Exponential Bounds and Analyticity for the Tree Builder Random Walk

EXPONENTIAL BOUNDS AND ANAL YTICITY F OR THE TREE BUILDER RANDOM W ALK CAIO AL VES AND RODRIGO RIBEIRO Abstract. In this w ork we in v estigate a class of random w alks that in teracts with its environmen t called T ree Builder Random W alk (TBR W). In our set- tings, at eac h step, the w alker adds a random n um b er of v ertices to its position sampled according to a distribution Q . Previous w orks show ed that the walk er is ballistic with a w ell-defined sp eed, and that the TBR W admits a renewal structure, meaning that the pro cess can be split into i.i.d epo chs. W e show that the first renew al time has exp onential tail. Moreov er, w e show tw o conse- quences of the light tail of the first renewal time: an exp onential upp er bound for the empirical sp eed of the w alker, and, for the case in which the walk er adds only one vertex with probabilit y p , we show that the limiting speed is an analytic function of the parameter p . In some of our pro ofs, we apply tech- niques from b ond p ercolation, which consist of extending probabilities to the complex num b ers and using the W eierstrass M -test. 1. Introduction The T ree Builder Random W alk (TBR W) is a class of random walks that inter- acts with its environmen t by modifying the underlying graph. A t each step, the w alker can add a random num b er of v ertices to its curren t p osition, and then takes a simple random walk step on the (p ossibly) up dated graph. This feedback mech- anism, in whic h the w alker activ ely shapes the geometry of its environmen t, sets the TBR W apart from classical Random W alk on Random Environmen t (R WRE) mo dels [12, 15], where the environmen t is fixed, and from reinforced random w alk mo dels [9], where the walk er mo difies transition probabilities but not the graph itself. More broadly , the interaction betw een random w alks and random media remains a highly activ e area. Recent progress includes fundamen tal adv ances in the random conductance mo del [2], connections betw een reinforced random walks and super- symmetric spin models [11], the analysis of once-reinforced random walks on trees [4], and la w of large num b ers for random w alks in dynamic random en vironments [1]. The TBR W was introduced in [6], where the authors show ed that, in the Bernoulli case, where a new vertex is added with probabilit y p at each step, the w alk er is transien t and ballistic with a well-defined sp eed v ( p ), that is, lim n →∞ dist( X n , o ) n = v ( p ) > 0 , almost surely . The general case, where at eac h step the num b er of added vertices follo ws an arbitrary distribution Q , was addressed in [8], where the authors es- tablished transience and ballisticit y under a uniformly elliptic condition on Q . In [10], this w as extended to a full Strong La w of Large Numbers with a well-defined R. Ribeiro is affiliated with IMP A T ec h, Rio de Janeiro, Brazil. E-mail: ro- drigo.ribeiro@impatech.edu.br C. Alves is affiliated with FH T echnikum Wien, Vienna, Austria. Email: caio teo doro.de magalhaes alves@tec hnikum-wien.at. 1 2 CAIO AL VES AND RODRIGO RIBEIR O sp eed v ( Q ), a Cen tral Limit Theorem, and a Law of the Iterated Logarithm, via the construction of a renewal structure for the TBR W. Whereas in [3], the authors in vestigated a biased v ersion of the TBR W, sho wing that the walk er can b e either ballistic or recurrent dep ending on the bias parameter. The key to ol b ehind these limit theorems is the renew al structure itself: the w alker’s tra jectory decomp oses in to i.i.d. ep o chs, which allows one to transfer clas- sical results for i.i.d. sequences to the study of the w alk er. This approach has a long and fruitful history in random walk theory; notably , Sznitman and Zerner [14] used renewal structures to prov e ballisticit y and a SLLN for R WRE in Z d under condition ( T ′ ), and Sznitman [13] later obtained a CL T under momen t conditions on the first regeneration time. A recurring theme in this program is that the finer prop erties of the w alker, CL T, large deviations, regularity of the sp eed, are all gov erned b y the tail b ehavior of the first renewal time τ 1 . In [10], the authors show ed a stretc hed exp onential b ound P Q ( τ 1 ≥ n ) ≤ e − cn 1 / 8 , which already suffices for the CL T and the LIL. Ho w ev er, sharp er tails unlo ck stronger results. In this paper, we substantially impro ve the tail bound on τ 1 and exploit it to deriv e t wo new results for the TBR W: (i) Exp onen tial concen tration of the sp eed. W e pro ve that the prob- abilit y that the empirical sp eed dist( X n , o ) /n exceeds the limiting sp eed v ( Q ) b y some ε is exp onentially small in n , uniformly o ver the family of uniformly elliptic distributions (Theorem 2). (ii) Analyticit y of the sp eed. In the Bernoulli case Q = Ber( p ), w e sho w that v ( p ) is an analytic function of p ∈ (0 , 1] (Theorem 3). Our approach adapts techniques from b ond p ercolation developed in [7], extending prob- abilities to the complex plane and applying the W eierstrass M -test, with the exp onential tail of τ 1 pro viding the necessary summability . In order to state our results precisely , let us in tro duce the mo del and the necessary notation. 1.1. The T ree Builder Random W alk (TBR W). The TBR W generates a se- quence of pairs ( T n , X n ), where T n denotes the ro oted tree at time n and X n is a v ertex of T n whic h represen ts the p osition of the w alker at time n . The model is defined in a mark ovian fashion as follows: fix an initial state ( T 0 , X 0 ), where T 0 is a ro oted finite tree of ro ot o and X 0 is a v ertex of T 0 , and a sequence of probability distribution ov er the natural n um b ers {L n } n , then for eac h n • obtain T n +1 from ( T n , X n ) b y adding random n um b er of vertices to X n . This random num b er of v ertices is sampled according to L n +1 ; • obtain X n +1 b y moving X n to a uniformly selected neigh b or of it in T n +1 ; In words, we first add a random num b er of vertices to X n according to the distri- bution L n +1 , and then let X n tak e a single step of a simple random walk on the, p ossibly up dated, tree T n +1 . T o illustrate the mo del and ho w the parameter p , in the Bernoulli case Q = Ber( p ), shapes the tree, Figure 1 sho ws three indep endent realizations of the TBR W for p = 0 . 