological order.The key insight is that two instances of three favorite sites with no spatialoverlap are almost independent. Before giving the bound for the secondmoment, we discuss some useful concepts and tools that characterize theindependence of different three favorite sites.Let D ( t ) = ( D ( t, x ) , x ∈ Z ) ∈ N Z be the random vector that records thenumber of downcrossings of each site by the time t . For (cid:96) ∈ N Z , we use (cid:96) ( i ), i ∈ Z to denote the i -th component of (cid:96) . For (cid:96) ∈ N Z , define B x ( (cid:96) ) = {∃ t < ∞ : D ( t ) = (cid:96), S ( t − 1) = x − , S ( t ) = x } . Note that if B x ( (cid:96) ) happens, thereexists a unique t ∈ N such that D ( t ) = (cid:96), S ( t − 1) = x − S ( t ) = x .Sometimes we abuse the terminology “after B x ( (cid:96) ) happens” by meaning“after the unique t with D ( t ) = (cid:96), S ( t − 1) = x − , S ( t ) = x ”. We also say“ B x ( (cid:96) ) happens before B x (cid:48) ( (cid:96) (cid:48) )” by meaning the unique t (corresponding to B x ( (cid:96) )) is less than the unique t (cid:48) (corresponding to B x (cid:48) ( (cid:96) (cid:48) )).Let P = { (cid:96) : P ( B x ( (cid:96) )) > x } . Clearly for any (cid:96) ∈ P , (cid:96) hascompact support. For Q ⊂ P , denote B x ( Q ) = (cid:83) (cid:96) ∈Q B x ( (cid:96) ). Then we have A ( k ) x,h = B x ( P ( k ) x,h ) where P ( k ) x,h is the collection of (cid:96) ∈ P such that (cid:96) ( x − 1) = k, (cid:96) ( x ) = h − k − , (cid:96) ( x + 1) = k + 1 , (cid:96) ( x + 2) = h − k − (cid:96) ( i − 1) + (cid:96) ( i ) < h for all i (cid:54) = x, x + 1 , x + 2 . Our main intuition on bounding the correlation between two instances ofthree favorite sites is the following: Suppose at some time (say T ) we havean instance of three favorite points at x, x +1 , x +2 with edge local time (i.e.,downcrossings) given by (cid:96) . Our crucial observation is that conditioning on B x ( (cid:96) ) does not increase much of the probability for producing an instanceof three favorite sites in a future time (say T ) which are spatially differentfrom those of (cid:96) . To this end, we let (cid:96) (cid:48) be one of many local perturbations of (cid:96) (which are obtained from (cid:96) by decreasing the values at x + 1 and x + 2).We note that (see Figure 1 for an illustration) • The event B x ( (cid:96) ) (respectively, B x ( (cid:96) (cid:48) )) corresponds to that the edge lo-cal time is (cid:96) (respectively, (cid:96) (cid:48) ) when the random walk cross the directededge ( x − , x ) for the ( (cid:96) ( x − (cid:96) ( x − 1) = (cid:96) (cid:48) ( x − T in Figure 1). Conditioned on B x ( (cid:96) ) (respectively, B x ( (cid:96) (cid:48) )), the edge local time at a later time (whichcorresponds to T in Figure 1) is (cid:96) (respectively, (cid:96) (cid:48) ) superposed withan independent edge local time field which we denote by ˜ (cid:96) . By thestrong Markov property for random walks, the law of ˜ (cid:96) is the sameregardless of conditioning on B x ( (cid:96) ) or B x ( (cid:96) (cid:48) ). J. DING