On transience of frogs on Galton--Watson trees

We consider a random interacting particle system, known as the frog model, on infinite Galton-Watson trees allowing offspring zero and one. The system starts with one awake particle (frog) at the root of the tree and a random number of sleeping particles at the other vertices. Awake frogs move according to simple random walk on the tree and as soon as they encounter sleeping frogs, those will wake up and move independently according to simple random walk. The frog model is called transient, if there are almost surely only finitely many particles returning to the root. In this paper we prove a zero-one law for transience of the frog model and show the existence of a transient phase for certain classes of Galton-Watson trees.


Introduction
The frog model FM(X, η, P) is a random interacting particle system, consisting of three parts: a graph X with a dedicated root, a (random) configuration η := (η) x∈X of sleeping frogs on each vertex and the path measure P describing the movement of the particles -also called frogs. The model starts with one awake frog at the root o of graph X and sleeping particles according to η at the other vertices. The awake frogs move independently on the graph with respect to P. When a vertex with sleeping frogs is visited the first time, the sleeping frogs at this vertex wake up and start to move according to P independently of the other frogs. The different frog models can vary in the underlying graph, the initial distribution of the sleeping frogs (deterministic or random) and the path measure of the awake frogs. Unless it is not specified differently, we assume that the frogs move according to simple random walk, write from now on SRW instead of P and shorten the notation from FM(X, η, SRW) to FM(X, η). More precisely, for v, w ∈ X we consider the transition probabilities p(v, w) = 1 deg(v) if v is a neighbour of w and 0, otherwise. In 1999 the frog model was originally introduced as "egg model" in [27] and later on Rick Durrett established the name "frog model". One main point of interest since its introduction was studying the recurrence and transience of the frog model. Let FM := FM X := SRW × η denote the probability measure on paths of all frogs (following the dynamics of a SRW) given by choosing i.i.d. initial frog configuration according to η on the graph X. Moreover, we define the random variable ν := # of frogs returning to the root which is the number of visits to the root in the frog model. Then, we define recurrence and transience in the following way: Studying transience and recurrence of the frog model is only interesting when the single random walk is transient. The first result concerning the question about recurrence was in the aforementioned article [27], where Telcs and Wormald showed that FM(Z d , 1, SRW) is recurrent for all d ∈ N. Later Gantert and Schmidt showed conditions for recurrence for the frog model with drift on the integers in [10]. This was generalized to higher dimensions and a drift in the direction of one axis by Döbler and Pfeifroth [8] and Döbler et al. [7].
In 2002, Alves, Machado and Popov [1] studied the frog model on trees with the modification, that the frogs can die with a certain probability p in each step. Let p c denote the smallest p such that the frog model survives with positive probability. In [1] they are proving in which cases there exists a phase transition, that is 0 < p c < 1, on homogeneous trees and integer lattices. Moreover, they have proven phase transitions between transience and recurrence with respect to the surviving probability. In 2005 there was the first improvement of the upper bound of p c by Lebensztayn, Machado and Popov [18]. Recently, Lebensztayn and Utria improved the result again in [20] and proved an upper bound for p c on biregular trees in [19]. Another modification of the frog model was considered by Deijfen, Hirscher and Lopes in [5] and by Deijfen and Rosengren in [6]. These two papers work on a two-type frog model performing lazy random walk. They show that two populations of frogs on Z d can coexist under certain conditions on the path measure of the frogs. Moreover, the coexistence of the frog model does not depend on the shape of the initially activated sets and their frog configuration.
The question if FM(T d , 1, SRW) on the homogeneous tree T d is recurrent or transient remained open for quite some time. In 2017 Hoffmann, Johnson and Junge could show in [12], that FM(T d , 1, SRW) is recurrent for d = 2 and transient for d ≥ 5. This result was extended by Rosenberg [26] showing that the alternating tree T 3,2 with degree 3 and 2 is recurrent. Studying the frog model on trees was continued by modifying the frog configuration η to pois(µ)-distributed frogs. Hoffmann, Johnson and Junge proved in [11] the existence of a critical parameter µ c , bounded by Cd < µ c (d) < C ′ d log d with C, C ′ constants, such that FM(T d , pois(µ), SRW) is recurrent for µ > µ c and transient for µ < µ c . Johnson and Junge improved the bounds to 0.24d ≤ µ c (d) ≤ 2.28d for sufficiently large d in [15].
The subtlety of the question of recurrence and transience is also reflected in the result by Johnson and Rolla [16]. In fact, transience and recurrence are sensitive not just to the expectation of the frogs but to the entire distribution of the frogs. This is in contrast to closely related models like branching random walk and activated random walk.
Very recently, Michelen and Rosenberg proved in [24] the existence of a phase transition between transience and recurrence on Galton-Watson trees. This was done for trees of at least offspring two. In this paper we want to answer an open question, which appeared in [24] and extend their result. We will prove the existence of a transient phase for supercritical Galton-Watson trees with bounded offspring but also allowing offspring zero and one. As in the references above we assume that the initial distribution is random according to a distribution η with finite first moment. We start with showing a 0-1-law for transience. Michelen and Rosenberg recently proved a stronger 0-1-law for recurrence and transience in [24]. We learned about their proof after writing our first version. While both proofs rely on the stationarity of the augmented Galton-Watson measure, our proof differs in the connection between the ordinary Galton-Watson measure and the augmented Galton-Watson measure. In [17] Kosygina and Zerner proved another 0 − 1-law for transience and recurrence of the frog model on quasi-transitive graphs.
The main result of the paper is the existence of a transient phase while allowing offspring zero and one: Theorem 1.3 Let GW be a Galton-Watson measure defined by (p i ) i≥0 . We assume that d max = max{i : p i > 0} < ∞ and denote d min := min{i ≥ 2 : p i > 0}. Then, for any choice of p 0 and p 1 there exists some constants c d = c d (p 0 , p 1 ) and c η = (p 0 , p 1 , d max ) such that for d min ≥ c d the frog model FM(T, η, SRW) is transient GW-almost surely (conditioned on T to be infinite) if E[η] < c η .