1 , p = 0 . 5 and p = 0 . 9. In each case, the n um b er of steps is chosen as n = ⌊ 99 /p ⌋ so that the exp ected num b er of vertices is approximately 100, for a fair comparison across different regimes. 1.1.1. Basic Notation and Conventions. W e will make use of the notation F n for the natural filtration, that is, (1) F n := σ ( T 0 , X 0 , T 1 , X 1 , T 2 , X 2 , . . . , T n , X n ) . EXPONENTIAL BOUNDS AND ANAL YTICITY FOR THE TREE BUILDER RANDOM W ALK 3 p = 0.1, vertices = 87 p = 0.5, vertices = 96 p = 0.9, vertices = 98 Figure 1. TBR W trees with ∼ 100 expected vertices for p = 0 . 1, p = 0 . 5, and p = 0 . 9. Notice that there is no need to add information ab out v alue of the n umber of vertices added at each step, since they are obtained by keeping track on the changes on the trees from one step to the other. W e will reserve the notation F to the smallest sigma algebra that mak es the whole pro cess measurable. And the usual notation (Ω , F ) for the measurable space. In this pap er we will inv estigate the instance of the mo del when the sequence of la ws {L n } n is set to b e L n = Q for all n . In this case, we will write P T 0 ,X 0 ; Q (and E T 0 ,X 0 ; Q for the asso ciated exp ectation), for the distribution of the TBR W with initial state ( T 0 , X 0 ) and L n = Q for all n . W e refer to the distribution Q as the le af distribution since it controls the n umber of leav es added at each step. When the initial state is an edge with X 0 at the non-ro ot tip of it, w e omit the initial state for simplicity , writing P Q (resp ec. E Q ). W e will leverage the tree structure of T n quite often, and it will be useful to apply genealogy terms to refer to vertices. In this direction, giv en a vertex x on a tree T , we will write f ( x ) for its father , that is, for the neighbor of x on T that is closer to the ro ot o . Also, we will denote by dist( X n , o ) the (graph) distance of the w alker at time n from the ro ot o . W e will write | T | for the n umber of vertices on T and h ( T ) for its height. And finally , deg( X n ) denotes the degree of X n in T n , that is, the num b er of neighbors of X n at time n . 1.2. The Uniformly Elliptic Condition. In the TBR W, the notion of ballisticity is connected to a condition on the sequence of laws L . W e say a sequence of prob- abilit y distributions L = {L n } n o ver the natural n um b ers is said to b e uniformly el liptic if there exists κ ∈ (0 , 1] such that (2) inf n ∈ N L n ( { 1 , 2 , . . . } ) = κ. A TBR W with a uniformly elliptic sequence L has the feature that at each step the w alker adds at least one leaf to its position with probability at least κ . In [8], the authors prov ed that under this settings, the TBR W is ballistic. F or a given κ ∈ (0 , 1], we let Q κ b e the family of probability distribution o v er N suc h that Q ( { 1 , 2 , . . . } ) ≥ κ . 1.3. Main Results. In [10], the authors constructed a renewal structure for the TBR W where the first renewal time τ 1 is defined as follo ws. τ 1 = inf  n > 0 : deg( X n ) = 1 , dist( X s , o ) < dist( X n , o ) ≤ dist( X t , o ) , ∀ s < n, ∀ t > n  . (3) In words, τ 1 is the first time the w alker reac hes a certain distance dist( X n , o ) by jumping to a leaf and then it never visits the father of that leaf. 4 CAIO AL VES AND RODRIGO RIBEIR O Our first main result shows that τ 1 has exponentially ligh t tail. Moreov er, the rate can be chosen uniformly across all distributions Q ∈ Q κ for some κ ∈ (0 , 1]. More precisely , we hav e the follo wing. Theorem 1 (Exp onential tail of τ 1 ) . Consider a TBR W with le af distribution Q and fix κ ∈ (0 , 1] , then ther e exists a c onstant c dep ending on κ only, such that sup Q ∈Q κ P Q ( τ 1 ≥ n ) ≤ e − cn . In [10], the authors show ed a stretc hed exp onential tail b ound for τ 1 , that is, their Theorem 1 guaran tees that (4) sup Q ∈Q κ P Q ( τ 1 ≥ n ) ≤ e − cn 1 / 8 , for some constan t c dep ending on κ only . The wa y the ab o v e b ound is obtained in volv es k eeping track on the times the w alker visits leav es at a further lev el and the times it returns to the father of those lea v es. Then the even t { τ 1 > n } is rewritten as a disjoint union of even ts in volving how many times the w alker visited leav es b efore time n . An union bound is then applied to obtain the ab ov e b ound. Our approac h to prov e Theorem 1 ov ercomes the union b ound done in [10]. This new faster decay is p ossible due to a new decomposition of τ 1 whic h we describe at Section 2. The key asp ects of our decomp osition are: whenever the walk er do es an excursion and returns to the father of a v ertex, these excursions cannot be to o long, otherwise, it is too exp ensive for the walk er to make its wa y to the father of a v ertex; th us, the walk er is not to o far wa y from a leaf, which means that it do es not hav e to w ait too long in order to ha ve a new chance to regenerate from a leaf again. The upshot is that it is then possible to write τ 1 as a random sum of random v ariables with exp onentially ligh t tails, where this random index ob eys a geometric distribution. In [6] the authors show ed that TBR W with Q = Ber( p ) for p ∈ (0 , 1] has a w ell-defined speed, that is, there exists a p ositive constant v = v ( p ) such that lim n →∞ dist( X n , o ) n = v ( p ) , almost surely . In [10], the authors generalize the abov e result and sho w that a TBR W with a leaf distribution Q ∈ Q κ for some p ositive κ has a w ell-defined sp eed as well, that is, (5) lim n →∞ dist( X n , o ) n = v ( Q ) , almost surely , for some constan t v that dep ends only on Q . In our second main result, we leverage the b ound giv en b y our Theorem 1 and sho w an exp onential concentration result for dist( X n , o ) whic h also works uniformly on Q κ . That is, we show that the probability that dist( X n , o ) deviates from v ( Q ) n b y εn decays exp onentially fast in n . More specifically , w e sho w the following result. Theorem 2 (Exp onential concen tration) . Consider a TBR W with le af distribution Q ∈ Q κ for a fixe d κ ∈ (0 , 1] . Then, ther e exists a p ositive c onstant c = c ( ε ) such