Range an spee of rotor wals on trees Wilfrie Huss an Ecaterina Sava-Huss May 15, 1 Abstract We prove a law of large numbers for the range of rotor wals with ranom initial configuration on regular trees an on Galton-Watson trees. We also show the existence of the rate of escape for such rotor wals. More precisely, we show that on the classes of trees uner consieration, even in the case when the rotor wal is recurrent, the range grows at linear spee. Key wors an phrases. rotor wal, range, rate of escape, Galton-Watson tree, generating function, law of large numbers, recurrence, transience, contour function. Contents 1 Introuction Preliminaries 5.1 Rotor wals.................................... 5. Galton-Watson trees............................... 3 Range on regular trees 7 3.1 Recurrent rotor wals.............................. 3.1.1 Positive recurrent rotor wals..................... 3.1. Null recurrent rotor wals....................... 13 3. Transient rotor wals.............................. 1 3.3 Uniform rotor wals............................... 1 Rate of escape on regular trees 5 Rotor wals on Galton-Watson trees 5.1 Range an rate of escape on Galton-Watson trees.............. 5 5. Simulations on Galton-Watson trees...................... 7 1
1. Introuction A The contour of a subtree 9 A.1 Contour of the range: recurrent case...................... 9 1 Introuction For let T be the roote regular tree of egree + 1, an enote by r the root. We attach an aitional sin vertex o to the root r. We use the notation T = T {o} to enote the tree without the sin vertex. For each vertex v T we enote its neighbors by v (), v (1),..., v (), where v () is the parent of v an the other neighbors, the chilren of v, are orere counterclocwise. Figure 1: The binary tree T. Each vertex v is enowe with a rotor ρ(v) {,..., }, where ρ(v) = j, for j {,..., } means that the rotor in v points to neighbor v (j). Let (X n ) n N be a rotor wal on T starting in r with initial rotor configuration ρ = (ρ(v)) v T : for all v T, let ρ(v) {,..., } be inepenent an ientically istribute ranom variables, with istribution given by P[ρ(v) = j] = r j with j= r j = 1. The rotor wal moves in this way: at time n, if the waler is at vertex v, then it first rotates the rotor to point to the next neighbor in the counterclocwise orer an then it moves to that vertex, that is X n+1 = v (ρ(v)+1) mo (+1). If the initial rotor configuration is ranom, then once a vertex has been visite for the first time, the configuration there is fixe. A chil v (j) of a vertex v T is calle goo if ρ(v) < j, which means that the rotor wal will first visit the goo chilren before visiting the parent v () of v. Remar that v has ρ(v) goo chilren. The tree of goo chilren for the rotor wal (X n ), which we enote T goo, is a subtree of T, where all the vertices are goo chilren. Let us enote by R n = {X,..., X n } the range of the rotor wal up to time n, that is, the set of istinct visite points by the rotor wal (X n ) up
1. Introuction to time n. Its carinality, enote by R n represents then the number of istinct visite points by the waler up to time n. We enote by (r, X n ) := X n the istance from the position X n at time n of the rotor waler to the root r. The spee or the rate of escape of the rotor wal (X n ) is the almost sure limit (if it exists) of Xn. We say that R n n satisfies a law of large numbers if Rn converges almost surely to a constant. The aim of this wor is to n prove a law of large numbers for X n an R n, that is, to fin constants l an α such that X n lim n n R n lim n = l, almost surely, = α, almost surely, n when (X n ) is a rotor wal with ranom initial rotor configuration on a regular tree, an on a Galton-Watson tree, respectively. On regular trees, these constants epen on whether the rotor wal (X n ) is recurrent or transient on T, property which epens only on the expecte value E[ρ(v)] of the rotor configuration at vertex v, as shown in [AH11, Theorem ]: if E[ρ(v)] 1, then the rotor wal (X n ) is recurrent, an if E[ρ(v)] < 1 then it is transient. Since in the case E[ρ(v)] = 1, the expecte return time to the root is infinite, we shall call this case a critical case, an we say that the rotor wal is null recurrent. Otherwise, if E[ρ(v)] > 1, we say that (X n ) is positive recurrent. The tree T goo of goo chilren for the rotor wal is a Galton-Watson tree with mean offspring number E[ρ(v)] an generating function f(s) = j= r js j. The main results of this paper can be summarize into the two following theorems. Theorem 1.1 (Range of the rotor wal). If (X n ) n N is a rotor wal with ranom initial configuration of rotors on T,, then there exists a constant α >, such that R n lim = α, almost surely. n n The constant α epens only on an on the istribution of ρ an is given by: (i) If (X n ) n N is positive recurrent, then (ii) If (X n ) n N is null recurrent, then α = 1 E[ρ(v)]. α = 1. 3
1. Introuction (iii) If (X n ) n N is transient, then conitione on the nonextinction of T goo, α = q f (q)(q q + 1) q + q f (q)(q q + 1) where q > is the extinction probability of T goo. Remar that, even in the recurrent case, the range of the rotor wal grows at linear spee, which is not the case for simple ranom wals on regular trees. The methos of proving the above result are completely ifferent for the transient an for the recurrent case, an the proofs will be one in separate sections. While in the recurrent case, it is clear that the rate of escape is almost surely, in the transient case we have the following result. Theorem 1. (Rate of escape of the rotor wal). If (X n ) is a transient rotor wal with ranom initial configuration of rotors on T,, then conitione on nonextinction of T goo, there exists a constant l >, such that where l is given by X n lim n n l = = l, almost surely, (q f (q))(1 q) q + q f (q)(q q + 1). The constant l is in the following relation with the constant α from Theorem 1.1 in the transient case: α l = 1. (1) We call this equation the Einstein relation for rotor wals. This relation between α an l hols also in the case of null recurrent rotor wals, where l = an α = 1/. We prove similar results for the case of rotor wals on Galton-Watson trees T with ranom initial configuration of rotors, an we show that in this case, the constants α an l epen only on the istribution of the configuration ρ an on the offspring istribution of T. Range of rotor wals an its shape was consiere also in [FLP1] on comb lattices an on Eulerian graphs. On combs, it is proven that the size of the range R n is of orer n /3, an its asymptotic shape is a iamon. It is conjecture in [KD9], that on Z, the range of uniform rotor wals is asymptotically a is, an its size if of orer n /3.
