The branching-ruin number as critical parameter of random processes on trees

The branching-ruin number of a tree, which describes its asymptotic growth and geometry, can be seen as a polynomial version of the branching number. This quantity was defined by Collevecchio, Kious and Sidoravicius (2018) in order to understand the phase transitions of the once-reinforced random walk (ORRW) on trees. Strikingly, this number was proved to be equal to the critical parameter of ORRW on trees. In this paper, we continue the investigation of the link between the branching-ruin number and the criticality of random processes on trees. First, we study random walks on random conductances on trees, when the conductances have an heavy tail at $0$, parametrized by some $p>1$, where $1/p$ is the exponent of the tail. We prove a phase transition recurrence/transience with respect to $p$ and identify the critical parameter to be equal to the branching-ruin number of the tree. Second, we study a multi-excited random walk on trees where each vertex has $M$ cookies and each cookie has an infinite strength towards the root. Here again, we prove a phase transition recurrence/transience and identify the critical number of cookies to be equal to the branching-ruin number of the tree, minus 1. This result extends a conjecture of Volkov (2003). Besides, we study a generalized version of this process and generalize results of Basdevant and Singh (2009).


Introduction
Let us consider a random process on a tree which is parametrized with one parameter p. We say that this process undergoes a phase transition if there exists a critical parameter p c such that the (macroscopic) behavior of the random process is significantly different for p < p c and for p > p c . This is, for instance, the case of Bernoulli percolation on trees, biased random walks (see [19,20,21]) or linearly edge-reinforced random walks [22] on trees. In [19], R. Lyons proved the following beautiful result. Bernoulli percolation and biased random walks (among others) share the same critical parameter which is equal to the branching number of the tree. The branching number, defined by Furstenberg [15], is, roughly speaking, a quantity that provides a precise information on the asymptotic growth and geometry of a tree, at the exponential scale (see (2.1) for a definition). For instance, for trees that are "well-behaved" (such as spherically symmetric trees) and whose spheres of diameter n have size m n , the branching number is equal to m. This description is actually not accurate as some trees have a peculiar geometry, and the size of their spheres is not a good indicator of their asymptotic complexity. The phase transition of the once-reinforced random walk was studied in [8]. In order to see a phase transition, one needs to consider trees that grow polynomially fast (see [16]), and therefore the branching number is not the quantity that would provide a relevant information in this case. Indeed, the branching number does not allow us to distinguish among trees with polynomial growth as the branching number of any tree with sub-exponential growth is equal to 1. In [8], it was proved that the critical parameter for the once-reinforced random walk on trees is equal to the branching-ruin number of the tree (see (2.2)). The branching-ruin number of a tree is best described as the polynomial version of the branching number: if a well-behaved tree has spheres of size n b , then the branching-ruin number of this tree is b. Again, this fact is not true in general because of the possible complex asymptotic geometry of trees. The purpose of the current paper is to emphasize two other examples where the branching-ruin number appears as the critical parameter of a random process, as it was done for the branching number. We study random walks on random conductances with heavy-tails and a model of excited random walks called the M-digging random walk. In the next two subsections, we describe our results. In the first one, we relate the branching-ruin number to the critical weight of the tails of the conductances. In the second result, we relate the critical number of cookies per site to the branching-ruin number and, in particular, our result extends a conjecture of Volkov [25].
1.1. Random walk on heavy-tailed random conductances. First, we study random walks on random conductances in the case where the conductances have heavy tails at zero. Consider an infinite, locally finite, tree T with branching-ruin number b (see (2.2) for a definition). Even though our results hold for any branching-ruin number, for the sake of the following explanations, let us temporarily assume that b > 1, so that simple random walk is transient on this tree (see Theorem 2,or [8]). Assign i.i.d. conductances, or weights, to each edge of T and let us define a nearest-neighbor random walk which jumps through an edge with a probability proportional to the conductance of this edge. This model is very classical and has been extensively study on various graph, including Z and Z d . The behavior of the walk depends on the common law of the conductances. For instance, if the conductances are bounded away from 0 and from the infinity, the behavior of the walk is close to the one of simple random walk and it will therefore be transient on T , moving at a speed similar to that of simple random walk. If the conductances can be very large, i.e. unbounded and for instance with an heavy-tail at infinity, this should not affect the transience of the walk. Nevertheless, this would have an important impact on the time that the random walk spends on small areas of the environment. We do not prove anything in this direction in this paper as our main interest is in the recurrence/transience of the walk, but we would like to describe here what should happen. If the conductances can be extremely large with a not-so-small probability, then the walker will meet, here and there, an edge with an overwhelmingly large conductance and will cross this edge back-and-forth for a very large number of times before moving on. The consequence of this mechanism is that the random walker will spend most of its time on these traps and will move at a speed much smaller than simple random walk on the same tree. This phenomenon is reminiscent of Bouchaud's trap model, see [13,10,11,12], or [14] where an explicit link is made between Bouchaud's trap model and biased random walk on random conductances. The last possible scenario is when the conductances could be extremely small, which is what we are mainly interested in here. The extreme case would be percolation where the random walk is recurrent as soon as the percolation is subcritical. In our case, the conductances remain positive but have an heavytail at 0. This creates "barriers" of edges with atypically small conductances that can make the walker come back to the root infinitely often, even when the tree is transient for simple random walk. Let us now describe our results.
Recall that T is an infinite, locally finite, tree and let E be the set of all its edges. Let (C e ) e∈E be a collection of i.i.d. random conductances that are almost surely positive. Moreover, assume that where L : R → R is a slowly-varying function. For simplicity, we will also assume that P (C e ≥ 1) > 0 without loss of generality. For a realisation of the environment (C e ), we can define a random walk on these conductances which jumps through an edge e with a probability proportional to C e . For a formal definition of this random walk on random conductances (RWRC), we refer to Section 2.3.1. In the following, we say that a walk is transient if it does not return to its starting point with positive probability. If a walk is not transient, it comes back to the root almost surely and it is called recurrent. We also give a formal definition of recurrence and transience in Section 2.3.1. Finally, the branching -ruin number of T , formally defined in (2.2), is denoted by br r (T ). Theorem 1. Fix an infinite, locally finite, tree T and let b = br r (T ) ∈ [0, ∞] be its branching-ruin number. If b < 1, then RWRC is recurrent. Assuming b > 1, if mb > 1 then RWRC is transient and if mb < 1 then it is recurrent.