AND J. SHEN • If the field ( (cid:96) + ˜ (cid:96) ) produces three favorite sites which are spatiallydifferent from those of (cid:96) , then the field ( (cid:96) (cid:48) + ˜ (cid:96) ) also produces threefavorite sites. Fig 1 . The black bars represent vertex local times at T and the grey bars represent onesat T . When we decrease the edge local times at x + 1 and x + 2 , descent of vertex localtimes happens at x + 1 , x + 2 and x + 3 . After the local time perturbation at time T , wewill still get “three favorite sites” at T . In summary, we see that the conditional probability of producing an in-stance of three favorite sites which are spatially different from those of (cid:96) given B x ( (cid:96) ) is the same as the conditional probability given B x ( (cid:96) (cid:48) ). But theprobability for the union of B x ( (cid:96) (cid:48) )’s when (cid:96) (cid:48) ranging over all legitimate per-turbations is much larger than that of B x ( (cid:96) ) — in fact larger by a factor oforder h = (cid:96) ( x − 1) + (cid:96) ( x ) + 1 (see Lemma 3.4 below). This is a (quantitative)manifestation that the event B x ( (cid:96) ) is uncorrelated with a spatially differentinstance of three favorite sites in the future.Our formal proof does not exactly follow the discussion above on control-ling the conditional probability, as it turns out slightly simpler to directlycompute the joint probability for two instances of three favorite sites (butthe intuition is the same). For the precise implementation, we let A be theset of all subsets of P and define a map ϕ x : P (cid:55)→ A mapping an (cid:96) ∈ P toa collection of vectors where we locally push down the values at locations x + 1 and x + 2. More precisely, we define ϕ x ( (cid:96) ) to be { (cid:96) ∗ ∈ P : (cid:96) ∗ ( i ) < (cid:96) ( i ) for i = x + 1 , x + 2, (cid:96) ∗ ( i ) = (cid:96) ( i ) for i (cid:54) = x + 1 , x + 2 } . Lemma . For i = 1 , and (cid:96) ∗ i ∈ ϕ x i ( (cid:96) i ) with (cid:96) i ∈ P ( k i ) x i ,h , we havethat B x ( (cid:96) ∗ ) ∩ B x ( (cid:96) ∗ ) = ∅ if ( x , (cid:96) ) (cid:54) = ( x , (cid:96) ) . Further, we have B x ( (cid:96) ∗ ) ∩ B x ( (cid:96) ∗ ) = ∅ if ( x , (cid:96) ) = ( x , (cid:96) ) but (cid:96) ∗ (cid:54) = (cid:96) ∗ . AVORITE SITES OF RANDOM WALKS Proof. Case (i): Suppose x (cid:54) = x . Since clearly B x ( (cid:96) ∗ ) and B x ( (cid:96) ∗ )cannot happen at the same time t , we can then assume without loss ofgenerality that B x ( (cid:96) ∗ ) happens first. Then when B x ( (cid:96) ∗ ) happens the vertexlocal time at x is at least h , arriving at a contradiction.Case (ii): Suppose that x = x but (cid:96) (cid:54) = (cid:96) . In this case, we have (cid:96) ∗ (cid:54) = (cid:96) ∗ .Since clearly B x ( (cid:96) ∗ ) and B x ( (cid:96) ∗ ) cannot happen at the same time t , we canthen assume without loss of generality that B x ( (cid:96) ∗ ) happens first. In orderfor B x ( (cid:96) ∗ ) to happen, the random walk has to leave x (= x ) and revisit x . As a result, the vertex local time at x will be