The proof of Theorem 1.3 gives bounds on the constants. These bounds can certainly be improved in refining the involved estimates, see Figure 1 for some explicit values. If p 1 = 0 then there always exists a transient regime on infinite Galton-Watson trees, see Lemma 5.2. However, we believe that a different approach or a new perspective is needed to prove the following conjecture. For proving Thorem 1.3 we compare the frog model with a branching Markov chain (BMC). In contrast to the frog model, the particles in the BMC branch at every vertex, regardless if they visited the vertex already or not. Therefore, there are more particles in the BMC than in the frog model and we can couple the two models. In this way, transience of the BMC implies transience of the frog model. The same kind of approach was already used for example in the proofs of transience in [11] and [15]. While on homogeneous trees the existence of a transient branching Markov chain is guaranteed, this is no longer true in general for Galton-Watson trees. Namely, allowing the particle to have 0 and 1 offspring creates stretches and finite bushes in the family tree. Such trees have a spectral radius equal to 1 and therefore the branching Markov chain is always recurrent on such trees, see [9]. To tackle this problem, we first modify the Galton-Watson trees and then adapt the branching Markov chain to get a dominating process. In the case of appearing finite bushes the procedure is fairly straightforward. Dealing with arbitrary long stretches turns out to be the more difficult part, since a direct coupling of the frog model and the branching Markov chain is not possible. For this reason we compare the expected number of returns to the root of the frog model with the expected number of returns of an appropriate branching Markov chain.
The paper is structured in the following way. In Section 2 we give an introduction to Galton-Watson trees and state some useful structural results. Then, we will define the branching Markov chain together with the above stated transience criterion in Section 3. The 0 − 1-law is proved in Section 5. The proof of Theorem 1.3 will be split in three parts. We will treat firstly Galton-Watson trees with bushes (p 0 > 0, p 1 = 0), then Galton-Watson trees with stretches (p 1 > 0, p 0 = 0) and in the end we treat the general case.

Galton-Watson trees
The Galton-Watson tree (GW-tree) is the family tree of a Galton-Watson process. This latter process starts with one particle at time 0 and at each discrete time step every particle generates new particles independently of the previous history and the other particles of the same generation. More formally, let Y be a non-negative integer valued random variable with p k := P[Y = k] for each k ∈ N and let m := k≥0 k p k be the mean of Y . Moreover, let Y (n) i , i, n ∈ N, be independent and identically distributed random variables with the same distribution as Y . Then, the Galton-Watson process is defined by Z 0 := 1 and for n ≥ 1. The random variable Z n represents the number of particles in the n-th generation. A GW-process with p 0 > 0 will survive with positive probability, that is P[Z n > 0 for all n > 0] > 0, if and only if m > 1. We introduce T as the random variable for the family tree of the GW-process and its corresponding measure by GW. Moreover, we denote by T := T(ω) a fixed realization of T. In the remaining paper we only consider GW-trees with bounded number of offspring: There exists a d max ∈ N, such that dmax k=0 p k = 1. For a more detailed introduction on GW-processes and trees we refer to Chapter 5 in [22].
In the case where p 0 > 0 the GW-tree contains a.s. finite bushes. We will distinguish between two types of vertices.
Definition 2.1 We call a vertex v ∈ T of type g if it lies on an infinite geodesic starting from the root. Otherwise we call vertex v of type b.
If a vertex of type b is the descendant of a type g vertex we call it of type b r and speak of it as the root of the finite bush that consists of its descendants.
We denote f (r) := E r Z = k≥0 r k p k the generating function of the GW-process and q the smallest solution of f (r) = r.
Let us consider the case where p 0 > 0. Conditioned on nonextinction the tree T is distributed as a treeT generated as follows, e.g. see Proposition 5.28 in [22]: We start with a tree T * generated according to the generating function This tree will serve as the backbone ofT and looks like a supercritical GW-tree without leaves. All vertices in this tree are of type g. To each of the vertices of T * we attach a random number of independent copies of a sub-critical GW-tree generated according tõ These are finite bushes consisting of vertices of type b. The resulting treeT has the same law as T, conditioned on nonextiction and is a multitype GW-tree with vertices of type b and g. We denote the measure generatingT by GW mult .
Let (Z sub n ) n≥0 denote the subcritical Galton-Watson process with probability generating functionf and T sub its family tree. We know that E[Z sub 1 ] < 1 and moreover it holds, e.g., Theorem 2.6.1 in [14] that Now, if we assume that p 1 > 0 the resulting GW-tree may contain arbitrary long stretches. We want to show that this tree generated by GW is equivalent to a tree generated in three steps where firstly the tree without stretches is generated, secondly the location of the stretch is determined and thirdly the stretches are inserted. Therefore we define a new GW-measure using the modified offspring distribution for k = 0, 2, . . . , N and let GW bg be the measure generating a tree with this distribution. Let us denote a tree generated by GW bg with T bg . In the next step every vertex will be independently labeled with bs with probability p 1 , which denotes the starting point of a stretch. If such a vertex has no offspring we attach one vertex, otherwise insert a vertex with offspring one in between the vertex and its descendants, see Figure 2. We write for such a tree T p×bg . In the next step, the length of the stretch attached to a vertex with label bs will be distributed according to L were L is geometrically distributed geo(p 1 ) and we obtain a tree T s×p×bg . This yields in 1 + geo(p 1 ) distributed vertices with offspring one in a row. The length of the stretches will be determined for each stretch starting point  independently and identically distributed. We will call this measure of selecting a stretch point PER and the one of choosing the length of the stretch by ST. We denote by T s×p×bg the tree constructed in the three steps according to ST × PER × GW bg . The resulting tree has the same distribution as the tree constructed as follows: We start with a root and proceed inductively. Every new vertex • is the starting point of a stretch with length ℓ + 1 and the end of the stretch has k = 0, 2, . . . , N descendants with probability p 1 p ℓ Moreover, it holds for any finite tree T, that GW(T) = ST × PER × GW bg (T) and therefore that GW(T) = ST × PER × GW bg (T) for all trees T.