1.2.
The M-digging random walk. Our second main result concerns a model of multi-excited random walks on trees, also known as cookie random walks. Excited random walks were introduced by Benjamini and Wilson in [3] on Z d , and have been extensively studied (see [1,4,17,18,24]). Zerner [26,27] introduced a generalization of this model called multi-excited random walks (or cookie random walk). These walks are well understood on Z, but not much is known in higher dimensions. Here, we study an extreme case of multi-excited random walks on trees, introduced by Volkov [25], called the M-digging random walk (M-DRW). We also study its biased version and generalize a result by Basdevant and Singh [2], see Theorem 5, who studied it on regular trees.
Assign to each vertex M cookies, where M is a non-negative integer. Define a nearest-neighbor random walk X as follows. Each time it visits a vertex, if there is any cookie left there, it eats one of them and then jumps to the parent of that vertex. If no cookies are detected, then it jumps to one of the neighbors with uniform probability. We refer to section 2.3.2 for a formal definition of this process.
Volkov [25] conjectured that this process is transient on any tree containing the binary, which was proved by Basdevant and Singh [2]. Here, we obtain a much finer description of the process and we can prove that this random walk actually undergoes a phase transition on trees with polynomial gowth, i.e. on trees T where the branching-ruin number br r (T ) is finite.
Theorem 2. Let T be an infinite, locally-finite, rooted tree, and let M ∈ N. If br r (T ) < M + 1 then M-DRW is recurrent and if br r (T ) > M + 1 then M-DRW is transient.
We refer to Theorem 5 for the more general result on the biased case and Theorem 7 for the case where the number of cookies on each vertex is inhomogeneous over the tree.

The models
In this section, we define relevant vocabulary and conventions. We then recall the definition of the branching number and branching-ruin number of a tree, and finally we formally define the models.
2.1. Notation. Let T = (V, E) be an infinite, locally finite, rooted tree with set of vertices V and set of edges E. Let ̺ be the root of T . Two vertices ν, µ ∈ V are called neighbors, denoted ν ∼ µ, if {ν, µ} ∈ E. For any vertex ν ∈ V \ {̺}, denote by ν −1 its parent, i.e. the neighbour of ν with shortest distance from ̺. For any ν ∈ V , let |ν| be the number of edges in the unique self-avoiding path connecting ν to ̺ and call |ν| the generation of ν. In particular, we have |̺| = 0. For any edge e ∈ E denote by e − and e + its endpoints with |e + | = |e − | + 1, and define the generation of an edge as |e| = |e + |. For any pair of vertices ν and µ, we write ν ≤ µ if ν is on the unique selfavoiding path between ̺ and µ (including it), and ν < µ if moreover ν = µ. Similarly, for two edges e and g, we write g ≤ e if g + ≤ e + and g < e if moreover g + = e + . For two vertices ν < µ ∈ V , we will denote by [ν, µ] the unique self-avoiding path connecting ν to µ. For two neighboring vertices ν and µ, we use the slight abuse of notation [ν, µ] to denote the edge with endpoints ν and µ (note that we allow µ < ν). For two edges e 1 , e 2 ∈ E, we denote e 1 ∧ e 2 the vertex with maximal distance from ̺ such that e 1 ∧ e 2 ≤ e + 1 and e 1 ∧ e 2 ≤ e + 2 .

