Range and Speed of Rotor Walks on Trees

We prove a law of large numbers for the range of rotor walks with random initial configuration on regular trees and on Galton–Watson trees. We also show the existence of the speed for such rotor walks. More precisely, we show that on the classes of trees under consideration, even in the case when the rotor walk is recurrent, the range grows at linear speed.


Introduction
For d ≥ 2, let T d be the rooted regular tree of degree d + 1, and denote by r the root. We attach an additional sink vertex o to the root r . We use the notation T d = T d \{o} to denote the tree without the sink vertex. For each vertex v ∈ T d , we denote its neighbors by v (0) , v (1) , . . . , v (d) , where v (0) is the parent of v and the other d neighbors, the children of v, are ordered counterclockwise (Fig. 1).
Each vertex v ∈ T d is endowed with a rotor ρ(v) ∈ {0, . . . , d}, where ρ(v) = j, for j ∈ {0, . . . , d} means that the rotor in v points to neighbor v ( j) . Let (X n ) n∈N be a rotor walk on T d starting in r with initial rotor configuration ρ = (ρ(v)) v∈ T d : for all v ∈ T d , let ρ(v) ∈ {0, . . . , d} be independent and identically distributed random variables, with distribution given by P[ρ(v) = j] = r j with d j=0 r j = 1. The rotor walk moves in this way: at time n, if the walker is at vertex v, then it first rotates the rotor to point to the next neighbor in the counterclockwise order and then it moves to that vertex, that is X n+1 = v (ρ(v)+1) mod (d+1) . If the initial rotor configuration is random, then once a vertex has been visited for the first time, the configuration there is fixed. A child v ( j) of a vertex v ∈ T d is called good if ρ(v) < j, which means that the rotor walk will first visit the good children before visiting the parent v (0) of v. Remark that v has d − ρ(v) good children. The tree of good children for the rotor walk (X n ), which we denote T good d , is a subtree of T d , where all the vertices are good children. Let us denote by R n = {X 0 , . . . , X n } the range on T d = T d \{o} of the rotor walk (X n ) up to time n, that is, the set of distinct visited points by the rotor walk (X n ) up to time n, excluding the sink vertex o. Its cardinality, denoted by |R n | represents then the number of distinct visited points by the walker up to time n.
We denote by d(r , X n ) := |X n | the distance from the position X n at time n of the rotor walker to the root r . The speed or the rate of escape of the rotor walk (X n ) is the almost sure limit (if it exists) of |X n | n . We say that |R n | satisfies a law of large numbers if |R n | n converges almost surely to a constant. The aim of this work is to prove a law of large numbers for |X n | and |R n |, that is, to find constants l and α such that when (X n ) is a rotor walk with random initial rotor configuration on a regular tree and on a Galton-Watson tree, respectively. On regular trees, these constants depend on whether the rotor walk (X n ) is recurrent or transient on T d , a property which depends  of good children for the rotor walk is a Galton-Watson tree with mean offspring number d − E[ρ (v)] and generating function f (s) = d j=0 r d− j s j . The main results of this paper can be summarized into the two following theorems. The constant α depends only on d and on the distribution of ρ and is given by: .
(iii) If (X n ) n∈N is transient, then conditioned on the non-extinction of T where q > 0 is the extinction probability of T (q − f (q))(1 − q) where q > 0 is the extinction probability of T good d .
The constant l is in the following relation with the constant α from Theorem 1.1 in the transient and null recurrent case: We call this equation the Einstein relation for rotor walks. We state similar results for rotor walks on Galton-Watson trees T with random initial configuration of rotors, and we show that in this case, the constants α and l depend only on the distribution of the configuration ρ and on the offspring distribution of T . Range of rotor walks and its shape was considered also in [5] on comb lattices and on Eulerian graphs. On combs, it is proven that the size of the range |R n | is of order n 2/3 , and its asymptotic shape is a diamond. It is conjectured in [7] that on Z 2 , the range of uniform rotor walks is asymptotically a disk, and its size is of order n 2/3 . In the recent paper [3], for special cases of initial configuration of rotors on transient and vertex-transitive graphs, it is shown that the occupation rate of the rotor walk is close to the Green function of the random walk.
Organization of the paper. We start by recalling some basic facts and definitions about rotor walks and Galton-Watson trees in Sect. 2. Then in Sect. 3 we prove Theorem 1.1, while in Sect. 4 we prove Theorem 1.2. Then we prove in Theorem 5.3 and in Theorem 5.5 a law of large numbers for the range and the existence of the speed, respectively, for rotor walks on Galton-Watson trees. Finally, in "Appendix A" we look at the contour function of the range of recurrent rotor walks and its recursive decomposition.

