Spread of visited sites of a random walk along the generations of a branching process

In this paper we consider a null recurrent random walk in random environment on a super-critical Galton-Watson tree. We consider the case where the log-Laplace transform $\psi$ of the branching process satisfies $\psi(1)=\psi'(1)=0$ for which G. Faraud, Y. Hu and Z. Shi in \cite{HuShi10b} show that, with probability one, the largest generation visited by the walk, until the instant $n$, is of the order of $(\log n)^3$. In \cite{AndreolettiDebs1} we prove that the largest generation entirely visited behaves almost surely like $\log n$ up to a constant. Here we study how the walk visits the generations $\ell=(\log n)^{1+ \zeta}$, with $0<\zeta<2$. We obtain results in probability giving the asymptotic logarithmic behavior of the number of visited sites at a given generation. We prove that there is a phase transition at generation $(\log n)^2$ for the mean of visited sites until $n$ returns to the root. Also we show that the visited sites spread all over the tree until generation $\ell$.


Introduction
We start giving an iterative construction of the environment. Let (A i , i ≥ 1) a positive random sequence and N an independent N-valued random variable following a distribution q, in other words P(N = i) = q i for i ∈ N. Let φ the root of the tree and (A(φ i ), i ≤ N φ )) an independent copy of (A i , i ≤ N ). Then, we draw N φ children to φ: these individuals are the first generation. Each child φ i is associated with the corresponding A(φ i ) and so on. At the n-th generation, for each individual x we pick (A(x i ), i ≤ N x ) an independent copy of (A i , i ≤ N ) where N x is the number of children of x and A(x i ) is the random variable attached to x i . The set T, consisting of the root and its descendants, forms a Galton-Watson tree (GW) of offspring distribution q and where each vertex x = φ is associated with a random variable A(x). We denote by |x| the generation of x, ← x the parent of x, and for convenience reasons we add ← φ, the parent of φ. The set of environments denoted by E is the set of all sequences ((A(x i ), i ≤ N x ), x ∈ T), with P and E respectively the associated probability measure and expectation.
We assume that the distribution of (A i , i ≤ N ) is non-degenerate and, to obtain a supercritical GW, that E[N ] > 1. Moreover we add uniform ellipticity conditions ∃ 0 < ε 0 < 1, P − a.s ∀i, ε 0 ≤ A i ≤ 1/ε 0 , (1.1) ∃ N 0 ∈ N, P − a.s N ≤ N 0 . (1.2) Given E ∈ E, we define a T-valued random walk (X n ) n∈N starting from φ by its transition probabilities, Note that our construction implies that (p(x, .), x ∈ T) is an independent sequence. We denote by P E the probability measure associated to this walk, the whole system is described under the probability P, the semi-direct product of P and P E .
To study asymptotical behaviours associated to (X n ) n∈N , a quantity appears naturally: the potential process V associated to the environment which is actually a branching random walk. It is defined by V (φ) := 0 and where φ, x is the set of vertices on the shortest path connecting φ to x and φ, x = φ, x \{φ}. We put ourself in the non lattice case so log A i can not be written as b + cZ, and introduce the moment-generating function characterizing the environment. Note that the hypothesis we discuss above implies that ψ is defined on R, and ψ(0) > 0. In fact the hypothesis (1.1) and (1.2) are not always needed for our work and they could be replaced by the existence of ψ in (−δ, 1 + δ) with δ > 0 together with the existence of a moment larger than 1 for N . In Section 2 for example we could lighten the hypothesis this way, but it would be much more complicated in Section 4.
Thanks to the work of M.V. Menshikov and D. Petritis, see [15] and the first part of [8] by G. Faraud, if then X is null recurrent, with ψ ′ (1) = −E |x|=1 V (x)e −V (x) . In [9] (see also [12]), G. Faraud, Y. Hu and Z. Shi study the asymptotic behavior of max 0≤i≤n |X i | = X * n , i.e. the largest generation visited by the walk. Assuming (1.3), they prove the existence of a positive constant a 0 (explicitely known) such that P a.s. on the set of non-extinction of the GW lim n→+∞ X * n (log n) 3 = a 0 . (1.4) In [3] we were interested in the largest generation entirely visited by the walk, that is to say the behavior of R n := sup{k ≥ 1, ∀|z| = k, L(z, n) ≥ 1}, with L the local time of X defined by L(z, n) := n k=1 1 X k =z . More precisely, if (1.3) is realized, P a.s. on the set of non-extinction Although in [3] all recurrent cases are treated, here we focus only on the hypothesis (1.3). According to (1.4) and (1.5), until generation log n /γ all the points are visited but X does not visit generations further than a 0 (log n) 3 . The aim of this paper is to study the asymptotic of the number of visited sites at a given generation (log n) 1+ζ with 0 < ζ < 2. For this purpose we define the number of visited sites at generation m ∈ N until the instant n M n (m) := #{|z| = m, L(z, n) ≥ 1}, and before n returns to the root K n (. x , X k = x} for n ≥ 1 and T 0 x = 0 for x ∈ T. Let Z m the number of descendants at generation m ∈ N, we have Z 1 = N . Our first results quantify the number of visited points at a given generation ℓ := (log n) 1+ζ . Thanks to the hypothesis of ellipticity, ψ can be written as a power series in particular, for any x small enough, , these are called cumulants and here u 1 := ψ ′ (1) = 0, u 2 := ψ ′′ (1) = σ 2 . Let us define the function f , for any x small enough λ is the Cramér's series depending on the cumulants of ψ(1 − x) (for more details on the Cramér's series see for example [17] p. 219-223).
