Rotor-routing on Galton-Watson trees

A rotor-router walk on a graph is a deterministic process, in which each vertex is endowed with a rotor that points to one of the neighbors. A particle located at some vertex first rotates the rotor in a prescribed order, and then it is routed to the neighbor the rotor is now pointing at. In the current work we make a step toward in understanding the behavior of rotor-router walks on random trees. More precisely, we consider random i.i.d. initial configurations of rotors on Galton-Watson trees, i.e. on a family tree arising from a Galton-Watson process, and give a classification in recurrence and transience for rotor-router walks on these trees.


Introduction
A rotor-router walk on a graph is a deterministic process in which the exits from each vertex follow a prescribed periodic sequence. For an overview and other properties, see the expository paper [HLM + 08]. Rotor-router walks capture in many aspects the expected behavior of simple random walks, but with significantly reduced fluctuations compared to a typical random walk trajectory; for more details see [CS06,FL11,HP10]. However, this similarity breaks down when one looks at recurrence or transience of the walks, where the rotor-router walk may behave differently than the corresponding random walk.
By an unpublished argument of Schramm the rotor-router walk is recurrent on any graph where the simple random walk is recurrent. The converse is not true; there exist graphs where the simple random walk is transient, but the rotor-router walk is still recurrent; see [LL09] for homogeneous trees and [FGLP14] for initial rotor configurations with all rotors aligned on Z d . For random initial configuration of rotors on homogeneous trees, the issue of transience and recurrence was studied in [AH11], and by [HS12] on directed covers of graphs.
In this note we investigate the recurrence and transience of rotor-router walks with random initial rotor configuration ρ on Galton-Watson trees. In order to measure how transient this configuration is, we run n rotor-walks starting from the root and we record whether each walk returns to the root or escapes to infinity. A rotor-router walk is called transient if the relative density of the number of escapes is positive. The main result Theorem 3.2 gives a criterion for recurrence and transience of the rotor-router walk and states that in the transient regime the relative density of escapes of the rotor-router walk equals almost surely the return probability of the simple random walk on Galton-Watson trees. As a consequence of the main result we get the following.
Theorem 1.1. Let ρ be a uniformly distributed rotor configuration on a Galton-Watson tree T with mean offspring number m. The rotor-router walk on T is recurrent if and only if m ≤ 2.

Galton-Watson trees
Let {Z h } h∈N 0 be a Galton-Watson process with offspring distribution ξ given by p k = P[ξ = k] for k ∈ N 0 . Throughout the paper we assume p 0 = 0; this assumption is made for presentational reasons. Informally, a Galton-Watson process is defined as follows: we start with one particle Z 0 = 1, which has k children with probability p k . Then each of these children also have children with the same offspring distribution, independently of each other and of their parent. This continues forever. Formally, let {ξ h i } i,h∈N be i.i.d. random variables with the same distribution as ξ. Define Z h to be the size of the h-th generation, that is, and Z 0 = 1. Our assumption p 0 = 0 means that each vertex has at least one child, and the process survives almost surely. Also, the mean offspring number m = E[ξ] ≥ 1. Here and thereafter we denote by E[ξ] = k≥1 kξ k for a stochastic vector ξ. Observe hereby that a supercritical Galton-Watson process conditioned on survival is a Galton-Watson process with p 0 = 0 plus some finite bushes. The existence of finite bushes does not influence recurrence and transience properties.
We will not be interested only in the size Z h of the h-th generation, but as well in the underlying family trees. Let T be the family tree of this Galton-Watson process, with vertex set j is a descendant of x h i there is an edge in the family tree between these two vertices. For ease of notation we will always identify T with its vertex set. For technical reasons we will add one additional vertex o to the tree, which will act as the parent of the root vertex r = x 1 0 . The vertex o is called the sink of the tree. We always consider T together with its natural planar embedding, that is for each generation h we draw the vertices x h i from right to left, for i = 1, . . . , Z h . In particular this means that each possible embedding of a given combinatorial tree has the same probability.

Rotor-router walks
For each vertex x ∈ T , denote by d x the number of children of x. Given the planar embedding described above, we denote by x (k) , k = 0, . . . , d x , the neighbours of x in T in counterclockwise order beginning at the parent x (0) of x.