Rotor Walks
Let T d be the regular infinite rooted tree with degree d+1, with root r , and an additional vertex o which is connected to the root and is called the sink, and let T d = T d \{o}. Every vertex v ∈ T d has d children and one parent. For any connected subset V ⊂ T d , define the set of leaves in T d as ∂ o V = {v ∈ T d \V : ∃u ∈ V s. t. u ∼ v} as the set of vertices outside of V that are children of vertices of V , that is, ∂ o V is the outer boundary of V . On T, the size of ∂ o V depends only on the size of V : A rotor configuration ρ on T d is a function ρ : T d → N 0 , with ρ(x) ∈ {0, . . . , d}, which can be interpreted as following: each vertex v ∈ T d is endowed with a rotor ρ(v) (or an arrow) which points to one of the d + 1 neighbors. We fix from the beginning a counterclockwise ordering of the neighbors v (0) , v (1) , . . . , v (d) , which represents the order in which the neighbors of a vertex are visited, where v (0) is the parent of v, and v (1) , . . . , v (d) are the children. A rotor walk (X n ) on T d is a process where at each time step n, a walker located at some vertex v ∈ T d first increments the rotor at v, i.e., it changes its direction to the next neighbor in the counterclockwise order, and then the walker moves there. We start all our rotor walks at the root r , X 0 = r , with initial rotor configuration ρ 0 = ρ. Then X n represents the position of the rotor walk at time n, and ρ n the rotor configuration at time n. The rotor walk is also used as rotor-router walk in the literature. At each time step, we record not only the position of the walker, but also the configuration of rotors, which changes only at the current position. More precisely, if at time n the pair of position and configuration is (X n , ρ n ), then at time n + 1 we have and X n+1 = X (ρ n+1 (X n )) n . As defined above, (X n ) is a deterministic process once ρ 0 is determined. Throughout this paper, we are interested in rotor walks (X n ) which start with a random initial configuration ρ 0 of rotors, which makes (X n ) a random process that is not a Markov chain.

Random initial configuration.
For the rest of the paper, we consider ρ a random initial configuration on T d , in which (ρ(v)) v∈ T d are independent random variables with distribution on {0, 1, . . . , d} given by with d j=0 r j = 1. If ρ(v) is uniformly distributed on the neighbors, then we call the corresponding rotor walk uniform rotor walk. Depending on the distribution of the initial rotor configuration ρ, the rotor walk can exhibit one of the following two behaviors: either the walk visits each vertex infinitely often, and it is recurrent, or each vertex is visited at most finitely many times, and it escapes to infinity, and this is the transient case. For rotor walks on regular trees, the recurrence-transience behavior was proven in [1,Theorem 6], and the proof is based on the extinction/survival of a certain branching process, which will also be used in our results. Similar results on recurrence and transience of rotor walks on Galton-Watson trees have been proven in [6].
For the rotor configuration ρ on T d , a live path is an infinite sequence of vertices v 1 , v 2 , . . . each being the parent of the next, such that for all i, the indices k for which . is a live path if and only if all v 1 , v 2 , . . . are good, and a particle located at v i will be sent by the rotor walker forward to v i+1 before sending it back to the root. An end in T d is an infinite sequence of vertices o = v 0 , v 1 , . . ., each being the parent of the next. An end is called live if the subsequence (v i ) i≥ j starting at one of the vertices o = v 0 , v 1 , . . . is a live path. The rotor walk (X n ) can escape to infinity only via a live path.

Galton-Watson Trees
Consider a Galton-Watson process (Z n ) n∈N o with offspring distribution ξ given by p k = P[ξ = k]. We start with one particle Z 0 = 1, which has k children with probability p k ; then each of these children independently has children with the same offspring distribution ξ , and so on. Then Z n represents the number of particles in the n-th generation. If (ξ n i ) i,n∈N are i.i.d. random variables distributed as ξ , then and Z 0 = 1. Starting with a single progenitor, this process yields a random family tree T , which is called a Galton-Watson tree. The mean offspring number m is defined as the expected number of children of one particle m = E[ξ ]. In order to avoid trivialities, we will assume p 0 + p 1 < 1. The generating function of the process is the function f (s) = ∞ k=0 p k s k and m = f (1). If is well known that the extinction probability of the process, defined as q = lim n→∞ P[Z n = 0], which is the probability the process ever dies out, has the following important property. With probability 1, we have Z n → 0 or Z n → ∞ and lim n P[Z n = 0] = 1 − lim n P[Z n = ∞] = q. For more information on Galton-Watson processes, we refer to [2]. When m < 1, = 1, or > 1, we shall refer to the Galton-Watson tree as subcritical, critical, or supercritical, respectively.