The Branching Number and The Branching-Ruin Number.
In order to define the branching number and the branching-ruin number of a tree, we will need the notion of cutsets. Let T be an infinite, locally finite and rooted tree. A cutset in T is a set π of edges such that, for any infinite self-avoiding path (ν i ) i≥0 started at the root, there exists a unique i ≥ 1 such that [ν i−1 , ν i ] ∈ π. In other words, a cutset is a minimal set of edges separating the root from infinity. We use Π to denote the set of cutsets.
The branching number of T is defined as branching-ruin number of T is defined as These quantities provide good ways to measure respectively the exponential growth and the polynomial growth of a tree. For instance, a tree which is spherically symmetric (or regular) and whose n generation grows like b n , for b ≥ 1, has a branching number equal to b. On the other hand, if such a tree grows like n b , for some b ≥ 0, its branching-ruin number is equal to b. We refer the reader to [21] for a detailed investigation of the branching number and [8] for discussions on the branching-ruin number.
2.3. Formal definition of the models.

2.3.1.
The random walk on heavy-tailed random conductances. In this section, we provide a formal definition of the random walk on random conductances (RWRC). First let us define the environment of the walk. To the edges of T , we associate i.i.d. random conductances C e ∈ (0, ∞), e ∈ E, with common law P, where E denotes the corresponding expectation. We will assume that where L : R → R is a slowly varying function. Given a realisation of the environment (C e ) e∈E , we define a reversible Markov chain X = (X n ) n . We denote P ω ν the law of this Markov chain when it is started from a vertex ν ∈ V . Under P ω ̺ , we have that X 0 = ̺ and, if X n = ν and µ ∼ ν, we have that .
We call P ω · the quenched law of the random walk and denote E ω · the corresponding expectation. We define the annealed law of X started at ̺ as the semi-direct product P ̺ = P × P ω ̺ , that is the random walk averaged over the environment. We denote E ̺ the corresponding annealed expectation. For a vertex v ∈ V , T (v) stands for the return time to v, that is A RWRC is said to be recurrent if it returns to ̺, P ̺ -almost surely. This process is transient if it is not recurrent, that is Finally, as X is a Markov chain under P ω · , we have that it is transient if and only if the walk returns finitely often to the root ̺ and, using a zero-one law on the environment, we can prove that this happens with probability 0 or 1. Therefore, the notions of recurrence and transience are well defined in the quenched and annealed sense.

2.3.2.
The M-digging random walk. Let T = (V, E) be an infinite, locallyfinite, tree rooted at a vertex ̺. We are going to define a biased version of the M-DRW described above, which will also allow for an inhomogeneous initial number of cookies. Let M = (m ν , ν ∈ V ) be a collection of non-negative integers, with m ̺ = 0, and fix λ > 0. For convenience, for e ∈ E, we denote m e = m e + . Let us define a random walk X = (X n ) n≥0 as follows. For any vertex ν ∈ V , define (2.4) ℓ n (ν) = |{k ∈ {0, . . . , n} : X k = ν}| .
For each edge e ∈ E and each time n ∈ N, we associate the following weight: As can be seen in (2.6) below, the model remains unchanged if, in the above definition, we use λ −|e| instead of λ −|e|+1 . Our choice turns out to be convenient in the proofs. For a non-oriented edge [ν, µ], we will simply write W n (ν, µ) = W n (µ, ν) = W n ([ν, µ]) We start the random walk at X 0 = ̺. At time n ≥ 0, for any ν ∈ V , on the event {X n = ν}, we define, for any µ ∼ ν, where F n = σ(X 0 , . . . , X n ) is the σ-field generated by the history of X up to time n. We call this walk an M-digging random walk with bias λ and denote it M-DRW λ . It will be very convenient to observe X only at times when it is on vertices with no more cookies. For this purpose, let us define X = ( X n ) n a nearest-neighbor random walk on T as follows. Let σ 0 = 0 and, for any n ∈ N, We define, for all n ∈ N, X n = X σn . Next, we want to define notions of recurrence and transience for X. As above, we define the return time of X, or X, to a vertex ν ∈ V by In words, we consider that a vertex ν is hit by X when it is hit by X in the usual sense. The fact to choose this time to be greater than 1 will be convenient technically to accommodate with the particularities of the root. We say that X, or X, is transient if Otherwise, we say that X, or X, is recurrent.
Note that if we choose m ν = M ∈ N for all ν ∈ V \ {̺} and λ = 1, then X is the M-DRW described in Section 1.2.

