Transience and recurrence of rotor-router walks on directed covers of graphs

The aim of this note is to extend the result of Angel and Holroyd concerning the transience and the recurrence of transfinite rotor-router walks, for random initial configuration of rotors on homogeneous trees. We address the same question on directed covers of finite graphs, which are also called trees with finitely many cone types or periodic trees. Furthermore, we provide an example of a directed cover such that the rotor-router walk can be either recurrent or transient, depending only on the planar embedding of the periodic tree.


Introduction
Suppose we are given a finite connected and directed graph G with adjacency matrix D = (d ij ) i,j∈G . Using G, one can construct a labelled rooted tree T in the following way. Start with a root vertex which is labelled with some i ∈ G. Then define the tree recursively such that, if x is a vertex in T with label i ∈ G, then x has d ij successors with label j. The tree T is called the directed cover of G. Random walks on directed covers of graphs have been studied by Takacs [Tak97], Nagnibeda and Woess [NW02]. On infinite graphs, their methods have been extended by Gilch and Müller [GM11].
Rotor-router walks have been first introduced into the physics literature under the name Eulerian walks by Priezzhev, D.Dhar et al [PDDK96] as a model of self organized criticality, a concept established by Bak, Tang and Wiesenfeld [BTW88]. To define a rotor-router walk on a graph consider on each vertex of the graph an arrow (the rotor) pointing to one of the neighbours of the vertex. A particle performing a rotor-router walk first changes the rotor at its current position to point to the next neighbour, in a fixed order chosen at the beginning, and then moves to the neighbour the rotor is now pointing at. These walks have received increased attention in the last years, and in many settings there is remarkable agreement between the behaviour of rotor-router walks and the expected behaviour of random walks. For example, see Holroyd and Propp [HP10], who showed that many quantities associated to rotor-router walks such as normalized hitting frequencies, hitting times and occupation frequencies, are concentrated around their expected values for random walks.
For a bibliographical picture in this context, see also Cooper and Spencer [CS06], Doerr and Friedrich [DF06], Angel and Holroyd [AH11], and also Cooper, Doerr et al. [CDFS06]. On the other hand, rotor-router walks and random walks can also have striking differences. For example, in questions concerning recurrence and transience of rotor-router walks on homogeneous trees, this has been proven by Landau and Levine [LL09]. For random initial configurations on homogeneous trees, see Angel and Holroyd [AH11]. Furthermore, one can use rotor-router walks in order to solve questions regarding the behaviour of random walks: for example, in [HS11] we have used a special rotor-router process in order to determine the harmonic measure, that is, the exit distribution of a random walk from a finite subset of a graph.
In this note, we extend the result of Angel and Holroyd [AH11, Theorem 6] for rotor-router walks with random initial configuration of rotors on directed covers of graphs. The proofs are a generalization of [AH11] and are based on the extinction/survival of an appropriate multitype branching process (MBP). Such a MBP encodes the subtree on which rotor-router particles can reach infinity. We also give several examples where different phase transitions may appear. We give a graph G with two types of vertices and consider its directed cover T with all its possible planar embeddings in the plane. For the same random initial configuration of rotors on these trees, we show that the behaviour of the rotor-router walk depends dramatically on the planar embedding. This corresponds to the fact that different rotor sequence gives rise to different behaviour of the rotor-router walk.

Preliminaries
Graphs and Trees. Let G = (V, E) be a locally finite and connected directed multigraph, with vertex set V and edge set E. For ease of presentation, we shall identify the graph G with its vertex set V , i.e., i ∈ G means i ∈ V . If (i, j) is an edge of G, we write i ∼ G j, and write d(i, j) for the graph distance. Let D = (d ij ) i,j∈G be the adjacency matrix of G, where d ij is the number of directed edges connecting i to j. We write d i for the sum of the entries in the i-th row of D, that is d i = j∈G d ij is the degree of the vertex i.
