The speed of the tagged particle in the exclusion process on Galton-Watson trees

We study two different versions of the simple exclusion process on augmented Galton-Watson trees, the constant speed model and the varying speed model. In both cases, the simple exclusion process starts from an equilibrium distribution with non-vanishing particle density. Moreover, we assume to have initially a particle in the root, the tagged particle. We show for both models that the tagged particle has a positive linear speed and we give explicit formulas for the speeds.


Introduction
The simple exclusion process is a classical example of an interacting particle system which was intensively studied over the last decades, see [11,12,13] for an overview. It serves as one of the standard models in order to describe random movements of particles in a gas. Intuitively, it can be described as follows: For a given graph, we place indistinguishable particles on the sites of the graph such that each vertex is occupied by at most one particle. The particles then independently perform simple random walks under an exclusion rule. This means that whenever a particle would move to an occupied site, this move is suppressed. In this article, the underlying graph will be chosen randomly, as a supercritical, augmented Galton-Watson tree without extinction. After the choice of the tree, we consider a stationary starting distribution where we condition on initially having a particle in the root. Our goal is to study the evolution of this tagged particle over time.

The model
Let G = (V, E) be a locally finite graph. Let p : V × V → R + 0 be a function which satisfies p(v, v) = 0 for all v ∈ V . The exclusion process with transition rates p(., .) is defined as the Feller process (η t ) t≥0 with state space {0, 1} V generated by the closure of for all cylinder functions f . We denote by η x,y ∈ {0, 1} V the configuration where we exchange the values at positions x and y in η ∈ {0, 1} V . For a given configuration η, a site x is occupied by a particle if η(x) = 1 and vacant otherwise. Moreover, we write deg(x) for the degree of x. Whenever sup y∈V x∈V p(x, y) < ∞ holds, we have that (1) indeed gives rise to a Feller process, see [12,Theorem 3.9]. We consider two different ways of defining the simple exclusion process on G. For p(x, y) = 1 {x,y}∈E , we refer to the resulting process (η v t ) t≥0 as the varying speed model, for p(x, y) = deg(x) −1 1 {x,y}∈E , we call (η c t ) t≥0 the constant speed model of the simple exclusion process. Note that condition (2) holds for both models. In words, each particle in the varying speed model at a site x has an exponential waiting time with parameter deg(x) independently of all other particles. When the time is up, it jumps to one of its neighbors uniformly at random under an exclusion rule. In the constant speed model, the particles wait according to i.i.d. exp(1)-random variables. They then choose one neighbor uniformly at random and jump to the selected site if it is vacant. Note that the two models of the simple exclusion process agree for regular graphs up to a deterministic time change.
In the following, the underlying graph G will be given as an augmented Galton-Watson tree (T, o) with vertex set V (T ), edge set E(T ) and a distinguished vertex o ∈ V (T ) called the root. More precisely, let (p k ) k∈N 0 be a sequence of non-negative numbers with ∞ k=0 p k = 1, which defines the offspring distribution µ of the tree by µ(k) = p k for all k ∈ N 0 . We construct (T, o) in such a way that each site has precisely k + 1 neighbors with probability p k for all k ∈ N 0 independently of all other sites. To do so, define a starting vertex o and recursively, starting from o, let every site have a number of descendants drawn independently according to µ. The resulting tree is called Galton-Watson tree. Since in this construction, the root has on average one neighbor less than all other sites, we add one additional descendant to o and apply the same recursion in order to obtain an augmented Galton-Watson tree. In this article, we assume that the underlying Galton-Watson branching process is supercritical and without extinction, i.e. we have that p 0 = 0 and 1 < k≥1 kp k < ∞ holds. In particular, this ensures that the corresponding augmented Galton-Watson tree is almost surely locally finite. Moreover, there exists a k > 1 with p k > 0 and hence almost every such tree has infinitely many ends.
For a given realization (T, o) of an augmented Galton-Watson tree, we describe a parametrized set of invariant measures with respect to both models of the simple exclusion process. For ρ ∈ [0, 1], let π ρ,T be the Bernoulli-ρ-product measure on for all x ∈ V (T ). The measures π ρ,T are invariant for the simple exclusion process (η v t ) t≥0 in the varying speed model whenever ρ ∈ [0, 1], see [12, Chapter VIII, Theorem 2.1]. Similarly, the measures ν α,T are invariant for the simple exclusion process (η c t ) t≥0 in the constant speed model for all α ∈ [0, ∞). When we condition to initially have a particle in the root, we call the resulting measures the Palm measures π * ρ,T and ν * α,T on {0, 1} V (T ) given by for all ρ ∈ (0, 1) and α ∈ (0, ∞). For ρ = 0 and α = 0, the simple exclusion process started from π * 0,T , respectively ν * 0,T , is the simple random walk on T in the respective model starting in the root o. When we choose π * ρ,T , respectively ν * α,T , as an initial distribution of the simple exclusion process, the particle initially placed in the root is called the tagged particle. We denote by (X v t ) t≥0 the position of the tagged particle in (T, o) in the varying speed model and by (X c t ) t≥0 its position in (T, o) in the constant speed model of the simple exclusion process.