Main results
We are about to state a sharp criterion of recurrence/transience in terms of a quantity RT (T , X), first introduced in [8]. For a function ψ : E → R + , we define the quantity As we will see, for the relevant function ψ, the recurrence or transience of the walks will be related to this quantity being smaller or greater than 1.

3.1.
Main results about RWRC. It is straightforward to see that the two following results together imply Theorem 1. The proof of Proposition 3 is given in Section 5.
Let us define, for any e ∈ E, ψ RC (e) = 1 if |e| = 1 and, if |e| > 1, Proposition 3. Fix an infinite, locally finite, tree T and let b = br r (T ) ∈ [0, ∞] be its branching-ruin number. If b < 1 then RT (T , ψ RC ) < 1, P-almost surely. Assuming b > 1, we have that The following result is a direct consequence of Theorem 5 of [8], recalling the discussion at the end of Section 2.3.1 and noting that condition (2.5) in [8] is trivially satisfied by Markov chains, which in that context is translated into non-reinforced environments. Therefore, we will omit its proof.

3.2.
Main results about the M-DRW λ . The following Theorem is more general than Theorem 2 in the introduction and deals with the homogeneous Let us emphasize that, in item (1) below, the phase transition is given in terms of branching-ruin number whereas, in item (2), the phase transition is given in terms of branching number.
Theorem 5. Let T be an infinite, locally-finite, rooted tree, and let M ∈ N, λ > 0. Denote X the M-DRW λ on T with parameters λ > 0 and M = (m ν ; ν ∈ V ) such that m ̺ = 0 and m ν = M for all ν ∈ V \ {̺}. We have that (1) in the case λ = 1, if br r (T ) < M + 1 then X is recurrent and if br r (T ) > M + 1 then X is transient; If, for a tree T , br(T ) > 1, then we have that br r (T ) = ∞, as proved of Case V of the proof of Lemma 8. Therefore, the items (1) and (2) in Theorem 5 are not contradictory.
Note that, for a b-ary tree, br(T ) = b and our result therefore agrees with Corollary 1.7 of [2]. In [2], the authors prove that the walk is recurrent at criticality on regular trees, but this is not expected to be true in general.
We are about to state a sharp criterion of recurrence/transience in terms of a quantity RT (T , ·) as defined in (3.1), which will apply to the general case M = (m ν ; ν ∈ V ) ∈ N V . We will then prove that Theorem 5 is a simple corollary of this general result.
For this purpose, we need some notation. Let us define a function ψ M,λ on the edges of E such that, for any e ∈ E, ψ M,λ (e) = 1 if |e| = 1 and, for any e ∈ E with |e| > 1, As we will see in Section 7, ψ M,λ (e) corresponds to the probability that X, or X, when restricted to [̺, e + ] (i.e. the path from the root to e + ), hits e + before returning to ̺, after having hit e − . We will prove the following result in Section 8.
then X is transient.
The following result concerns the homogeneous case. Theorem 5 is a straightforward consequence of Theorem 7 and Lemma 8.

Lemma 8.
Consider an M-DRW λ X on an infinite, locally finite, rooted tree T , with parameters λ > 0 and M = (m ν ; ν ∈ V ) such that m ̺ = 0 and m ν = M for all ν ∈ V \ {̺}. We have that The proofs of Theorem 7 and Lemma 8 are given in Section 6.