A tree T is a connected, cycle-free graph. A rooted tree is a tree with a distinguished vertex r, called the root. For a vertex x ∈ T , denote by |x| the height of x, that is the graph distance from the root to x. For h ∈ N, define the truncated tree T h = {x ∈ T : |x| ≤ h} to be the subgraph of T induced by the vertices at height smaller or equal to h. For a vertex x ∈ T \ {r}, denote by x (0) its ancestor, that is the unique neighbour of x closer to the root r. It will be convenient to attach an additional vertex r (0) to the root r, which will be considered in the following as a sink vertex. Additionally we fix a planar embedding of T and enumerate the neighbours of a vertex x ∈ T in counter-clockwise order x (0) , x (1) , . . . , x (dx−1) beginning with the ancestor. We will call a vertex y a descendant of x, if x lies on the unique shortest path from y to the root r. A descendant of x, which is also a neighbour of x, will be called a child. The principal branches of T are the subtrees rooted at the children of the root r. The wired tree T h of height h is the multigraph obtained from T h by collapsing all vertices y ∈ T with d(r, y) = h, together with the ancestor r (0) of the root to a single vertex s, the sink. We do not collapse multiple edges. Let us introduce the down and up sinks s´= and s = s´∪ sˆ.
Directed Covers of Graphs. Suppose now that G is a finite, directed and strongly connected multigraph with adjacency matrix D = (d ij ). Let m be the cardinality of the vertices of G, and label the vertices of G by {1, 2, . . . , m}. The directed cover T of G is defined recursively as a rooted tree T whose vertices are labelled by the vertex set {1, 2, . . . , m} of G. The root r of T is labelled with some i ∈ G. Recursively, if x is a vertex in T with label i ∈ G, then x has d ij descendants with label j. We define the label function τ : T → G as the map that associates to each vertex in T its label in G. The label τ (x) of a vertex x will be also called the type of x. For a vertex x ∈ T , we will not only need its type, but also the types of its children. In order to keep track of the type of a vertex and the types of its children we introduce the generation function χ = (χ i ) i∈G with χ i : {1, . . . , d i } → G. For a vertex x of type i, χ i (k) represents the type of the k-th child x (k) of x, i.e., if τ (x) = i then χ i (k) = τ (x (k) ), for k = 1, . . . , d i . As the neighbours x (0) , . . . , x (d τ (x) ) of any vertex x are drawn in clockwise order, the generation function χ also fixes the planar embedding of the tree and thus defines T uniquely as a planted plane tree. The tree T constructed in this way is called the directed cover of G. Such trees are also known as periodic trees, see Lyons [LP], or trees with finitely many cone types in Nagnibeda and Woess [NW02]. The graph G is called the base graph or the generating graph for the tree T . We write T i for a tree with root r of type i, that is τ (r) = i.
Example 2.1. The Fibonacci tree is the directed cover of the graph G on two vertices with adjacency matrix 0 1 1 1 . The (α, β) bi-regular tree with parameters α, β ∈ N is the directed cover of the graph G on two vertices with adjacency matrix 0 α β 0 .
Rotor-Router Walks. On a locally finite and connected graph G, a rotor-router walk is defined as follows. For each vertex x ∈ G fix a cyclic ordering c(x) of its neighbours: c(x) = x (0) , x (1) , . . . , x (dx−1) , where x ∼ G x (i) for all i = 0, 1, . . . , d x − 1 and d x is the degree of x. The ordering c(x) is called the rotor sequence of x. A rotor configuration is a function ρ : G → G, with ρ(x) ∼ G x, for all x ∈ G. By abuse of notation, we write ρ(x) = i if the rotor at x points to the neighbour x (i) . A rotor-router walk is defined by the following rule. Let x be the current position of the particle, and ρ(x) = i the state of the rotor at x. In one step of the walk the following happens. First the position of the rotor at x is incremented to point to the next neighbour x (i+1) in the ordering c(x), that is, ρ(x) is set to i + 1 (with addition performed modulo d x ). Then the particle moves to position x (i+1) . The rotor-router walk is obtained by repeatedly applying this rule.