Main result
Our main result is to establish a law of large numbers for the tagged particle in the simple exclusion process when starting from a Palm measure on an augmented Galton-Watson tree. For a rooted tree (T, o), we write |x| for the shortest path distance from the root for all x ∈ V (T ). Theorem 1.1. Let Z be distributed according to the offspring distribution µ. Then for almost every augmented Galton-Watson tree (T, o), the following holds: almost surely.
In particular, the tagged particle has almost surely a strictly positive speed.
The tagged particle shows a similar behavior as in related works on exclusion processes on regular graphs and random walks on Galton-Watson trees.  [14].
(ii) For the varying speed model, we see a linear scaling in the particle density ρ. Similar results are known for an exclusion process with drift on Z d and without drift on the regular tree, see [4,17].
(iii) In the constant speed model, we have that holds, i.e. in general the scaling of the speed is lower than linear in the averaged particle density.

Related work
In the last decades, many results for random walks on Galton-Watson trees were obtained. The study of random walks on Galton-Watson trees goes back to Grimmett and Kesten who proved that the simple random walk on supercritical Galton-Watson trees conditioned on non-extinction is almost surely transient [7]. Lyons et al. showed that the random walk has then almost surely a positive linear speed and calculated the velocity explicitly [14]. The case of a random walk on Galton-Watson trees with bias was studied by Lyons et al. in [15]. More recent treatments of the speed of random walks on Galton-Watson trees include [1,2,6] among others. An introduction to this topic can be found in the book of Lyons and Peres [16,Chapter 17].
Studying the behavior of the tagged particle in an exclusion process is a classical problem [19]. When the underlying graph is Z d , many results were obtained. In the case of translation invariant transition probabilities, a law of large numbers for the position of the tagged particle is known in all regimes [9,17]. For the d-dimensional ladder graph, the speed of a tagged particle was studied by Zhang [20]. For the exclusion process on regular trees, Chen et al. established a law of large numbers [4]. We obtain their results as a special case. For random environments of the exclusion process, less results are known. In [3], Chayes and Liggett consider the case of the exclusion process in a one-dimensional i.i.d. random environment. For Z d when the transition probabilities are symmetric and not concentrated on the nearest neighbors in the one-dimensional case, Kipnis and Varadhan established a central limit theorem for the tagged particle in their famous paper [10]. Their result was the starting point for a sequence of papers showing central limit theorems for the position of the tagged particle, see [11,13] for an overview.

Outline of the paper
This paper is organized as follows. In Section 2, we define a common probability space for locally finite, rooted trees and the respective exclusion processes on them. This will allow us to study the environment process in Section 3, which can be interpreted as the exclusion process "seen from the tagged particle". We provide stationary measures of the environment process in both models of the simple exclusion process. The arguments in this section are based on the ideas of Lyons et al. for studying random walks on Galton-Watson trees. The results of Section 3 will be used in Section 4 in order to establish transience of the tagged particle. We follow the arguments of Liggett [13,Section III.4] in this part. In Section 5, we show ergodicity for the environment process. This will be achieved by combining the ideas of Saada in [17] for the exclusion process on Z d with drift and arguments of Lyons and Peres in [16,Chapter 17]. From this, we deduce a law of large numbers for the position of the tagged particle in Section 6. We conclude with an outlook on related open problems.

Spaces and measures for trees
In this section, we introduce spaces and measures for rooted trees which allow us to study the simple exclusion process and locally finite, rooted trees on a common probability space. We write (T, o) ∈ T for a tree T with root o, where T denotes the space of all rooted, locally finite trees. We denote by B r (T, o) the ball of radius r around the root of T with respect to the graph distance. The space T will be equipped with the local topology, that is the topology introduced by the pseudo-metricd loc on T given byd and ∼ = denoting the isomorphism relation between finite, rooted graphs. In order to turn T together withd loc into a metric space, we consider isomorphism classes of trees. We say that two trees (T, o), (T ′ , o ′ ) ∈ T are isomorphic ifd loc ((T, o), (T ′ , o)) = 0 and write [T ] for the set of isomorphism classes. It is a well-known result that ([T ],d loc ) forms a Polish space, see [14]. Let the space Ω of 0/1-colored, locally finite, rooted trees be defined as We let B r (T, o, η) denote the ball of radius r around the root o of T where each site receives a color 0 or 1 according to η. The space Ω will be equipped with the topology induced by for all (T, o, η), (T ′ , o ′ , η ′ ) ∈ Ω. Again, we will restrict ourselves to isomorphism classes of 0/1-colored trees in order to obtain a Polish space ([Ω], d loc ), see Lemma 2.4 in [18]. For a fixed tree (T, o) ∈ T , we define to be the space of 0/1-configurations on (T, o) and to be the space of 0/1-configurations on (T, o) with shifted roots. Moreover, let be the set of configurations in Ω with occupied root and define Ω * T andΩ * T similarly. Note that forming the above subspaces is consistent under taking isomorphism classes of trees. From now on we only work on isomorphism classes of trees and drop the brackets in the notation. Let us stress that we define all probability measures on the subspaces of (T ,d loc ) and (Ω, d loc ) with respect to the Borel-σ-algebra.