. Preliminaries Organization of the paper. We start by recalling some basic facts an efinitions about rotor wals an Galton-Watson trees in Section. Then in Section 3 we prove Theorem 1.1, while in Section we prove Theorem 1.. Then we prove in Theorem 5.3 an in Theorem 5.5 a law of large numbers for the range an the existence of the rate of escape, respectively for rotor wals on Galton-Watson trees. Finally in Appenix A we loo at the contour function of the range of recurrent rotor wals an its recursive ecomposition. Preliminaries.1 Rotor wals Let T be the regular infinite roote tree with egree + 1, with root r, an an aitional vertex o which is connecte to the root an is calle the sin, an let T = T {o}. Every vertex v T has chilren an one parent. For any connecte subset V T containing the sin o, efine the set of leaves o V = {v V : u V s. t. u v} as the set of vertices outsie of V that are chilren of vertices of V, that is, o V is the outer bounary of V. On T, the size of o V epens only on the size of V : o V = 1 + ( 1) V. () A rotor configuration ρ on T is a function ρ : T N, with ρ(x) {,..., }, which can be interprete as following: each vertex v T is enowe with a rotor ρ(v) (or an arrow) which points to one of the +1 neighbors. We fix from the beginning a counterclocwise orering of the neighbors v (), v (1),..., v (), which represents the orer in which the neighbors of a vertex are visite, where v () is the parent of v, an v (1),..., v () are the chilren. A rotor wal (X n ) on T is a process where at each timestep n, a waler locate at some vertex v T first increments the rotor at v, i.e. it changes its irection to the next neighbor in the counterclocwise orer, an then the waler moves there. We start all our rotor wals at the root r, X = r, with initial rotor configuration ρ = ρ. Then X n represents the position of the rotor wal at time n, an ρ n the rotor configuration at time n. The rotor wal is also referee to as rotor-router wal in the literature. At each timestep, we recor not only the position of the waler, but also the configuration of rotors, which changes only at the current position. More precisely, if at time n the pair of position an configuration is (X n, ρ n ), then at time n + 1 we have { ρ n (x) + 1 mo ( + 1), if x = X n ρ n+1 (x) = ρ n (x), otherwise. 5
. Preliminaries an X n+1 = X (ρ n+1(x n)) n. As efine above, (X n ) is only a eterministic process, where at each step, the waler follows a prescribe rule. Throughout this paper we are intereste in rotor wals (X n ) which start with a ranom initial configuration of rotors, which mae (X n ) a ranom process, that is not a Marov chain. Ranom initial configuration. For the rest of the paper, we consier ρ a ranom initial configuration on T, in which (ρ(v)) v T are inepenent ranom variables with istribution on {, 1,..., } given by P[ρ(v) = j] = r j, (3) with j= r j = 1. If ρ(v) is uniformly istribute on the neighbors, then we call the corresponing rotor wal uniform rotor wal. Depening on the istribution of the initial rotor configuration ρ, the rotor wal can exhibit one of the following two behaviors: either the wal visits each vertex infinitely often, an it is recurrent, or each vertex is visite finitely many times, an it escapes to infinity, an this is the transient case. For rotor wals on regular trees, the recurrence-transience behavior was proven in [AH11, Theorem ], an the proof is base on the extinction/survival of a certain branching processes, which will be also use in our results. Similar results on recurrence an transience of rotor wals on Galton-Watson trees have been proven in [HMSH15]. For the rotor configuration ρ on T, a live path is an infinite sequence of vertices r v 1, v,... each being the parent of the next, such that for all i, the inices for which v i+1 = v () i satisfy ρ(v) <. In other wors, v 1, v,... is a live path if an only if all v 1, v,... are goo, an a particle locate at v i will be sent by the rotor waler forwar to v i+1 before sening it bac to the root. An en in T is an infinite sequence of vertices o = v, v 1,..., each being the parent of the next. An en is calle live if the subsequence (v i ) i j starting at one of the vertices o = v, v 1,... is a live path. The rotor wal (X n ) can escape to infinity only via a live path.. Galton-Watson trees Consier a Galton-Watson process (Z n ) n No with offspring istribution ξ given by p = P[ξ = ]. We start with one particle Z = 1, which has chilren with probability p ; then each of these chilren inepenently have chilren with the same offspring istribution ξ, an so on. Then Z n represents the number of particles in the n-th generation. If (ξi n ) i,n N are i.i.. ranom variables istribute as ξ, then Z n Z n+1 = ξi n, i=1
3. Range on regular trees an Z =. Starting with a single progenitor, this process yiels a ranom family tree T, which is calle a Galton-Watson tree. The mean offspring number m is efine as the expecte number of chilren of one particle m = E[ξ]. In orer to avoi trivialities, we will assume p + p 1 < 1. The generating function of the process is the function f(s) = = p s an m = f (1). If is well nown that the extinction probability of the process, efine as q = lim n P[Z n = ], which is the probability the the process ever ies out, has the following important property. Theorem.1 (Theorem 1, page 7, in [AN7]). The extinction probability of (Z n ) is the smallest nonnegative root of s = f(s). It is 1 if m 1 an < 1 if m > 1. With probability 1, we have Z n or Z n an lim n P[Z n = ] = 1 lim n P[Z n = ] = q. For more information on Galton-Watson processes, we refer to [AN7]. When m < 1 = 1, or > 1, we shall refer to the Galton-Watson tree as subcritical, critical, or supercritical, respectively. We shall use Seneta-Heye Theorem which we state here. Unifie information on this results can be foun in [LP1]. Theorem. (Seneta-Heye Theorem). If 1 < m <, then there exist constants c n such that (i) lim n Z n c n exists almost surely in [, + ). (ii) P[lim n Z n n (iii) c n+1 c n m. = ] = q. 3 Range on regular trees For a simple ranom wal on a regular tree T,, which is transient, if we enote by S n its range, then it is nown [CYZ97, Theorem 1.] that S n satisfies a law of large numbers: S n lim n n = 1 almost surely. () We prove a similar result for the range of any rotor wal with ranom initial configuration on a regular tree T. Recall the efinition (3) for the ranom initial configuration ρ. Then, from [AH11, Theorem ], (X n ) is recurrent if E[ρ(v)] = j=1 jr j 1 an transient if E[ρ(v)] < 1. 7
3. Range on regular trees The tree of goo chilren for the rotor wal, enote T goo an efine in Section.1, is then a Galton-Watson process, with offspring istribution ξ = number of goo chilren of a vertex, given by P[ξ = j] = r j, for j =, 1,.... Each vertex has, inepenently of all the others, a number of goo chilren with the same istribution ξ. The mean offspring number of T goo is m = E[ρ(v)]. Let f(s) = j r js j be the generating function for T goo. For proving Theorem 1.1, we shall treat the three cases separately: the positive recurrent, null recurrent an transient case. 3.1 Recurrent rotor wals In this section, we consier recurrent rotor wals (X n ) on T, that is, once again from [AH11, Theorem ] E[ρ(v)] 1. Then the tree of goo chilren T goo has mean offspring number m 1, which by Theorem.1 ies out with probability one. While in the case m < 1, where the rotor wal is positive recurrent, the expecte size of T goo is finite, this is not the case when m > 1. For this reason we have to hanle these two cases separately. In orer to prove a law of large numbers for the range R n = {X 1,..., X n } of the rotor wal up to time n, we first loo at the behavior of the rotor wal at the times when it returns to the sin o. Define the times (τ ) of the -th return to the sin o, by: τ = an for 1 let It is easy to see that τ = inf{n > τ 1 : X n = o}. (5) τ τ 1 = R τ, () where R τ represents the range of (X n ) up to the time of the -th return to the sin o. Since we are in the recurrent case, where the rotor wal returns to the sin infinitely many times, it is clear that the stopping times are almost surely finite. 3.1.1 Positive recurrent rotor wals If E[ρ(v)] > 1, we prove the following. Theorem 3.1. For a positive recurrent rotor wal (X n ) on T, with we have R τ lim = 1 τ E[ρ], almost surely.