Range on Regular Trees
For a simple random walk on a regular tree T d , d ≥ 2, which is transient, if we denote by S n its range, then it is known [4, Theorem 1.2] that |S n | satisfies a law of large numbers: We prove a similar result for the range of any rotor walk with random initial configuration on a regular tree T d . From [1,Theorem 6], The tree of good children for the rotor walk, denoted T The lemma below is a key observation that is crucial for the main results of this paper. For a proof, we refer to [1]. For proving Theorem 1.1, we shall treat the three cases separately: the positive recurrent, null recurrent and transient case.

Recurrent Rotor Walks
In this section, we consider recurrent rotor walks (X n ) on T d , that is, once again from [1, has mean offspring number m ≤ 1, which by Theorem 2.1 dies out with probability one. While in the case m < 1, where the rotor walk is positive recurrent, the expected size of T good d is finite, this is not the case when m > 1. For this reason, we handle these two cases separately. In order to prove a law of large numbers for the range R n = {X 0 , X 1 , . . . , X n } of the rotor walk up to time n, we first look at the behavior of the rotor walk at the times when it returns to the sink o. Define the times (τ k ) of the k-th return to the sink o, by: τ 0 = 0 and for k ≥ 1 let At time τ k , the walker is at sink, all rotors in the visited set R τ k point toward the root, while all other rotors still are in their initial configuration. Between the two consecutive stopping times τ k−1 and τ k , the rotor walk performed a depth first search in the finite subtree induced by R τ k , by visiting every child of a vertex in right to left order. For every vertex in v ∈ R τ k , we can uniquely associate the edge (v, v (0) ), with v (0) being the unique ancestor of v, which implies that |R τ k | equals the number of edges in the tree induced by R τ k . In a depth first search of R τ k , each edge is visited exactly two times, and in view of the bijection above, it requires exactly 2|R τ k | steps to return to the origin. We can then deduce that Since we are in the recurrent case, where the rotor walk returns to the sink infinitely many times, these stopping times are almost surely finite.

Positive Recurrent Rotor Walks
we prove the following.

Theorem 3.2 For a positive recurrent rotor walk
, almost surely.
Proof For simplicity of notation, we write R k = R τ k . The tree of good children, T good d = R 1 is a subcritical Galton-Watson tree with mean offspring number m = d − E[ρ] < 1, that is, it dies out almost surely. The expected size of the range up to time τ 1 is given by At the time τ k of the k-th return to the sink o, all rotors in the previously visited set R k point toward the root, and the remaining rotors are still in their initial configuration. Thus, during the time interval (τ k , τ k+1 ], the rotor walk visits all the leaves of R k from right to left, and at each leaf it attaches independently a (random) subtree that has the same distribution as R 1 is a supercritical Galton-Watson process with mean offspring number ν (the mean number of leaves of |R 1 |), which in view of (2), is given by Moreover, P[L k = 0] = 0. Since ν > 1, it follows from the Seneta-Heyde Theorem (see [8]) applied to the supercritical Galton-Watson process (L k ) that there exists a sequence of numbers (c k ) k≥1 and a nonnegative random variable W such that and dividing by c k gives It then follows that Since τ k c k and τ k−1 c k−1 either both diverge or have the same limit τ , it follows that Thus, τ k c k converges almost surely to an almost surely positive random variable τ Hence, and this proves the claim.
Passing from the range along a subsequence (τ k ) to the range R n at all times requires additional work, because of the exponential growth of the increments (τ k+1 − τ k ). We next prove that the almost sure limit |R n | n exists.