Let GW denote the Galton-Watson measure on T which is induced by the Galton-Watson branching process, see [16,Chapter 4]. Similarly, we define AGW to be the augmented Galton-Watson measure on T which we obtain by joining two trees according to GW at the root of the first sample, see [16,Chapter 17]. The simple exclusion process in the varying speed model can be seen as a process on Ω with initial distribution P v ρ for P v ρ := AGW × π * ρ,T being a semi-direct product of AGW on T and π * ρ,T . Similarly, we refer to the simple exclusion process in the constant speed model as a process on Ω with initial distribution P c α for being a semi-direct product of AGW on T and ν * α,T . In contrast to P v ρ , we can not define P c α for 0/1-colored balls of radius r in a direct way. To remedy this problem, we condition according to the number of children in the (r + 1) th generation for each site at level r. For the resulting balls of radius r + 1 with colors only up to level r, it is straightforward to give sense to the measure P c α . Note that for a fixed augmented Galton-Watson tree (T, o) ∈ T , the simple exclusion process on (T, o) is a Markov process with values in the space Ω T for both models.

Stationarity for the environment process
In this section, we study the simple exclusion process on augmented Galton-Watson trees "seen from the tagged particle". For an exclusion process on Ω with transition rates p(., .), we define the corresponding environment process to be the Feller process with state space Ω * generated by the closure of for all cylinder functions f . We write L v and L c for the generator of the environment process of the simple exclusion process in the varying speed model and in the constant speed model, respectively. Note that the generator can be split into two parts, namely into transitions which do only exchange particles and do not change the underlying tree as well as into transitions which involve the root of the tree. More precisely, we define the generators as well as for the environment process in the constant speed model for (T, o, ζ) ∈ Ω * and all cylinder functions f . The generators L v ex and L v sh for the environment process in the varying speed model are defined analogously.
We want to investigate the invariant measures of the environment process. We provide two classes of reversible measures for the environment process, Q v ρ for ρ ∈ (0, 1) and Q c α for α ∈ (0, ∞), such that Q v ρ and P v ρ , respectively Q c α and P c α , are equivalent for all ρ ∈ (0, 1) and α ∈ (0, ∞). Let us stress once again that we work on isomorphism classes of trees in order to properly define stationary measures for the environment process. For the environment process in the varying speed model, we will use the ideas of Aldous and Lyons [2]. Consider the unimodular Galton-Watson measure UGW which we obtain from AGW by weighting a tree according to the reciprocal of the degree of its root, i.e.
for (T, o) ∈ T , where Z is distributed according to the offspring distribution µ. We define Q v ρ on Ω * to be the probability measure given as the semi-direct product for all ρ ∈ (0, 1). As pointed out by the authors of [2], the measure AGW on T gives a natural bias to trees proportional to the degree of the root. This bias is compensated by the Radon-Nikodym derivative in (10). For the environment process in the constant speed model with parameter α ∈ (0, ∞), we let Q c α denote the probability measure on Ω * which is absolutely continuous with respect to P c α and satisfies for all (T, o, ζ) ∈ Ω * , where Z is distributed according to the offspring distribution µ. We want to provide some intuition for the Radon-Nikodym derivative in (11).
Observe that the semi-direct product AGW × ν α,T satisfies Since the root is always occupied in the environment process, we expect to see a similar weighting of the degree of o within Q c α . Recall that we have Since AGW provides a natural bias proportional to the degree of the root o, it remains to include the factor of 1 αk+1 for Q c α . We now show that Q v ρ and Q c α are indeed reversible measures for the environment process for all ρ ∈ (0, 1) and α ∈ (0, ∞), respectively. For an introduction to reversibility of Feller processes, we refer to Liggett [12, Chapter II.5].
Proposition 3.1. Fix parameters ρ ∈ (0, 1) and α ∈ (0, ∞) for the measures Q v ρ and Q c α , respectively. Then the following statements hold. (i) The measure Q v ρ is reversible for the environment process generated by L v . (ii) The measure Q c α is reversible for the environment process generated by L c . In particular, the measures Q v ρ and Q c α are invariant for the environment process in the varying speed model and the constant speed model, respectively.