Also for all n large enough, there exist two positive constants C 1 and C 2 such that (1.6) shows that, at each generation ℓ, the cardinal of visited sites is at least n ψ(0)(1−ε)/γ for any ζ, that is to say like the last generation entirely visited R n (ψ(0)/γ < 1, by convexity of ψ and the fact that ψ(1) = 0). Also the upper bound of M n (ℓ) is at most of the order of ne −C 3 (log n) 1−ζ /(log n) C 4 , with C 3 , C 4 > 0. This suggests that it may have a phase transition when ζ = 1. Although we are not able to show this for M n (ℓ) the existence of a phase transition is proved in (1.7) for the mean of K n (ℓ). Indeed by definition of f , We can see that in the neighborhood of generation (log n) 2 that is to say when ζ = 1, the asymptotic behavior of N ζ := E[K n (ℓ)] changes. We easily check that for all 0 < ζ < ζ ′ ≤ 1, lim n→+∞ N ζ ′ /N ζ = +∞ whereas for all 1 ≤ ζ < ζ ′ < 2, lim n→+∞ N ζ ′ /N ζ = 0. So the generations of order (log n) 2 are, in mean, the most visited generation (in term of distinct site visited) until n returns to the origin. Finally notice that when ζ > 1/2 we are in a Gaussian behavior as In order to establish our second result, recall Neveu's notation to introduce a partial order on our tree. In [16], to each vertex x at generation m ∈ N, Neveu associates a sequence x 1 . . . x m where x i ∈ N, to simplify we write x = x 1 . . . x m . This sequence gives the complete "genealogy" of x: if y = x 1 . . . x i with |y| = i < m, y is the unique ancestor of x at generation i and we write y < x.
To extend this partial order for |x| = |z|, we write x < z if there exists i < m such that x k = z k for k < i and x i < z i . Hence we can number individuals at a given generation "from the left to the right" and for A a subset of {z ∈ T, |z| = m}, inf A and sup A are respectively the minimum and maximum associated to this numbering. Our last result gives an idea of the way the visited points spread on the tree, for this purpose we introduce clusters: let z ∈ T and m ≥ |z|, we call cluster issued from z at generation m denoted C m (z), the set of descendants u of z such that |u| = m, in other words (1.8) At some point we need to quantify the number of individuals between two disjoint clusters with common generations. For given initial and terminal generations, denote C a set of disjoint clusters. Let (D j , 1 ≤ j ≤ |C |), with |C | the cardinal of C , an ordered sequence of (disjoint) clusters belonging to C , that is to say for all j, sup D j < inf D j+1 . We define the minimal distance between clusters in the following way D(C ) := min 1≤j≤|C |−2 (inf D j+2 − sup D j ), where, by definition, inf D j+2 − sup D j is the number of individuals between sup D j and inf D j+2 . Notice that we do not look at two successive clusters, but two successive separate by one. We now state a second result Let k n , h n and r n positive sequences of integers such that k n r n + (k n − 1)h n = ℓ. For all 1 ≤ i ≤ k n , let us denote C i a set of clusters initiated at generation (i − 1)(r n + h n ) and with end points at generation ir n + (i − 1)h n (see Figure 3), also define the following event for all m > 0 and q > 0 There exist 0 < k < 1∧ζ, 0 < r < 1 with 0 < k+r ≤ 1 and for k n = (log n) k , r n = (log n) r lim n→+∞ P kn i=2 A i (e ψ(0)hn /2 , e ψ(0)rn(i−1)/2 ) = 1.
(1.11) (1.9) implies the existence of a cluster starting at a generation ℓ − log n/γ completely visited (see Figure 1). As conditionnaly on the tree until generation |z|, |C ℓ (z)| is equal in law to Z ℓ−|z| = Z ψ(0) log n/γ , this cluster is large and, in particular, (1.9) implies the lower bound in (1.6).
(1.10) tells that we can find visited individuals at generation ℓ = (log n) 1+ζ , with a common ancestor to a generation close to the root, that is to say before generation εℓ 1/3 (see Figure  4). Thus, with a probability close to one, at least e ε(1−ε)ψ(0)ℓ 1/3 /2 individuals of generation ℓ separate by at least e ψ(0)ℓ/2 individuals of the same generation ℓ, are visited. Finally (1.11) tells that if we make cuts regularly on the tree we can find many visited clusters (which number increases with the generation) well separated. In particular these visited clusters can not be in a same large visited clusters as they are separated by at least e ψ(0)hn/2 ∼ e ψ(0)(log n) 1+ζ−k /2 > n individuals (see also Figure 3).
To obtain these results we show that K n (ℓ) can be linked to a random variable depending only on the random environment and n. For all z ∈ T, all integer k and all real a, we define the random variable . We obtain the following Proposition 1.3 Let ε > 0 and Φ a sequence such that (1.12) (1.14) We use the notation a n ≍ b n when there exists two positive constants c 1 and c 2 such that c 1 b n ≤ a n ≤ c 2 b n for all n large enough. The lack of precision for the first result shows no difference between R log n (ℓ) and M n (ℓ) (see (1.6)), unlike between the means of R log n (ℓ) and K n (ℓ).
The rest of the paper is organized as follow: in Section 2 we study R Φ(n) (ℓ) and prove Proposition 1.3. In Section 3 we link R Φ(n) (ℓ) and M n (ℓ), which leads to Theorem 1.1 and (1.9) of Theorem 1.2. In Section 4 we prove the end of Theorem 1.2. Also we add an appendix where we state known results on branching processes and local limit theorems for sums of i.i.d. random variables. Note that for typographical simplicity, we do not distinguish a real number and its integer part throughout the article.