Proof of Theorem 1.1(i)
k be the range of d independent recurrent rotor walks on the tree T d with i.i.d. initial rotor configurations, at the k-th return to the sink vertex. Moreover, let ξ k be the times of the k-th visit to the sink vertex by these rotor walks. One can couple the original rotor walk with these d independent rotor walks in such a way that the dynamics of the original rotor walk in the component of the tree rooted at x i is given by the dynamics in the i-th rotor walk, for i = 1, . . . , d.
Let n be an arbitrary positive number in (τ k , τ k+1 ). Then the original rotor walk (X n ) at time n is located in the component of the tree rooted at x i for some i. The coupling above then gives us these two inequalities: Note that the addition of k in the inequalities above is to account for the time spent at the sink vertex in the original rotor walk. The contribution of k is negligible as n → ∞ as far as we are concerned. It then follows that where k := k(n) and i := i(n) depend on n. By Theorem 3.2, it then follows that Recall from the proof of Theorem 3.2 that there exists c k > 0 and an integrable random variable W such that, for any i The same coupling can be applied not only to the d children of the root, but also to the vertices of T d and any given level j. This means that, for any j ≥ 0, we have Since W is an integrable random variable, it follows from the law of large numbers By symmetry, we can also conclude that lim inf n→∞ R n n ≥ α. Theorem 1.1(i) now follows.
If ν is the mean offspring number of the supercritical Galton-Watson process (L k ),

Null Recurrent Rotor Walks
In this section, we consider null recurrent rotor walks, that is Recall the stopping times τ k as defined in (5). The proofs for the law of large numbers for the range will be slightly different, arising from the fact that the expected return time to the sink for the rotor walker is infinite. We first prove the following.

Theorem 3.3 For a null recurrent rotor walk
Proof Rewriting Eq. (6), we get and we prove that the quotient τ k−1 τ k goes to zero almost surely. We write again R k = R τ k and we first show that τ k τ k−1 → ∞ almost surely, by finding a lower bound which converges to ∞ almost surely. From (6), we have If ∂ 0 R k−1 is the set of leaves of R k−1 , then in the time interval τ k − τ k−1 , the i.i.d critical Galton-Watson trees rooted at the leaves ∂ 0 R k−1 will be added to the current range R k−1 .
Recall that from the proof of Theorem 3.2 that L k = |∂ 0 R k |. For each k = 1, 2, . . ., we partition the time interval (τ k , τ k+1 ] into finer intervals, on which the behavior of the range can be easily controlled. The vertices in ∂ o R τ k = {x 1 , x 2 , . . . , x L k } are ordered from right to left. We introduce the following two (finite) sequences of stopping times That is, for each leaf x i , the time η i k represents the first time the rotor walk reaches x i , and θ i k represents the last time the rotor walk returns to x i after making a full excursion in the critical Galton-Watson tree rooted at x i . Then almost surely. It is easy to see that the increments (θ i k − η i k ) i are i.i.d and distributed according to the distribution of τ 1 , which is the time a rotor walk needs to return to the sink for the first time. Once the rotor walk reaches the leaf x i for the first time at time η i k , the subtree rooted at x i was never visited before by a rotor walk. Even more, the tree of good children with root x i is a critical Galton-Watson tree, which becomes extinct almost surely. Thus, the rotor walk on this subtree is (null) recurrent, and it returns to , the rotor walk leaves the leaf x i−1 and returns to the confluent between x i−1 and x i , from where it continues its journey until it reaches is the time the rotor walk needs to reach the new leaf x i after leaving x i−1 . In this time interval, the range does not change, since (X n ) makes steps only in R τ k . We have, as a consequence of (6) By the strong law of large numbers, we have on the one side (8) and (10) yields almost surely, where the last inequality follows from (6). By letting l = lim inf τ k τ k−1 and taking limits, the previous equation yields Unless l = 1, the right-hand side above goes to infinity almost surely, which implies τ k = 0, almost surely. Suppose now l = 1, almost surely. Once at the root at time τ k−1 , until the next return at time τ k , the rotor walk visits everything that was visited before plus new trees where the configuration is in the initial status. As a consequence of Lemma 3.1, for visiting the previously visited set, it needs time τ k−1 , therefore τ k − τ k−1 > τ k−1 , which gives that lim inf k τ k τ k−1 > 2, which contradicts the fact that l = 1. Therefore lim k τ k τ k−1 = ∞, almost surely. Finally, we show that indeed lim k→∞ |R k | τ k exists. On the one hand, from (6), it is easy to see that |R k | τ k ≤ 1 2 . We have almost surely, and the claim follows.
where (R j 1 ) j and (τ j 1 ) j are i.i.d. random variables distributed like R 1 = R τ 1 and τ 1 , respectively. From (6) we have τ j 1 = 2|R j 1 |, and from the previous two equations we obtain the following upper bound on |R n | n : Since for every i, τ i 1 is almost surely finite and τ k → ∞ as k → ∞, the term converges almost surely to 0 as k → ∞. Moreover, since in the null recurrent case, from the proof of Theorem 3.3, τ k−1 τ k → 0 almost surely, as k → ∞, we also get the almost sure convergence to 0 of τ k−1 , and by the same reasoning as above, we obtain which together with the upper bound on limsup proves that lim n→∞ |R n | n = 1 2 almost surely.