Branching Markov chain
One method for proving transience of the frog model relies on the comparison of the frog model to a branching Markov chain (BMC). This is a labelled Galton-Watson process or tree-indexed Markov chain, [2], where the labels correspond to the position of the particles. In our setting the particles will move on a tree T according to the transition operator P = (p(v, w)) v,w∈T of a simple random walk (SRW). We note p (n) (v, w) for the n-step probabilities. If T is connected, the SRW is irreducible and the spectral radius is well-defined and takes values in (0, 1]. We add the branching mechanism that in every vertex v ∈ T the particle branches according to a branching distribution µ(v); i.e. each µ(v) is a measure on N. We denote by µ the whole sequence (µ(v)) v∈T . The expected value of each branching distribution is k is the probability that a particle jumping in v branches into k ∈ N particles. We denote BMC(T, P, µ) for this branching Markov chain.
Similarly to the frog model, the BMC is called transient if the root will be visited almost surely only by finitely many particles. Otherwise it is called recurrent. A particular case of the transience criterion for BMC given in [9] is the following.

0-1-law for transience
Before proving the existence of a transient phase for the frog model we want to show that the existence of a transient phase does not depend on the specific realization of the GW-tree. In other words, we show that the frog model is either transient for GW-almost all infinite trees or recurrent for GW-almost all infinite trees. The proof of this 0-1-law, Theorem 1.2, relies on the concept of the environment viewed by the particle. We prove that the events of transience and recurrence are invariant under re-rooting and hence the 0-1-law follows from the ergodicity of the augmented GWmeasure.
The augmented Galton-Watson measure, denoted by AGW is a stationary version of the usual Galton-Watson measure. This measure is defined just like GW except that the number of children of the root has the law of Y + 1; i.e. the root has k + 1 children with probability p k . The measure AGW can also be constructed as follows: choose two independent copies T 1 and T 2 with roots o 1 and o 2 according to GW and connect the two roots by one edge to obtain the tree T with the root o 1 . We write We consider the Markov chain on the state space of rooted trees. If we change the root of a tree T to a vertex v ∈ T , we denote the new rooted tree by MoveRoot(T, v). We define a Markov chain on the space of rooted trees as: By Theorem 3.1 and Theorem 8.1 in [21] it holds that this Markov chain with transition probabilities p SRW and the initial distribution AGW is stationary and ergodic conditioned on non-extinction of the Galton-Watson tree. Proof As the case of finite trees is trivial we consider an infinite rooted tree (T, o) and let v ∼ o. We proof that transience of (T, o) implies transience of (T, v) by assuming the opposite. If FM(MoveRoot((T, v), η) is recurrent, then there exists some k ∈ N such that with positive probability infinitely many frogs visit v conditioned on η o = k. In the frog model FM((T, o), η) conditioned on η v = k the starting frog in o jumps to v with positive probability. Again with positive probability at the second step all frogs awaken in v jump back to o while the frog that came from o is assumed to stay put in v for one time step.
Note that this has no influence on transience or recurrence of the process. This recreates the same initial configuration of FM((T, v), η) conditioned on η o = k with the difference that more frogs are already woken up. By assumption in this process infinitely many particles visit v with positive probability, and hence, by Borel-Cantelli, also o is visited infinitely many times with positive probability. A contradiction. The claim for arbitrary v now follows by induction and irreducibility of the tree.
Proof (Theorem 1.2) By the ergodicity of the Markov chain with transition probabilities p SRW and Lemma 4.1 it holds that We prove first that Let T 1 be a realization on which the frog model is transient. Then, there exists some ball B around the root o 1 such that no frog awaken outside this ball B will visit the origin o 1 . Let T 2 be an independent realization according to GW and let T : In the frog model on (T, o 1 ) the starting frog jumps into T 1 at time n = 1 with positive probability. Now, since every frog is transient, with positive probability all frogs in the set B that are woken up will never cross the additional edge (o 1 , o 2 ) and we ob- It remains to show that Let T 1 and T 2 be two recurrent realizations of GW ∞ and let T : , is recurrent with positive probability, hence, we have to verify that the possibility that frogs can change from one T i to the other does not change this property. Let us say that every frog originally in T 1 wears a red T-shirt and every frog in T 2 wears a blue T-shirt. Now, every frog that jumps from o 1 to o 2 leaves its red T-shirt in a stack in o 1 . In the same way every frog leaving o 1 to o 2 leaves its blue T-shirt in a stack in o 2 . A frog arriving from o 1 to o 2 takes a blue T-shirt from the stack. If the stack is empty, the frog "creates" a new blue shirt. We proceed similarly for the frogs that arrive in o 1 coming from o 2 . The frog model FM(T, η) starts with one awoken frog in a red T-shirt in o 1 . Once a frog visits o 2 , the blue frog model FM(T 2 , η) is started and a red shirt is left in o 1 . Conditioned on the event that FM(T 2 , η) is recurrent a blue frog will eventually jump from o 2 to o 1 and put on the red shirt. In this way, every red shirt is finally put on and the distribution of the red frogs in FM(T, η) equals the distribution of the frogs in FM(T 1 , η) with possible additional frogs. In other words, FM(T, η) is recurrent with positive probability. Finally, we can conclude 5 Transience of the frog model

No bushes, no stretches
We start with considering GW-trees T with p 0 + p 1 = 0. By Lemma A.5 we know that ρ(T) < 1 and hence Theorem 3.1 guarantees a transient phase for BMC on such GWtrees T. Coupling the frog model with an appropriate branching Markov chain implies a transience phase for the frog model.
