P Andreoletti

P Debs

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

Keywords: random walks, random environment, trees, branching random walk

 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 ℓ = (log n) 1+ζ , with 0 < ζ < 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 ℓ.

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, Pa.s ∀i, ε 0 ≤ A i ≤ 1/ε 0 , (1.1) ∃ N 0 ∈ N, Pa.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,

p(x, x i ) = A(x i ) Nx j=1 A(x j ) + 1 , p(x, ← x) = 1 - Nx j=1 p(x, x j ), p( ← φ, φ) = 1.
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

V (x) := - z∈ φ,x log A(z), x ∈ T\{φ},
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

ψ(t) := log E   |x|=1 e -tV (x)   ,
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 [START_REF] Menshikov | On random walks in random environment on trees and their relationship with multiplicative chaos[END_REF] and the first part of [START_REF] Faraud | A central limit theorem for random walk in a random environment on marked galton-watson trees[END_REF] by G. Faraud, if x) . In [START_REF] Faraud | Almost sure convergence for stochastically biased random walks on trees[END_REF] (see also [START_REF] Hu | Slow movement of recurrent random walk in random environment on a regular tree[END_REF]), 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 .

ψ(1) = ψ ′ (1) = 0 (1.3) then X is null recurrent, with ψ ′ (1) = -E |x|=1 V (x)e -V (
(1.4)

In [START_REF] Andreoletti | The number of generations entirely visited for recurrent random walks on random environment[END_REF] 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

lim n→+∞ R n log n = 1 γ , (1.5)
where γ := sup{a ∈ R, J(a) > 0} with J(a) := inf t≥0 {ψ(-t) -at}.

Although in [START_REF] Andreoletti | The number of generations entirely visited for recurrent random walks on random environment[END_REF] 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 (.) := M T n φ (.) where

T n x = inf{k > T n-1
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, ψ(1x) = +∞ j=1 u j x j , where u j = ψ (j) (1), 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 f (x) := 1 - x 2σ 2 + x 2 λ(x).
λ is the Cramér's series depending on the cumulants of ψ(1x) (for more details on the Cramér's series see for example [START_REF] Petrov | Sums of Independent Random Variables[END_REF] p. 219-223).

Theorem 1.1 For all 0 < ζ < 2, ε > 0 independent of ζ there exists C 0 > 0 such that

lim n→+∞ P ψ(0) γ (1 -ε) ≤ log M n (ℓ) log n ≤ 1 -C 0 log log n log n ∨ 1 (log n) ζ = 1. (1.6)
Also for all n large enough, there exist two positive constants C 1 and C 2 such that

C 1 (log n) ε e (log n)•f [(log n) -ζ ] (log n) (1+ ζ)/2 ≤ E[K n (ℓ)] ≤ C 2 e (log n)•f [(log n) -ζ ] (log n) (1+ ζ)/2 , (1.7
)

with ζ := 1 0<ζ<1 + ζ1 1≤ζ<2 .
(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 ,

(log n)f [(log n) -ζ ] = log n - (log n) 1-ζ 2σ 2 + (log n) 1-2ζ λ((log n) -ζ )
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 e

(log n)f [(log n) -ζ ] ∼ ne -(log n) 1-ζ 2σ 2
, and when

ζ ≥ 1, e (log n)f [(log n) -ζ ] ∼ n.
In order to establish our second result, recall Neveu's notation to introduce a partial order on our tree. In [START_REF] Neveu | Arbres et processus de galton-watson[END_REF], 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. For instance

← x = x 1 . . . x m-1 and 1 ≤ x m ≤ N← x , in other words x is the x m -th child of ← 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

C m (z) := {u > z, |u| = m}. (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 3), also define the following event for all m > 0 and q > 0

) := min 1≤j≤|C |-2 (inf D j+2 - sup D j ),
A i (m, q) := C i    {|C i | ≥ q, D(C i ) ≥ m} D∈C i {∀z ∈ D, L(z, n) ≥ 1}    .
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 ℓlog 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

R z a (k) := #{u > z, |u| = k, V (z) ≤ a},
where V (z) = max u∈ φ,z V (u). For notational simplicity, we write R a (k) for R φ a (k). We obtain the following Proposition 1.3 Let ε > 0 and Φ a sequence such that

(1 -ε) log n ≤ Φ(n) ≤ log n + o(log n).
(1.12)

Then, for all 0 < ζ < 2 there exists

C ′ 0 > 0 lim n→+∞ P ψ(0) γ (1 -ε) ≤ log R Φ(n) (ℓ) Φ(n) ≤ 1 -C ′ 0 log log n Φ(n) ∨ Φ(n) ℓ = 1, (1.13)
E[R Φ(n) (ℓ)] ≍ (ℓ -1 1 0<ζ<1 + Φ(n)ℓ -3/2 1 1≤ζ<2 )e Φ(n)f (Φ(n)/ℓ) . (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).

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 appendix), E[R Φ(n) (ℓ)] = E e S ℓ 1 Sℓ ≤Φ(n)
where S j is a centered random walk, we only have to prove

Lemma 2.1 For all ε > 0, Φ satisfying (1.12), for all 0 < ζ < 2 E e S ℓ 1 Sℓ ≤Φ(n) ≍ e Φ(n)f (Φ(n)/ℓ) ℓ -1 1 0<ζ≤1 + Φ(n)ℓ -3/2 1 1≤ζ<2 .

Proof.