3. Range on regular trees Proof. For simplicity of notation, we write R = R τ. Using the efinition of the tree of goo chilren, T goo = R 1 is a subcritical Galton-Watson tree with mean offspring number m = E[ρ] < 1, that is, it ies out almost surely. The expecte size of the range up to time τ 1 is given by E[ R 1 ] = 1 > 1. At the time τ 1 m of the -th return to the sin o, all rotors in the previously visite set R point towars the root, an the remaining rotors are still in their initial configuration. Thus uring the time interval (τ, τ +1 ], the rotor wal visits all the leaves of R from right to left, an at each leaf it attaches inepenently a (ranom) subtree that has the same istribution as R 1. If we enote by L = o R, then L +1 = L i=1 L 1,i, where L 1,i are inepenent copies of L 1, an (L ) is a supercritical Galton-Watson process with mean offspring number ν (the mean number of leaves of R 1 ), which in view of (), is given by ν = 1 + ( 1)E[ R 1 ] = 1 + 1 1 m > 1. Moreover P[L = ] =. Since ν > 1 it follows from Theorem. applie to the supercritical Galton-Watson process (L ) that there exists a sequence of numbers (c ) 1 an a nonnegative ranom variable W such that (i) L c W, almost surely. (ii) P[W = ] equals the probability of extinction of (L ). (iii) c +1 c ν. By () we have R = L 1, which together with equation () yiels 1 an iviing by c gives ( 1) ( τ τ 1 ) + = ( 1) R + = L, ( 1) It then follows that ( τ τ ) 1 c 1 = L c c 1 c 1 c ( τ τ ) 1 c 1 + = L. c c 1 c c c ( 1)c 1 W, since c, an the limit W is an almost surely positive ranom variable. 1 Since τ c an τ 1 c 1 either both iverge or have the same limit τ, it follows that ( τ τ ) = ν 1 W. 9
3. Range on regular trees Thus τ c converges almost surely to an almost surely positive ranom variable τ ( τ τ = lim = 1 1 ) 1 W >. c ν 1 Hence τ 1 τ = τ 1 c 1 c c 1 c τ τ 1 ν 1 τ = 1 ν, a.s. Now from () we get R lim τ 1 = lim an this proves the claim. ( 1 τ ) 1 = 1 ( 1 1 ) = 1 τ ν ( m) = 1 E[ρ(v)], Passing from the range along a subsequence (τ ) to the range R n at all times requires aitional wor, because of the exponential growth of the increments (τ +1 τ ). We next prove that the almost sure limit Rn exists. n Proof of Theorem 1.1(i). Theorem 3.1 gives the law of large numbers along the subsequence (τ ) of return times to the sin, as efine in (5). For every n, let = max{j : τ j < n}, so that τ < n τ +1 an X n R τ+1. Since the increments (τ +1 τ ) have exponential growth, the inequality R τ τ +1 R n n R τ +1 τ, is not sharp enough in orer to guarantee the almost sure existence of the limit Rn. n From this inequality we only get, 1 1 lim Rn ν 1, where ν > 1 is as ν E[ρ(v)] n E[ρ(v)] in the proof of Theorem 3.1. We partition the time interval (τ, τ +1 ] into finer intervals, on which the behavior of the range can be easily controlle. Recall, from the proof of Theorem 3.1, that by L = o R τ we have enote the number of leaves of R τ, an (L ) is a supercritical Galton-Watson tree with mean offspring number ν > 1. As usual the vertices in o R τ = {x 1, x,..., x L } are orere from right to left. We introuce the following two (finite) sequences of stopping times (η i ) an (θi ) of ranom length L + 1, as following: let θ = τ an η L +1 = τ +1 an for i = 1,,..., L η i = min{j > θ i 1 : X j = x i } θ i = min{j > η i : X j = x i } 1 (7)
3. Range on regular trees That is, for each leaf x i, the time η i represents the first time the rotor wal reaches x i, an θ i represents the secon an the last time the rotor wal returns to x i after maing a full excursion in the subcritical Galton-Watson tree roote at x i. Then { ( (τ, τ +1 ] = L +1 i=1 θ i 1, η] } { ( ] } i L i=1 η i, θ i It is easy to see that the increments (θ i ηi ) i are i.i., an istribute accoring to the istribution of τ 1, which is the time a rotor wal nees to return to the sin for the first time. Once the rotor wal reaches the leaf x i for the first time at time η i, the subtree roote at x i was never visite before by a rotor wal. Even more, the tree of goo chilren with root x i is a subcritical Galton-Watson tree, which ies out almost surely. Thus, the rotor wal on this subtree is recurrent, an it returns to x i at time θ i. Then (θi ηi ) represents the length of this excursion which has expectation E[τ 1 ] = E[ R τ1 ] =. In the time intervals (θi 1 1 m, η i ], the rotor wal leaves the leaf x i 1 an returns to the confluent between x i 1 an x i, from where it continues its journey until it reaches x i. Then η i θi 1 is the time the rotor wal nees to reach the new leaf x i after leaving x i 1 In this time intervals, the range oes not change, since (X n ) maes steps only in R τ. Depening on the position of X n, we shall istinguish two cases: Case 1: There exists an i {1,,..., L } such that n (η i, θi ]. Case : There exists an i {1,,..., L } such that n (θ i 1, η i ]. Case 1: if n (η i, θi ] for some i, then ηi < n θi R η i θ i R n n R θi η i, () an we show that, as, the ifference R θ i η i use the following facts R θ i = R η i + R τ i 1 θ i = η i + τ i 1, R η i θ i, almost surely. We where τ i 1 is a ranom variable which is ientically istribute as τ 1, which is finite. Then R τ i 1 is almost surely finite an ientically istribute with R τ1. Thus, R θ i η i R η i θ i = τ i 1 R τ i 1 η i (ηi + τ i 1) + ηi R τ i 1 η i (ηi + τ i 1) + τ i 1 R ηi η i (ηi + τ i 1). As, η i as well as R η i, but τ i 1 an R τ i 1 are finite, almost surely, therefore the first two terms on the right han sie of the equation above converge 11