Preliminary results
Proposition 10 below can be proved following line by line the argument in Section 8 of [8]. For the sake of completeness, we give an outline of the proof in the Appendix A. It relies on the concept of quasi-independent percolation defined as below (see also [21], page 144). In the following, we denote by C(̺) the cluster of open edges containing the root ̺. Definition 9. An edge-percolation is said to be quasi-independent if there exists a constant C Q ∈ (0, ∞) such that, for any two edges e 1 , e 2 ∈ E with common ancestor e 1 ∧ e 2 , we have that This previous notion is useful when one tries to prove the super-criticality of a correlated percolation.
Proposition 10. Consider an edge-percolation (not necessarily independent), such that edges at generation 1 are open almost surely and, for e 1 ∈ E with |e 1 | > 1, where e 0 ∼ e 1 and e 0 < e 1 . If RT (T , ψ) < 1 then C(̺) is finite almost surely. If the percolation is quasi-independent and if RT (T , ψ) > 1 then C(̺) is infinite with positive probability.
The proof of Proposition 10 above is postponed in Appendix A.
Let us first apply this to a particular percolation in order to obtain a sufficient criterion for subcriticality.
Corollary 11. Let T be a tree with branching ruin number br r (T ) = b ∈ [0, ∞]. Fix a parameter δ > 0 and perform a percolation (not necessarily independent) on T such that (4.2) holds and assume moreover that ψ(e) = 1 − δ|e| −1 as soon as |e| > n 0 , for some integer n 0 > 1. If δ > b then the percolation is subcritical.
Next, we use Proposition 10 and Corollary 11 to prove the following result.
Proposition 12. Let T be a tree with branching ruin number br Fix a parameter δ > 0 and perform a quasi-independent percolation on T such that (4.2) holds and assume moreover that ψ(e) ≥ 1 − δ|e| −1 as soon as |e| > n 0 , for some integer n 0 > 1. Let T δ be the connected cluster containing the root ̺. We have that (1) if δ < b then T δ is infinite with positive probability; (2) for any δ ∈ (0, b) we have that, with positive probability, br r (T δ ) ≥ b−2δ.
Next, we turn to the proof of (2). Consider the previous percolation, with δ < b and fix p < b − δ.
On the event {T δ is infinite}, which has positive probability, we perform an independent percolation on T δ for which an edge e stays open with probability (1 − p|e| −1 ). We proved that if p < br r (T δ ) then the percolation is supercritical and if p > br r (T δ ) then it is subcritical. We denote T ′ δ+p the resulting cluster of the root. On the other hand, performing this percolation on T δ is equivalent to performing a quasi-independent percolation on the whole tree T where an edge e stays open with probability ψ(e)(1 − p|e| −1 ). As p+δ is infinite with positive probability. This implies that, on the event {T δ is infinite}, the cluster T ′ δ+p is infinite with positive probability. Therefore, by Corollary 11, br r (T δ ) ≥ p with positive probability. As this holds for any p < b − δ, we obtain the conclusion.

Proof of Proposition 3 and Theorem 1
First, note that Theorem 1 is a straightforward consequence of Proposition 3 and Proposition 4. Therefore, it remains to prove Proposition 3.

5.1.
Transience: proof of the first item of Proposition 3. In this section, we will prove that RT (T , ψ RC ) > 1, where we recall that this quantity is defined in (3.1) and ψ RC is defined in (3.2). In particular, we can rewrite Besides, notice that ψ(e) represents the probability that a one-dimensional random walk on the conductances (C e ) e∈E , restricted to the ray connecting ̺ to e + and started at e − , hits e + before returning to ̺.
Proposition 13. For any p ∈ N, and for any τ > 0, there exists a positive finite constant K p,τ such that , for all n ∈ N.
Proof. Recall that for any non-negative random variable Z we have, for a > 1, For any b > 0 we have that any slowly varying function L(u) is o(u b ), as u → ∞. Hence, for any τ > 0, there exists a constant K τ , i 0 > 0 depending only on L and τ , such that, for i ≥ i 0 , For simplicity we drop τ from the notation, and use , that is there exists K a > 0 depending only on L, a and τ such that for all i ∈ N. In order to prove the proposition, we proceed by double induction. First we prove that (5.2) holds for p = 1 and all n ∈ N. In fact, for m > 0, we have (5.4) Note that, in the previous inequality, we use that P[C e ≥ 1] > 0 for any e ∈ E, so that the conditional probability on the left-hand side is well-defined. Assume that (5.2) holds for all p ≤ β − 1 and for all n ∈ N. Notice that (5.2) is trivially true for n = 1 and p = β. Suppose it is true for all n ≤ N and for p = β. To simplify the notation, set η = (1+τ ) 2 m ∨ 1. Next we prove the result for N + 1. We can suppose that K β is larger than In the step before the last one, we used independence between C N +1 and (C i ) i≤N . As we can choose K β to be larger than (5.5), we have It remains to prove that the right-hand side of (5.7) is less than K β (N + 1) βη .
Notice that the right-hand side of (5.7) equals where we used (1 − x) a ≤ 1 − x for all x ∈ (0, 1) and a > 1.