strictly larger than h ,arriving at a contradiction.Case (iii): Suppose that x = x , (cid:96) = (cid:96) but (cid:96) ∗ (cid:54) = (cid:96) ∗ . This follows from thesame reasoning as in Case (ii). Lemma . There exist a constant c > such that for any (cid:96) ∈ P ( k ) x,h with k ∈ I h , P ( B x ( ϕ x ( (cid:96) ))) ≥ ch P ( B x ( (cid:96) )) . Proof. We consider (cid:96) ∗ ∈ ϕ x ( (cid:96) ) such that (cid:96) ∗ ( x + 1) ∈ [ k + 1 − √ h, k + 1)and (cid:96) ∗ ( x + 2) ∈ [ h − k − − √ h, h − k − c > P ( B x ( (cid:96) ∗ )) P ( B x ( (cid:96) )) = π ( (cid:96) ∗ ( x ) , (cid:96) ∗ ( x + 1)) π ( (cid:96) ∗ ( x + 1) , (cid:96) ∗ ( x + 2)) π ( (cid:96) ∗ ( x + 2) , (cid:96) ( x + 3)) π ( h − k − , k + 1) π ( k + 1 , h − k − π ( h − k − , (cid:96) ( x + 3)) ≥ c. Note that there are about h of such (cid:96) ∗ ∈ ϕ x ( (cid:96) ) that satisfy the inequality.By Lemma 3.3, we get that P ( B x ( ϕ x ( (cid:96) ))) ≥ ch P ( B x ( (cid:96) )). Proposition . We have that E N H = O (log H ) · E N H . Proof. We decompose the second moment into the following three parts: E N H = 2 (cid:88) ≤ h First we estimate I. By the Strong Markov Property, P ( A ( k ) x,h , A ( k (cid:48) ) x (cid:48) ,h (cid:48) ) = (cid:88) (cid:96) ∈P ( k ) x,h (cid:88) (cid:96) (cid:48) ∈P ( k (cid:48) ) x (cid:48) ,h (cid:48) P (cid:0) B x ( (cid:96) ) , B x (cid:48) ( (cid:96) (cid:48) ) (cid:1) = (cid:88) (cid:96) ∈P ( k ) x,h (cid:88) ˜ (cid:96) : (cid:96) +˜ (cid:96) ∈P ( k (cid:48) ) x (cid:48) ,h (cid:48) P ( B x ( (cid:96) )) · P x ( B x (cid:48) (˜ (cid:96) )) , where the x in P x indicates the starting point of the random walk. For any x (cid:48) ∈ Z + and k (cid:48) ∈ I h (cid:48) , using Lemma 3.4, we get (cid:88) k ∈ I h (cid:88) x : | x − x (cid:48) | > P (cid:16) A ( k ) x,h , A ( k (cid:48) ) x (cid:48) ,h (cid:48) (cid:17) = (cid:88) k ∈ I h (cid:88) x : | x − x (cid:48) | > (cid:88) (cid:96) ∈P ( k ) x,h (cid:88) ˜ (cid:96) : (cid:96) +˜ (cid:96) ∈P ( k (cid:48) ) x (cid:48) ,h (cid:48) P ( B x ( (cid:96) )) · P x ( B x (cid:48) (˜ (cid:96) )) ≤ (cid:88) k ∈ I h (cid:88) x : | x − x (cid:48) | > (cid:88) (cid:96) ∈P ( k ) x,h (cid:88) ˜ (cid:96) : (cid:96) +˜ (cid:96) ∈P ( k (cid:48) ) x (cid:48) ,h (cid:48) O (1) h − P ( B x ( ϕ x ( (cid:96) ))) · P x ( B x (cid:48) (˜ (cid:96) )) ≤ O (1) h − (cid:88) k ∈ I h (cid:88) x : | x − x (cid:48) | > (cid:88) (cid:96) ∈P ( k ) x,h (cid:88) (cid:96) ∗ ∈ ϕ x ( (cid:96) ) (cid:88) ˜ (cid:96) : (cid:96) +˜ (cid:96) ∈P ( k (cid:48) ) x (cid:48) ,h (cid:48) P ( B x ( (cid:96) ∗ ) , B x (cid:48) ( (cid:96) ∗ + ˜ (cid:96) )) . The last inequality follows from Lemma 3.3 and Strong Markov Property.By Lemma 3.3, all events B x ( (cid:96) ∗ ) for x ∈ N , (cid:96) ∗ ∈ ϕ x ( (cid:96) ), k ∈ I h and (cid:96) ∈ P ( k ) x,h are disjoint. Note that | x − x (cid:48) | > ϕ x only reduces the downcrossingnumber at x + 1, x + 2. So (cid:96) ∗ + ˜ (cid:96) ∈ P ( k (cid:48) ) x (cid:48) ,h (cid:48) . Hence we have (cid:88) k ∈ I h (cid:88) x : | x − x (cid:48) | > P (cid:16) A ( k ) x,h , A ( k (cid:48) ) x (cid:48) ,h (cid:48) (cid:17) ≤ O (1) h − (cid:80) (cid:96) (cid:48) ∈P ( k (cid:48) ) x (cid:48) ,h (cid:48) P (cid:0) B x (cid:48) ( (cid:96) (cid:48) ) (cid:1) . As a result, we obtain thatI ≤ O (1) (cid:88) ≤ h 1) = x (cid:48) − S ( t ) = x (cid:48) and D ( t, x (cid:48) − 1) = k (cid:48) , D ( t, x (cid:48) ) = h (cid:48) − k (cid:48) − k (cid:48) exists,it is unique).