3. Range on regular trees to almost surely, an we still have to prove convergence to of the thir term. But we now that τ < η i < τ +1, therefore τ ( 1 R i η i η i (ηi + τ 1) τ 1 i i R τ j=1 + (θj ηj ) ), almost surely. (9) i τ + τ1 i τ τ Since τ1 i is finite almost surely, an τ, it is clear that the first fraction goes to, while Rτ τ converges to 1 by Theorem 3.1. The last term in the E[ρ] equation above goes again to since the sum in the numerator is a finite sum of i.i. ranom variables, with istribution the same as τ 1 which is almost surely finite. This shows that R θ i number α = lim R θ i R η i η i θ i η i converges to almost surely, therefore there exists a almost surely, an both the right an the left han sie in () converge to α almost surely, which implies that Rn converges to α n almost surely, as well. But, along the subsequence (τ ), we have from Theorem 3.1, that R τ τ 1 almost surely, as, therefore α E[ρ(v)] = 1. E[ρ(v)] Case : if n (θ i 1 interval (θ i 1 have Since lim R θ i η i, η i ] for some i, then θi 1 < n η i, an since in the time, η i ), the rotor wal visits no new vertices an moves only in R τ, we R θ i 1 n = α = 1 E[ρ(v)] an = R n n Rηi = n. (1) R θ i η i = R θ i θ i τ i 1 = R θ i θ i 1 1 τ i 1 θ i, an τ i 1 θ i, it follows that R θ i θ i can be applie for the almost sure convergence of R η i η i (1) R η i R η i n = R n n = R i 1 θ n η i converges almost surely to α. The same argument R θ i 1 to α. Finally, in view of an both the lower boun an the upper boun in the previous equation converge almost surely to α = 1 Rn. Thus, also in this case converges almost surely E[ρ(v)] n to, an the claim follows. 1 E[ρ(v)] θ i 1, 1
3. Range on regular trees If ν is the mean offspring number of the supercritical Galton-Watson process (L ), with L = o R τ, then we can also write R n lim n n = 1 ( 1 1 ), almost surely. ν Lemma 3.. The variance of L 1 = o R τ1 is finite. Proof. Since by () L 1 = ( 1) R τ1 + 1, we have Var(L 1 ) = ( 1) Var( R τ1 ). Thus, it suffices to prove that Var( R τ1 ) is finite. Denote by Z n the n-th generation of the tree T goo = R τ1 of goo chilren, which is in the positive recurrent case, a subcritical Galton-Watson tree. Then (Z n ) is a Galton-Watson process with E[Z 1 ] = m < 1 an Var(Z 1 ) = σ <. Thus the variance of Z n is equal to Var(Z n ) = σ n 1 1 mn m 1 m, see (Athrea, Ney, I..). We have R τ1 = n=1 Z n. Thus Var( R τ1 ) = Cov(Z, Z l ) Var(Z )Var(Z l ) which finishes the proof. =1 l=1 =1 l=1 1 = σ m m 1 1 m 1 ml σ ml 1 1 m =1 l=1 σ m 1 (1 m ) m l 1 (1 m l ) 1 m =1 l=1 = σ m 1 (1 m ) m l 1 (1 m l ) 1 m =1 = σ 1 m 1 (1 m ) <, l=1 3.1. Null recurrent rotor wals In this section we consier null recurrent rotor wals, that is E[ρ(v)] = 1. Recall the stopping times τ as efine in (5), which represent the times the rotor wal returns to the sin, which are almost surely finite, since (X n ) is recurrent. The proofs for the law of large numbers for the range will be slightly ifferent, arising from the fact that the expecte return time to the sin for the rotor waler is infinite. We first prove the following. 13
3. Range on regular trees Theorem 3.3. For a null recurrent rotor wal (X n ) on T, we have R τ lim τ = 1, almost surely. Proof. The first part of the proof follows the same lines of the proof of Theorem 3.1. Rewriting equation (), we get ( 1 1 τ ) 1 = R τ, τ τ an we prove that the quotient τ 1 τ goes to zero almost surely. Intuitively, this is a clear assertion, since the tree T goo of goo chilren for the null recurrent rotor wal is a critical Galton-Watson tree, which ies out almost surely, but its expecte size (which is half of the expecte first return time to the sin) is infinite. If at time τ 1, the rotor wal is at the sin o, until the next return to the sin, the waler has to visit all the vertices of the previously visite set R τ 1, together with the leaves of R τ 1. At each leaf it maes a new excursion in a subtree roote at the corresponing leaf, an it as a new critical Galton-Watson tree to the range. For simplifying the notation, we write again R = R τ an we first show that τ τ 1 almost surely, by fining a lower boun which converges to almost surely. From (), we have τ > R. (11) τ 1 τ 1 If R 1 is the set of leaves of R 1, then in the time interval τ τ 1, to the current range R 1 will be ae the i.i. critical Galton-Watson trees roote at the leaves R 1. Recall now the efinition of the sequences of stopping times (η i ) an (θ i ) as efine in (7). That is, for a leaf x i R 1, i = 1,,..., L 1, with L 1 = R 1, the times η 1 i an θi 1 represent the time of the first an of the last visit to x i, after τ 1, respectively. Thus L 1 R = R 1 + 1 ( ) θ i 1 η 1 i. (1) i=1 The increments ( θ 1 i ) ηi 1 are i.i. since the subtrees roote at x i i were never visite before time η 1 i an the rotor configuration there is in its initial ranom state. Therefore the rotor wal (X n ) maes inepenent excursions of length θ 1 i ηi 1 in each subtree, an prouces inepenent critical Galton-Watson trees. The excursions are istribute lie τ 1, which in the null recurrent case has infinite expectation. Nevertheless, by the strong law of large numbers we have on one sie L 1 i=1 ( θ i 1 η i 1) L 1 E[τ 1 ] =, almost surely. (13) 1
3. Range on regular trees Putting now equations (11) an (1) together an using that L 1 1) R 1, we obtain By letting l = lim inf τ τ 1 > = L 1 i=1 L 1 i=1 L 1 i=1 ( θ i 1 η i 1) L 1 ( θ i 1 η i 1) = 1 + ( 1 + ( 1) R 1 τ 1 (1) R 1 (15) L 1 τ 1 ( θ i 1 η 1) i 1 ( 1 τ ). (1) L 1 τ 1 τ τ 1, an taing limits in the previous equation we get l 1 E[τ 1] ( 1 1 ). l Unless l = 1, the right han sie above converges to infinity almost surely, which τ implies l = = lim inf τ 1 lim sup τ τ τ 1, therefore lim τ τ 1 = an lim 1 τ =. Suppose now l = 1. Once at the root at time τ 1, until the next return at time τ, the rotor wal visits everything that was visite before plus new trees where the configuration is in the initial status. For visiting the previously visite set, it nees time τ 1, therefore τ τ 1 > τ 1, which gives that lim inf τ contraicts the fact that l = 1. Therefore lim τ 1 =. Finally, we show that inee lim R τ to see that R τ 1. We have 1 R τ an the claim follows. lim inf R τ τ τ 1 >, which exists. On one han, from (), it is easy = 1 lim inf ( 1 τ ) 1 = 1 τ, Proof of Theorem 1.1(ii).. The proof is the same as the one of Theorem 1.1(i), with the only ifference that the almost sure convergence of the mile fraction R τ τ in equation (9), in the null recurrent case follows by Theorem 3.3. The rest is unchange, since in the null recurrent case E[ρ(v)] = 1, an in this case, the constant α = 1. While the proofs for the strong law of large numbers for the range of the rotor wal (X n ) are quite similar in the positive recurrent an in the null recurrent case, for the transient case below we shall use another proof iea. 15