Proof. It suffices to show reversibility with respect to the different parts of the generators L v and L c . By construction, the processes on Ω * associated to the generators L v ex and L c ex , respectively, leave the underlying tree unchanged and ignore all moves involving the root. Recall that the measures π ρ,T and ν α,T are reversible for the simple exclusion process on (T, o) for all (T, o) ∈ T and ρ ∈ (0, 1) for the varying speed model as well as for all α ∈ (0, ∞) in the constant speed model, see [12, Chapter VIII, Theorem 2.1]. Since the Palm measures π * ρ,T and ν * α,T have the same law as π ρ,T and ν α,T except at the root, this shows reversibility of the measures Q v ρ and Q c α for the processes generated by L v ex and L c ex , respectively. We now show reversibility with respect to the processes generated by L v sh and L c sh following the ideas of Lyons et al. in [14] for the random walk on Galton-Watson trees.
where we join the roots of T and T ′ by an edge and let the resulting tree have its root at o. For Borel sets C, D ⊆ Ω, we define Observe that the processes generated by L v sh and L c sh on Ω * give rise to transition rates for the varying speed model and for the constant speed model for all (T, o, ζ) ∈ Ω * , respectively. We define for Borel sets A, B ⊆ Ω * . Note that it suffices to show that holds for almost all Borel sets A, B ⊆ Ω * in order to prove reversibility. Without loss of generality, we assume that A and B have the form More precisely, observe that two independent samples according to AGW have almost surely no non-trivial automorphisms between them. Hence, we see that these sets generate the Borel-σ-algebra on (Ω * , d loc ) up to nullsets. A visualization of the sets C •-D and D •-C is given in Figure 1. In the varying speed model, observe that In particular, we see that

Transience of the tagged particle
Recall that the position of the tagged particle in the simple exclusion process is denoted by (X v t ) t≥0 in the varying speed model and by (X c t ) t≥0 in the constant speed model. Let P P v ρ be the law of the simple exclusion process started from P v ρ in the varying speed model for some ρ ∈ (0, 1). Similarly, let P P c α denote the law of the simple exclusion process started from P c α in the constant speed model for some α ∈ (0, ∞). For both models, we say that the tagged particle is transient if (X v t ) t≥0 , respectively (X c t ) t≥0 , hits the root P P v ρ -almost surely, respectively P P c α -almost surely, only finitely many times.
Proposition 4.1. The tagged particle is transient for the simple exclusion process in the varying speed model with initial distribution P v ρ and in the constant speed model with initial distribution P c α for all ρ ∈ (0, 1) and α ∈ (0, ∞). In order to show Proposition 4.1, we use a similar notation as introduced by Lyons et al. in [14] for the study of random walks on Galton-Watson trees. Fix a tree (T, o) ∈ T . We write → x for a path x 0 , x 1 , . . . in (T, o) and say that it is a ray ξ if it never backtracks, i.e. we have that x i = x j holds for all i = j. The boundary ∂(T, o) of a tree (T, o) is defined to be the set of all rays ξ starting at the root o. Note that ∂(T, o) consists AGW-almost surely of infinitely many elements. We say that a path → x converges to ξ ∈ ∂(T, o) if ξ is the only ray which is intersected infinitely often and → x visits every site at most finitely many times. We let [x, ξ] denote the unique ray starting at a site x ∈ V (T ) and converging to ξ ∈ ∂(T, o). For sites x, y ∈ V (T ) and a ray ξ, let x ∧ ξ y be the first site at which [x, ξ] and [y, ξ] meet.
For two sites x, y ∈ V (T ), we define their horodistance with respect to some given ray ξ of (T, o) to be the signed distance where | . | denotes the shortest path distance in (T, o). We set x ξ := x− o ξ . In the following, we assume that for every infinite tree Similarly, the local drift at the root in the constant speed model is denoted by Using these notions, we rewrite the position of the tagged particle as a martingale and a function depending only on the environment process in a ball of radius 1 around its root. This follows the ideas of Proposition 4.1 in [13, Chapter III].
Lemma 4.2. Fix ρ ∈ (0, 1) and α ∈ (0, ∞). Then the following two statements hold: be the environment process in the varying speed model with initial distribution Q v ρ and natural filtration (F v t ) t≥0 . We have that be the environment process in the constant speed model with initial distribution Q c α and natural filtration (F c t ) t≥0 . We have that Proof. We only show part (i) of Lemma 4.2 as for part (ii) the same arguments apply. For a given tree (T, o) ∈ T , we define Note that we can write for all Q v ρ -measurable sets A. Thus, it suffices to show that holds for all t > s ≥ 0 and UGW-almost every tree (T, o) ∈ T , where E Q v ρ,T denotes the expectation with respect to the environment process started from Q v ρ,T . Recall that the choice of the ray for a tree in T is consistent under performing shifts of the root. Moreover, note that a environment process started from Q v ρ,T remains inΩ * T almost surely. Using the Markov property of the environment process as well as the fact that Q v ρ is stationary for the environment process by Proposition 3.1, we see that holds for all s ≥ 0. Hence, it suffices to show that for UGW-almost every (T, o) ∈ T , we have that is satisfied for all t ≥ 0. For a tree (T, o) ∈ T , let g be the function onΩ * T given by Plugging g into the generator in (9) and using that the ray of (T, o) is consistent under performing shifts of the root, we see that holds for all (T, x, ζ) ∈Ω * T . Thus, we obtain (18) by applying Dynkin's formula.