Recurrence and Transience Suppose now that the graph G is infinite and connected, and let o be a fixed vertex in G, the root. Start with a particle at o and let it perform a rotor-router walk stopped at the first return to o. Either the particle returns to o after a finite number of steps (recurrence), or it escapes to infinity without returning to o, and visiting each vertex only finitely many times (transience). In both cases, the positions of the rotors after the walk is complete are well defined. Before starting a new particle at the root, we do not reset the configuration of rotors. We then start a new particle at the root o, and repeat the above procedure, and so on. This type of rotor-router walk is called transfinite rotor-router walk, see Holroyd and Propp [HP10]. Let E n = E n (G) be number of particles that escape to infinity after n rotor-router walks are run from o in this way. The following result, due to Schramm states that a rotor-router walk is no more transient than a random walk. See [HP10, Theorem 10] for a proof.
where E denotes the probability that a simple random walk on G started at the origin o never returns to o.

Recurrence and Transience of Rotor-Router Walks
We study now the behaviour of transfinite rotor-router walks on directed covers of graphs for random initial rotor configurations. In particular, we generalize a theorem of Holroyd and Angel [AH11, Theorem 6] which proves a transition between recurrent and transient phases for transfinite rotor-router walks on homogeneous trees For the tree T with root r, the quantity E n (T , ρ) represents the number of particles that escape to infinity after n rotor-router walks are run from r with initial rotor configuration ρ, and E(T ) represents the probability that a simple random walk on T started at r, never returns to r. The theorem then states the following.
The discontinuous phase transition above is related to a branching process. The main idea of the proof is to model the connected subtree of vertices, along which particles may move to infinity, as a Galton-Watson tree. In the case of directed covers, since we have vertices of different types, we need to model a multitype branching process (MBP).
Multitype Branching Processes. A multitype branching process (MBP) is a generalization of a Galton-Watson process, where one allows a finite number of distinguishable types of particles with different probabilistic behaviour. The particle types will coincide with the different types of vertices in the direct covers under consideration, and will be denoted by {1, . . . , m}.
A multitype branching process is a Markov process (Z n ) n∈N0 such that the states Z n = (Z 1 n , . . . , Z m n ) are m-dimensional vectors with non-negative components. The initial state Z 0 is nonrandom. The i-th entry Z i n of Z n represents the number of particles of type i in the n-th generation. The transition law of the process is as follows. If Z 0 = e i , where e i is the m-dimensional vector whose i − th component is 1 and all the others are 0, then Z 1 will have the generating function and 0 ≤ z 1 , . . . , z m ≤ 1, where p i (s 1 , . . . , s m ) is the probability that a particle of type i has s j children of type j, for j = 1, . . . , m. For i = (i 1 , . . . , i m ) and j = (j 1 , . . . , j m ), the one-step transition probabilities are given by For vectors z, s, we write z s = (z s1 1 , . . . , z sm m ). Let M = (m ij ) be the matrix of the first moments: represents the expected number of offsprings of type j of a particle of type i in one generation. If there exists an n such that m (n) ij > 0 for all i, j, then M is called strictly positive and the process Z n is called positive regular. If each particle has exactly one child, then Z n is called singular. The following is well known; see Harris [Har63].
Theorem 3.2. Assume Z n is positive regular and nonsingular, and let r(M ) be the spectral radius of M . It r(M ) ≤ 1, then the process Z n dies with probability one. If r(M ) > 1, then Z n survives with positive probability.