3. Range on regular trees 3. Transient rotor wals We finally consier the transient case on regular trees, when E[ρ(v)] < 1. Then each vertex is visite only finitely many times, an the wal escapes to infinity along a live path. The tree T goo of goo chilren for (X n ) is a supercritical Galton- Watson tree, with mean offspring number m = E[ρ(v)] > 1. Thus, T goo survives with positive probability (1 q), where q (, ) is the extinction probability, an is the smallest nonnegative root of the equation f(s) = s, where f is the generating function of T goo. Notation 3.. For the rest of this section, we will always conition on the event of nonextinction, so that T goo is an infinite ranom tree. We enote by P non an by E non the associate probability an expectation conitione on nonextinction, respectively. That is, if P is the probability for the rotor wal in the original tree T goo, then for some event A, we have P non [A] = goo P[A T is infinite]. 1 q In orer to unerstan how the rotor wal (X n ) escapes to infinity, we will ecompose the tree T goo with generating function f conitione on nonextinction. Consier the generating functions g(s) = f ((1 q)s + q) q 1 q an h(s) = f(qs). (17) q Then the f-galton-watson tree T goo can be generate by: (i) growing a Galton-Watson tree T g with generating function g, which has the survival probability 1. (ii) attaching to each vertex v of T g a ranom number n v of h-galton Watson trees, acting as traps in the environment T goo. The tree T g is equivalent (in the sense of finite imensional istributions) with a tree in which all vertices have an infinite line of escent. T g is calle the bacbone of T goo. The ranom variable n v has a istribution epening only on (v) in T goo an given T g an n v the traps are i.i.. The supercritical Galton-Watson tree T goo conitione to ie out is equivalent to the h-galton-watson tree with generating 1
3. Range on regular trees function h, which is subcritical. For more etails on this ecomposition an the equivalence of the processes involve above, see [Lyo9] an [AN7, Chapter I, Part D]. Since we are in the transient case, the rotor wal visits the origin only finitely many times. Denote by t < the number of times the origin was visite. After the t -th visit to the origin, there are no returns to the origin, an there has to be a leaf γ belonging to the range of the rotor wal up to the t -th return, along which the rotor waler escapes to infinity, that is, there is a live path starting at γ. Denote by n the first time the rotor wal arrives at γ. The tree roote at γ was not visite previously by the waler, an the rotors are in their ranom initial configuration. The tree of goo chilren T goo roote at γ has the same istribution as the initial tree of goo chilren roote at the root of the original tree. At time n, we have alreay a finite visite subtree an its carinality R n, which will play no role in the limit behavior for the range. When computing the limit for the size of the range, we have to consier also this irrelevant finite part. On the event of nonextinction, let γ = (γ, γ 1,...) be the rightmost infinite ray in T goo roote at γ, that is, the rightmost infinite live path in T, which starts at γ. On the event of nonextinction, recalling the ecomposition of a supercritical Galton-Watson tree, this is the rightmost ray in the tree T g. Since all vertices in T g have an infinite line of escent, such a ray exists. The ray γ is then a live path, along which the rotor wal (X n ) escapes to infinity, without visiting the vertices to the left of γ. In orer to unerstan the behavior of the range of (X n ) an to prove a law of large numbers, we introuce the sequence of regeneration times (τ ) for the ray γ. Let τ = n an for 1: τ = inf{n n : X n = γ }. (1) Note that, for each the ranom times τ an τ +1 1 are the first an the last hitting time, of γ, respectively. Inee, once we are at vertex γ, since γ +1 is the rightmost chil of γ with infinite line of escent, the rotor wal visits all goo chilren to the right of γ, an maes finite excursions in the trees roote at those goo chilren, an then returns to γ at time τ +1 1. Then, at time τ +1, the wal moves to γ +1 an never returns to γ. We first prove the following. Theorem 3.5. Let (X n ) be a transient rotor wal on T. If the tree T goo of goo chilren for the rotor wal (X n ) has extinction probability q, then conitione on nonextinction, there exists a constant α >, which epens only on q an, such that R τ lim = α, almost surely, τ 17
3. Range on regular trees an α is given by α = q f (q)(q q + 1) q + q f (q)(q q + 1). Proof. We write again R := R τ, an R for the range of the rotor wal up to time n, which is finite almost surely. Since γ is the rightmost infinite live path on which (X n ) escapes to infinity, to the right of each vertex γ in T goo there is a ranom number of vertices, an in the tree roote at those vertices (which are h-galton-watson trees), the rotor wal maes only finite excursions. Then, at time τ +1, the wal reaches γ +1 an never returns to γ. For each, γ +1 is a goo{ chil of γ an} the rotor at γ points to the right of γ +1. For =, 1,..., let γ (1),..., γ(n ) be the set of vertices which are goo chilren of γ, an are situate to the right of γ +1 ; enote by N the carinality of this set. Aitionally, enote by T (j) the tree roote at γ (j), j = 1,..., N an by T (j) = T (j) (γ, γ (j) ). That is, the tress T (j), for j = 1,..., N have all common root γ, an T (j) = T (j) + 1. Claim 1. Conitionally on the event of nonextinction, (N ) are i.i.. Proof of Claim 1. For each, the istribution of N epens only on the offspring istribution (which is the number of goo chilren for the rotor wal) of T goo, an the last one epens only on the initial rotor configuration ρ. Since the ranom variables (ρ(v)) v are i.i., the claim follows. Claim. Conitionally on the event of nonextinction, ( T (j) ) 1 j N are i.i.. Proof of Claim. This follows immeiately from the efinition of the Galton- Watson tree, since each vertex in T goo has chilren with probability r, inepenently of all other vertices, an all these chilren all have inepenent Galton- Watson escenant subtrees. All the subtrees T (j) roote at γ (j) are then inepenent h-galton-watson subtrees (subcritical), therefore ( T (j) ) 1 j N are i.i.. Since T (j) = T (j) + 1, the claim follows. Claim 3. Given nonextinction, the increments (τ +1 τ ) between regeneration times are i.i... Proof of Claim 3. Given nonextinction, the time (τ +1 τ ) epens only on N (the number of goo chilren to the right of γ ) an on T (j), which by Claim 1 an Claim above, are all i.i.. Thus the inepenence of (τ +1 τ ) follows as well. 1