Expectation and bounds of R Φ(n) (ℓ)
In this section we only work with the environment more especially with what we call number of accessible points R Φ(n) (ℓ).
2.1 Expectation of R Φ(n) (ℓ) (proof of (1.14)) According to Biggins-Kyprianou identity (also called many-to-one formula, see part A of where S j is a centered random walk, we only have to prove Writing S as a sum of i.i.d. random variables, we easily see that (S i ) 0≤i≤j and (S i ) 0≤i≤j have the same law. Then, conditioning on σ{S k , k ≤ j} We now need to distinguish the cases 0 < ζ < 1 and 1 ≤ ζ < 2. When 0 < ζ < 1, D 1 j = 0 as k > Aℓ 1/2 . Also using Lemma B.6 for A large enough For the upper bound, we can assume without loss of generality that ε is small enough to ensure that for |x| ≤ ε, λ(x) converges and f ′ (x) is negative. Therefore, the derivative of F defined by F (x) : and finally Note that for s > 0 small enough, ψ(1 − s) ≤ s/2. So the exponential Markov inequality applied to P (sS j > sk) and the identity E[e sS j ] = e jψ(1−s) yield (2.4) follows. When 1 ≤ ζ < 2, we prove that the main contribution comes from D 1 j . As for any n large enough, Φ(n) ≤ Aℓ 1/2 for some A > 0, for any j ≥ (Φ(n)/A) 2 using Lemma B.6

Bounds for log
The upper bound is a direct consequence of Markov inequality and (1.14).
For the lower bound, we first need an estimation on the deviation of min |z|=m V (m), this topic has been studied in details in [9], Proposition 2.2 Let a n a positive sequence such that a n ∼ n 1/3 , there exists b 0 > 0 such that for any A useful consequence of the above Proposition is the following Lemma 2.3 Assume that a n is a positive increasing sequence such that a n ∼ n 1/3 , there exists a constant µ > 0 such that for any n large enough where λ n = e −c 1 e c 1 an if q 0 + q 1 = 0, and λ n = e −c 1 an otherwise, also c 1 > 0 depends only on the distribution P .
Let us denote z 1 , . . . , z k , . . . the ordered points at generation ℓ − w n satisfying V (z i ) ≤ y n . Conditionally on {R yn (ℓ − w n ) = k}, z 1 exists and Furthermore, by stationarity R z 1 Φ(n)−yn (ℓ) and R Φ(n)−yn (w n ) have the same law, so As y n = o(Φ(n)), the first probability tends to one thanks to a result of Mac-Diarmid [14] (see also [3] Lemma 2.1), so does the second one as a consequence of Lemma 2.3.
3 Expectation of K n (ℓ), bounds for log K n (ℓ) and log M n (ℓ) 3.1 Proof of (1.7) We start with general upper and lower bounds for the annealed expectation of K n (ℓ).
Lemma 3.1 For n ∈ N: C − and c − (respectively C + and c + ) are positive constants that may decrease (respectively increase) from line to line. Proof.