Rotor-router walks
A rotor configuration ρ is a map ρ : T → N 0 such that ρ(x) ∈ {0, . . . , d x } for all x ∈ T . For a given rotor configuration ρ 0 and a starting vertex x 0 ∈ T , a rotor-router walk is a sequence of pairs (x i , ρ i ) i≥0 such that for all i ≥ 1 we have the transition rule . Informally this means that a particle performing a rotor-router walk, when reaching the vertex x first increments the rotor at x and then moves to the neighbour of x the rotor is now pointing at.
We are interested in the recurrence or transience of transfinite rotor-router walks, as defined in [HP10], for random initial configuration of rotors. For a given rotor configuration, a rotor walk either visits each vertex infinitely often or visits each vertex only a finite number of times, (see e.g. [HP10, Lemma 6]). Thus in the second case, the rotor-router walk escapes to infinity, leaving behind a well defined limit rotor configuration. A transfinite rotor-router walk with initial rotor configuration ρ, is a sequence of rotor-router particles starting at the root r. Denote by ρ n the initial rotor configuration of the n-th particle, and let ρ 1 = ρ. For all n ≥ 1, run the n-th rotor-router particle until it returns to o for the first time. If this occurs after a finite number of steps, let ρ n+1 be the rotor configuration left behind by this particle. In case the n-th particle never returns to o we define ρ n+1 to be the limit configuration created by the particle escaping to infinity. Let now e k = 1, if the n-th particle escaped to infinity 0, otherwise, and let E n (T , ρ) = n k=1 e k count the number of escapes of the first n walks. The next result, due to Schramm states that a rotor-router walk is no more transient than a random walk. A proof of this result can be found in [HP10, Theorem 10].
Theorem 2.1. For any locally finite graph G, any starting vertex, any cyclic order of neighbours and any initial rotor configuration ρ where γ(G) represents the probability that the simple random walk on the graph G never returns to the starting point.

Random initial rotor configuration
We construct random initial rotor configurations on Galton-Watson trees T introduced above. In order to do this, 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 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.: Definition 3.1. A random rotor configuration ρ on the Galton-Watson tree T is Qdistributed, if for each realization T (ω), and for each v ∈ T (ω) the rotor ρ(v) is a random variable with the following properties: We are now ready to state our main result.
Theorem 3.2. Let ρ be a random Q-distributed rotor configuration on a Galton-Watson tree T with offspring distribution ξ, and let ν = ξ · Q. Then we have almost surely: where γ(T ) represents the probability that simple random walk started at the root of T never returns to o. Proof of Theorem 1.1.
Remark 3.3. Notice here the difference between simple random walk on Galton-Watson trees which is transient for mean offspring number m ∈ (1, 2], while the rotor-router walk with uniformly distributed initial rotors is recurrent.

Recurrent part
Proof of Theorem 3.2(a). Let ρ be a random Q-distributed rotor configuration. Recall that for a vertex x ∈ T we denote by x (k) , k = 1, . . . , d x , the children of x. We call a child This means that the rotor walk at x will visit the good children before visiting the parent Therefore the distribution of the number of good children of a vertex x in T is given by which is the l th component of the vector ν = ξ · Q. Thus, for each vertex x the set of descendants of x which are connected to x by a path consisting of only good children forms a Galton-Watson tree with offspring distribution ν.