Proof The proof relies on the fact that the BMC(T, P, µ) with µ(v) ∼ η + 1 for each v ∈ T stochastically dominates the frog model. We use a coupling of the frog model with a BMC such that at least as many frogs (in the frog model) as particles (in the BMC) visit the root. More precisely, in both models we start with one frog, respectively particle, at the root and couple them. A particle of the BMC that is coupled to a frog x in the frog model is denoted by x ′ . The "additional" particles in the BMC, in the meaning that there is no correspondent in the frog model, will move and branch without having any influence on the coupling. Let (f v ) v∈T be a realization of the sleeping frogs. If a first coupled particle arrives at v it branches into f v + 1 particles. The newly created frogs and particles are coupled. If more than one coupled particle arrive at v for the first time at the same moment, we choose (randomly) one of these, let it have f v + 1 offspring and couple the resulting particles with the frogs as above. The offspring of the other particles (those that are coupled to the remaining frogs arriving at v) are chosen i.i.d. according to µ(v) and one of them (randomly chosen) is coupled to each corresponding frog. Similarly, if a vertex v ∈ T will be visited a second time by a frog, no new frogs will wake up but the particle will branch again into random µ(v) particles and we couple the frog arriving at v with one (randomly chosen) of the particles. In this way every awake frog is coupled with a particle of the BMC. Hence if the BMC is transient, then also the frog model is transient. The mean offspringμ of BMC(T, P, µ) is constant for any v ∈ T as µ(v) are independent and identically distributed. Using Theorem 3.1 it follows that the BMC is transient if and only if

Bushes, no stretches
In the case p 0 + p 1 > 0 a direct coupling as in the proof of Theorem 3.1 does not allow to prove transience since every non-trivial BMC is recurrent. This is due to the existence of bushes or stretches in the Galton-Watson tree and the fact that the spectral radius of such trees is a.s. equal to 1, see Lemma A.5. We will deal with bushes and stretches separately. The case of bushes is rather straightforward: Once a frog visits the root of a bush for a first time all frogs in the bush are woken up and are placed at the root. Since every excursion inside the bush has to end at the root we can erase the bush without changing the transience behaviour of the process. Then, we construct a transient BMC dominating this modified process. Proof Every infinite GW-tree can be seen as a multitype GW-treeT with types g and b, see Section 2. We note T for a realization of GW conditioned to be infinite.
To start with, we modify the original frog model FM. If a frog enters a bush for the first time by stepping on v ∈ T of type b r , then immediately all frogs from the bush with root v wake up and are placed at the root v of the bush. More formally, let G v denote the random bush attached to v ∈ T. Then, there will be η ′ v := w∈Gv η w frogs in a vertex v of type b r and η ′ v := η v frogs in a vertex of type g. The bushes G v are i.i.d. distributed like a subcritical GW-process with generating functionf , see Section 2, and the expected size of G v is finite. Moreover, we restrict the movement of frogs to vertices of type g and b r . Frogs sitting at a vertex v of type b r step almost surely back to its predecessor. This new model actually lives on a new state space T ′ that arises from T by erasing all vertices of type b, see also Figure 3. Then we identify the frog configuration of the two models on T and T ′ and call the new model FM ′ := FM(η ′ , T ′ ). We denote by ν ′ the number of visits to the root in FM ′ and by ν ′ the number of visits to the root in FM ′ . Using a coupling as in the proof of Lemma 5.1 we obtain that Thus transience of FM ′ implies transience of FM.
In the next step, we define a BMC on T ′ that dominates FM ′ . For finding a similar coupling as in Lemma 5.1, let G ∼ |T sub | and (X i ) i≥1 an i.i.d. sequence of η distributed random variables, and choose the branching for all v ′ ∈ T ′ . Then, we can couple FM ′ with BMC(T ′ , P, µ) similar to Lemma 5.1. This yields where ν BMC is the number of visits to the root of BMC(T ′ , P, µ). By Wald's equation it follows with positive probability where T d is the homogeneous tree of degree d min + 1 and d min := min{k > 1 : p k > 0} and K the maximum degree of the tree T ′ . Theorem 3.1 says that Hence, we can choose E[η] sufficiently small such that the BMC(T ′ , P, µ ′ ) is transient. Together with (2) and (3) we conclude that we can find a η such that the frog model FM(T, η) is transient.

No bushes, but stretches
The case of stretches is more evolved. Similar to before we wake up all frogs in a stretch, if the beginning of a stretch is visited for the first time. The awoken frogs are placed according to the first exit measures at the ends of this stretch. Moreover we send every frog entering a stretch immediately to one of the ends of the stretch; again according to the exit measures. This makes it possible to consider the stretch as one vertex. However, the original length of the stretch is important for the path measure. We treat the path measure by introducing two step probabilities. Since a direct coupling between the frog model and a BMC is no longer possible, we compare the expected number of returns to the root of the frog model with those in a suitable BMC. Proof Let T be an infinite realization of GW. As 0 < p 1 < 1 we can consider T constructed according to ST × PER × GW bg , see Section 2. Using this construction we label its vertices in the following way, see also Figure 4: • label bs: a vertex of degree 2 with a mother vertex of degree strictly larger than 2; • label es: a vertex of degree 2 with a child of degree strictly larger than 2; • label s: a vertex of degree 2 with all two neighbours of degree 2; • label n: a vertex of degree higher than 2.
These labels help us to identify the stretches and their starting and end points. More precisely, a stretch is a path [v 1 , . . . , v n ] where v 1 has label bs and v n has label es and all vertices v i , i ∈ {2, . . . , n − 1}, are labeled with s. As mentioned above a BMC on a GW-tree with 0 < p 1 < 1 would a.s. be recurrent. To find a dominating BMC, which has a transient phase, we consider two modified state spaces T ′ and T ′ N .