Transient Rotor Walks
We consider here the transient case on regular trees, when E[ρ(v)] < d − 1. Then each vertex is visited only finitely many times, and the walk escapes to infinity along a live path. The tree T

Notation 3.4
For the rest of this section, we will always condition on the event of nonextinction, so that T good d is an infinite random tree. We denote by P non and by E non the associated probability and expectation conditioned on non-extinction, respectively.
That is, if P is the probability for the rotor walk in the original tree T good d , then for some event A, we have In order to understand how the rotor walk (X n ) escapes to infinity, we will decompose the tree T good d with generating function f conditioned on non-extinction. Consider the generating functions Then the f -Galton-Watson tree T good d can be generated by: (i) growing a Galton-Watson tree T g with generating function g, which has the survival probability 1. conditioned to die out is equivalent to the h-Galton-Watson tree with generating function h, which is subcritical. For more details on this decomposition and the equivalence of the processes involved above, see [9] and [2, Chapter I, Part D].
The exposition in this paragraph is a non-trivial consequence of the key Lemma 3.1(b). Denote by t 0 < ∞ the number of times the origin was visited. After the t 0 -th visit to the origin, there are no returns to the origin, almost surely, and there has to be a leaf γ 0 belonging to the range of the rotor walk up to the t 0 -th return, along which the rotor walker escapes to infinity, that is, there is a live path starting at γ 0 , almost surely. Denote by n 0 the first time the rotor walk arrives at γ 0 . The tree rooted at γ 0 was not visited previously by the walker, and the rotors are in their random initial configuration. The tree of good children T good d rooted at γ 0 in the initial rotor configuration has the same distribution as the supercritical Galton-Watson tree conditioned on non-extinction. At time n 0 , we have already a finite visited subtree and its cardinality |R n 0 |, which is negligible for the limit behavior of the range. When computing the limit for the size of the range, we have to consider also this irrelevant finite part.
On the event of non-extinction, let γ = (γ 0 , γ 1 , . . .) be the rightmost infinite ray in T good d rooted at γ 0 , that is, the rightmost infinite live path in T d , which starts at γ 0 . This is the rightmost ray in the tree T g . Since all vertices in T g have an infinite line of descent, such a ray exists. The ray γ is then a live path, along which the rotor walk (X n ) escapes to infinity, without visiting the vertices to the left of γ ; see again Lemma 3.1(b). In order to understand the behavior of the range of (X n ) and to prove a law of large numbers, we introduce the sequence of regeneration times (τ k ) for the ray γ . Let τ 0 = n 0 and for k ≥ 1: Note that, for each k the random times τ k and τ k+1 − 1 are the first and the last hitting time of γ k , respectively. Indeed, once we are at vertex γ k , since γ k+1 is the rightmost child of γ k with infinite line of descent, the rotor walk visits all good children to the right of γ k , and makes finite excursions in the trees rooted at those good children, and then returns to γ k at time τ k+1 − 1. Then, at time τ k+1 , the walk moves to γ k+1 and never returns to γ k . We first prove the following.