Corollary 14.
For any ε ∈ (0, 1), any t > 0, there exist C ε,t > 0 such that, for any e ∈ E, we have that Proof. Using Proposition 13 and Markov's inequality gives that, for any p ∈ N, This gives the conclusion by choosing p = ⌈t/ε⌉ and by noting that 1 ∨ (1+ε) 2 m + ε ≤ 1 ∨ 1 m + m+3 m ε Next, we will define a quasi-independent percolation on the tree T . Let us fix ε ∈ (0, 1 ∧ b) small enough, such that the following conditions are satisfied Let us define the percolation such that, for e ∈ E with |e| = 1, e is open almost surely and if |e| > 1 then (5.11) {e is open} : We will denote by T C the cluster of open edges containing the root. Let us define the function ψ C on edges such that ψ C (e) = 1 if |e| = 1 and, if |e| > 1 and e 0 is the parent of e, that is the unique edge such that e + 0 = e − , then (5.12) ψ C (e) := P (e ∈ T C | e 0 ∈ T C ) .
Proof. Let us prove that there exists a constant p 0 > 0 such that, for any e ∈ E, Indeed, the conditioning in the above expression is equivalent to picking a sequence of independent conductances (C j ) j≥1 under a measure P such that C j is picked under the conditioned law P(·|C −1 j ≤ j 1+ε m ), and looking at the events corresponding to the second event on the right hand side of (5.11), that is By Corollary 14 (applied with t = 2 for instance) and Borel-Cantelli Lemma, there exists k ∈ N (deterministic) such that P (∩ n≥k A n ) > 0. Now, if one replaces C j byC j = max(C j , 1) for 1 ≤ j ≤ k, and letÃ n be the the same event as A n but where C j is replaced byC j , thenÃ 1 , . . . ,Ã k always happen and P ∩ n≥1Ãn ≥ P (∩ n≥k A n ) > 0. Finally, we can choose which proves the claim (5.13). Let us prove that the percolation is quasi-independent. Let e 1 , e 2 ∈ E and let e be their common ancestor with highest generation. We have that P e 1 , e 2 ∈ T C e ∈ T C = P (e 1 , e 2 ∈ T C ) P (e ∈ T C ) = e<g≤e 1 or e<g≤e 2 where the first equality simply uses the definition of conditional probability, the second uses (5.13) and bounds the probability in the numerator by 1, the third is a simple re-writing, the fourth uses again (5.13) and bounds the probability in the denominator by 1 and, finally, the fifth one is just using the definition of conditional probability. This proves that the percolation is quasi-independent. Let e be a generic edge with |e| > 1, and denote by e 0 its parent. Using (5.13), (5.11) and again Corollary 14, we have that, there exists c 0 > 0 such that Therefore, there exists n 0 > 1 such that, for any e ∈ E with |e| > n 0 , we have that By Proposition 12, as the percolation defined by (5.11) is quasi-independent and ε < b, we have that br r (T C ) ≥ b − ε with positive probability.
Let us consider different cases and prove that RT (T , ψ RC ) > 1, where we refer to (5.1) for a definition of this quantity.
Proof. Recall the percolation T C defined in (5.11). Let us denote Π C the set of all the cutsets in T C . By Proposition 15, we have that br r (T C ) ≥ b − ε with positive P-probability. On this event, we have that inf π∈Π e∈π where we used (5.9). This implies that RT (T , ψ RC ) > 1 with positive Pprobability, as defined in (5.1).
Proof. Recall the percolation T C defined in (5.11). By Proposition 15, we have that br r (T C ) ≥ b − ε with positive probability. Let us denote Π C the set of all the cutsets in T C . On this event, we have that, if b > 1, inf π∈Π e∈π where we used (5.10). This implies that RT (T , ψ RC ) > 1 with positive Pprobability, as defined in (5.1).

5.2.
Recurrence: proof of the second item of Proposition 3. We will again consider different cases and prove this time that RT (T , ψ RC ) < 1, where we refer to (5.1) for a definition of this quantity.
Proof. Fix two positive parameters δ and ε such that (1/m) − δ > 0 and The latter is possible as mb < 1.
We have that By the definition of branching-ruin number, there exists a sequence of cutsets (π n , n ≥ 1) such that for any n > 0, On the other hand, for any n > 0 we have, Note that there exists n 0 such that for any n > n 0 , we have, In virtue of the first Borel Cantelli Lemma, all edges e ∈ n≥1 π n , with the exception of finitely many, satisty Hence, for n large enough where we used (5.19). Hence, The next result concludes the proof of Theorem 1.
Proof. First, fix δ ∈ (0, 1) such that The latter is possible as b < 1. Then, note that, for any ε ∈ (0, 1), there exists η > 0 such that In the following, we denote (C j ) j≥0 a sequence conductances distributed like a generic conductance C e . There exists a constant c δ,ε > 0 such that, for any e ∈ E, Indeed, to prove the first inequality above, note that By the definition of branching-ruin number, there exists a sequence of cutsets (π n , n ≥ 1) such that for any n > 0, We use (5.29) and (5.31) to obtain P e∈πn g≤e Therefore, by Borel-Cantelli Lemma, as soon as n is large enough, we have that where we used (5.27). Hence, following a strategy similar to (5.25), (5.26), we have that RT (T , ψ RC ) ≤ 1 − δ, P-almost surely.