Nondeterministic Rotor Configurations on Directed Covers
Recall the setting we are working on: G is a finite graph with vertices labelled by {1, . . . , m}; for i ∈ G, T i is the directed cover of G with root r of type i. For a vertex x ∈ T i with τ (x) = j ∈ G, we have denoted by x (0) its parent, and by x (1) , . . . , x (dj ) its d j children. We choose the cyclic ordering c(x) of the neighbours of x to be (x (0) , x (1) , . . . , x (dj ) ), and we allow the initial rotor to point at an arbitrary neighbour in this order. We will embed the tree in the plane in such a way that the rotors turn in counter-clockwise order, when following this rotor sequence. Recall also that for some rotor configuration ρ on T i , we write ρ(x) = k if the rotor at x points to the neighbour x (k) .
Let now D = (D 1 , . . . , D m ) be a vector of probability distributions: for each i ∈ G, D i is a probability distribution with values in {0, . . . , d i }. Consider a random initial configuration ρ of rotors on T i , such that ρ(x) x∈Ti are independent random variables, and ρ(x) has distribution D j if the vertex x is of type j. Shortly If (4) is satisfied, we shall say that the rotor configuration ρ is D = (D 1 , . . . , D m )-distributed, and we write ρ d ∼ D. Performing transfinite rotor-router walks on T i with D-distributed initial rotor configuration ρ, we observe a phase transition between the recurrent and transient regimes, similar to the case of homogeneous trees in Theorem 3.1. For defining the critical point of this phase transition, we need to introduce some additional definitions.
Consider a general tree T with rotor configuration ρ. For a vertex x ∈ T define the set of good children as x (k) : ρ(x) < k ≤ d τ (x) . This means that a rotor-router particle starting at a vertex will first visit all its good children before visiting its ancestor. An infinite sequence of vertices x n n∈N with each vertex being a child of the previous one, is called a live path if for every n ≥ 0 the vertex x n+1 is a good child of x n . An end of T is an infinite sequence of vertices x 1 , x 2 , . . . each being the parent of the next. An end is called live if the subsequence (x i ) i≥j starting at one of its vertices is a live path.
Denote by E ∞ (T , ρ) = lim n→∞ E n (T , ρ) the total number of particle escaping to infinity, when one launches an infinite number of particles. Recall now an useful result for general trees T , whose proof can be found in [AH11, Proposition 8].  We have that j∈G C j i (k) = d i − k. Using this definition we can now define a MBP which models connected subtrees consisting of only good children. In this MBP, p i (s 1 , . . . , s m ) represents the probability that a vertex of type i has s j good children of type j, with j = 1, . . . , m. Define the generating function of the MBP as in (2) and the probabilities p i by The quantity E i represents the probability that a simple random walk starting at the root r of T i never returns to r, and r M (D) is the spectral radius of M (D).

Proof of Theorem 3.5(a).
For any fixed x ∈ T i with τ (x) = j ∈ G the set of descendants of x that can be reached via a path of good vertices forms a multitype branching process with offspring distributions p j (s 1 , . . . , s m ) defined as in (5). The survival/extinction of this MBP is controlled by the matrix of the first moments M (D) = (m ij ) i,j∈G .
Since r(M (D)) ≤ 1, the extinction probability is 1 by Proposition 3.2 and the MBP dies almost surely, hence there are no live paths. Therefore by Proposition 3.3 there are no escapes almost surely and E n (T i , ρ) = 0. This gives the recurrence of the rotor-router walk with random initial configuration ρ which is D = (D 1 , . . . , D m )-distributed.
Transience. The transience part in Theorem 3.5 requires some additional work. Consider now the wired tree T h i of height h, with sink s = s´∪ sˆas defined in (1). Let ρ be some rotor configuration on T h i . The proof of the transience uses the abelian property of rotor-router walks. Denote by e r (ρ) the rotor configuration of the rotor-router group of T h i resulting from starting a particle at the root r and letting it perform a rotor-router walk until it reaches the sink s. Write e n r (ρ) for the corresponding rotor configuration when we let n particles perform a rotor-router walk starting at the root until hitting the sink. In addition, let σ n i (ρ) be the particle configuration on T h i obtained by routing n particles from the root r to the sink. As we route all n particles to the sink, the support of σ n i (ρ) is contained in s. Define Hence n i (h) is the minimal number of particles needed such that every vertex of sˆ, that is, every vertex at level h, has been hit by one rotor-router particle, maximized over all possible rotorconfigurations of T h i . The next result gives an upper bound for n i (h).