Theorem 3.5 Let (X n ) be a transient rotor walk on T d . If the tree T
good d of good children for the rotor walk (X n ) has extinction probability q, then conditioned on non-extinction, there exists a constant α > 0, which depends only on q and d, such that lim k→∞ |R τ k | τ k = α, almost surely, and α is given by .
Proof We write again R k := R τ k , and R 0 for the range of the rotor walk up to time n 0 , 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 k in T good d we have a random number of vertices, and in the tree rooted at those vertices (which are h-Galton-Watson trees), the rotor walk makes only finite excursions. Then, at time τ k+1 , the walk reaches γ k+1 and never returns to γ k .
For each k, γ k+1 is a good child of γ k and the rotor at γ k points to the right of γ k+1 . For k = 0, 1, . . . let γ (1) k , . . . , γ (N k ) k be the set of vertices which are good children of γ k , and are situated to the right of γ k+1 ; denote by N k the cardinality of this set. Additionally, denote by T k ( j) the tree rooted at γ ( j) k , j = 1, . . . , N k and by That is, the trees T k ( j), for j = 1, . . . , N k have all common root γ k , and |T k ( j)| = | T k ( j)| + 1.   Proof of Claim 3. Given non-extinction, the time (τ k+1 − τ k ) k≥0 depends only on N k (the number of good children to the right of γ k ) and on |T k ( j)|, which by Claims 1 and 2 above, are all i.i.d. Thus, the independence of (τ k+1 − τ k ) k≥0 follows as well.
Clearly, each γ k is visited exactly N k + 1 times, and all vertices to the left of γ are never visited. We write R (k− 1,k] for the range of the rotor walk in the time interval (τ k−1 , τ k ]. For r = s, the path of the rotor walk in the time interval (τ r −1 , τ r ] has empty intersection with the path in the time interval (τ s−1 , τ s ], and we have Moreover, for i = 1, . . . k, is an i.i.d. sequence of random variables with the same distribution as | T i−1 ( j)|, then, using Claim 1 and Claim 2 we obtain Similarly, if we write τ k = n 0 + k i=2 (τ i − τ i−1 ) and use the fact that, for i ≥ 2 as a consequence of (6) then again by Claims 1 and 2 we get Putting Eqs. (19) and (20) together, we finally get , almost surely.
By the strong law of large numbers, we have which in terms of the generating function f (s) can be written as In computing E non [N 0 ]: is infinite|γ 0 has j good children] P[γ 0 has j good children] In terms of the generating function f (s), using f (q) = q, E non [N 0 ] can be written as Putting the two expectations together, we obtain the constant α in terms of the generating function f (s) given by In Theorem 3.5, we have proved a law of large numbers for the range of the rotor walk along a subsequence (τ k ). With very little effort, we can show that we have indeed a law of large numbers at all times.
By Claim 3 from the proof of Theorem 3.5, the increments (τ k+1 − τ k ) are i.i.d, and finite almost surely, therefore τ k+1 −τ k τ k → 0 almost surely as k → ∞. Then, since we have that τ k+1 τ k → 1 almost surely, as k → ∞. This, together with Theorem 3.5 implies that the left-hand side of Eq. (23) converges to α almost surely. By the same argument, we obtain that also the right-hand side of (23) converges to the same constant α almost surely, and this completes the proof.

Uniform Rotor Walks
We discuss here the behavior of rotor walks on regular trees T d , with uniform initial rotor configuration ρ, that is, for all v ∈ T d , the random variables (ρ(v)) are i.i.d with uniform distribution on the set {0, 1, . . . , d}, i.e., P[ρ(v) = j] = 1 d+1 , for j ∈ {0, 1, . . . , d}. Such walks are null recurrent on T 2 and transient on all T d , d ≥ 3. As a special case of Theorem 1.1(iii), we have the following.

Corollary 3.6 If (X n ) is a uniform rotor walk on T d , d ≥ 3, then the constant α is given by
in Theorem 3.5, we get the result.
The following table shows values for the constants α in comparison with the limit (d − 1)/d for the simple random walk on trees. Note that only in the case of the binary tree T 2 , the limit values for the range of the uniform rotor walk and of the simple random walk are equal, even though the uniform rotor walk is null recurrent and the simple random walk is transient. In the transient case, that is, for all d ≥ 3 we always have α > (d − 1)/d.

Speed on Regular Trees
In this section, we prove the existence of the almost sure limit |X n | n , as n → ∞, where |X n | represents the distance from the root to the position X n at time n of the walker.

Proof of Theorem 1.2(i)
We show that in this case, l = lim n→∞ |X n | n = 0, almost surely. For arbitrary n, let k = max{i : τ i < n}, which implies that τ k < n ≤ τ k+1 , and up to time n we have k returns to the root. Let D k be the maximum distance from the root reached after k returns to the root.
Positive recurrent rotor walks. We have In view of [1, Theorem 7(ii)], the maximal depth grows linearly with the number of returns to the root, that is, D k+1 k+1 is almost surely bounded. On the other hand, τ k grows exponentially in k, therefore k+1 τ k converges almost surely to 0 as k → ∞, that is l = lim n→∞ |X n | n = 0 almost surely. Null recurrent rotor walks. In the null recurrent case, the situation is a bit different, since even though all particles return to the root, they reach very great depths; see again [1, Theorem 7(i)]. Write again R k := R τ k . For the position of the rotor walk X n , we distinguish the following three cases: and the right-hand side above converges to 0 almost surely in view of Theorem 3.3 together with the fact τ k−1 τ k → 0 almost surely, proven again in Theorem 3.3. Therefore lim n→∞ where τ i 1 is a random variable, independent and identically distributed to τ 1 which is finite almost surely. The quantity |R k−1 | τ k converges to 0 by the same argument as in case (i), whereas τ i 1 τ k converges also to 0 almost surely, as k = k(n) → ∞.