Construction of dominating frog model FM ′ on T and T ′ We modify similarly to Lemma 5.2 the frog model in the following way. Frogs in the new FM ′ behave as in FM on vertices that are not in stretches. Once a frog enters a stretch we add more particles in the following way. Let [v 1 , . . . , v ℓ ] be a stretch of length ℓ v 1 and u the mother vertex of v 1 and w the child of v ℓv 1 , see in stretch. The modified frog model then basically moves on a new state space T ′ , constructed as follows: Let [v 1 , . . . , v ℓv 1 ] ⊂ T be a stretch and w ∈ T the successor of v l . Then, we merge the stretch into the vertex v 1 (with label bs). Hence, there is a single vertex of degree 2 left in between vertices with higher degree, see Figure 4. We identify each vertex v ′ ∈ T ′ with its corresponding vertex v ∈ T due to this construction. We can distinguish the vertices of T ′ into V n := {v ′ ∈ T ′ | v ′ with label n} and V s := {v ′ ∈ T ′ | v ′ with label bs}. This modified state space T ′ corresponds to the first two stages, namely PER × GW bg , in the construction of ST × PER × GW bg . In other words, its law is the same as T p×bg . Moreover, the third step, i.e. ST, in the construction of the measure is encoded in the length of each stretch. We introduce the following quantities. Let ν ′ (w ′ ) be the number of visits to w ′ and ν ′ n (w ′ ) the number of particles in w ′ at time n. Then, for a fixed realization ] be the expected number of visits to w ′ ∈ T ′ , when the frog started in v ′ ∈ T ′ . We also denote this as depends on the state space T ′ and we can look at the expected value with respect to ST for v ′ , w ′ ∈ T ′ . Note here, that the measure ST has no impact on the underlying tree but only on the number of frogs and the exit measure from the stretches. Moreover, it holds that Construction of dominating BMC ′ on T ′ In the next step we are going to define a branching Markov chain BMC ′ on T ′ such that where ν BMC ′ is the number of returns to the root of the BMC ′ . We recall that the length L of a stretch in the original tree T is geometrically distributed; L ∼ geo(p 1 ). Let L v , v ∈ T, denote this random stretch attached to a vertex v with label bs. The presence of arbitrary long stretches prevents the existence of transient BMC on T, see Lemma A.4. For this reason, let N ∈ N (to be chosen later) and define the tree T N as a copy of T where each stretch of length larger than N is replaced by a stretch of length N . We define a BMC, called BMC N , on T N , with driving measure SRW and offspring distribution µ ∼ η + 1. The BMC N defined on T N defines naturally a BMC BM C ′ N on T ′ , where once a particle enters a stretch, it produces offspring particles according to the exit-measures. This quantity is given by F ℓ+1 (1, 0|µ) and F ℓ+1 (1, ℓ + 1|µ) defined as in Lemma A.1 and A.8, where ℓ is the length of the stretch. The aim is now to find some N such that BMC ′ N is still transient and dominates (in ST-expectation) the frog model FM ′ .
In order to find such a domination we compare the mean number of returns "pathwise" in FM ′ and BMC ′ N . More precisely, we want to express the quantity ν ′ n (o ′ ) in terms of frogs following a specific path. Let p ′ be a path starting and ending at o ′ . A path of length n ∈ N looks like p ′ = [o ′ , p ′ 1 , p ′ 2 , . . . , p ′ n−1 , o ′ ] with p ′ i ∈ T ′ and p ′ i ∼ p ′ i+1 for each i. Let θ k denote the k-th cut of a path, that is θ(p ′ ) := [p ′ k , . . . , o ′ ]. We call a frog sleeping at some p i , 1 ≤ i ≤ n − 1, activated by frogs following the path p ′ (affb p ′ ), if inductively the frog was activated from a frog in p i−1 that was activated by frogs following the path p ′ or started at p 1 and followed p ′ . Additionally, for i, j ∈ N let S j (v ′ , i) denote the position of the i-th frog initially placed at v ′ ∈ T ′ after j time steps after waking up. (Here we assume an arbitrary enumeration of the frogs at each vertex.) We write affb p ′ (v ′ , i) for the event that the ith frog in v ′ is affb p ′ . Using this, ν ′ n (o ′ ) is equal to

Now, we can rewrite
by using the monotone convergence theorem. For a given path p ′ the term equals the expected number of frogs that were activated following the path and that follow the paths after their activation. In the same way as for the frog process we can define the expected number of particles ν BMC (p ′ ) for a BMC following a path p ′ . In the remaining part of the proof we construct a branching Markov chain BMC ′ such that for all paths p ′ . Transience of the BMC then induces transiences of the frog model. The paths p ′ are concatenations of three different types of vertex sequences. type 1 is a sequence that does not see any stretches. A sequence of type 2 traverses a stretch, whereat a sequence of type 3 visits a stretch but does not travers it. We will split each path p ′ into these three types and give upper bounds for (8) for each type separately. We have to take into account that multiple visits of the same sequence of vertices are not independent from each other. Here the frogs face in every visit the same length of a stretch. Hence while taking the expectation over the length of the stretches, multiple visits of the same vertices have to be considered at the same time. Therefore we give upper bounds of (8) for each combination of multiple visits. Then we combine the results for a final upper bound of a mixed path. For this purpose we consider for the BMC the mean number of particles created in stretches in T N . We consider the situation described in Figure 4. Let ℓ be the length of a stretch generated according to ST. Such a stretch appears in T N with probability ) the expected number of particles arriving in p ′ i+1 while starting in p ′ i . Again we can look at the expectation with respect to ST where ST impacts only the number of created particles and not the underlying tree. We define the vertices u and w as absorbing and denote by η N (u) (resp. η N (w)) the number of particles absorbed in u (resp. in w), see also Section A.1.