3. Range on regular trees Clearly, each γ is visite exactly N + 1 times, an all vertices to the left of γ are never visite. We write R ( 1,] for the range of the rotor wal in the time interval (τ 1, τ ]. For r s, the paths of the rotor wal in the time interval (τ r 1, τ r ] has empty intersection with the path in the time interval (τ s 1, τ s ], an we have Moreover, for i = 1,..., R = R + R (i 1,i]. i=1 R (i 1,i] = 1 + N i 1 j=1 T i 1 (j), an if we enote by α = 1 i= N i an if ( t j ) is an i.i.. sequence of ranom variables with the same istribution as T i 1 (j), then, using Claim 1 an Claim we obtain α R = R + + t j. (19) Similarly, if we write τ = n + i= (τ i τ i 1 ) an use the fact that, for i j=1 j=1 N i 1 τ i τ i 1 = 1 + T i 1 (j), then again by Claim 1 an Claim we get α τ = R + + t j. () j=1 Putting equations (19) an () together, we finally get R = R + + α t j=1 j τ R + + α t. j=1 j By the strong law of large numbers, we have which implies R + α 1 E non [N ], α j=1 t j α E non [ T (1) ] almost surely R 1 + E non[n ]E non [ T (1) ] τ 1 + E non [N ]E non [ T, almost surely. (1) ] 19
3. Range on regular trees We have to compute now the two expectations E non [N ] an E non [ T (1) ] involve in the equation above, in orer to get the formula from the statement of the theorem. Conitione on nonextinction, for all =, 1,... an j = 1,... N the trees T (j) are i.i.. subcritical Galton-Watson trees with generating function h(s) = f(qs) q an mean offspring number h (1). That is, they all ie out with probability one, an the expecte number of vertices is E non [ T (j) ] = 1. On the other han 1 h (1) h(s) = 1 q j= r j(qs) j, which implies that h (1) = 1 q j= jr jq j, an E non [ T (1)] = q q j= jr jq j, which in terms of the generating function f(s) can be written as In computing E non [N ]: 1 E non [N ] = = 1 1 q = 1 1 q = i= j 1 j= j= q (1 q) ( i= E non [ T (1)] = ip non [N = i] = 1 1 q ip[n = i, T goo r j j 1 1 iq i (1 q) = i= j= 1 i= q q f (q). (1) ip[n = i, T goo is infinite] is infinite γ has j goo chilren]p[γ has j goo chilren] j= r j q j 1 q q r j j 1 iq i = i= ) jr j q j j= (j 1)q j+1 jq j + q r j (1 q) ( ) 1 jr j q j 1 j= = q 1 q In terms of the generating function f(s), using f(q) = q, E non [N ] can be written as E non [N ] = j=1 q 1 q (1 f (q)). () Putting the two expectations together, we obtain the constant α in terms of the generating function f(s) given by α = q f (q)(q q + 1) q + q f (q)(q q + 1) = q (q q + 1) j=1 jr jq j 1 q + q (q q + 1) j=1 jr jq. j 1
3. Range on regular trees In Theorem 3.5, we have prove a law of large numbers for the range of the rotor wal along a subsequence (τ ). With very little effort, we can show that we have inee a law of large numbers at all times. Proof of Theorem 1.1(iii). For n N, the infinite ray γ an the stopping times (τ ) as efine in (1) let = max{j : τ j < n}. Then τ < n τ +1 an R τ R n R τ+1 which in turn gives τ +1 τ R τ τ R n n R τ +1 τ. (3) τ +1 τ +1 By Claim 3 from the proof of Theorem 3.5, the increments (τ +1 τ ) are i.i., an finite almost surely, because the rotor wal is transient, therefore τ +1 τ τ almost surely as. Then, since τ +1 τ = τ +1 τ τ + 1, we have that τ +1 τ 1 almost surely, as. This, together with Theorem 3.5 implies that the left han sie of equation (3) converges to α almost surely. By the same argument we obtain that also the right han sie of (3) converges to the same constant α almost surely, an this completes the proof. 3.3 Uniform rotor wals We iscuss here the behavior of rotor wals on regular trees T, with uniform initial rotor configuration ρ, that is, for all v T, the ranom variables (ρ(v)) are i.i. with uniform istribution on the set {, 1,..., }, i.e. P[ρ(v) = j] = 1, for j +1 {, 1,..., }. Such wals are null recurrent on T an transient on all T, 3. As a special case of Theorem 1.1(iii), we have the following. Corollary 3.. If (X n ) is a uniform rotor wal on T, 3, then the constant α is given by α = q(1 )(1 q+1 ) + q (1 q 1 ). ( + 1)(q 1) 3 (q 1) 3 Proof. Using that P[ρ(v) = j] = 1 = r +1 j an putting f (q) = q Theorem 3.5 we get the result. q 1 q+1 1 (+1)(q 1) in 1
. Rate of escape on regular trees The following table shows values for the constants α in comparison with the limit ( 1)/ for the simple ranom wal on trees. α ( 1)/.5.5 3.77..7.75 5.5..53.33 7.7.57..75 9.99. 1.99.9 Note that only in the case of the binary tree T, the limit values for the range of the uniform rotor wal an of the simple ranom wal are equal, even though the uniform rotor wal is (critically) recurrent an the simple ranom wal is transient. In the transient case, that is, for all 3 we always have α > ( 1)/. Rate of escape on regular trees In this section, we want to prove the existence of the almost sure limit Xn, as n n, where X n represents the istance from the root to the position X n at time n of the waler. In the recurrent case, the tree T goo is finite a.s., an X n is boune, thus X n /n goes to almost surely. Therefore, for the rest of this part we will assume (X n ) is a transient rotor wal on T. Since in this case, there is a positive probability of extinction of T goo, it clearly maes sense to conition on the event of nonextinction, an the notation remains the same as in Section 3.. Proof of Theorem 1.. Recall the efinition of the infinite ray γ along which the rotor wal (X n ) escapes to infinity, an the stopping times τ as efine in (1). We first prove the existence of the rate of escape l along the sequence (τ ) of stopping times. This is rather easy, since (r, γ ) = X n +, where n < is the first time the wal reaches γ, from where it escapes without returning to the root. Recall now from the proof of Theorem 3.5, that τ = n + + α j=1 t j an α =