Lemma 3.6. For all h ≥ 1 and i ∈ G, we have Proof. Using the results in [GLPZP12], in particular Lemma 13 and Lemma 19, it is enough to take the maximum over all recurrent rotor configurations. We prove the statement by induction over the height h.
For h = 1, after one full turn of the rotor in T 1 i , every vertex of s has been visited, hence n i (1) = d i + 1. For h > 1, start with n particles at the origin, where n = (d i + 1)n'. By the abelian property, if we first take one step for each of the particles and then route each particle from its new location to the sink, we also reach the same final particle configuration σ n i . Hence, we have which completes the proof.
We shall see below that it suffices to have at least one particle which stopped at each sink vertex of sˆ, instead of exactly one particle like in [AH11] for homogeneous trees. They can specify there the exact numbers of particles needed at the origin. For our case, it suffices to have an exponential upper bound. In our setting, we are not only interested in the number of particles stopped at some level h, but also on the type of particles. This will be done next.
Number of Vertices of a Given Type on a Level. Denote by w i,j (n) the number of vertices of type j at height n in T i and let w(n) be the matrix with entries (w i,j (n)) i,j∈G . In addition, let W i,j (z) be the generating function of w i,j (n): Factorizing w i,j (n) with respect to the first level, we have Then W i,j (z) can be written as where δ i (j) equals 1 if i = j and 0 otherwise. If we write W (z) = (W i,j (z)) i,j∈G , then the above implicit equation can be written in matrix form as W (z) = I + zDW (z), where D is the adjacency matrix of the generating graph G and I is the identity matrix. Hence Having explicitly the matrix W (z) of generating function, we can compute the coefficients w(n) In a tree T i with root of type i, the total number of vertices at level n is w i (n) := j∈G w i,j (n).
Example 3.7. In the case of Fibonacci tree, we have where F n represents the n-th Fibonacci number. In a tree with root of type 1, the total number of vertices at level n is F n and a tree with root of type 2 has F n+1 vertices at level n. Proof. We shall prove the bound for n ∈ N of the form (D max + 1) h , where h is an integer and D max = max i∈G d i , since this implies the claimed result for all n, but with different constants.
The MBP with probabilities p i as in (5) and r (M (D)) > 1 survives with positive probability. Hence, for each i ∈ G, with positive probability there exists a live path starting at the root of T i . Existence of a live path implies that the first particle escapes, hence For each i, j ∈ G, denote by X i,j the set of vertices v ∈ T i of type j at level h + 1 such that there is a live path starting at v. Let X i = j∈G X i,j and #X i be the cardinality of this set. For all j, the random variables #X i,j are independent, with distribution Binomial w i,j (h), p j . Let us first prove that and n = (D max + 1) h . From [HP10,Lemmas 18,19], it suffices to prove (7) for the truncated tree Here E n (T H i , sˆ, ρ H ) represents the number of particles that stop at sˆ(the vertices at level H of T H i ) when we start n rotor-router walks at the root of T i and rotor configuration ρ H (the restriction of ρ on T H i ). In the truncated tree T H i , start n = (D max + 1) h particles at the root, and stop them when they either enter the level h + 1 or return to the root. By Lemma 3.6, there is at least one particle at each vertex on level h + 1, and the rest of them are located at the root. Moreover, the vertices at distance greater than h + 1 were not reached, and the rotors there are unchanged. Now for every vertex v in X i restart one particle. Since there is a live path a v the particle will reach the level H without leaving the cone of v, at which point the particle is stopped again. Hence if we restart all particles which were stopped at level h + 1 at least #X i of them will reach level H before returning to the root. Because of the abelian property of rotor-router walks, (8) follows, therefore also (7). Now fix i ∈ G and choose k ∈ G such that there are vertices of type k on the h-level of T i , that is w i,k (h) > 0. We can then bound #X i from below by #X i,k which has distribution Binomial w i,k (h), p k . Hence for all δ i > 0, using the Markov inequality, we have In the last inequality we define C k = 1 − p k + p k e −1 < 1 and use that n = (D + 1) h and w i,k (h) ≤ n. Hence for δ i small enough we can choose c i > 0 such that which proves the statement.