that sup Q ∈Q κ P Q      dist( X n , o ) n − v ( Q )     > ε  ≤ e − cn . Our final main result is ab out the regularity of the sp eed v ( p ) as a function of p in the Bernoulli case. In [10], the authors sho wed that v ( p ) is contin uous on p . Here, w e substantially improv e their result by showing that v ( p ) is actually an analytic function of p . More precisely , we hav e the following result. EXPONENTIAL BOUNDS AND ANAL YTICITY FOR THE TREE BUILDER RANDOM W ALK 5 Theorem 3 (The speed is analytic) . Consider a TBR W with Q = Ber( p ) , for p ∈ (0 , 1] . Then, the sp e e d v ( p ) is analytic on p . The rest of this pap er is organized as follo ws: at Section 2 we introduce the renew al structure formally and show Theorem 1. A t Section 3, we sho w Theorem 2. A t Section 4, w e sho w Theorem 3. W e then conclude in Section 5 with a discussion of op en problems arising from this work. 2. Proof of Theorem 1: Exponential t ail bounds f or τ 1 In this part, we will define the renewal structure and show that the first renewal time has exp onential tail. Before w e define the renewal structure, let us say a couple w ords ab out the mark ovian nature of the TBR W. Observ e that the TBR W itself is a Marko v pro cess on the space of rooted trees, even though each of its comp onents separately are not. In some proofs, it will b e useful to mak e use of the (Strong) Mark ov Prop erties. F or that, w e will denote the shift op erator of length n by θ n . In light of the Mark ov Prop erties, it is important to p oint out that when the TBR W is shifted b y n steps and L n = Q for all n , the shifted pro cess is a TBR W that starts from a random initial state ( T n , X n ) and with the exact same sequence of la ws {L n } n . More formally , if ϕ : Ω → R is F -measurable, then the Simple Mark ov Prop erty w orks as follows E T 0 ,X 0 ; Q [ ϕ ◦ θ n | F n ] = E T n ,X n ; Q [ ϕ ] . 2.1. The Renewal Structure. Recall the definition of τ 1 in (3). The renew al structure is then defined inductively (6) τ k := τ 1 ◦ θ τ k − 1 + τ k − 1 . Giv en a vertex x , we will write H x for the hitting time to x , that is, the first time n when X n = x . W e will k eep track on the times the w alk er jumps to leav es at maximal distance from the ro ot and also when it backtrac ks to the father of these sp ecial lea ves. In order to do so, w e will introduce tw o sequences of stopping times whose definition dep end on each other. W e start defining ζ 0 ≡ 0; e H 0 = H f ( X 0 ) . Then, we let ζ 1 b e the following stopping time ζ 1 = inf { n > e H 0 : deg( X n ) = 1 , dist( X n , o ) = h ( T n ) , dist( X n , o ) > dist( X 0 , o ) } . In w ords, ζ 1 happ ens when the following conditions take place: after the w alk er visits the father of X 0 , it visits a leaf with maximal distance from the root, this leaf also having a greater distance from the root than the previous leaf, X 0 . Note that never visiting the father of X 0 again would imply that τ 1 happ ens at time 0. W e then put e H 1 to b e (7) e H 1 := inf { n > ζ 1 : X n = f ( X ζ 1 ) } , with the conv ention that the infimum ov er the empty set is infinit y . In words, e H 1 is the first time the walk er returns to the father of X ζ 1 after stepping on X ζ 1 for the first time. When the w alk er is transient, e H 1 can b e infinity . Then, we define ζ k and e H k for k > 1 inductively as follows (8) ζ k = ( ζ 1 ◦ θ e H k − 1 + e H k − 1 , if e H k − 1 < ∞ ∞ , e H k − 1 = ∞ , again with the con ven tion that inf ∅ = ∞ . (9) e H k = H f ( X 0 ) ◦ θ ζ k + ζ k . 6 CAIO AL VES AND RODRIGO RIBEIR O In w ords, ζ k happ ens when, after visiting the father of X ζ k − 1 , the walk er visits a leaf at maximal distance from the ro ot, and even further aw a y from the ro ot then the previous leaf visit at ζ k − 1 . F urthermore, e H k is the first time the w alker returns to the father of X ζ k . Recall that τ 1 is defined only at times for whic h a unique new maximum for the distance b et ween the walk er and the root is achiev ed. The times ( ζ k ) k ≥ 1 are the ones in which a regeneration could happen, but if ˜ H k is finite, then the regeneration fails. With that in mind, we define K to b e the random index (10) K = inf { k ≥ 0 : e H k = ∞} , and notice that w e can define τ 1 as (11) τ 1 := ζ K = K X j =1 ζ j − ζ j − 1 . The fact that w e can write τ 1 as the ab ov e sum will b e crucial to us. This new standp oin t is already telling us that the tail of τ 1 should b e as ligh t as the tails of ζ j − ζ j − 1 and K pro vided they b ehav e w ell enough. The next t wo lemmas go exactly in this direction dealing with the distributions of K and ζ j − ζ j − 1 . Lemma 1. L et K b e as in (10) . Then, for al l Q  = δ 0 , K ob eys a ge ometric distribution of p ar ameter P Q ( H o = ∞ ) . T o a v oid clutter with the notation, we will mak e use of the ∆ notation for the difference b etw een t wo random v ariables in a sequence. That is, for all k ∈ N , w e write (12) ∆ ζ k := ζ k − ζ k − 1 . Lemma 2. Fix κ ∈ (0 , 1] . Then ther e exists t 0 = t 0 ( κ ) > 0 such that for al l t < t 0 and j ≥ 1 sup Q ∈Q κ E Q h exp n t ∆ ζ j 1 { e H j − 1 < ∞} o    F ζ j − 1 i ≤ e t [ P κ ( H o < ∞ ) + ϵ ( t )] , wher e ϵ ( t ) is a function of t dep ending on κ only, satisfying ϵ ( t ) → 0 , when t → 0 . W e will p ostp one the pro of of the ab o v e lemmas to the end of this section. F or no w, w e will fo cus on showing how Theorem 1 follows from them. Pr o of of The or em 1 (Exp onential T ail Bounds for τ 1 ). Since τ 1 = ζ K and the even ts { K = k } partition Ω, since the walk er is transient in our settings, w e can decompose the probability of { τ 1 ≥ m } as (13) P Q ( τ 1 ≥ m ) = ∞ X k =1 P Q   k X j =1 ∆ ζ j ≥ m, K = k   . On the ev ent { K = k } , w e hav e e H 0 < ∞ , . . . , e H k − 1 < ∞ and e H k = ∞ , so in particular ζ k < ∞ (since in these settings the walk er is transient) and the walk er is at a leaf at time ζ k , by definition of ζ k . Using these observ ations and shifting the pro cess by ζ k , Strong Marko v prop erty yields (14) P Q  e H k = ∞    F ζ k  1 { ζ k < ∞} = P Q ( H o = ∞ ) 1 { ζ k < ∞} , P Q -a.s. EXPONENTIAL BOUNDS AND ANAL YTICITY FOR THE TREE BUILDER RANDOM W ALK 7 Applying the abov e identit y at ζ k on the ev en t { K = k } (where ζ k < ∞ ) and then dropping the conditions e H 0 < ∞ , . . . , e H k − 1 < ∞ gives the upp er b ound (15) P Q   k X j =1 ∆ ζ j ≥ m, K = k   ≤ P Q ( H o = ∞ ) P Q   k X j =1 ∆ ζ j 1 { e H j − 1 < ∞} ≥ m   . Notice that for an y t > 0, an application of Mark ov’s inequalit y together with Lemma 2 k times yields P Q   k X j =1 ∆ ζ j 1 { e H j − 1 < ∞} ≥ m   = P Q   exp    k X j =1 t ∆ ζ j 1 { e H j − 1 < ∞}    ≥ e tm   ≤ e − tm E Q   exp    k X j =1 t ∆ ζ j 1 { e H j − 1 < ∞}      ≤ e − tm e tk [ P κ ( H o < ∞ ) + ϵ ( t )] k . (16) Cho osing t small enough so that e t [ P κ ( H o < ∞ ) + ϵ ( t )] < 1, we obtain ∞ X k =1 P Q   k X j =1 ∆ ζ j 1 { e H j − 1 < ∞} ≥ m   ≤ e − tm ∞ X k =1 e tk [ P κ ( H o < ∞ ) + ϵ ( t )] k ≤ e − tm e t [ P κ ( H o < ∞ ) + ϵ ( t )] 1 − e t [ P κ ( H o < ∞ ) + ϵ ( t )] . (17) Plugging the ab ov e inequality bac k into (15) and using (13) yields P Q ( τ 1 ≥ m ) ≤ e − tm e t P κ ( H o = ∞ ) [ P κ ( H o < ∞ ) + ϵ ( t )] 1 − e t [ P κ ( H o < ∞ ) + ϵ ( t )] , whic h is enough to pro ve the theorem for a suitable adjustment of the constants. □ 2.2. Pro ofs of Lemmas 1 and 2. W e begin with pro of of Lemma 1 which is a consequence of Strong Mark ov prop ert y . Pr o of of L emma 1. F or any fixed k ∈ N , the definition of K in (10) giv es us that (18) P Q ( K = k ) = P Q ( e H 1 < ∞ , . . . , e H k − 1 < ∞ , e H k = ∞ ) . Since the e H ’s are stopping times, using (14) leads us to P Q ( e H 1 < ∞ , . . . , e H k − 1 < ∞ , e H k = ∞ ) = P Q ( e H 1 < ∞ , . . . , e H k − 1 < ∞ ) P Q ( H o = ∞ ) . T o finish the proof, recall that at time ζ k − 1 , the w alk er is at a leaf of maximal distance from the ro ot in T ζ k − 1 . Also that e H i < e H i +1 on the even t where e H i is finite. Thus, using Strong Marko v Prop erty again by shifting the pro cess b y ζ k − 1 P Q ( e H 1 < ∞ , . . . , e H k − 1 < ∞ ) = P Q ( e H 1 < ∞ , . . . , e H k − 2 < ∞ ) P Q ( H o < ∞ ) . Com bining the tw o ab ov e equations and using induction finishes the pro of. □ Lemma 2 will follo w from an exp onential tail b ound for ∆ ζ j on the even t that the walk er returns to the father of ζ j − 1 . W e will first state the result we need and sho w how Lemma 2 follows from it. Then we will finish this section sho wing the lemma. Lemma 3. Fix κ ∈ (0 , 1] and j ≥ 1 . Then ther e exists c = c ( κ ) > 0 such that sup Q ∈Q κ sup ( T ,x ) ∈T ∗ P T ,x ; Q  ∆ ζ j > m, e H j − 1 < ∞    F ζ j − 1  ≤ 2 e − cm 8 CAIO AL VES AND RODRIGO RIBEIR O W e can now prov e Lemma 2 Pr o of of L emma 2. W e will fix t > 0 and c ho ose it properly latter. Then w e hav e that (19) E Q h exp n t ∆ ζ j +1 1 { e H j < ∞} o    F ζ j i ≤ ∞ X m =1 P Q  ∆ ζ j +1 ≥ log( m ) t , e H j < ∞     F ζ j  . Notice that if m is smaller than e t , then by the Strong Marko v Prop erty P Q  ∆ ζ j +1 ≥ log( m ) t , e H j < ∞     F ζ j  = P Q  e H j < ∞    F ζ j  = P Q ( H o < ∞ ) 1 { ζ j < ∞} , (20) where we ha v e used that e H j is infinite if ζ j is infinite. On the other hand, if m is greater than e t , we can use Lemma 3 to obtain (21) P Q  ∆ ζ j +1 ≥ log( m ) t , e H j < ∞     F ζ j  ≤ 2 e − c log( m ) /t = 2 m − c/t , where c depends only on the constan t κ . Cho osing t < c , using the ab ov e t wo iden tities on (19), and using [10, Lemma 1] which states that P Q ( H o < ∞ ) ≤ P κ ( H o < ∞ ) , ∀ ∈ Q κ , w e obtain the following b ound E Q h exp n t ∆ ζ j +1 1 { e H j < ∞} o    F ζ j i ≤ e t P Q ( H o < ∞ ) + 2 ∞ X m =1 m − c/t ≤ e t P κ ( H o < ∞ ) + 2 t/ ( c − t ) = e t P κ ( H o < ∞ )  1 + 2 te − t ( c − t ) P κ ( H o < ∞ )  . Finally , setting ϵ ( t ) to b e ϵ ( t ) := 2 t e t ( c − t ) P κ ( H o < ∞ ) , w e conclude the pro of. □ W e are no w left with showing Lemma 3. As usual, w e need to in tro duce addi- tional notation and a few auxiliary results. On the even t where ζ k is finite, we define M k to b e as follo ws (22) M k = sup ζ k ≤ n ≤ e H k { dist( X n , o ) − dist( X ζ k , o ) } . If e H k is infinite, then w e set M k to b e infinite as well. In w ords, on the even t where e H k is finite, M k returns how far X wen t b efore returning to the father of ζ k . In [6] the authors constructed a coupling of the distance process { dist( X n , o ) } n with a right-biased random w alk { S k } k on Z so that the right-biased random w alk is alwa ys closer to 0. More formally , in Lemma 5.12 of [6], the authors show ed that fixed q ∈ (1 / 2 , 1), there exists a natural num b er r that might depend on Q and a sequence of stopping times { σ ( r ) k } k suc h that (1) | σ ( r ) k − σ ( r ) k − 1 | ≤ e √ r , almost surely for all k (2) | dist( X σ ( r ) k , o ) − dist( X σ ( r ) k − 1 , o ) | ≤ r , almost surely for all k (3) P  dist( X σ ( r ) k , o ) ≥ rS k  = 1, EXPONENTIAL BOUNDS AND ANAL YTICITY FOR THE TREE BUILDER RANDOM W ALK 9 where { S k } k is a random walk on Z with probability q of jumping to the right. In Lemma 5 of [10], the authors sho wed that the constant r can b e c hosen uniformly across all Q ∈ Q k , for κ ∈ (0 , 1]. This wa y , the constant r dep ends only on the constan t κ and on q and do es not depend on the initial state ( T 0 , x 0 ) or on other prop erties of the probability Q . Roughly sp eaking, this coupling says that all TBR W having a probabilit y at least κ of adding at least one v ertex at each step are all being pushed aw a y from the ro ot at least as fast as the right-biased random w alk { S k } k is moving aw ay from its initial p osition. W e will lev erage this coupling to first sho w the lemma b elow which states that M k has exp onential tails when the w alk er returns to the father of X ζ k . Then, w e will show Lemma 3. Lemma 4. Fix κ ∈ (0 , 1] . Then ther e exists c = c ( κ ) > 0 such that sup Q ∈Q κ sup ( T ,x ) ∈T P T ,x ; Q ( M k > m, e H k < ∞ ) ≤ e − cm Pr o of. Let H ∗ m b e the first time the walk er reaches distance m from its initial p osition x . That is, (23) H ∗ m := inf { n > 0 : dist( X n , o ) − dist( X 0 , o ) > m } . Then, shifting the pro cess by ζ k and using Strong Mark ov Prop ert y , we ha v e that (24) P T ,x ; Q ( M k > m, e H k < ∞ ) = E T ,x ; Q h 1 { ζ k < ∞} P T ζ k , X ζ k ; Q  H ∗ m < H f ( X 0 ) < ∞  i , since the w alk er must reac h distance m from its initial p osition then visit the father of its initial p osition after that. Then, shifting the process b