Proof of Theorem 5 and Lemma 8
In this section, we prove Lemma 8. With this in hand, Theorem 2 and Theorem 5 will then trivially follow from Theorem 7 (proved in Section 8) by noting that (3.4) is satisfied when m ν = M ∈ N for all ν ∈ V \ {̺}. For any e ∈ E, we define (6.1) Ψ M,λ (e) := g≤e ψ M,λ (g).
As we will see in Section 7, Ψ M,λ (e) corresponds to the probability that X, or X, when restricted to [̺, e + ] and started from ̺, hits e + before returning to ̺.
We have that there exists δ > 0 such that Therefore, by (6.3), we have that Therefore, we have that RT (T , ψ M,λ ) < 1.

Extensions
Here, we define the same construction as in [7] and [8], which is a particular case of Rubin's construction. A large part of this section is a verbatim of Section 5 of [8].
The following construction will allow us to emphasize useful independence properties of the walk on disjoint subsets of the tree.

Remark 20.
Recall that a Gamma random variable with parameters m µ + 1 and 1 has the same distribution as the sum of m µ +1 i.i.d. exponential random variables with mean 1.
Below, we use these collections of random variables to generate the steps of X. Moreover, we define a family of coupled walks using the same collection of 'clocks' Y.
Define, for any ν, µ ∈ V with ν ∼ µ, the quantities We are now going to define a family of coupled processes on the subtrees of T . For any rooted subtree T ′ of T , we define the extension X (T ′ ) = (V ′ , E ′ ) on T ′ as follows. Let the root ̺ ′ of T ′ be defined as the vertex of V ′ with smallest distance to ̺. For a collection of nonnegative integersk = (k µ ) µ:[ν,µ]∈E ′ , let Note that the event A (T ′ ) k,n,ν deals with jumps along oriented edges. Set X (T ′ ) 0 = ̺ ′ and, for ν, ν ′ such that [ν, ν ′ ] ∈ E ′ and for n ≥ 0, on the event where the function r is defined in (7.2) and the clocks Y 's are from the same collection Y fixed in (7.1).
Thus, this defines X (T ) as the extension on the whole tree. It is easy to check, from properties of independent exponential and Gamma random variables, the memoryless property and Remark 20, that this provides a construction of X on the tree T . This continuous-time embedding is classical: it is called Rubin's construction, after Herman Rubin (see the Appendix in [9]). Now, if we consider proper subtrees T ′ of T , one can check that, with these definitions, the steps of X on the subtree T ′ are given by the steps of X (T ′ ) (see [7] for details). As it was noticed in [7], for two subtrees T ′ and T ′′ whose edge sets are disjoint, the extensions X (T ′ ) and X (T ′′ ) are independent as they are defined by two disjoint sub-collections of Y.
Of particular interest will be the case where T ′ = [̺, ν] is the unique selfavoiding path connecting ̺ to ν, for some ν ∈ T . In this case, we write X (ν) instead of X ([̺,ν]) , and we denote T (ν) (·) the return times associated to X (ν) . For simplicity, we will also write X (e) and T (e) (·) instead of X (e + ) and T (e + ) (·) for e ∈ E. Finally, it should be noted that, for any e ∈ E and any g ≤ e, Ψ M,λ (e) = P T (e) (e + ) < T (e) (̺) , (7.5) where θ is the canonical shift on the trajectories.
Remark 21. Note that, for any vertex ν, only the clocks Y (ν, µ, 0) with µ ∼ ν, ν < µ, have a particular law. They follow a Gamma distribution instead of following an Exponential distribution. This resembles what would happen for a once-reinforced random walk (see [8]). In this case, these clocks would still have an Exponential distribution but with a different parameter than the other ones (related to the reinforcement). This means that an M-DRW λ is, in nature, very close to a once-reinforced random walk.