Only visits of type 1: We assume that p ′ = [p ′ 0 , p ′ 1 , p ′ 2 , . . . , p ′ n−1 , p ′ n ] only consists of sequences of type 1. Using the Markov property we can bound due to the choice of the BMC ′ , see the paragraph after Equation (7). Multiple visits of a stretch in sequences of type 2: We assume that the path also has some sequences of type 2, see Figure 5. An important observation is that every path from o ′ to o ′ that traverses a stretch in one direction has to traverse it in the other direction as well. Hence, such a path of length n has for example the form where v ′ has degree 2. We start with the case that a sequence of vertices is visited only once. The case of more visits will be an immediate consequence. In order to bound the expected number of frogs along this path we define m T ′ FM ′ (u ′ → v ′ → w ′ ) as the expected number of frogs that follow the path [u ′ , v ′ , w ′ ] in FM ′ starting with one frog in u ′ . The modified frog model FM ′ is defined such that all frogs in the stretch are woken up and distributed at the end of the stretches if the starting vertex of the stretch is visited. In the case of traversing a stretch, this is dominated by the following modification: if the frog jumps on v 1 from u the first time we start a BMC in v 1 with offspring distribution η + 1 and absorbing states u and w. The mean number of frogs absorbed in u and w can be calculated using Lemmata A.1 and A.8. This dominates FM ′ since we consider a path traversing the stretch. This means that all vertices in the stretch were visited in the new model, since some particles arrived in w ′ and we can couple the sleeping frogs in FM ′ with the created particles in the stretch. We conclude by Lemma where ℓ = ℓ v ′ + 1 is the length of the total stretch (including the start of the stretch). To take into account that at the second traversal of the stretch no sleeping frogs are left in the stretch we define m T ′ ,2+ as the expected number of particles following [w ′ , v ′ , u ′ ] with no frogs between w ′ und u ′ . Hence, .

Note that the last term equals the mean number of particles in BMC
is monotone decreasing in ℓ. We can now integrate with respect to ST to obtain: and If we have more visits of type 2, there are no new frogs waking up and we have as transition probability through the stretch 1 ℓ+1 for each visit. In the case of k visits we obtain We notice again that f k (ℓ) := F ℓ (1, ℓ|E[η] + 1) (1/ℓ) 2k−1 is monotone decreasing in ℓ and we can integrate with respect to ST in the same manner as above.
Define m ST BMC ′ (u ′ → v ′ → w ′ ) as the expected number of particles that follow the path [u ′ , v ′ , w ′ ] in BMC ′ starting with one particle in u ′ . Using the aforegoing we can bound Moreover the stretches are independently generated. We obtain by induction for different sequences of type 2 that: Multiple visits of a stretch in sequences of type 2 and 3: We split this section in three parts. In the first we assume, that a sequence of vertices is only visited once in the manner of type 3. Secondly, we treat a sequence of a path which visits a stretch more than once in the manner of type 3. Lastly, we study sequences which are visited by type 2 and type 3 sequences. There, we have to distinguish between the type of the first visit of the sequence.
We start with the first part. We assume that the path p of length n contains a sequence of type 3, that is p ′ i j = v ′ ,i j ∈ {1, . . . , n}, of degree 2 and p i j −1 = p i j +1 = u ′ , see Figure 6. This means that the frogs in FM ′ did not pass the stretch completely. We call these parts of the path stretchbits. A typical path p in this case can be for example We define m T ′ FM ′ (u ′ → v ′ → u ′ ) as the expected number of frogs that follow the path [u ′ , v ′ , u ′ ] in FM ′ starting with one frog in u ′ . Then Recall that the distribution of the total stretch length L = ℓ v 1 + 1 is exponential: Hence, integrating (21) with respect to ST yields Let d = min{i ≥ 2 : p i > 0}. A stretch of length ℓ is equivalent to an unbranched path of length ℓ + 1 in Section A.2 of the Appendix. As we only allow a maximum stretch length N in case of BMC N , we obtain at maximum an unbranched path of length N + 1. Then, using Lemma A.5, Theorem A.6, and Lemma A.3 the spectral radius ρ(P N +1 ) on the absorbing stretch piece of length N + 1 satisfies Furthermore, We choose nowμ = cos for some sufficiently small ε > 0 and define Observe here that, sinceμ < 1 ρ(T ′ ) , the BMC ′ with mean offspringμ is not only transient but it also holds that E BMC ′ [ν] < ∞, see Chapter 5.C in [29]. Now, integrating equation (25) with respect to ST yields We now look for E[η] sufficiently small and N sufficiently large such that It suffices to find an N such that By Lemma A.9 we can bound the right side from below by where ϕ = arccos(μ −1 ). This reduces (30) to: Due to the choice of ϕ the left hand side of (32) decays exponentially in N while the first part of the right hand side has polynomial decay in N . Therefore, there exists some N such that (32) is verified. We continue with the second part, where a sequence of the path faces multiple type 3 visits. If a frog makes a second type 3 visit to an already woken up stretch, this frog encounters no new frogs and returns to u ′ almost surely. This follows for every other visit of type 3. Hence, conditioning the frog, that it will not make another type 3 visit to a stretch, has no influence on the possible frogs returning to the root and consequently on transience and recurrence. We will call this model FM ′′ . But we notice that the path measure changes when we change to FM ′′ : where y ′ is any neighbour of u ′ apart from v ′ . Since the path measure of BMC N is unchanged we have to compare as u ′ were visited already by assumption and obtain We conclude for the mean offspringμ of BMC N that is a necessary condition for our majorization. Using the new model FM ′′ we are left with only the first visit of type 3 to the stretch. As we have seen before, there is a N such that (32) holds. Now, we will treat the third part, where we allow multiple visits of type 2 and 3 to a sequence of vertices. We want to erase again multiple visits of type 3 of a stretch and assume, that (35) holds, such that the BMC N dominates the conditioned path. Then it is left to deal with either a first vistit of type 2 or a first visit of type 3 and multiple visits of type 2. If the first visit is of type 2, we can bound the frog model by using (19) additionally to (35).