y H ∗ m w e ha v e a process starting at distance m from its initial position that manages to visit the father of its initial position in finite time. Ho w ever, by the coupling with the right-biased random walk (fixing q as, sa y , 2 / 3) there exists a constant c depending on κ only suc h that P T ζ k , X ζ k ; Q  H ∗ m < H f ( X 0 ) < ∞  ≤ e − cm , P T ,x ; Q -a.s. Then, using the ab o ve b ound in (24), obtain P T ,x ; Q ( M k > m, e H k < ∞ ) ≤ e − cm . whic h is what we desired. □ The reader ma y already ha ve an idea on how Lemma 3 follows from the ab ov e b ound and the coupling with the right-biased random walk. The random incre- men t ∆ ζ j measures the time it takes from the w alker to tak e an excursion on the tree below ζ j − 1 and clim b back to the father of ζ j − 1 . On the other hand, b y Lemma 4, if the w alk er made that return, it did not go too far from X ζ j − 1 . Th us, after stepping on the father of ζ j − 1 , the right-biased random w alk will push the w alk er all the w ay down again at linear sp eed, making the w alker co v er the excursion it constructed b efore in linear time. With the ab o v e in mind, w e are finally ready to sho w Lemma 3 whic h was the last result left to conclude the pro of of Theorem1. Pr o of of L emma 3. W e would like to b ound from ab ov e the following probability P T ,x ; Q  ∆ ζ j > m, e H j − 1 < ∞    F ζ j − 1  . In order to do that, we will intersect the even t { ∆ ζ j > m, e H j − 1 < ∞} with { M j − 1 ≤ εm } and its complement. Notice that b y Strong Marko v Prop erty and Lemma 4, it follows that (25) P T ,x ; Q  M j − 1 > εm, e H j − 1 < ∞    F ζ j − 1  ≤ e − cm , P T ,x ; Q -a.s. , 10 CAIO AL VES AND RODRIGO RIBEIR O for some p ositive constant c dep ending on ε and κ . The next step is to b ound P T ,x ; Q  ∆ ζ j > m, M j − 1 ≤ εm, e H j − 1 < ∞    F ζ j − 1  . Notice that by the construction of ζ j , when the pro cess is shifted b y ζ j − 1 , we hav e that the ab ov e probability equals P T ζ j − 1 ,X ζ j − 1 ; Q ( ζ > m, M ≤ εm, H f ( X 0 ) < ∞ ) , where ζ is the first time the walk er reac hes maximal distance after visiting the father of its initial position ( f ( X 0 )) for the first time and M is ho w far the walk er trav eled b efore visiting f ( X 0 ) for the first time. By shifting the pro cess again by H f ( X 0 ) , we ha ve that ζ measures the time to hit the b ottom of T H f ( X 0 ) , kno wing that f ( X 0 ) is at distance at most εm from the bottom of T H f ( X 0 ) . Using the coupling with the biased random walk, and mak e ε sufficien tly small, w e kno w that the biased random w alk cannot take to o long to co v er a distance of at most εm . This discussion gives us that there exists another p ositive constant dep ending on ε and κ only , such that (26) P T ,x ; Q  ∆ ζ j > m, M j − 1 ≤ εm, e H j − 1 < ∞    F ζ j − 1  ≤ e − cm , P a.s. Com bining the ab ov e b ound with (25) gives us that P T ,x ; Q  ∆ ζ j > m, e H j − 1 < ∞    F ζ j − 1  ≤ 2 e − cm , for some constant c depending on ε and κ only . This concludes the pro of. □ 3. Proof of Theorem 2: Exponential concentra tion Before showing Theorem 2, we will need t wo auxiliary lemmas and in tro duce some additional notation to av oid clutter. Throughout this section, c, c ( ε ) denote p ositiv e constan ts dep ending only on ε and κ whose v alue may change from line to line. W e let µ τ b e (27) µ τ := E Q [ τ 1 | H o = ∞ ] , and µ D is defined as (28) µ D := E Q [dist( X τ 1 , o ) | H o = ∞ ] It is also helpful to recall the definition of τ m giv en at page 5 Equation (6). Lemma 5. Fix κ ∈ (0 , 1] . Then, for al l ε > 0 , ther e exists c = c ( ε, κ ) such that for al l m ∈ N the fol lowing b ound holds (29) sup Q ∈Q κ P Q ( | τ m − mµ τ | > εm ) ≤ e − c ( ε ) m . Pr o of. This lemma is a consequence of the renew al structure for the TBR W con- structed in [10] and our Theorem 1. By Theorem 2 of [10], w e hav e that the random v ariables τ 1 , τ 2 − τ 1 , . . . , τ k − τ k − 1 , . . . are independent with τ k − τ k − 1 distributed as τ 1 conditioned on { H o = ∞} . Notice that for eac h m , we can write τ m = τ 1 + m X k =2 ( τ k − τ k − 1 ) , whic h is a sum of indep endent random v ariables with finite MGF by our Theorem 1. The exponential deca y then follo ws from Chernoff bounds. The fact that the bound holds uniformly ov er Q ∈ Q κ also follows from Theorem 1, whic h holds uniformly o ver Q ∈ Q κ . □ EXPONENTIAL BOUNDS AND ANAL YTICITY FOR THE TREE BUILDER RANDOM W ALK 11 F or the second auxiliary lemma, w e will need an extra definition. F or each t ∈ N , let N ( t ) be as follows (30) N ( t ) = sup { k : τ k ≤ t } . That is, N ( t ) coun ts the n umber of regenerations before time t . Our next result guaran tees an exp onential concentration for N ( t ). Lemma 6. Fix κ ∈ (0 , 1] . Then, for al l ε > 0 , ther e exists a p ositive c onstant c = c ( ε, κ ) such that (31) sup Q ∈Q κ P Q      N ( t ) − t µ τ     > εt  ≤ e − ct . Pr o of. W e start by defining (32) m + = m + ( t, ε ) =  1 µ τ + ε  t and therefore, (33) N ( t ) >  1 µ τ + ε  t = ⇒ τ ⌈ m + ⌉ ≤ t = µ τ ⌈ m + ⌉  1 − εµ τ + O ( ε 2 µ 2 τ )  , whic h is possible since, by Theorem 1 and the definition of τ 1 , for all Q ∈ Q κ , µ τ is b ounded from ab ov e and from b elow by p ositive constants dep ending on κ only . Finally , using Lemma 5, we can make ε small enough so that (34) P Q  N ( t ) >  1 µ τ + ε  t  ≤ exp {− c ( ε ) m + } = exp  − c ( ε )  1 µ τ + ε  t  , since the even ts { N ( t ) > m + } and { τ ⌈ m + ⌉ ≤ t } are the same. The pro of for the upp er b ound on N ( t ) follows in the same manner. □ W e can finally show Theorem 2 using the ab o v e t w o lemmas and Theorem 1. Pr o of of The or em 2. Note that P Q ( N ( t ) = 0) = P Q ( τ 1 > t ) ≤ e − ct b y Theorem 1, so it suffices to work on the ev en t { N ( t ) ≥ 1 } . In this ev en t, dist( X t , o ) can b e written as dist( X t , o ) = dist( X τ 1 , o ) + N ( t ) − 1 X k =1  dist( X τ k +1 , o ) − dist( X τ k , o )  (35) + dist( X t , o ) − dist( X τ N ( t ) , o ) . W e prov e the bound for the probabilit y of dist( X t , o ) /t exceeding v ( Q ) by ε , since the matching bound is prov ed in a similar wa y: on the ev en t { dist( X t , o ) < ( v ( Q ) − ε ) t } , drop the nonnegative first and last summands on the right-hand side of (35) to reduce to b ounding the sum of i.i.d. increments from ab ov e, and then apply the same Chernoff argument. By the union b ound, w e can write P Q (dist( X t , o ) > ( ε + v ( Q )) t ) ≤ P Q  dist( X τ 1 , o ) > ε 3 t  (36) + P Q   N ( t ) − 1 X k =1  dist( X τ k +1 , o ) − dist( X τ k , o )  >  ε 3 + v ( Q )  t   + P Q  dist( X t , o ) − dist( X τ N ( t ) , o ) > ε 3 t  Since dist( X t , o ) − dist( X s , o ) ≤ t − s (the w alker mo v es at most one step p er unit time), we hav e dist( X τ 1 , o ) ≤ τ 1 + 1 , dist( X t , o ) − dist( X τ N ( t ) , o ) ≤ τ N ( t )+1 − τ N ( t ) . 