Proof of Proposition 4.1. We will only prove transience for the tagged particle in the varying speed model of the simple exclusion process. For the tagged particle in the constant speed model of the simple exclusion process, similar arguments apply. We will show that the tagged particle has P P v ρ -almost surely a strictly positive speed with respect to the horodistance, which implies transience. Observe that the martingale (M v t ) t≥0 defined via the relation (15) has stationary increments by Proposition 3.1 and thus satisfies a law of large numbers. Using that Q v ρ is stationary for the environment process, we obtain by Lemma 4.2 that holds, where Z is distributed according to the offspring distribution µ. Since we have a supercritical, augmented Galton-Watson tree without extinction and ( o t (T 0 ,o 0 ) ) t≥0 describes the horodistance of the tagged particle from the root within the environment process, we see that with positive Q v ρ -probability, the tagged particle has a strictly positive speed. In order to show that the tagged particle is transient with respect to the initial distribution P v ρ , suppose that there exists an initial set of 0/1-colored trees B ⊆ Ω * with Q v ρ (B) > 0 for which the tagged particle has speed zero. Note that B can be chosen such that it forms an invariant set for the environment process. Let E Q v ρ (.|B) denote the expectation of the environment process started from Q v ρ (.|B). Using the arguments of the proof of Lemma 4.2, the environment process must satisfy for all t ≥ 0. From the construction of the horodistance, we see that holds for all k ≥ 2. Moreover, note that the conditional probability does not depend on the particular choice of x ∼ o s . Hence, we see that must hold for all t ≥ 0 and x ∼ o s . However, this gives a contradiction as the term in (20) is non-negative for all k ≥ 2 and strictly positive for at least one k ≥ 3.
Otherwise, the underlying augmented Galton-Watson tree would almost surely be restricted to a copy of Z. Thus, the tagged particle has a strictly positive speed Q v ρ -almost surely. We conclude since P v ρ and Q v ρ are equivalent for all ρ ∈ (0, 1).
Corollary 4.3. The tagged particles in two independent environment processes with the same initial configuration (T, o, ζ) ∈ Ω * according to Q v ρ , respectively Q c α , converge almost surely to two distinct rays in ∂(T, o).
Proof. In the proof of Proposition 4.1, we only require the ray of a tree to be consistent under performing shifts of the root and to be fixed at the beginning. Since the tagged particle has almost surely a positive speed, the tagged particle in the first sample converges almost surely to a unique ray in ∂(T, o). Since the two environment processes move independently, we can use this boundary element of the first process as the ray for the tree in the second environment process.
Remark 4.4. Using the expressions for the averaged speed of the tagged particle in the environment process given in (19) and similar for Q c α , we see that as well as holds for Z ∼ µ and all ρ ∈ (0, 1), α ∈ (0, ∞). We will show in Section 6 that (21) and (22) give the speed of the tagged particle P P v ρ -almost surely, respectively P P c α -almost surely, using an ergodicity argument for the environment process.

Ergodicity for the environment process
In this section, we show that the environment process started from Q v ρ in the varying speed model and from Q c α in the constant speed model, respectively, is ergodic for all ρ ∈ (0, 1) and α ∈ (0, ∞). The proof will have two main ingredients. First, we show that every invariant set A can be represented by a set of trees, which we obtain by dropping the 0/1-coloring in every configuration of A. This follows the arguments of Saada for the exclusion process on Z d with drift [17]. We then deduce ergodicity for the environment process using regeneration points. This follows the ideas of Lyons and Peres in [16,Chapter 17] for the simple random walk on Galton-Watson trees.
Proposition 5.1. Fix parameters ρ ∈ (0, 1) and α ∈ (0, ∞) for the measures Q v ρ and Q c α , respectively. The following two statements hold. (i) The measure Q v ρ is ergodic for the environment process generated by L v . (ii) The measure Q c α is ergodic for the environment process generated by L c .
We will only show part (i) of Proposition 5.1, i.e. we have that Q v ρ (A) ∈ {0, 1} holds for any set A which is invariant under the environment process in the varying speed model. For part (ii) of Proposition 5.1 the same arguments apply. The following lemma says that in order to determine if (T, o, ζ) ∈ A holds, it suffices Q v ρ -almost surely to know the underlying tree (T, o) ∈ T .