We shall also need [AH11, Lemma 25] which holds for general trees.
Lemma 3.9. For a graph G with m vertices and T i its directed cover with root r of type i ∈ G, let T χi(1) , . . . , T χi(di) be its principal branches rooted at the children r (k) of the root, with k = 1, . . . , d i . Let ρ be some rotor configuration on T i and ρ k be its restriction on the tree T χi(k) . For each i ∈ G, let Then The probability E i that a simple random walk (X t ) on T i never returns to the root satisfies an relation similar to (9). If r is the root of T i , then E i = P r [X t = s´, ∀t ≥ 0]. Factorizing the random walk on T i with respect to the first step, we get which gives We are now able to prove the transience part in Theorem 3.5.
Proof of Theorem 3.5(b). For each i ∈ G, let Using Borel-Cantelli Lemma for the events in Corollary 3.8, it follows that for each i, there exists δ i such that Let a i be some positive constants such that l i ≥ a i for all i.
Applying this repeatedly gives that for all i, l i is greater or equal to the fixed point of the iteration a i → 1 − 1 1 + j∈G d ij a j from R m → R m . The return probabilities E i are also solutions of the same fixed point equation (10). Hence l i ≥ E i . On the other hand, by Theorem 2.2 we have l i ≤ E i , which implies if r(M (D)) > 1, which proves the desired.

Examples
Example 3.11. Let us first consider a generalized Fibonacci tree depending on the parameter α ∈ N. Consider the graph G with adjacency matrix and T i its directed cover with root of type i = 1, 2. Depending on the values of α, the rotor-router walk on T i can be either transient or recurrent. If α = 1 we get the Fibonacci tree, and for α = 2 we get the binary tree. On such trees we take a random initial configuration of rotors which is uniformly distributed on the neighbours. Since these trees have 2 types of vertices (first type 1 with α children of type 2, and second type 2 with 1 child of type 1 and one of type 2), we have D = (D 1 , D 2 ), with D 1 = Uniform(0, . . . , α) and D 2 = Uniform(0, 1, 2). Consider the following generation function χ i : . . = χ 1 (α) = 2 and χ 2 (1) = 1 and χ 2 (2) = 2.
The behaviour of rotor-router walks on directed covers of graphs is controlled by the matrix M (D) of the first moments of the MBP defined above. The entries of M (D) can be computed using (3), and for this particular example we have M (D) = 0 α/2 2/3 1/3 .
In particular, the rotor-router walk on the Fibonacci tree and on the binary tree is recurrent. Note here the contrast with the simple random walk which is transient.
The next example shows a case where different planar embeddings of the same tree, gives rise to changes in the recurrence/transience of the rotor-walk with random initial rotor configuration.
There are three possible planar embeddings χ a , χ b and χ c , which are shown in Figure 2 together with the directed covers they generate. On these trees we perform transfinite rotor-router walks where the rotors are initially distributed according to the uniform distribution on both types on vertices. The following table shows the generating functions, the first moment matrix and spectral radius of the associated MBP in all cases.    Hence depending only on the planar embedding the rotor-router walk is either recurrent (for χ a ), recurrent in the critical case (for χ b ) or transient (for χ c ). Another interpretation of this example is: on the same tree (forgetting now about the planar embedding) different rotor sequence gives rise to different behaviour of the rotor-router walk, for random initial configuration of rotors.