12 CAIO AL VES AND RODRIGO RIBEIRO Recall that τ m +1 − τ m is distributed as τ 1 conditioned on { H o = ∞} . By [10, Lemma 1], P Q ( H o < ∞ ) ≤ P κ ( H o < ∞ ) < 1 for all Q ∈ Q κ , so inf Q ∈Q κ P Q ( H o = ∞ ) ≥ P κ ( H o = ∞ ) > 0. Therefore the conditional tail satisfies P Q ( τ 1 ≥ n | H o = ∞ ) ≤ P κ ( H o = ∞ ) − 1 e − cn , and Theorem 1 gives exponential b ounds for b oth the unconditional and conditional distributions. Thus the first and third summands in the right-hand side of (36) are b ounded from ab ov e by exp {− c ( ε ) t } . Therefore, we no w focus on b ounding the second summand. By Lemma 6, w e ha v e (37) P Q  N ( t ) t − 1 µ τ > ε 6 µ 2 τ  ≤ exp {− c ( ε ) t } Therefore, again by the union b ound, the result will follow if we provide a suitable upp er b ound for (38) P Q   ⌈ ( t/µ τ )(1+ ε/ 6 µ τ ) ⌉ X k =1  dist( X τ k +1 , o ) − dist( X τ k , o )  >  ε 3 + v ( Q )  t   . By [10, Theorem 3], we hav e v ( Q ) = µ D /µ τ and the ab ov e b ecomes (39) P Q   ⌈ ( t/µ τ )(1+ ε/ 6 µ τ ) ⌉ X k =1  dist( X τ k +1 , o ) − dist( X τ k , o )  > t µ τ  εµ τ 3 + µ D    . W riting m t = ⌈ ( t/µ τ )(1 + ε/ 6 µ τ ) ⌉ , w e note that µ D ≤ µ τ (since dist( X τ 1 , o ) ≤ τ 1 + 1 and µ τ ≥ 1) and that µ τ is b ounded ab ov e uniformly ov er Q ∈ Q κ (b y the uniform exp onen tial tail of τ 1 ). Therefore, for sufficiently small ε , the abov e probability is b ounded from ab ov e by (40) P Q m t X k =1  dist( X τ k +1 , o ) − dist( X τ k , o )  > m t  εµ τ 10 + µ D  ! . But now dist( X τ k +1 , o ) − dist( X τ k , o ) are i.i.d. random v ariables with mean µ D and a momen t generating function, since they are stochastically dominated b y τ k +1 − τ k . Since m t is linear in t , a Chernoff b ound finishes the pro of of the result. □ 4. Proof of Theorem 3: v ( p ) is anal ytic F or each p ∈ (0 , 1], w e know that the walk er has a well-defined speed v ( p ), which is given b y the limit of dist( X n , o ) /n as n go es to infinit y . In this section, we will sho w that the function p 7→ v ( p ) is analytic on (0 , 1]. The main to ol for showing this result is complex analysis. W e will follo w the tec hnique given by [7] in the context of b ond p ercolation in Z d . In a n utshell, the argument consists of seeing probabilities as functions of the parameter p , and then extending them to the complex num b ers where the W eierstrass Theorem (Theorem 5) and his M -test (Theorem 4) can b e applied. The goal is then to prop erly b ound the modulus of the complex extensions so the M -test can b e used. With the abov e discussion in mind, w e will mak e the following abuse of notation: giv en an ev ent A , we denote by P z ( A ) the analytic complex extension of the real function p 7→ P p ( A ). Although w e kno w that P z ( A ) is meaningless in terms of probabilit y , this notation has the adv antage of being more visually appealing and making some expressions nicer. Also, it sa ves us the burden of in tro ducing a new notation for ev ery new extension that appears in our arguments. F or instance, if P p ( A n ) = P q n (1 − p ) n − i p i , then P z ( A n ) = P q n (1 − z ) n − i z i for z ∈ C . W e will denote the complex ball of radius r around z as B C ( z , r ). EXPONENTIAL BOUNDS AND ANAL YTICITY FOR THE TREE BUILDER RANDOM W ALK 13 F or the pro of of Theorem 3, we will need the exp onential tail b ounds on τ 1 giv en b y Theorem 1 and the following technical lemmas Lemma 7. Fix p ∈ (0 , 1] , n ∈ N and 0 < r < p . Then, for any z ∈ B C ( p, r ) the fol lowing b ound holds | P z ( A n ) | ≤ exp { (2 r / ( p − r )) n } P p − r ( A n ) for al l A n ∈ F n . Lemma 8. Ther e exists an analytic extension of the function p 7→ P p ( H o = ∞ ) to an op en neighb orho o d of C c ontaining the interval (0 , 1) . We denote such function by P z ( H o = ∞ ) . Lemma 9. L et B n ∈ F n , then ther e exists A n ∈ F n , such that (41) P p ( B n , τ 1 = n, H o = ∞ ) = P p ( A n ) P p ( H o = ∞ ) and that (42) P p ( A n ) ≤ P p ( τ 1 ≥ n ) . W e will p ostp one the pro of of the ab o v e lemmas to the end of this section. F or no w, w e will fo cus on how Theorem 3 follows from them. Pr o of of The or em 3: v ( p ) is analytic. Lev eraging the renewal structure, in Theo- rem 3 of [10], the authors show ed the following expression for v ( p ) (43) v ( p ) = E p [dist( X τ 1 , o ) ; H o = ∞ ] E p [ τ 1 ; H o = ∞ ] . W e will show that b oth n umerator and denominator are analytic functions. E p [ τ 1 ; H o = ∞ ] is analytic. W e will use Theorems 5 and 4 to show that the func- tion defined b elow is analytic E z [ τ 1 ; H o = ∞ ] := ∞ X n =1 nP z ( τ 1 = n, H o = ∞ ) . (44) By Lemma 9, there exists an F n -measurable even t A n suc h that P p ( τ 1 = n, H o = ∞ ) := P p ( A n ) P p ( H o = ∞ ) . W e can then extend P p ( τ 1 = n, H o = ∞ ) to a complex neigh b orhoo d of (0 , 1] as the pro duct of t wo functions (45) P z ( τ 1 = n, H o = ∞ ) := P z ( A n ) P z ( H o = ∞ ) . Note that the pro duct ab ov e is an analytic function by Lemma 8 and the fact that P p ( A n ) is a p olynomial function of p (see the pro of of Lemma 7 b elo w). No w, fix p ∈ (0 , 1] and r > 0 with 0 < r < p . Then, b y Lemma 7, it follows that for all z ∈ B C ( p, r ) | P z ( A n ) | ≤ exp  2 r n p − r  P p − r ( A n ) , whic h com bined with (42) of Lemma 9 and Theorem 1 yields | P z ( A n ) | ≤ exp  2 r n p − r  P p − r ( τ 1 ≥ n ) ≤ exp  2 r n p − r  e − cn , for a constant c whic h can b e chosen uniformly across all p ∈ ( p 0 , 1], for p 0 > 0. This implies that b y making r smaller if needed, we obtain the following upp er b ound for all z ∈ B C ( p, r ) (46) | P z ( A n ) | ≤ e − cn/ 2 . 