Lemma 5.2. Let A ⊆ Ω * be an invariant set for the environment process started from Q v ρ . Then for UGW-almost every tree (T, o) ∈ T , we have that holds. Moreover, we can find a Borel set of rooted trees U ⊆ T which is invariant under the environment process such that is satisfied.
In order to show Lemma 5.2, we follow the arguments of Saada in [17]. A similar approach can be found in [4] for the simple exclusion process on regular trees. Our arguments will be based on the fact that we have ergodicity for the simple exclusion process started from π ρ,T for AGW-almost every initial tree (T, o) ∈ T . More precisely, by Theorem 2.1 of [8], we have that the measures π ρ,T are extremal invariant for the simple exclusion on {0, 1} V (T ) for AGW-almost every tree (T, o) ∈ T , for all ρ ∈ (0, 1). Since the simple exclusion process is a Markov process, we obtain ergodicity for the simple exclusion process on a given tree by applying Theorem B52 of [13]. In order to show that the environment process onΩ * T with initial law Q v ρ,T is ergodic for UGW-almost every tree (T, o) ∈ T , we proceed with a proof by contradiction. Suppose that for some ρ ∈ (0, 1), we have that holds. Since the set A is invariant for the environment process with starting distribution Q v ρ , it has to be invariant for the environment process onΩ * T with initial law Q v ρ,T for UGW-almost every tree (T, o) ∈ T . Define B :=Ω * T \ (A ∩Ω * T ) and note that B is a non-trivial, invariant set for the environment process started from Q v ρ,T . Moreover, we let the setsÃ,B ⊆ Ω T be given as In words,Ã is the set of all 0/1-colorings of (T, o) which we obtain by taking all 0/1-colored trees in A ∩Ω * T and considering their coloring of (T, o). Observe that the setsÃ andB are invariant for the simple exclusion process with initial distribution where δ (T,o) denotes the Dirac measure on T with respect to (T, o). Moreover, since Q v ρ,T is absolutely continuous with respect to P v ρ,T for all ρ ∈ (0, 1), we have that is satisfied for AGW-almost every tree (T, o) ∈ T . Using ergodicity for the simple exclusion process on Ω T , we conclude that In particular, the setsÃ andB are not disjoint. From this, we want to deduce that   Proof. Using (26), there almost surely exist sites y, z ∈ V (T ) such that (T, y, η) ∈ A and (T, z, η) ∈ B holds. Without loss of generality, the shortest path connecting y and z can be assumed to consist only of vacant sites. To see this, observe that (T, a, η) ∈ (A ∩Ω * T ) · ∪B =Ω * T holds for all a ∈ V (T ) with η(a) = 1. Thus, we can choose y and z to be the closest such sites which satisfy (i). By our assumptions on the augmented Galton-Watson tree, there almost surely exists a site x in a branch of y different from the one containing z with degree at least 3. Let C(x, y) and D(x, y) denote the vertices of two distinct branches of x which do not intersect the path [x, y]. Using a Borel-Cantelli argument, we see that C(x, y) and D(x, y) both contain P v ρ -almost surely a ray starting at x with infinitely many vacant sites. Let w be the first vacant site along that ray in C(x, y) and let v be the first site along that ray in D(x, y) such that there are |x − y| + 1 empty sites along the path [x, v]. We will now provide two ways of transforming η into η w,z which only involve the sites in N as depicted in Figure 2. This also provides two ways of changingη intõ η w,z for any fixed t 0 > 0. At the beginning, we assume for both transformations that all particles in [x, y] \ {y} are moved into the empty sites within [v, x] \ {x} in an arbitrary way using only nearest neighbor moves within N . In a next step, the two transformations differ in performing the following transitions.
(a) Move the particle at y to v along the path [v, y], i.e. for {v i , 1 ≤ i ≤ k} being successive vertices along [v, y] with η(v i ) = 1, move the particle from v 2 to v 1 = v, then from v 3 to v 2 and so on. Next, move the particle from z to w along [w, z] in the same way. Afterwards, move the particle at v back to y.
(b) Move the particle from y to w along the path [w, y] and then the particle from z to y along the path [y, z].
At the end, in both transformations all particles which were moved to the empty sites in [v, x] \ {x} at the beginning are moved back to their original positions.
Following the transformation according to (a), we see that (T, y,η w,z ) ∈ A holds P v ρ,T -almost surely using that A is invariant for the environment process and (i) of Lemma 5.3. For the transformation according to (b), note that (T, y,η w,z ) ∈ B holds following the trajectory of the particle originally at z and using that B is invariant for the environment process. Observe that at time t 0 , the simple exclusion process started from (T, o, η) agrees with (T, o,η w,z ) with positive probability using the graphical representation. Hence, we obtain the desired contradiction of A and B not being disjoint.
For the second statement in Lemma 5.2, we let S denote the set of trees which we obtain by deleting all 0/1-colorings in the elements of A, i.e.