The frontier process
To prove the transient part of Theorem 3.2, we will use the frontier process introduced in [HSH14]. For sake of completeness we state the definition of the process here.
Fix an infinite tree T with root r and without leaves and a rotor configuration ρ on T . As before we attach an additional vertex o to the root r of the tree. Consider the following process which generates a sequence F ρ (n) of subsets of vertices of the tree T . F ρ (n) is constructed by a rotor-router process consisting of n rotor-router walks starting at the root r, such that each vertex of F ρ (n) contains exactly one particle. In the first step put a particle at the root r and set F ρ (1) = {r}. Inductively given F ρ (n) and the rotor configuration that was created by the previous step, we construct the next set F ρ (n + 1) using the following rotor-router procedure. Perform rotor-router walk with a particle starting at the root r, until one of the following stopping conditions occurs: (a) The particle reaches o. Then set F ρ (n + 1) = F ρ (n).
(b) The particle reaches a vertex x, which has never been visited before. Then set F ρ (n + 1) = F ρ (n) ∪ {x}.
(c) The particle reaches an element y ∈ F ρ (n). We delete y from F ρ (n), i.e., set F ′ (n) = F ρ (n)\{y}. Now there are two particles at y, both of which are restarted until stopping condition (a), (b) or (c) for the set F ′ (n) applies to them. Note that since we are on a tree at least one particle will stop at a child of y after one step, due to halting condition (b).
We will call the set F ρ (n) the frontier of n particles. Basic properties of this process can be found in [HSH14]. Remark that the frontier process F ρ (n) depends on the underlying tree T .
Following [HSH14], let us introduce where |x| is the distance of x to the root r. M (n) represents the maximal height of the frontier F ρ (n). In order to get an upper bound for M (n), we will use the anchored expansion constant.
Let G be an infinite graph with vertex set V = V (G) and edge set E = E(G). For S ⊂ V (G) we denote by |S| the cardinality of S and by ∂S the vertex boundary of S, that is, the set of vertices in V (G) \ S that have one neighbor in S. We say that the set S is connected if the induced subgraph on S is connected. Fix the root r ∈ V (G). The anchored expansion constant of G is For the remainder of this section, we assume that the tree T has positive anchored expansion constant ι * E (T ) > 0. The next result is a generalization of [HSH14, Lemma 1.5]. Proof. Consider the frontier process F ρ (n) on T . Let x be an element of F ρ (n) with maximal distance M = |x| to the root r. Denote by p = (r = x 0 , x 1 , . . . , x M = x) the shortest path between r and x. Since F ρ (1) = {r} and by the iterative construction of F ρ (n), there exist 1 = n 0 < n 1 < · · · < n M = n, such that x i ∈ F ρ (n i ) for all i ∈ 0, . . . , M .
We want to find a lower bound for n i+2 − n i , that is, for the number of steps needed to replace x i by x i+2 in the frontier. At time n i , the vertex x i is added to the frontier. The next time after n i that a particle visits x i halting condition (c) occurs, thus the rotor at x i is incremented two times. As long as not all children of x i are part of the frontier, every particle can visit x i at most once, since it either stops immediately at a child of x i on stopping condition (b) or is returned to the ancestor of x i . This means that at subsequent visits the rotor at x i is incremented exactly once. In order for x i+2 to be added to the frontier, the rotor at x i has to point at direction x i+1 twice. Thus replacing x i with x i+2 in the frontier, needs at least d x i + 1 particles which visit x i . Hence, n i+2 − n i ≥ 1 + d x i , for i = 0, . . . , M − 2. Denote byp = (x 0 , x 1 , . . . , x M −3 , x M −2 ). By assumption, we have ι * E (T ) > 0. Thus, for M big enough there exists a constant κ > 0, such that |∂p| ≥ κ|p|. Sincep is a path of a tree we get We have therefore (3)
We aim now at getting a lower bound for the size of F ρ (n). Following [HSH14], we first estimate the number of particles stopped at o during the formation of the frontier F ρ (n). This is accomplished using Theorem 1 from [HP10]. For a tree T , define ℓ(n) = {x ∈ T : |x| = M (n) and the path from r to x contains no vertex of F ρ (n)}, (5) where M (n) is defined in (1). By construction, the set F ρ (n) may have "holes". We can fill these holes by adding additional vertices ℓ(n) on the maximal level M (n). All these additional vertices were not touched by a rotor particle during the formation of F ρ (n). Fix n and a rotor configuration ρ, and let be the sink determined by the frontier process F ρ (n). Denote by T S the finite tree which is obtained by truncating T at S.
Let (X t ) be the simple random walk on T . Let τ o = inf{t ≥ 0 : X t = o} and τ S = inf{t ≥ 0 : X t ∈ S} be the first hitting time of o and S respectively. Consider now the hitting probability that is, the probability to hit o before S, when the random walk starts in x. We have h(o) = 1 and h(x) = 0, for all x ∈ S. For x ∈ T \ T S , we set h(x) = 0.
Start now n rotor particles at the root r, and stop them when they either reach o or S. By the Abelian property of rotor-router walks (see [AH12,Lemma 24]) and by the construction of the frontier process F ρ (n) we will have exactly one rotor particle at each vertex of F ρ (n), no particles at ℓ(n), and the rest of the particles are at o. In order to estimate the proportion of rotor particles stopped at o we use Theorem 1 from [HP10], which we state here adapted to our case.
Theorem 3.5 (Theorem 1, [HP10]). Consider the sink S as above, and let (X t ) be the simple random walk on T . Let E be the edge set of T and suppose that the quantity is finite. If we start n rotor particles at the root r, then where n o represents the number of particles stopped at o.
Proof. The function h is harmonic away from the sink: For a x ∈ T S \ S and its ancestor x (0) , we have h x (0) ≥ h(x), and Then where S k = {y ∈ T : |y| = k} represents the k-th level of the tree T . For a fixed k Summing up over all levels the claim follows.
We shall use the following result of Chen and Peres [CP04].
Therefore if we perform the frontier process F ρ (n) on a Galton-Watson tree T , we get by Corollary 3.7 that there exists an almost surely positive random variableκ such that #F ρ (n) >κn almost surely.