. Rate of escape on regular trees 1 i= N i, with the involve quantities again as compute in Theorem 3.5. Then X τ τ = X n + τ = 1 + Xn 1 + n + α t j=1 j α. α By the strong law of large numbers for sums of i.i. ranom variables, we have α j=1 t j α E non [ T (1) ] an α X n an n are both finite, the almost sure limits Xn This implies E non [N ] almost surely, as, Also, since an n are, as. X τ τ 1 1 + E non [ T, almost surely, as. (1) ]E non [N ] By equation (1) an (), which give E non [ T (1) ] an E non [N ] in terms of the generating function f of the tree T goo, we obtain X τ τ (q f (q))(1 q) q + q f (q)(q q + 1) = l, almost surely as. () In orer to prove the almost sure convergence of Xn, not only along a subsequence n (τ ), but for all n, we tae = max{i : τ i < n}. We now that τ < n τ +1, X τ = X n +, X τ+1 = X n + + 1 an X n X n +. Moreover, between times τ an τ +1, the istance can increase with no more than τ +1 τ, an we have X n + τ +1 X n n X n + + (τ +1 τ ) τ Since Xτ τ τ τ +1 = τ τ +1, together with equation () an the facts that τ +1 1 an Xn τ +1 almost surely, we obtain that the left han sie of the equation above converges to l almost surely, as. For the right han sie, we use again equation (), together with the fact that τ +1 τ τ almost surely, since the increments (τ +1 τ ) are i.i. an almost surely finite. This yiels then the almost sure convergence of the right han sie of the equation above to the constant l, which implies that Xn l almost surely as n. n 3
5. Rotor wals on Galton-Watson trees 5 Rotor wals on Galton-Watson trees The methos we have use in proving the law of range numbers an the existence of the rate of escape for rotor wals (X n ) with ranom initial configuration on regular trees can be, with minor moifications, aapte to the case when the rotor wal (X n ) moves initially on a Galton-Watson tree T. We get very similar results to the ones on regular trees, which we will state below. We will not write own the proofs again, but only mention the ifferences which appear on Galton-Watson trees. Let T be a Galton-Watson tree with offspring istribution ξ given by p = P[ξ = ], for an we assume that p =, that is T is supercritical an survives with probability 1. Moreover the mean offspring number μ = E[ξ] is also greater than 1. We recall the notation an the main result from [HMSH15]. For each we choose a probability istribution Q supporte on {,..., }. That is, we have the sequence of istributions (Q ) N, where Q = ( q,j ) j with q,j an j= q,j = 1. Let Q be the infinite lower triangular matrix having Q as row vectors, i.e.: q... q 1 q 11... Q = q q 1 q.... q 3 q 31 q 3 q 33.......... Below, we write x for the (ranom) egree of vertex x in T. Definition 5.1. A ranom rotor configuration ρ on T is Q-istribute, if for each x T, the rotor ρ(x) is a ranom variable with the following properties: 1. ρ(x) is Q x istribute, i.e., P[ρ(x) = x l x = ] = q,l, with l =,... x,. ρ(x) an ρ(y) are inepenent if x y, with x, y T. We write R T for the corresponing probability measure. Then RGW = R T GW represents the probability measure given by choosing a tree T accoring to the GW measure, an then inepenently choosing a rotor configuration ρ on T accoring to R T. Recall that to the root r T, we have
5. Rotor wals on Galton-Watson trees ae an aitional sin vertex s. If we start n rotor particles, one after another, at the root r of T, with ranom initial configuration ρ, an we enote by E n (T, ρ) the number of particles out of n, that escape to infinity, then the main result of [HMSH15] is the following. Theorem 5.. [HMSH15, Theorem 3.] Let ρ be a ranom Q-istribute rotor configuration on a Galton-Watson tree T with offspring istribution ξ, an let ν = ξ Q. Then we have for RGW-almost all T an ρ: (a) E n (T, ρ) = for all n 1, if E[ν] 1, (b) lim n E n (T, ρ) n = γ(t ), if E[ν] > 1, where γ(t ) represents the probability that simple ranom wal starte at the root of T never returns to s. Let us enote m := E[ν]. That is, if m 1, then (X n ) is recurrent an if m > 1, then (X n ) is transient. 5.1 Range an rate of escape on Galton-Watson trees Tree of goo chilren. If ρ is Q-istribute, P[x has l goo chilren x = ] = q,l, then the istribution of the number of goo chilren of a vertex x in T is given by P[x has l goo chilren] = p q,l =: r l, (5) which is the l th component of the vector ν = ξ Q. The tree of goo chilren T goo is in this case a Galton-Watson tree with offspring istribution ν = ξ Q whose mean was enote by m. Denote by f T the generating function of T goo an by q its extinction probability. For the range Rn T of rotor wals (X n ) on Galton-Watson trees, we get the following result. Theorem 5.3. Let T be a Galton-Watson tree with offspring istribution ξ an mean offspring number E[ξ] = μ > 1. If (X n ) is a rotor wal with ranom Qistribute initial configuration on T, an ν = ξ Q, then there is a constant α T > such that Rn T lim n n = α T, RGW -almost surely. If we write m = E[ν], then the constant α T is given by: 5 =l
5. Rotor wals on Galton-Watson trees (i) If (X n ) n N is positive recurrent, then (ii) If (X n ) n N is null recurrent, then α T = μ 1 (μ m). α T = 1. (iii) If (X n ) n N is transient, then conitione on the nonextinction of T goo, α T = q f T (q)(q q + 1) q + q f T (q)(q q + 1) where q > is the extinction probability of T goo. The proof follows the lines of the proof of Theorem 1.1, with minor changes which we state below. The offspring istribution of the tree of goo chilren T goo will be here replace with (5), an the corresponing mean offspring number is m. If L an R τ are the same as in the proof of Theorem 3.1, then in case when (X n ) moves on a Galton-Watson tree T, L = 1 + ξ v R, () v R where ξ v enotes the (ranom) number of chilren of the vertex v T. Since T is the initial Galton-Watson tree with mean offspring number μ, by Wal s ientity we get E[L 1 ] = 1 + 1 1 m μ 1 1 m = μ m 1 m := λ > 1. Then ν in Theorem 3.1 will be replace with λ an the rest wors through. In the μ 1 proof of Theorem 1.1(i), α will be replace with. Theorem 3.3, wors as (μ m) well here, with the minor change in equation (1), where the last term will be 1 + v R 1 ξ v R τ 1, while the following relations stay the same. In the transient case, in the proof of Theorem 3.5, when computing the expectation E non [N ], the egree 1 has to be replace by the offspring istribution ξ, which in terms of the generating function f T of T goo, an its extinction probability q prouces the same result.