From the construction of the σ-algebras on Ω and T , we see that U forms a Borel set of trees. Using the first statement of Lemma 5.2, we obtain that U is invariant for the environment process started from Q v ρ .
Next, we show that UGW(U ) ∈ {0, 1} holds for the set U defined in (28). This will yield Proposition 5.1 since we have that holds by Lemma 5.2. We follow the arguments in the proof of Theorem 17.13 in [16] which were used in order to establish ergodicity for the environment process of the simple random walk on supercritical Galton-Watson trees.
Lemma 5.4. Let (T t , o t , ζ t ) t≥0 denote the environment process with state space Ω * and initial distribution Q v ρ . The corresponding dynamical system is mixing in the tree-component, i.e. we have that holds for all Borel-sets C, D ⊆ T . In particular, we have that UGW(Ũ ) ∈ {0, 1} holds for any set of treesŨ which is invariant for the environment process.
Recall that the σ-algebras on T and Ω are generated by sets of trees which agree within a ball of finite radius around the root. Including all finite unions and intersections of these balls, we see that the balls generating the σ-algebra on T form a semi-algebra. Hence, using a well-known result from ergodic theory, it suffices to show mixing in the tree component for the sets C and D which take into account only a finite range of the tree around its root, see [5,Exercise 2.7.3(1)]. Let EX × Q v ρ denote the probability measure on PathsInTrees given by choosing a starting configuration on Ω * according to Q v ρ and then performing two independent environment processes. We let ↔ x be the composition of the tagged particle trajectories and note that ↔ x ∈ ↔ T holds almost surely by Corollary 4.3. The path space is equipped with the σ-algebra F induced by the environment processes. Since the environment process is a reversible Markov process with respect to Q v ρ , observe that forms a measure-preserving system, i.e. we have that holds for all F ∈ F. Define the event of having a regeneration point at x 0 to be Regen := ( ↔ x , T ) ∈ PathsInTrees s.t. ∀n ≤ 0 : x n = x 1 and ∀n ≥ 1 : x n = x 0 .
In words, x 0 is a regeneration point if the edge {x 0 , x 1 } is traversed precisely once.
The following lemma is the analogue of Proposition 17.12 in [16] for the simple random walk on Galton-Watson trees. Using reversibility of the environment process with respect to Q v ρ together with (30), we see that holds. Using that the tagged particle is transient together with Corollary 4.3, we have EX × Q v ρ -almost surely infinitely many fresh points, i.e.
holds for any m ≥ 0. Since the probability of the event of having a fresh point at x 0 is invariant under shifts according to S, we conclude that the probabilities in (31) must be strictly positive. Moreover, this shows that we have EX × Q v ρ -almost surely infinitely many exit points. For m, n ∈ Z with m ≤ n, we define the event In words, H m,n is the event that x m is a fresh point, x n is an exit point and the shortest path connecting x m and x n does not intersect the remaining trajectory.
Proof. Observe that for EX × Q v ρ -almost every ( ↔ x , T ) ∈ PathsInTrees, the tagged particles in the corresponding environment processes converge to distinct rays ξ 1 , ξ 2 ∈ ∂(T, x 0 ). Let a ∈ V (T ) be the last common vertex of ξ 1 and ξ 2 . Using transience, we observe that a is hit almost surely only finitely often. We choose m such that S m ( ↔ x , T ) ∈ Fresh with a / ∈ {. . . , x m−1 , x m } and n such that S n ( ↔ x, T ) ∈ Exit with a / ∈ {x n , x n+1 , . . . }. For these choices of m and n, we have that ( ↔ x , T ) ∈ H m,n . Note that this construction holds for EX × Q v ρ -almost every element of PathsInTrees and so EX × Q v ρ (H m,n ) > 0 must hold for some deterministic choice of m and n. Set k = n − m and use that we have a measure-preserving system to conclude. Figure 3: Construction of a regeneration point at x 0 . The tagged particle is drawn in red.
For a given configuration ( ↔ x , T ) ∈ H 0,k with k ≥ 0 of Lemma 5.6, let (T t , o t , ζ t ) t≥0 be the underlying environment process (T t , o t , ζ t ) t≥0 in positive time direction. We recursively define a sequence of almost surely finite stopping times by τ 0 := 0 and is contained in the set of regeneration points at x 0 with positive probability using a similar argument as in the proof of Lemma be the environment process with the same initial configuration as (T t , o t , ζ t ) t≥0 but where all moves involving the tagged particle are suppressed. Using the graphical representation, note that ζ ′ τ k and ζ τ k differ almost surely in at most finitely many values and let N denote the sites in the minimal spanning tree consisting of all sites in which ζ ′ τ k and ζ τ k differ. The proof of the following lemma uses similar arguments as the proof of Lemma 5.3.
Lemma 5.7. For almost every configuration ( ↔ x , T ) and 0/1-colorings ζ ′ τ k and ζ τ k differing only in sites within N , there exist v, w, x ∈ V (T ) with the following properties: (iii) x k , v and w are located in pairwise different branches with respect to x in (T, o).