Proof of Theorem 7
In this section, we follow the blueprint of Section 7 of [8]. In order to prove transience, the idea is to interpret the set of edges crossed before returning to ̺ as the open edges in a certain correlated percolation. A key step is to prove that this correlated percolation is quasi-independent, which will allow us to conclude its super-criticality from the super-criticality of some independent percolation. Note that we will prove the transience of X which is equivalent to the transience of X.
8.1. Link with percolation. Denote by C(̺) the set of edges which are crossed by X before returning to ̺, that is: This set can be seen as the cluster containing ̺ in some correlated percolation. Next, we consider a different correlated percolation which will be more convenient to us. Recall Rubin's construction and the extensions introduced in Section 7. We define: Proof. We can follow line by line the proof of Lemma 11 in [8], except that one should replace X by X. For simplicity, for a vertex v ∈ V , we write v ∈ C CP (̺) if one of the edges incident to v is in C CP (̺). Besides, recall that for two edges e 1 and e 2 , their common ancestor with highest generation is the vertex denoted e 1 ∧ e 2 .
Lemma 24. Assume that the condition (3.4) holds with some constant M. Then the correlated percolation induced by C CP is quasi-independent, as defined in Definition 9.
Proof. Here, we need to adapt the argument from the proof of Lemma 12 in [8]. Y (e + , e − , j) r(e + , e − ) , where |A| denotes the cardinality of a set A and θ is the canonical shift on trajectories. Note that L(e) is the time consumed by the clocks attached to the oriented edge (e + , e − ) before X (e) , X (e 1 ) or X (e 2 ) goes back to ̺ once it has reached e + . Recall that these three extensions are coupled and thus the time L(e) is the same for the three of them. For i ∈ {1, 2}, let v i be the vertex which is the offspring of e + lying the path from ̺ to e i . Note that v i could be equal to e + i . We define for i ∈ {1, 2}: Y (e + , e − , j) r(e + , e − ) .
Appendix A. Proof of Proposition 10 As above, we define a function Ψ on the set of edges such that, for e ∈ E, Taking the infimum over π ∈ Π allows to conclude that: A.2. Proof of Proposition 10 in the case RT (T , ψ) > 1. As we are considering a quasi-independent percolation, we are able to lower-bound the probability of this correlated percolation to be infinite by the probability that some independent percolation is infinite. We do this by proving that a certain modified effective conductance is positive. Define C eff the effective conductance of T when the conductance c(e) is assigned to every edge e ∈ E. For a definition of effective conductance, see [21] page 27.
Proposition 27. Let C(̺) be the cluster of the root in a percolation such that (4.2) holds. If the percolation is quasi-independent, then there exists C Q ∈ (0, ∞) such that 1 C Q × C eff 1 + C eff ≤ P(|C(̺)| = ∞).
Proof of Proposition 27. We can use the lower-bound in Theorem 5.19 (page 145) of [21] to obtain the result.
Recall that a flow (θ e ) on a tree is a nonnegative function on E such that, for any e ∈ E, θ e = g∈E:g − =e + θ g . A flow is said to be a unit flow if moreover e:|e|=1 θ e = 1. A usual technique in order to prove that some effective conductance is positive is to find a unit flow with finite energy. This is the content of the following statement, which is a simple consequence of classical results.
Lemma 28. Assume that (3.4) is satisfied. Consider the tree T with the conductances defined in Definition 26 and assume that there exists a unit flow (θ e ) e∈E on T from ̺ to infinity which has a finite energy, that is e∈E (θ e ) 2 c(e) < ∞.
Then, a quasi-independent percolation such that (4.2) holds is supercritical.
Proof. Using Proposition 27, if C eff > 0 then a quasi-independent percolation such that (4.2) holds is supercritical. By Theorem 2.11 (page 39) of [21], C eff > 0 if and only if there exists a unit flow (θ e ) e∈E on T from ̺ to infinity which has a finite energy.
The following result, from [8], is inspired by Corollary 4.2 of R. Lyons [19], which is itself a consequence of the max-flow min-cut Theorem. This result will provide us with a sufficient condition for the existence of a unit flow with finite energy. The proof is ended once we have proved the following proposition.
On one hand, we have that, for any e ∈ E, g≤e u g ≤ C γ . (A.7) Indeed, for each e ∈ E, we can apply Proposition 17 of [8] to functions f e defined by f e (0) = 1 and, for n ≥ 1, f e (n) = 1 − ψ(g) with g the unique edge such that g ≤ e and |g| = n ∧ |e|. We emphasize that (A.7) holds with a uniform bound. On the other hand, using (A.5), we have Therefore, there exists a unit flow with finite energy and Lemma 28 implies the result.