Transient part
In this section we will prove the transient part of Theorem 3.2. Let T be a tree with root r, and rotor configuration ρ. Denote by T 1 , . . . , T dr the principal branches of T , and by ρ j the restriction of ρ to T j . Write l(T ) = lim inf n→∞ E n (T, ρ) n and l j (T ) = lim inf n→∞ E n (T j , ρ j ) n , j = 1, . . . , d r ,

(10)
If T is a Galton-Watson tree we have that all l j = l j (T ), j = 1, . . . , d r are i.i.d. Furthermore l = l(T ) has the same distribution as the l j , but is not independent of the l j . Since (10) is valid for any tree, holds stochastically. We first show that under the conditions of Theorem 3.2(b) the random variable l is almost surely bounded away from zero.
Here, E n (T H , S H , ρ H ) represents the number of particles that stop at S H = {x ∈ T : |x| = H} when we start n rotor-router walks at the root r of T and rotor configuration ρ H (the restriction of ρ on T H ). In T S , truncated at the frontier S, start n particles at r, and stop them when they either reach S or return to o. The vertices at distance greater than M (n) were not reached, and the rotors there are unchanged. Now for every vertex x in X restart one particle. Since there is a live path a x the particle will reach the level H without leaving the cone of x, at which point the particle is stopped again. Hence if we restart all particles which are located in F ρ (n) at least #X of them will reach level H before returning to the root. Because of the Abelian property of rotor-router walks, (13) follows, therefore also (12). Using the standard Chernoff bound, there exists δ ∈ (0, 1) such that We can then choose c > 0 such that P E n (T , ρ) < δn ≤ e −cn , for n big enough. Applying Borel-Cantelli Lemma yields the claim.
Recall that γ(T ) is the probability that simple random walk started at the root of T will never visit o. Then γ(T ) in probability satisfies the recursion see [LPP97, Equation (4.1)].
Next, we want to prove that the random variable l stochastically dominates the random variable γ(T ). This requires some additional work. Denote by F γ = F γ (T ) and F l = F l (T ) the cumulative distribution function (c.d.f.) of γ and l respectively. The recursive structure of Galton-Watson trees with offspring distribution ξ and P[ξ = k] = p k gives that F γ satisfies Moreover, for the c.d.f. F l we have using (11) We shall use [LPP97, Theorem 4.1], which we state here. For any initial c.d.f. F with F (0) = 0 and F (1) = 1 other than the Heavyside function, we have weak convergence under iteration to F γ : Remark 3.11. In particular, for c.d.f.'s F 1 ≤ F 2 , for all n ≥ 1 it holds K n (F 1 ) ≤ K n (F 2 ).
Lemma 3.12. Let F l be the c.d.f. of the random variable l. For all n ≥ 1, we have that F l ≤ K n (F l ).
Proof. For n = 1, we have F l ≤ K(F l ) which holds by (16) and by the definition of the operator K in Theorem 3.10. For each n > 1, using Remark 3.11 for c.d.f.'s F l ≤ K(F l ) we get K n (F l ) ≤ K n+1 (F l ). The claim follows then easily by induction.
Proof. By Proposition 3.9, F l is not equal to the Heavyside function 1 [0,∞) . Hence, applying Theorem 3.10 to F l gives lim n→∞ K n (F l ) = F γ .
By Lemma 3.12 F l ≤ lim n→∞ K n (F l ).
Hence F l ≤ F γ , which is equivalent to the stochastic dominance of l over γ.
We are finally able to prove the transient part of Theorem 3.2.
Putting both parts together we get the stochastic inequalities γ ≤ lim inf n→∞ E n (T , ρ) n ≤ lim sup n→∞ E n (T , ρ) n ≤ γ, which implies that the limit L = lim n→∞ En(T ,ρ) n exists, and that L = γ(T ) in distribution.
It is now an easy exercise to show that L = γ almost surely. Since L = γ in distribution, we have E[γ − L] = 0. From Theorem 3.13, we know that γ − L ≥ 0 almost surely. The expectation of the nonnegative random variable γ − L can be zero only when γ − L = 0 almost surely. Therefore lim n→∞ E n (T , ρ) n = γ(T ), with probability one, which proves the transient part.