(2) Once (1) happens, both t and k (cid:48) are determined. The additional processafter B x ( (cid:96) ) need to satisfy: D ( t, x (cid:48) + 1) − (cid:96) ( x (cid:48) + 1) = h (cid:48) − k (cid:48) − − (cid:96) ( x (cid:48) + 1)and D ( t, x (cid:48) + 2) − (cid:96) ( x (cid:48) + 2) = k (cid:48) + 1 − (cid:96) ( x (cid:48) + 2).(3) L ( t, y ) < h (cid:48) for all y (cid:54) = x (cid:48) , x (cid:48) + 1 , x (cid:48) + 2.We omit the probability loss for (1) and (3) and only consider the probabilityfor (2). Formally, define T to be the time t such that S ( t − 1) = x (cid:48) − , S ( t ) = x (cid:48) , D ( t, x (cid:48) − 1) + D ( t, x (cid:48) ) = h (cid:48) − 1. Then, we have P ( ∃ k (cid:48) : A ( k (cid:48) ) x (cid:48) ,h (cid:48) (cid:12)(cid:12) B x ( (cid:96) )) isless equal to h (cid:48) (cid:88) k (cid:48) = (cid:96) ( x (cid:48) ) P (cid:0) T = T U ( k (cid:48) + 1 , x (cid:48) ) , D ( T, x (cid:48) ) = h (cid:48) − k (cid:48) − ,D ( T, x (cid:48) + 1) = k (cid:48) + 1 , D ( T, x (cid:48) + 2) = h (cid:48) − k (cid:48) − (cid:1) . Using the Ray-Knight representation for the random walk started at x after B x ( (cid:96) ), we have P (cid:16) ∃ k (cid:48) : A ( k (cid:48) ) x (cid:48) ,h (cid:48) (cid:12)(cid:12) B x ( (cid:96) ) (cid:17) is less equal to h (cid:48) (cid:88) k (cid:48) = (cid:96) ( x (cid:48) ) P (cid:0) T = T U ( k (cid:48) + 1 , x (cid:48) ) , D ( T, x (cid:48) ) = h (cid:48) − k (cid:48) − (cid:1) × π ∗ ( h (cid:48) − k (cid:48) − − (cid:96) ( x (cid:48) ) , k (cid:48) + 1 − (cid:96) ( x (cid:48) + 1)) representation × π ∗ ( k (cid:48) + 1 − (cid:96) ( x (cid:48) + 1) , h (cid:48) − k (cid:48) − − (cid:96) ( x (cid:48) + 2)) . where π ∗ ( · , · ) is either π ( · , · ) or ρ ∗ ( · , · ) depending on the relative position of x and x (cid:48) (see (2.4) and (2.6)). Since both ( h (cid:48) − k (cid:48) − − (cid:96) x ( x (cid:48) ))+( k (cid:48) +1 − (cid:96) ( x (cid:48) +1)) J. DING AND J. SHEN and ( k (cid:48) + 1 − (cid:96) ( x (cid:48) + 1)) + ( h (cid:48) − k (cid:48) − − (cid:96) ( x (cid:48) + 2)) are greater than or equal to h (cid:48) − h , by Lemma 2.2 (ii) and the relation ρ ∗ ( i, j ) = π ( i, j − π ∗ ( h (cid:48) − k (cid:48) − − (cid:96) ( x (cid:48) ) , k (cid:48) +1 − (cid:96) ( x (cid:48) +1)) · π ∗ ( k (cid:48) +1 − (cid:96) ( x (cid:48) +1) , h (cid:48) − k (cid:48) − − (cid:96) ( x (cid:48) +2))is at most O (1) h (cid:48) − h for any (cid:96) ( x (cid:48) ) ≤ k (cid:48) ≤ h (cid:48) . Therefore, P (cid:16) ∃ k (cid:48) : A ( k (cid:48) ) x (cid:48) ,h (cid:48) (cid:12)(cid:12) B x ( (cid:96) ) (cid:17) ≤ h (cid:48) (cid:88) k (cid:48) = (cid:96) ( x (cid:48) ) P (cid:0) T = T