If the first visit is of type 3, and we have apart from other visits of type 3 (which will be erased and bounded using (35)) k visits and returns of type 2, we obtain For the upcoming equations we omit the factors of the transitions probabilities from u ′ to v ′ and from w ′ to v ′ . These probabilities are the same for the BMC and do not play a role for the comparison between the frog model. Then we get: For the BMC we have the following identities as before: By Lemma A.9 (and again omitting the transitions probabilities) this is greater or equal to We want to show that we can choose for each p 1 and N an η such that the following holds for all k ≥ 1: The second part of the left hand side, (45), is equal to the first part, (47), on the right hand side. Next we compare the third part of the left, (46), to the third part on the right, (49). We notice that the function ℓ ℓ+1 1 ℓ+1 2k is monotonously decreasing in ℓ and thus ∞ ℓ=N ℓ ℓ + 1 Now, we consider the remaining term on the left hand side, (44), and the second of the right hand side, (48). We start with giving an upper bound for the second sum in (44): The second term of the right hand side, (48), can be transformed into We have that (44) < (48) if For all choices of p 1 and N ∈ N we can now find E[η] sufficiently small such that the latter inequality is verified for all k ∈ N.

Summary
We summarize all the conditions on η andμ such that we can find a dominating transient BMC for a given frog model FM in the case of appearing stretches: is small enough such that there exists a N such that (29) holds; 4. Choosing η such that for given p 1 and the previously selected N the inequality (44)-(49) holds; In other words, for every p 1 > 0 there exists some N such that if there exists some small E[η] > 0 and some BMC ′ with mean offspring larger than 1 such for all paths p ′ and GW-a.a. trees T ′ . Finally, we found that ν ′ < ∞ FM-a.s. for GWa.a. trees and hence ν < ∞ FM-a.s. for GW-a.a. trees. The existence of the constant c η follows from the 0-1-law of transience.

Bushes and stretches
It is left to prove the main theorem of this paper. The proof will be mainly a consecutive application of Lemma 5.2 and Lemma 5.3. Doing the consecutive application of removing bushes and then compressing stretches, we have to deal with a new structural case, see Figure 7. We call this structure geodesic stretch. A geodesic stretch does not appear in v w v w Figure 7: A possible stretch with finite bushes inbetween and its modification by removing the bushes.
the considered GW-trees before, as we exclude either offspring one or non offspring. Such a geodesic stretch would not be removed if we do the modification steps of Lemma 5.2 and afterwards of Lemma 5.3. But an arbitrary long sequence of vertices with only one child of type g in each generation will cause a spectral radius equal to 1, again. Therefore, we consider in the upcoming proof such a sequence of vertices more or less as a normal stretch.
Proof (Theorem 1.3) We assume that both p 0 > 0 and p 1 > 0. Otherwise the proof is finished directly by using Lemma 5.2 or 5.3. Thus, the tree has finite bushes, stretches and geodesic stretches. Stretches are actually a subset of geodesic stretches and we will continue by considering only bushes and geodesic stretches. We recall that our GW-tree has bounded offspring, that is there is a K < ∞ such that Y (n) i < K for all i, n ∈ N. Therefore, every vertex which is part of a geodesic stretch can have at most a by K bounded number of finite bushes attached. Hence, the model is dominated by the frog model on T, where we fill every geodesic stretch with according to T sub independently generated finite bushes, such that there are K − 1 finite bushes attached to every vertex of the geodesic stretch, see frogs to the two ends of the geodesic stretch. Here, the length ℓ of the geodesic stretch is distributed according to geo( p 1 ) + 1, where p 1 is the probability of having only one child of type g. For the construction of a dominating BMC let again N ∈ N and define the tree T N as a copy of T, where each filled up geodesic stretch of length larger than N is replaced by a filled up geodesic stretch of length N and the bushes are reduced to a single leave. On this tree we define again BMC N , on T N , with driving measure SRW and offspring distribution µ ∼ v∈Gy η y + 1, which defines naturally a BMC ′ = BMC ′ N on T ′ , where once a particle enters a former geodesic stretch, it produces offspring particles according to the exit-measures.
Defining the offspring of the BMC ′ by µ we treated the appearing bushes. Moreover, the spectal radius will change as described in Lemma A.7 and we have to take this into account in the end. Now, to find an N ∈ N such that BMC ′ is dominating for FM ′ we proceed like in the proof of Lemma 5.3 with the difference that in average to both sides of a geodesic stretch of length ℓ exit frogs. The frog which is waking up the stretch leaves the stretch to each side with the same probability as before. Moreover the length of the stretch is now distributed according to geo( p 1 ) + 1 and the probability that a vertex is dedicated as a starting vertex of a stretch is ber( p 1 )-distributed, as well.
Hence, we obtain as conditions for a transience criterion of the frog model that is small enough such that there exists a N such that • Choosing η such that for given p 1 and the previously selectedÑ equation Additionally, the BMC ′ N has to fulfill the transience criterion Theorem 3.1. The process lives originally on the tree T N . If T is a tree without stretches and reduces bushes, then the spectral radius of T N (tree with stretches of maximum length N ) is equal to ρ(T N ) = cos Adding also single leaves to this tree yields a maximal spectral radius of Hence we have to choose µ such that it holds additionally, that We can now conclude as in the end of the proof of Lemma 5.3. for many helpful discussions. Our grateful thanks are also extended to Nina Gantert for several advices and discussions. We would also like to thank Stefan Lendl for his great support in all numerical issues, as well as Ecaterina Sava-Huss for raising the idea for this project. The second author was supported by the Austrian Science Fund (FWF): W1230. Grateful acknowledgement is made for hospitality from the Institute of Discrete Mathematics of TU Graz, where the research was carried out during visits of the first author.

A.1 The relation with generating functions
At various places we use the connection between BMC and generating functions. This connection is a crucial tool in the study of BMC, e.g., see [3], [4], [13], [23], and [29]. Let M be a subset of the state space and modify the BMC in a way such that particles are absorbed in M and once they have arrived in M , they keep on producing one offspring a.s. In other words, particles arriving in M are frozen. Denote Z ∞ (M ) ∈ N ∪ {∞} the total number of frozen particles in M at time "∞". For M ⊆ Γ, we define the first visiting generating function: where Z n is the original SRW and P its corresponding probability measure The following lemma will be used several times in our proofs; a short proof can be found for example in [4,Lemma 4.2].

