The most visited sites of biased random walks on trees

We consider the slow movement of randomly biased random walk $(X_n)$ on a supercritical Galton--Watson tree, and are interested in the sites on the tree that are most visited by the biased random walk. Our main result implies tightness of the distributions of the most visited sites under the annealed measure. This is in contrast with the one-dimensional case, and provides, to the best of our knowledge, the first non-trivial example of null recurrent random walk whose most visited sites are not transient, a question originally raised by Erd\H{o}s and R\'ev\'esz [11] for simple symmetric random walk on the line.

There is an active literature on randomly biased walks on Galton-Watson trees; see, for example, a large list of references in [17].In this paper, we restrict our attention to a regime of slow movement of the walk in the recurrent case.
Clearly, the movement of the biased random walk (X n ) is determined by the law of the random environment ω.We assume that (ω(x, y), y ∼ x) for x ∈ T, are i.i.d.random vectors.It is convenient to view (ω, T) as a marked tree (in the sense of Neveu [23]).
The influence of the random environment is quantified by means of the random potential process (V (x), x ∈ T), defined by V (∅) := 0 and ] denoting the set of vertices (including x and ∅) on the unique shortest path connecting ∅ to x.There exists an obvious bijection between the random environment ω and the random potential V .
For any x ∈ T, let |x| denote its generation.Throughout the paper, we assume We also assume that the following integrability condition is fulfilled: there exists δ > 0 such that The random potential (V (x), x ∈ T) is a branching random walk as in Biggins [6]; as such, (1.2) corresponds to the "boundary case" (Biggins and Kyprianou [9]).It is known that, under some additional integrability assumptions that are weaker than (1.3), the branching random walk in the boundary case possesses some deep universality properties, see [25] for references.
For any vertex x ∈ T, let us define which is the (site) local time of the biased walk at x. Consider, for any n ≥ 1, the non-empty random set (1.6) In words, A n is the set of the most visited sites (or: favourite sites) at time n.The study of favourite sites was initiated by Erdős and Révész [11] for the symmetric Bernoulli random walk on the line (see a list of ten open problems presented in Chapter 11 of the book of Révész [24]).In particular, for the symmetric Bernoulli random walk on Z, Erdős and Révész [11] conjectured: (a) tightness for the family of most visited sites, and (b) the cardinality of the set of most visited sites being eventually bounded by 2.
Conjecture (b) was partially proved by Tóth [27], and is believed to be true by many.On the other hand, Conjecture (a) was disproved by Bass and Griffin [5]: as a matter of fact, inf{|x|, x ∈ A n } → ∞ almost surely for the one-dimensional Bernoulli walk.Later, we proved in [16] that it was also the case for Sinai's one-dimensional random walk in random environment.The present paper is devoted to studying both questions for biased walks on trees; our answer is as follows.
Corollary 2.2.Assume (1.2) and (1.3).There exists a finite non-empty set U min , defined in (2.5) and depending only on the environment, such that In particular, the family of most visited sites is tight under P.
So, concerning the tightness question for most visited sites, biased walks on trees behave very differently from recurrent one-dimensional nearest-neighbour random walks (whether the environment is random or deterministic).To the best of our knowledge, this is the first non-trivial example of null recurrent Markov chain whose most visited sites are tight.
In the next section, we give a precise statement of the main result of this paper, Theorem 2.1.
The main result of the paper is as follows.
Our results are not as strong as they might look like.For example, Theorem 2.1 does not claim that [27] proved for the symmetric Bernoulli random walk on Z: for example, it does not claim that P * -a.s., A n ⊂ U min for all sufficiently large n; we even do not know whether this is true.

is much weaker than what Tóth
For local time at fixed site of biased random walks on Galton-Watson trees in other recurrent regimes, see the recent paper [15].
An important ingredient in the proof of Theorem 2.1 is the following estimate on the local time of vertices that are away from the root: Proposition 2.3 is given in Section 3. Theorem 2.1 and Corollary 2.2 are proved in Section 4.
Throughout the paper, for any pair of vertices x and y, we write x < y or y > x if y is a (strict) descendant of x, and x ≤ y or y ≥ x if either y is either a (strict) descendant of x, or x itself.For any x ∈ T, we use x i (for 0 ≤ i ≤ |x|) to denote the ancestor of x in the i-th generation; in particular, x 0 = ∅ and x |x| = x.

Proof of Proposition 2.3
We start with some preliminaries.Define x) .Let S i − S i−1 , i ≥ 1, be i.i.d.random variables whose law is characterized by for any Borel function h : R → R + .
The following fact, quoted from [18], is a variant of the so-called "many-to-one formula" for the branching random walk.Fact 3.1.Assume (1.2) and (1.3).Let Λ(x) be as in (3.1).For any n ≥ 1 and any Borel function g : R n+1 → R + , we have
We now proceed to the proof of Proposition 2.3.Define In words, T x is the first hitting time at x by the biased walk, whereas T + ∅ is the first return time to the root ∅.
Let x ∈ T\{∅}.The probability P ω (T x < T + ∅ ) only involves a one-dimensional random walk in random environment (namely, the restriction at [[∅, x[[ of the biased walk (X i )), so a standard result for one-dimensional random walks in random environment (Golosov [14]) tells us that where x 1 is the ancestor of x in the first generation.
The rest of the section is devoted to the proof of (3.10).By (1.5), 1 (log n) 3 max 0≤i≤n |X i | converges P * -a.s. to a positive constant, and since n log n converges in P * -probability to a positive limit, we deduce that converges in P * -probability to a positive limit.So the proof of (3.10) is reduced to showing the following estimate: for some 1 < γ < 2, any c > 0 and any 0 < ε < 1, For k ≥ 1, we have z) .
It remains to prove Lemma 3.5.
Proof of Lemma 3.5.Since k i=1 e S i ≤ k S k , it suffices to check that Recall the law of S 1 from (3.2).By assumption (1.3) and Hölder's inequality, we have where δ > 0 is the constant in (1.3).In particular, E(e a|S 1 | ) < ∞ for all 0 ≤ a < δ.Since 0 < δ 2 < δ, we have E(e δ 2 (S k −S k ) ) ≤ e c 4 k for some constant c 4 > 0 and all k ≥ 1.So We make a change of indices Then ( S ℓ , ℓ ≥ 0) is a random walk having the law of (S ℓ , ℓ ≥ 0), and is independent of On the other hand, P(S ⌊(log n) 1/2 ⌋ ≥ −α) ≤ c 5 (log n) −1/4 for some constant c 5 > 0 and all n ≥ 2 (see Kozlov [19]); it suffices to prove that This will be a straightforward consequence of the following estimate (applied to λ := log n and b := δ 2 ; it is here we use the condition δ 2 < 1 16 ): for any 0 < b < δ, where τ λ := inf{i ≥ 1 : S i − S i > λ}.
To prove (3.15), we define the (strictly) ascending ladder times (H i , i ≥ 0): H 0 := 0 and for any i ≥ 1, We apply the strong Markov property, first at time H i−1 to see that and then successively at times We define σ −λ := inf{n ≥ 0 : for some constant c 6 > 0 and all sufficiently large λ, say λ ≥ λ 0 (for the last elementary inequality, see for example, Lemma A.1 in [17]).Thus we get that Finally, for all small b > 0, there exists some positive constant 4D ∞ e −U (x) > ε → 0, in P * -probability .