Markov property gives
Using successively the fact that c − ( x∈ φ,z e V (x) ) −1 ≤ p z ≤ c + e −V (z) and Biggins-Kyprianou identity (see Appendix A.1) Similar arguments show C − A − n ≤ E |z|=ℓ p z 1 npz<1 ≤ C + A + n . We now give upper bounds for B + n and A + n , and a lower bound for A − n . • For B + n , we use Lemma 2.1 taking Φ(n) = log n.

Proof.
Applying Corollary C.1, Using (1.13), E[R Φ 1 (n) (ℓ)] ≤ e Φ 1 (n) and the proof is achieved. The above Lemma together with (2.7) (taking Φ(n) = Φ 1 (n)), give for n large enough To obtain the lower bound in (1.9) we finally use the following result that can be deduced from [9] (see [3] Lemma 3.2 and what follows for details) For the lower bound in (1.6), we use Lemma 3.1, (3.14) and finally the lower bound in (1.13).

Visited points along the GW
In this paragraph we study the manner the random walk visits the tree.

Thus in both cases
When q 0 + q 1 = 0, the above choices give an even better rate of convergence for P (B). We now move back to (4.2), (4.3) together with the fact that Φ(n) ≤ log n + o(log n) implies According to (3.14), as P(n ≥ T n 1−ε φ ) tends to one we finally obtain So we can find set of clusters at regular cuts on the tree which are fully visited. To finish the proof of (1.11) we first show the existence of a lower bound for the number of visited clusters. Using successively that conditionally on B, |C i | is equal in law to Z (i−1)rn , Theorem A.2 and (4.3) Note that for the first term we have used the left tail of Z . with q 0 + q 1 > 0 as the other case provide an even better rate of convergence. Finally we prove that the previously defined visited clusters are very spaced out. Recalling the definition of D before Theorem 1.2, As conditionally on B, |C i(rn+hn) (z)| and Z hn are equal in law, on ≤ e rn log N 0 +2ψ(0)(i−1)rn P (Z hn ≤ e ψ(0)hn/2 ) + e −ψ(0)(i−1)rn ≤ e rn log N 0 +2ψ(0)(i−1)rn −νψ(0)hn/2 + e −ψ(0)(i−1)rn .
where (S i − S i−1 ) i≥1 are i.i.d. random variables, and the law of S 1 is determined by for any measurable function f : R → [0, +∞). A proof can be found in [5], see also [18].
We have the following identities In

B Results for sums of i.i.d. random variables
In this section we recall basic facts for sum of i.i.d. random variables applied to (S n ) n≥0 of Section A. Recall that for all x ∈ R, τ + x = inf{n ≥ 1, S n ≥ x} and τ − x = inf{n ≥ 1, S n ≤ x}. The following results are standard and can be found in [1] and [19].
Recalling that for all n ≥ 1, Y − (n) = n i=1 e −S i , we have Lemma B.2 There exists a constant C + > 1 such that for all a ≥ 0 and M > 0

Proof.
The upper bound can be found in [13] p.44, the lower bound can be obtained as follows: Recalling that for all n ≥ 1, Y − (n) = n i=1 e −S i , we have Lemma B.4 There exists a constant C + > 1 such that for all a ≥ 0 and M > 0 The following Lemma may be found in the literature, however as we can prove it easily for our case we present a short proof. Strong Markov property and homogeneity give: implying with Lemma B.1: A classical result of moderate deviations (see for instance [17], Chapter VIII, Theorem 1) implies Proof.
(B.4) is F. Caravenna [7] result and (B.5) can be obtained with [17] Chapter VIII, Theorem 2 and 10 and similar arguments than in the proof of Lemma B.5.

C Probability of hitting time
Lemma C.1 For x ′ ∈ φ, x : The result is classical (see for instance [3]) and a useful direct consequence of this latter is the following