U ( k (cid:48) + 1 , x (cid:48) ) , D ( T, x (cid:48) ) = h (cid:48) − k (cid:48) − (cid:1) · O (1) h (cid:48) − h = P (cid:0) ∃ k (cid:48) : T = T U ( k (cid:48) + 1 , x (cid:48) ) , D ( T, x (cid:48) ) = h (cid:48) − k (cid:48) − (cid:1) · O (1) h (cid:48) − h , which is bounded by O (1) h (cid:48) − h . As a consequence, we get that P (cid:16) A ( k ) x,h , {∃ k (cid:48) : A ( k (cid:48) ) x (cid:48) ,h (cid:48) } (cid:17) ≤ (cid:88) (cid:96) ∈P ( k ) x,h P ( B x ( (cid:96) )) · O (1) h (cid:48) − h = O (1) h (cid:48) − h · P (cid:16) A ( k ) x,h (cid:17) and thus II ≤ ∞ (cid:88) x =1 H (cid:88) h =1 (cid:88) k ∈ I h H (cid:88) h (cid:48) = h +1 O (1) h (cid:48) − h · P (cid:16) A ( k ) x,h (cid:17) (3.8) ≤ O (log H ) H (cid:88) h =1 (cid:88) k ∈ I h ∞ (cid:88) x =1 P ( A ( k ) x,h ) = O (log H ) E N H . (3.9)Combining (3.6), (3.7) and (3.8), we get that E N H = O (log H ) E N H .We are now ready to show that N = ∞ with positive probabiity. Proposition . There exists a constant δ > such that P ( N = ∞ ) ≥ δ where N = lim H →∞ N H . Proof. By Cauchy-Schwarz inequality, we get that E N H = E N H { N H > log log H } + E N H { N H ≤ log log H } ≤ (cid:113) E N H · P ( N H > log log H ) + log log H . AVORITE SITES OF RANDOM WALKS By Propositions 3.2 and 3.5, there exist constants c, δ > P ( N H > log log H ) ≥ ( E N H − log log H ) E N H ≥ c E N H log H E N H ≥ δ , for all sufficiently large H . Sending H → ∞ , we get that P ( N = ∞ ) ≥ δ .3.3. In this section, building on Proposition 3.6 we show that N = ∞ occurs with probability 1. There are a few possible approaches, andhere we choose to prove a 0-1 law taking advantage of the result on thetransience of favorite sites. Let V ( t ) be an arbitrary element in K ( t ). It wasshown in [3] that uniformly in all V ( t ) ∈ K ( t ) we have with probability 1(3.10) lim inf t →∞ | V ( t ) | t (log t ) − = ∞ . Denote ψ ( t ) = t (log t ) − and E = (cid:110) lim inf t →∞ | V ( t ) | ≥ ψ ( t ) (cid:111) . By (3.10), wehave P ( E ) = 1, and thus without loss of generality we can assume that E occurs in what follows. Our goal is to show that the event { f (3) = ∞} is atail event and it suffices to show that the event { f (3) = ∞} is independentof any σ -field F m (which is the σ -field generated by the first m steps of therandom walk) for all m ∈ N . To this end, for each m ∈ N we let M be the firsttime such that for all t ≥ M favorite sites occurs outside of [ − m, m ]. Wesee that M is not necessarily a stopping time but M < ∞ with probability1. Therefore, the event { f (3) = ∞} depends only on whether after M threefavorite sites occurs infinitely often. Now consider the event { f m (3) = ∞} where f m (3) is defined analogously to f (3) but for the random walk startedat time m . We claim that the symmetric difference between { f (3) = ∞} and { f m (3) = ∞} has probability zero since in the symmetric difference one musthave a favorite site (for the original random walk) in the interval [ − m, m ]after M . 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