(iv) The path [x, v] contains at least |x − x k | + k vacant sites.
Proof. Let C(x k ) be a branch of x k which does not contain x 0 . Using a Borel-Cantelli argument, there almost surely exists a site x ∈ C(x k ) with deg(x) ≥ 3 which is not contained in the set N . Let C(x) and D(x) be two different branches of x which are disjoint of [x, x k ] . Note that C(x) and D(x) are disjoint from the set N and contain almost surely an infinite number of vacant sites. Let w be the first site in C(x) which is empty. Similarly, let v be the first site in D(x) such that condition (iv) holds.
Proof of Lemma 5.5. Using Lemma 5.7, we now provide a way of transforming ζ ′ τ k into ζ τ k by finitely many transitions, see Figure 3 for a visualization. In this transformation, the tagged particle will not come back to x 0 once it has left its starting point. We start by moving all sites from [x 0 , x]\{x 0 } into the empty positions in [x, v] using only nearest neighbor transitions which do not involve x 0 . In a next step, move the tagged particle from x 0 to w via the path [x 0 , w]. We now move all particles to their correct positions in ζ τ k except for all particles which are placed at [x, x k ] in ζ τ k . These particle are moved to the empty positions in [v, x] or remain there if they were already moved to [v, x] in a previous step. Next, move the tagged particle to x k via the path [x k , w]. Finally, move the remaining particles from [x, v] to their positions in [x k , x] with respect to ζ τ k . Note that for almost every pair of configurations ζ ′ τ k and ζ τ k , this provides a way of transforming ζ ′ τ k into ζ τ k modifying the exclusion process only between times 0 and τ k on an almost surely finite set of vertices. Hence, we can use the graphical representation of the exclusion process and apply a coupling argument to see that EX × Q v ρ (Regen|H 0,k ) > 0 holds. Using Lemma 5.6, we conclude by applying Poincaré's recurrence theorem.

Speed of the tagged particle
Combining the results of the previous sections, we have all ingredients to prove Theorem 1.1. As pointed out in Remark 4.4, we will use the arguments given in Section 4 for showing transience of the tagged particle in order to determine the speed of the tagged particle with respect to P P v ρ and P P c α almost surely. Recall from Lemma 4.2 that we can rewrite the horodistance of the tagged particle in terms of the environment process in a ball of radius 1 around its root and a martingale. Using the results of Sections 3 and 5, we obtain the following lemma as an analogue of Corollaries 4.5 and 4.16 in [13, Chapter III]. Since the environment process is a stationary process when starting from Q v ρ , we have that holds for all s < t. From Propositions 3.1 and 5.1 we know that Q v ρ is a stationary and ergodic measure for the environment process and so the claimed statement follows.
Proof of Theorem 1.1. Using Proposition 5.1 and Lemma 6.1, we can apply the ergodic theorem for both terms on the right-hand side of (15), respectively, to see that holds almost surely for Q v ρ -almost every initial configuration in the varying speed model and ρ ∈ (0, 1). Similarly, holds almost surely for Q c α -almost every initial configuration in the constant speed model and α > 0. Recall that the measures Q v ρ and P v ρ , respectively Q c α and P c α , are equivalent for all ρ ∈ (0, 1) and α ∈ (0, ∞). Since (o t ) t≥0 describes the position of the tagged particle within the environment process, we conclude that holds P P v ρ -almost surely for P v ρ -almost every initial configuration in the varying speed model and all ρ ∈ (0, 1). Similarly, lim t→∞ X c t (T 0 ,o 0 ) t = E Z − 1 Z + 1 1 α(Z + 1) + 1 holds P P c α -almost surely for P c α -almost every initial configuration in the constant speed model and all α ∈ (0, ∞). Note that in both models of the simple exclusion process, the tagged particle converges almost surely to a ray ξ ′ ∈ ∂(T 0 , o 0 ) different from ξ = ξ(T 0 , o 0 ). Let a denote the last common vertex of ξ and ξ ′ in the varying speed model and observe that |X v t | = X v t (T 0 ,o 0 ) + 2|a| holds for all t ≥ 0 sufficiently large. A similar statement is true for the tagged particle in the constant speed model. We conclude since |a| does not depend on t.

Open problems
In this article, we consider the speed of a tagged particle when the particles perform simple random walks under an exclusion rule. It is a natural extension of our model to consider random walks with different transition probabilities.
Question 7.1. What is the speed of the tagged particle when the particles perform a biased random walk on augmented Galton-Watson trees under an exclusion rule?
A classical problem for exclusion processes is the question if the tagged particle satisfies a central limit theorem. In the case where the augmented Galton-Watson tree is a d-regular tree with d ≥ 3, a central limit theorem holds, see [4,Theorem 1.3].