5. Rotor wals on Galton-Watson trees If (X n ) is an uniform rotor wal on the Galton-Watson tree T, we have a particularly simple limit. In [HMSH15], it was shown that (X n ) is recurrent if an only if μ. Moreover, the tree of goo chilren T goo is a subcritical Galton-Watson tree with offspring istribution given by ν = ( l m = E[ν] = μ 1. 1 +1 p ) l, an mean offspring number Corollary 5.. For the range of uniform rotor wals on Galton-Watson trees T, we have Rn T lim n n = μ 1, RGW almost surely. μ Finally, we also have the existence of the rate of escape. Theorem 5.5. Let T be a Galton-Watson tree with offspring istribution ξ an mean offspring number E[ξ] = μ > 1. If (X n ) is a transient rotor wal with ranom Q-istribute initial configuration on T, an ν = ξ Q, then conitione on nonextinction of T goo, there exists a constant l T >, such that where l T is given by X n lim n n = l T, RGW almost surely, l T = (q f (q))(1 q) q + q f (q)(q q + 1). The constant l T is in the following relation with the constant α T from Theorem 5.3 in the transient case: α T l T = 1. 5. Simulations on Galton-Watson trees We present here some simulation ata about the growth of the range for a few one parameter families of Galton-Watson trees, for which the rotor wal is either recurrent or transient, epening on the parameter. In the table below, the Galton- Watson trees we use in the simulation are presente. T i (p 1,..., p ) μ i T p 1 = p, p = 1 p p T 3 p 1 = p, p 3 = 1 p 3 p T p 1 = p, p = 1 p 3p T 5 p 1 = p, p 5 = 1 p 5 p T p 1 = p, p = 1 p 5p 7
5. Rotor wals on Galton-Watson trees..7. ψ ϕ ψ 3 ϕ 3 ψ ϕ ψ 5 ϕ 5 ψ ϕ.5..3..1 1 3 5 Figure : Plots of the linear growth coefficients of the size of the range for rotor wal (soli lines) an simple ranom wal (ashe lines) on the Galton-Watson trees T,..., T. The x-axis epicts the branching number of the tree. The ots show the corresponing values for the regular trees. For each i =,..., an p [, 1) enote by ψ i (p) an φ i (p) the simulate values of the limits R lim n i n n = ψ S i (p) lim n i n n = φ i(p), (7) where Rn i an Sn i represent the range of the rotor wal an of the simple ranom wal up to time n on T i, respectively. To be able to compare the values we plot the constants ψ i (p) an φ i (p) against the mean offspring number μ i (p) of the offspring istribution on T i. That is, we loo at the functions ψ i = ψ i μ 1 i an φ i = φ i μ 1 i ; see Figure.
A. The contour of a subtree Appenix A The contour of a subtree We iscuss here some further ways one can loo at the range of rotor wals on regular trees T, an at their contour functions, which accoring to simulations, seem to have interesting fractal properties. For, let Σ = {,..., 1} an enote by Σ the set of finite wors over the alphabet Σ. We use ε to enote the empty wor. For w Σ we write w for the number of letters in w. If w, v Σ we write wv for the concatenation of the wors w an v. We ientify the tree T with the set Σ, since every vertex in T can be uniquely represente by a wor in Σ. For w = w 1... w n 1 w n Σ {ε}, the wor w 1... w n 1 is the preecessor of w in the tree, an for all w Σ the chilren of w are given be the wors w with Σ. Using the previous notation, for w = w 1... w n 1 w n we have w () = w 1... w n 1 an for =,..., 1, w ( ) = w. Let A T be a finite connecte subset (a subtree) of T containing the root ε. We ientify A with a piecewise constant function f A : [, 1] N as following. For each x [, 1], we ientify x with the infinite wor x 1 x... where the x i are the igits expansion of x in base, that is x = i=1 x i i. We then efine the function f A pointwise as following an we call f A the contour of the set A. f A (x) = min{n 1 : x 1... x n A}, () A.1 Contour of the range: recurrent case The range R n of the rotor wal (X n ) on T is a subtree of T. In what follows, we write f n = f Rn for the contour of the range of the rotor wal up to time n. Since we start with a ranom initial rotor-configuration, f n is a ranom càlàg-function. See Figure 3 for the contours of the rotor wal range for the first few steps of the process on the binary tree T. Figure a shows a typical contour of the rotor range on the binary tree for n = 1 steps. Figure b shows a numerical approximation of the expectation E[f 1 ]. As in the proofs of the law of large numbers for R n, we loo first at the times when the rotor wal returns to the root. Recall the efinition of the return times (τ ), as efine in (5), for recurrent rotor wals. Write again R = R τ for the range up to time τ of the -th return of the rotor wal (X n ) to the sin, an enote by g (x) = E [ f R (x) ] the expecte contour after the -th excursion, that is g (x) = m=1 mp[ f R (x) = m ]. Recall that we are in a case of a ranom initial 9