Transient Rotor Walks
Since in the transient case there is a positive probability of extinction of T good d , we condition on the event of non-extinction. The notation remains the same as in Sect. 3.2.

Proof of Theorem 1.2(ii)
Recall the definition of the infinite ray γ along which the rotor walk (X n ) escapes to infinity, and the regeneration times τ k as defined in (18). We first prove the existence of the speed l along the sequence (τ k ). This is rather easy, since d(r , γ k ) = |X n 0 | + k, a.s. where n 0 < ∞ is the first time the walk reaches γ 0 , from where it escapes without returning to the root; see again Lemma 3.1(b).
Recall now from the proof of Theorem 3.5, that τ k = n 0 + k + 2 α k j=1t j a.s. and α k = k−1 i=0 N i a.s., with the involved quantities again as computed in Theorem 3.5. Then By the strong law of large numbers for sums of i.i.d random variables, we have almost surely, as k → ∞, Also, since |X n 0 | and n 0 are both finite, the almost sure limits |X n 0 | k and n 0 k are 0, as k → ∞. This implies , almost surely, as k → ∞.
By Eqs.
In order to prove the almost sure convergence of |X n | n , for all n, we take We know that τ k < n ≤ τ k+1 a.s. |X τ k | = |X n 0 | + k, |X τ k+1 | = |X n 0 | + k + 1 a.s. and |X n | ≥ |X n 0 | + k a.s. Moreover, between times τ k and τ k+1 , the distance can increase with no more than τ k+1 − τ k , and we have , together with Eq. (24) and the facts that τ k τ k+1 → 1 and |X n 0 | τ k+1 → 0 almost surely, we obtain that the left-hand side of the equation above converges to l almost surely, as k → ∞. For the right-hand side, we use again Eq. (24), together with the fact that τ k+1 −τ k τ k → 0 almost surely, since the increments (τ k+1 −τ k ) are i.i.d and almost surely finite. This yields the almost sure convergence of the righthand side of the equation above to the constant l, which implies that |X n | n → l almost surely as n → ∞.

Rotor Walks on Galton-Watson Trees
The methods we have used in proving the law of large numbers and the existence of the rate of escape for rotor walks (X n ) with random initial configuration on regular trees can be, with minor modifications, adapted to the case when the rotor walk (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 down the proofs again, but only mention the differences which appear on Galton-Watson trees.
Let T be a Galton-Watson tree with offspring distribution ξ given by p k = P[ξ = k], for k ≥ 0, and we assume that p 0 = 0, that is T is supercritical and survives with probability 1. Moreover, the mean offspring number μ = E[ξ ] is also greater than 1. We recall the notation and the main result from [6]. For each k ≥ 0 we choose a probability distribution Q k supported on {0, . . . , k}. That is, we have the sequence of distributions (Q k ) k∈N 0 , where Q k = q k, j 0≤ j≤k with q k, j ≥ 0 and k j=0 q k, j = 1. Let Q be the infinite lower triangular matrix having Q k as row vectors, i.e., Below, we write d x for the (random) degree of vertex x in T .

Definition 5.1 A random rotor configuration
is a random variable with the following properties: We write R T for the corresponding probability measure.
Then RGW = R T × GW represents the probability measure given by choosing a tree T according to the GW measure, and then independently choosing a rotor configuration ρ on T according to R T . Recall that to the root r ∈ T , we have added an additional sink vertex s. If we start with n rotor particles, one after another, at the root r of T , with random initial configuration ρ, and we denote by E n (T , ρ) the number of particles out of n that escape to infinity, then the main result of [6] is the following.
where γ (T ) represents the probability that simple random walk started at the root of T never returns to s.
Let us denote m := E[ν]. That is, if m ≤ 1, then (X n ) is recurrent and if m > 1, then (X n ) is transient.

Range and Speed on Galton-Watson Trees
Tree of good children. If ρ is Q-distributed, then the distribution of the number of good children of a vertex x in T is given by which is the lth component of the vector ν = ξ · Q. The tree of good children T good is in this case a Galton-Watson tree with offspring distribution ν = ξ · Q whose mean was denoted by m. Denote by f T the generating function of T good and by q its extinction probability. For the range R T n of rotor walks (X n ) on Galton-Watson trees, we get the following result.