A.2 Spectral radius of trees
In order to study recurrence and transience of a BMC it is essential to understand the spectral radius of the underlying Markov chain. In this section, we collect several results on the spectral radius of SRW on trees.
Definition A.2 The isoperimetric constant ι(T ) of a tree with edges E and vertices V is defined by For the isoperimetric constant it holds that, ι(T ) = 0 if and only if the spectral radius ρ(T ) of the simple random walk equal to 1, see Theorem 10.3 in [28]. There is a more precise statement on finite approximation of the spectral radius, e.g., see [2] and [25]. Consider an infinite irreducible Markov chain (X, P ) and write ρ(P ) for its spectral radius. A subset Y ⊂ X is called irreducible if the sub-stochastic operator P Y = (p Y (x, y)) x,y∈Y defined by p Y (x, y) := p(x, y) for all x, y ∈ Y is irreducible. It is rather straightforward to show the next characterization. Lemma A.3 Let (X, P ) be an irreducible Markov chain. Then, where the supremum is over finite and irreducible subsets Y ⊂ X. Furthermore, ρ(P F ) < ρ(P G ) if F G.
A first observation is the following result, see [29,Lemma 9.86]. We say that a stretch (or unbranched path) of length N in a tree T is a path [v 0 , v 1 , . . . , v N ] of distinct vertices such that deg(v k ) = 2 for k = 1, . . . , N − 1.
Lemma A.4 Let T be a locally finite tree T . If T contains stretches of arbitrary length, then ρ(T ) = 1.
Moreover, we can give a precise characterization of the spectral radius of a simple random walk on a GW-tree Lemma A.5 Let ρ(T) be the spectral radius of the simple random walk on a Galton-Watson tree T with offspring distribution (p i ) i≥0 . Then, infinite realizations T, where d = min{i : p i > 0} and T d the homogeneous tree with offspring d.
Proof If T is finite, the simple random walk is recurrent and it holds that ρ(T) = 1, see Section 1 in [28]. Now, let us assume that T is infinite. In the case where p 1 > 0 the tree contains, for every choice of N ∈ N, GW-a.s. a stretch of length N ; this is a consequence of the lemma of Borel-Cantelli. Using Lemma A.4 we conclude that ρ(T) = 1. Now, we assume that p 1 = 0 but p 0 > 0. In this case the tree T contains, for every choice of N ∈ N, a finite bush of N generations, which we call bush B N . For such a bush B N it holds that |δE BN | V ol(BN ) ≤ 1 2N . Again, by finding an arbitrary large bush we obtain ι(T) = 0 and consequently using Theorem 10.3 in [28] we conclude ρ(T) = 1. In the case p 0 + p 1 = 0 Corollary 9.85 in [29] implies that ρ(T) ≤ ρ(T d ) = 2 √ d d+1 where d is the smallest offspring of the Galton-Watson tree and T d denotes the homogeneous tree with offspring d. The remaining equality follows by finding arbitrarily large balls of T d as copies in T as above and applying Lemma A.3.
We construct a new tree T by replacing each edge e of T with a stretch of length k = k(e). We call T a subdivision of T and max e {k(e)} the maximal subdivision length of T . We write T (N ) for the subdivision of T where k(e) = N for all edges e in T . We give a particular case of Theorem 9.89 in [29].
Theorem A.6 Let T be a locally finite tree and denote ρ(T ) (resp. ρ(T (N ) )) the spectral radius of the SRW on T (resp. T (N ) ). Then We denote by leaves vertices with degree 1. Let T be a locally finite tree with maximum degree K + 1 < ∞, let T be the tree where we attach to every vertex v a certain number (bounded by K) of single leaves. Let d be the degree of v in T . Then, we denote its original neighbored vertices by w 1 , . . . , w d and the attached leaves by y 1 , . . . , y k . We recall that d + k ≤ K + 1.
Lemma A.7 Let ρ(T ) < 1 be the spectral radius of a locally finite tree T with maximal degree K + 1. Then it holds that the spectral radius of the modified tree T satisfies where d + 1 is the minimal degree larger than 1.
Proof The lemma can be proved using generating functions only. However, we give a variant using branching Markov chains. Let T d,• be the homogeneous tree of degree d + 1 where in addition k = (K + 1) − (d + 1) leaves are attached to every vertex. The fact that ρ(T ) ≤ ρ(T d ), see [29,Corollary 9.85 ], together with Lemma A.3 imply that ρ( T ) ≤ ρ(T d,• ). Let m be the mean offspring of a BMC on T d,• . We know that the BMC on this tree is transient iff We want to determine how many particles arrive in average at one of the original neighbors w i if we start the BMC at v. We can reduce this problem to a reflected and absorbing BMC on the state space {y, v, w} like in Figure 9. The site v corresponds to the vertex v of the tree. Site w d d+k 1 k d+k 1 v w y Figure 9: The reduction of leaves to a three stage problem. corresponds to w 1 , . . . , w d . This consolidation is possible since reaching each one of these vertices in T is equally distributed. Similarly, the vertices y i are represented by y. For calculating how many particles arrive in w i we use the first visit generating F defined as where f (n) (x, y) := P[Z n = y, Z k = y ∀ 0 ≤ k < n|Z 0 = x] and Z n is the Makov chain described by Figure 9. Then, it holds and and we see that the corresponding BMC on T d is transient iff .
Transience of the BMC on T d,• mit mean m is equivalent to transience of the BMC with offspring F (v, m|m) on T d , and hence using Equation (67) we get that and obtain a relation between the two spectral radii This yields that the spectral radius of T d,• is A.3 Absorbing BMC on finite paths The convergence radius of the power series equals R N = 1/ρ(P N ). We give the following expressions of the generating function F N for two particular pairs of values of x and y, see Example 5.6 in [29].
The second part follows the exact same line as the first part of the proof.