14 CAIO AL VES AND RODRIGO RIBEIRO Also, by choosing a smaller r > 0 if needed, we guaran tee that z 7→ P z ( H o = ∞ ) is analytic in the compact ball ¯ B C ( p, r ). Therefore, there exists C > 0 suc h that (47) | P z ( τ 1 = n, H o = ∞ ) | ≤ C e − cn/ 2 . By W eierstrass Theorems 4 and then 5, w e ha ve that E z [ τ 1 ; H o = ∞ ] is indeed w ell defined and analytic on B C ( p, r ) (the M -test gives uniform conv ergence on the compact ball ¯ B C ( p, r ), and the W eierstrass theorem then yields analyticit y on the op en ball). Since p was arbitrary and it extends E p [ τ 1 ; H o = ∞ ], w e obtain the desired result. E p [dist( X τ 1 , o ) ; H o = ∞ ] is analytic. The strategy for the numerator of v ( p ) is the same as for the denominator, but the b o okkeeping differs. W e write E p [dist( X τ 1 , o ) ; H o = ∞ ] = ∞ X k =1 k P p (dist( X τ 1 , o ) = k , H o = ∞ ) . Expanding P p (dist( X τ 1 , o ) = k , H o = ∞ ) = P ∞ j = k P p (dist( X j , o ) = k , τ 1 = j, H o = ∞ ) (note that dist( X j , o ) = k requires j ≥ k ), we obtain E p [dist( X τ 1 , o ) ; H o = ∞ ] = ∞ X k =1 k ∞ X j = k P p (dist( X j , o ) = k , τ 1 = j, H o = ∞ ) . Since { dist( X j , o ) = k } ∈ F j , by Lemma 9 (applied with B j = { dist( X j , o ) = k } ), there exists an ev ent A j,k ∈ F j suc h that P p (dist( X j , o ) = k , τ 1 = j, H o = ∞ ) = P p ( A j,k ) P p ( H o = ∞ ) . Then, exactly as in (45), w e can extend P p (dist( X j , o ) = k , τ 1 = j, H o = ∞ ) as follo ws (48) P z (dist( X j , o ) = k , τ 1 = j, H o = ∞ ) = P z ( A j,k ) P z ( H o = ∞ ) , where P z ( A j,k ) is a polynomial function on z , as b efore. Pro ceeding as in the denominator case, b y Lemma 7 and (42), w e deduce that for r sufficien tly small and all z ∈ B C ( p, r ) (49) | P z (dist( X j , o ) = k , τ 1 = j, H o = ∞ ) | ≤ C e − c ′ j , for p ositive constants C , c ′ . T o apply the W eierstrass M -test we need ∞ X k =1 k ∞ X j = k C e − c ′ j = C ∞ X k =1 k · e − c ′ k 1 − e − c ′ < ∞ , whic h holds since P ∞ k =1 k e − c ′ k < ∞ . This is enough to conclude the proof b y applying Theorem 4 and then 5. □ 4.1. Pro of of Lemma 7. T o conclude the pro of the Theorem 3, we sho w Lemmas 7, 8 and 9. But before, let us in tro duce some additional notation that will be needed throughout the coming pro ofs. W e denote a realization of the first n steps of the TBR W as a sequence of tree-walk er tuples (50) (( T 1 , x 1 ) , . . . , ( T n , x n )) ≡ (  T ,  x ) n F or eac h n, i ∈ N and A n ∈ F n , let S i ( A n ) denote the realizations (  T ,  x ) n ∈ A n suc h that | T n | = i . W e then let P (  T ,  x ) n ( p ) the probabilit y of observing exactly (  T ,  x ) n with the coin parameter p . No w we are ready for the first proof. EXPONENTIAL BOUNDS AND ANAL YTICITY FOR THE TREE BUILDER RANDOM W ALK 15 Pr o of L emma 7. W e b egin observing that for each n , we hav e (51) P p ( A n ) = n X i =0 X (  T ,  x ) n ∈ S i ( A n ) P (  T ,  x ) n ( p ) . Notice that for an y pair (  T ,  x ) n ∈ S i ( A n ), it follows that (52) P (  T ,  x ) n ( p ) = q n (1 − p ) n − i p i , where q n = q n ((  T ,  x ) n ) is the pro duct of jump probabilities of the n steps of the w alker. Whereas (1 − p ) n − i p i accoun ts for the probability of adding exactly i new v ertices in n steps. No w, fixing p and choosing r suc h that 0 < r < p , we hav e that for z ∈ B C ( p, r ), (53) | z | ≤ p + r and | 1 − z | ≤ 1 − p + r. Indeed, the first inequality follows from the triangular inequality , and the second, from the fact that the the distance to 1 in B C ( p, r ) is maximized at z = p − r . Thus, using the ab ov e inequalities, for any sequence (  T ,  x ) n ∈ S i ( A n ), it follows that    P (  T ,  x ) n ( z )    = q n | 1 − z | n − i | z | i ≤ q n (1 − p + r ) n − i ( p + r ) i . (54) Dividing and multiplying by ( p − r ) i , the expression for P (  T ,  x ) n ( p − r ) appears, yielding (55) | P (  T ,  x ) n ( z ) | ≤ ( p + r ) i ( p − r ) i P (  T ,  x ) n ( p − r ) . Com bining the abov e inequalit y to (51) together with triangle inequality and that ( p + r ) / ( p − r ) > 1, w e obtain | P z ( A n ) | ≤ n X i =0 X (  T ,  x ) n ∈ S i ( A n ) ( p + r ) i ( p − r ) i P (  T ,  x ) n ( p − r ) ≤ ( p + r ) n ( p − r ) n n X i =0 X (  T ,  x ) n ∈ S i ( A n ) P (  T ,  x ) n ( p − r ) ≤  1 + 2 r p − r  n P p − r ( A n ) ≤ exp  2 r n p − r  P p − r ( A n ) , (56) concluding the pro of. □ 4.2. Pro of of Lemma 8: P z ( H o = ∞ ) is analytic. Pr o of of L emma 8. Since P p ( H o = ∞ ) = 1 − P p ( H o < ∞ ), it is enough to show that P p ( H o < ∞ ) is analytic. W e know that for any p (57) P p ( H o < ∞ ) = ∞ X n =0 P p ( H o = n ) . Notice that P p ( H o = n ) is a polynomial function on p . T o see that, just split the ev ent { H o = n } into all possible trees on at most n vertices and p ossible mo v emen ts the walk er can mak e leading it to o for the first time at time n . Moreov er, b y Lemma 7 applied to { H o = n } ∈ F n , it follows that (58) | P z ( H o = n ) | ≤ e 2 nr/ ( p − r ) P p − r ( H o = n ) . 16 CAIO AL VES AND RODRIGO RIBEIRO On the other hand, b y Theorem 1, w e ha ve that there is an absolute constan t c ′ suc h that (59) P p − r ( H o = n ) ≤ P p − r ( τ 1 > n ) ≤ e − c ′ n , for all p ∈ ( p 0 , 1]. T o see the inclusion of even ts, notice that if H o = n , due to the tree structure, for an y leaf the walk er migh t hav e p ossible visited b efore time n , it m ust return to its father in order to visit the ro ot at time n . Th us, none of the lea ves visited b efore time n can b e a regeneration p oint. Moreov er, at time n the w alker is at the ro ot, so it cannot regenerate from there either. This makes τ 1 > n . Finally , for each n , let M n b e M n := e 2 rn/ ( p − r ) P p − r ( H o = n ) . Then, for r sufficien tly small, w e hav e that (60) X n M n < ∞ . Recall that the radius r can b e made as small as we wan t and the constant c ′ do es not dep end on p as long as p ∈ ( p 0 , 1], for some fixed p 0 . Finally , by the W eierstrass M -test (Theorem 4) and (57) it follows that P z ( H o < ∞ ) is analytic, concluding the pro of. □ 4.3. Pro of of Lemma 9. Finally , we mo ve to wards the proof of the last lemma needed for the pro of of Theorem 3. Pr o of of L emma 9. W e start observing that for each n , { B n , τ 1 = n, H o = ∞} = { B n , ζ K = n, H o = ∞} = n [ j =1 { B n , ζ j = n, K = j, H o = ∞} = n [ j =1 n B n , ζ j = n, e H 1 < ∞ , . . . , e H j − 1 < ∞ , e H j = ∞ , H o = ∞ o (61) Notice that n B n , ζ j = n, e H 1 < ∞ , . . . , e H j − 1 < ∞ , e H j = ∞ , H o = ∞ o = n B n , ζ j = n, e H 1 < n, . . . , e H j − 1 < n, e H j = ∞ , H o > n o (62) Indeed, at time n the walk er is at a leaf and do es not return to its father anymore. Th us, due to the tree structure of the graph, in order to visit the father of X ζ i , for i < j , the w alker must visit it b efore time n , otherwise, it w ould need to visit the father of X ζ j in finite time. Also, for the walk er to never visit the ro ot o in the ab o v e even t, it is equiv alen t that it do es not visit it b efore time n , given that it will not return the father of X n after time n . Then, by (62) and the fact that ζ j , e H i ’s and H o are all stopping times, w e hav e that (63) n B n , ζ j = n, e H 1 < n, . . . , e H j − 1 < n, e H j = ∞ , H o > n o = A n,j ∩ n e H j = ∞ o , where A n,j ∈ F n is the even t b elow (64) A n,j = n B n , ζ j = n, e H 1 < n, . . . , e H j − 1 < n, H o > n o . EXPONENTIAL BOUNDS AND ANAL YTICITY FOR THE TREE BUILDER RANDOM W ALK 17 Ha ving the definition of e H j in mind, the iden tity b elo w holds P p -a.s. (65) 1 { H f ( X 0 ) = ∞} ◦ θ n · 1 A n,j = 1 { e H j = ∞} · 1 A n,j . Th us, Simple Marko v Prop erty yields P p  A n,j , e H j = ∞    F n  = 1 A n,j P T n ,X n ; p  H f ( X 0 ) = ∞  . (66) Recall that on A n,j , X n is on a leaf at the b ottom of T n . Th us, w e can couple the TBR W starting from ( T n , X n ) with ( T n , X n ) in A n,j with a TBR W starting from the nonro ot tip of an edge until the first time the former visits the father of X n and the latter visits the ro ot. In this w a y , the follo wing holds P p -almost surely (67) 1 A n,j P T n ,X n ; p  H f ( X 0 ) = ∞  = 1 A n,j P p ( H o = ∞ ) . Ha ving (63) in mind and com bining the ab ov e identit y with (66) gives us that (68) P p  B n , ζ j = n, e H 1 < n, . . . , e H j − 1 < n, e H j = ∞ , H o > n  = P p ( A n,j ) P p ( H o = ∞ ) . No w going back to (61) and using the ab ov e identit y yields (69) P p ( B n , τ 1 = n, H o = ∞ ) = P p ( H o = ∞ ) n X j =1 P p ( A n,j ) . Notice that by the definition of A n,j it follows that { A n,j } j is a sequence of disjoin t F n -measurable even ts. Thus, (70) P p ( B n , τ 1 = n, H o = ∞ ) = P p  ∪ n j =1 A n,j  P p ( H o = ∞ ) , whic h prov es the first part of the lemma by setting A n := ∪ n j =1 A n,j . T o conclude the proof, observe that for a fixed j b y the definition of τ 1 , on the even t A n,j , τ 1 ≥ ζ j = n . □ 5. Open pr oblems W e conclude with some op en questions arising from this work. 0.0 0.2 0.4 0.6 0.8 1.0 p 0.00 0.05 0.10 0.15 0.20 0.25 d ( X n , ) / n S p e e d o f t h e w a l k e r a s a f u n c t i o n o f p Figure 2. Estimated sp eed dist( X X n , o ) /n of the TBR W walk er as a function of the growth parameter p , a veraged ov er 100 inde- p enden t runs of 2000 steps each. The sp eed seems to b e increasing in p . 18 CAIO AL VES AND RODRIGO RIBEIRO (i) Monotonicit y of the sp eed. Theorem 3 establishes that the sp eed v ( p ) is an analytic function of p ∈ (0 , 1]. A natural follow-up question, prop osed b y Y. Peres to the second author in p ersonal comm unication, is whether v ( p ) is monotone increasing in p . Intuitiv ely , as p increases, the w alker adds leav es more frequently , pro ducing a stronger drift aw a y from the ro ot, whic h suggests that v ( p ) should be increasing, see Figure 2. Ho wev er, we do not hav e a formal pro of of this result; (ii) Smo othness of v ( p ) at p = 0 . One can try to study the right deriv ativ es of v ( p ) at 0, finding the correct smo othness class of the function. Figure 2 suggests a v alue of 0 for the righ t deriv ativ e at 0. (iii) Phase transition on elliptic case. F or the case in whic h L n = Ber( n − γ ), in [5], the authors show ed that when γ > 1 / 2, the w alker is recurrent. The regime γ ∈ (0 , 1 / 2] is b elieved to be transien t but it is still an op en problem. (iv) Large deviation principle for the empirical sp eed. In Theorem 2, we pro vide an upp er b ound for the large deviations of the empirical sp eed, it remains an op en problem to obtain a matc hing low er b ound, and c haracterizing the constant in the exp onential as a function of ε . Appendix A. Complex Anal ysis resul ts Theorem 4 (W eierstrass M -test) . L et f n b e a se quenc e of c omplex-value d functions define d on a subset Ω of the plane and assume that ther e exist p ositive numb ers M n with | f n ( z ) | ≤ M n for every z ∈ Ω , and P n M n < ∞ . Then, P n f n c onver ges uniformly on Ω . Theorem 5 (W eierstrass Theorem) . L et f n b e a se quenc e of analytic functions define d on an op en subset Ω of the plane, which c onver ges uniformly on the c omp act subsets of Ω to a function f . Then f is analytic on Ω . Mor e over, f ′ n c onver ges uniformly on the c omp act subsets of Ω to f ′ . A cknowledgements Caio Alves was partially supp orted by the CNPq gran t 447397/2024-9. References [1] Luca Av ena, F rank den Hollander, and F rank Redig. Law of large num b ers for a class of random walks in dynamic random environmen ts. Electr onic Journal of Pr obability , 16:587– 617, 2009. [2] Marek Biskup. Recen t progress on the random conductance model. Pr ob ability Surveys , 8:294–373, 2011. 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The A nnals of Prob ability , 3(1):1–31, 1975. [13] Alain-Sol Sznitman. Slowdo wn estimates and central limit theorem for random walks in ran- dom environmen t. Journal of the Eur op e an Mathematic al So ciety , 2(2):93–143, 2000. [14] Alain-Sol Sznitman and Martin P . W. Zerner. A law of large numbers for random walks in random environmen t. The Annals of Pr obability , 27(4):1851–1869, 1999. [15] Ofer Zeitouni. Random walks in random environmen t. In Le ctur es on Pr ob ability The ory and Statistics , volume 1837 of Le ctur e Notes in Mathematics , pages 189–312. Springer, 2004. (Caio Alves) FH Technikum Wien, Vienna, Austria Email address , Caio Alves: caio teodoro.de magalhaes alves@technikum-wien.at (Rodrigo Rib eiro) IMP A Tech, Rio de Janeiro, Brazil Email address , Ro drigo Rib eiro: rodrigo.ribeiro@impatech.edu.br