Theorem 5.3
Let T be a Galton-Watson tree with offspring distribution ξ and mean offspring number E[ξ ] = μ > 1. If (X n ) is a rotor walk with random Q-distributed initial configuration on T , and ν = ξ · Q, then there is a constant α T > 0 such that If we write m = E[ν], then the constant α T is given by: .
(ii) If (X n ) n∈N is null recurrent, then (iii) If (X n ) n∈N is transient, then conditioned on the non-extinction of T good , The proof follows the lines of the proof of Theorem 1.1, with minor changes which we state below. The offspring distribution of the tree of good children T good will be here replaced with (25), and the corresponding mean offspring number is m. If L k and R τ k are the same as in the proof of Theorem 3.2, then in case when (X n ) moves on a Galton-Watson tree T , where ξ v denotes the (random) number of children of the vertex v ∈ T . Since T is the initial Galton-Watson tree with mean offspring number μ, by Wald's identity we get Then ν in Theorem 3.2 will be replaced with λ and the rest works through. In the proof of Theorem 1.1(i), α will be replaced with μ−1 2(μ−m) . Theorem 3.3 works as well here, with a minor change in Eq. (12), where the last term will be 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 0 ], the degree d − 1 has to be replaced by the offspring distribution ξ , which in terms of the generating function f T of T good , and its extinction probability q produces the same result.
If (X n ) is an uniform rotor walk on the Galton-Watson tree T , we have a particularly simple limit. In [6], it was shown that (X n ) is recurrent if and only if μ ≤ 2. Moreover, the tree of good children T good is a subcritical Galton-Watson tree with offspring distribution given by ν =  Finally, we also have the existence of the rate of escape.
We then define the function f A pointwise as following and we call f A the contour of the set A.

A.1: Contour of the Range: Recurrent Case
The range R n of the rotor walk (X n ) on T d is a subtree of T d . In what follows, we write f n = f R n for the contour of the range of the rotor walk up to time n. Since we start with a random initial rotor configuration, f n is a random càdlàg-function. See  for the contours of the rotor walk range for the first few steps of the process on the binary tree T 2 . Figure 4a shows a typical contour of the rotor range on the binary tree for n = 10,000 steps. Figure 4b shows a numerical approximation of the expectation E[ f 10000 ].
As in the proofs of the law of large numbers for |R n |, we look first at the times when the rotor walk returns to the root. Recall the definition of the return times (τ k ), as defined in (5), for recurrent rotor walks. Write again R k = R τ k for the range up to time τ k of the k-th return of the rotor walk (X n ) to the sink, and denote by g k (x) = E f R k (x) the expected contour after the k-th excursion, that is g k (x) = ∞ m=1 mP f R k (x) = m . Recall that we are in a case of a random initial configuration ρ on T d , with E[ρ(v)] ≥ 1. The distribution of ρ is P ρ(v) = i = r i ≥ 0. Some additional notation will be needed. For i = 0, . . . , d let For each x = (x 1 , x 2 , . . .) ∈ [0, 1], we can now compute the probability P f R 1 (x) = m + 1 that the rotor walk visits the first m vertices of the ray represented by (x 1 , x 2 , . . .) before taking a step back toward the sink vertex. Once the rotor walk makes a step toward the sink, it cannot further explore the ray x without first returning to the sink vertex. Furthermore, the depth the walk can explore the ray before returning depends only on the initial rotor state along the vertices of the ray. Below, f R 1 (x) = 1 means that x 1 is not in the range of the walk after the first full excursion, x 1 being a vertex at level 1. We get the following . . .
For the general case m ≥ 1, we get For k ≥ 2 we have P f R k (x) = 1 = 0 since between each return the tree is explored at least for one additional level. When k ≥ 2 and m ≥ 2: For k ≥ 2 we have the shifted ray ← − x l always equals x. Hence, the number of steps the rotor walk descends into the left-and rightmost rays in the k-th excursion is i.i.d. In view of the two relations g k (0) = kg 1 (0) and g k (1) = kg 1 (1), it makes sense to look at the normalized versions g k (x) = g k k (x) (see Fig. 5b). Figure 5c, d shows corresponding plots for a positive recurrent rotor walk on T 2 with initial distribution given by r 0 = 1/4, r 1 = 1/4 and r 2 = 1/2.
We first look at the case k = 1 Since g 0 (x) = 0, by definition the case k = 1 follows. We now look at the case k ≥ 2. By the convolution formula (30), we have where in the last step we use the convolution formula (30) again in reverse. Thus, which proves the theorem in the general case.
Some comments. It may be interesting to understand on which graphs, other than regular and Galton-Watson trees, does the Einstein relation (1) hold. Other classes of trees such as periodic trees are definitely a good candidate. One can also look at finer estimates such as law of iterated logarithm, or central limit theorems for the range and the speed for rotor walks on regular trees.