For ε > 0:

E e S ℓ 1 Sℓ ∈I Φ(n) ≤ E e S ℓ 1 Sℓ ≤Φ(n) ≤ E e S ℓ 1 Sℓ ∈I Φ(n) + e Φ(n)(1-ε) . with I Φ :=]Φ(n)(1 -ε), Φ(n)].
For every sequence (u n ) n∈N , we denote ūj := max 1≤i≤j u i and u j := min 1≤i≤j u i , also let

S j := { Sj-1 < S j = Sℓ }. First, as S 0 = 0 ℓ j=1 E e S ℓ 1 S j ∈I Φ(n) ,S j ≤ E e S ℓ 1 Sℓ ∈I Φ(n) ≤ ℓ j=1 E e S ℓ 1 S j ∈I Φ(n) ,S j + 1.
For 0 ≤ i ≤ j, let Si := S j -S j-i , with this notation { Sj-1 < S j } = { Sj-1 > 0} and Sj = S j . Writing S as a sum of i.i.d. random variables, we easily see that (S i ) 0≤i≤j and ( Si ) 0≤i≤j have the same law. Then, conditioning on σ{S k , k ≤ j}

E e S ℓ 1 S j ∈I Φ(n) ,S j = D j F ℓ-j (2.1)
with F m := E e Sm 1 Sm≤0 and D j := E e S j 1 S j ∈I Φ(n) ,S j-1 >0 .

By (B.1), ∀j ≤ ℓ, F ℓ-j ≍ (ℓj + 1) -3/2 then it remains to estimate D j . For any A > 0

D j = Φ(n) k=Φ(n)(1-ε)+1 E e S j 1 k-1<S j ≤k,S j >0 ≍ Φ(n) k=Φ(n)(1-ε)+1 e k P (k -1 < S j ≤ k, S j > 0)(1 k≤Aj 1/2 + 1 k>Aj 1/2 ) =: D 1 j + D 2 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

H 2 j := Φ(n) k=Φ(n)(1-ε)+1 e k P (k -1 < S j ≤ k, S j > 0)1 Aj 1/2 <k<εj ≍ εj∧Φ(n) k=Φ(n)(1-ε)+1 e k P (k -1 < S j ≤ k, S j > 0) ≍ εj∧Φ(n) k=Φ(n)(1-ε)+1 e kf (k/j) j , (2.2)
recall that f (x) = 1 -x/(2σ 2 ) + x 2 λ(x)
where λ is the Cramér's serie associated to V .

(2.2) implies that H 2 j ≥ c -e (εj∧Φ(n))f ((εj∧Φ(n))/j) /j. 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) := e xf (x/j) /f (x/j) satisfies in the same interval

F ′ (x) ≥ e xf (x/j) -c + xe xf (x/j) /j ≥ e xf (x/j) (1 -c + ε), with c + > 0. Integrating this last inequality, for ε small enough εj∧Φ(n) Φ(n)(1-ε) e xf (x/j) ≤ [F (x)] εj∧Φ(n) Φ(n)(1-ε) /(1 -c + ε) ≤ C + e (εj∧Φ(n))f ((εj∧Φ(n))/j) , (2.3)
and finally

H 2 j ≤ C + j Φ(n) Φ(n)(1-ε) e xf (x/j) dx ≤ C + j e (εj∧Φ(n))f ((εj∧Φ(n))/j) .
Note that for s > 0 small enough, ψ(1s) ≤ 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

H2 j := Φ(n) k=Φ(n)(1-ε)+1 e k P (k -1 < S j ≤ k, S j > 0)1 k≥εj ≤ e Φ(n)(1-s/2) .
In particular H2 j = o(H 2 j ) for all j ≥ Φ(n) 1+u with u > 0. Finally, as for any ε small enough (εj

∧ Φ(n))f ((εj ∧ Φ(n))/j) is increasing in j, for any 0 < u < ζ ℓ j=1 D j F ℓ-j ≍ ℓ j=Φ(n) 1+u (ℓ -j + 1) -3/2 j -1 e (εj∧Φ(n))f ((εj∧Φ(n))/j) ≍ e Φ(n)f (Φ(n)/ℓ) /ℓ. (2.4) Indeed, writing ℓ j=Φ(n) 1+u (ℓ -j + 1) -3/2 j -1 e (εj∧Φ(n))f ((εj∧Φ(n))/j) := ℓ j=Φ(n) 1+u G j , ℓ j=Φ(n) 1+u G j ≥ G ℓ and as ℓ 2 j=Φ(n) 1+u G j ≤ C + e Φ(n)f (Φ(n)/ℓ) φ(n) 1+u ℓ 1 2 ≤ C + e Φ(n)f (Φ(n)/ℓ) ℓ , ℓ j= ℓ 2 +1 G j ≤ C + e Φ(n)f (Φ(n)/ℓ) ℓ ℓ j= ℓ 2 +1 (ℓ -j + 1) -3 2 ≤ C + e Φ(n)f (Φ(n)/ℓ) ℓ ,
(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

D 1 j ≍ Φ(n) k=Φ(n)(1-ε) ke k j 3/2 e -k 2 /(2σ 2 j) ≍ Φ(n) j 3/2 e Φ(n)f (Φ(n)/ℓ) .
When j < (Φ(n)/A) 2 , similar computations than for H j and Hj give

ℓ j=1 D 2 j F ℓ-j ≤ (Φ(n)/A) 2 j=1 F ℓ-j (H 2 j + H2 j ) ≤ C + e Φ(n)(1-(A 2 /2σ 2 ∧s/2))
.

Finally ℓ j=1 D j F ℓ-j ≍ Φ(n)e Φ(n)f (Φ(n)/ℓ) ℓ -3/2
, this together with (2.4) finishes the proof.

Bounds for log

R Φ(n) (ℓ) (proof of (1.13))
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 [START_REF] Faraud | Almost sure convergence for stochastically biased random walks on trees[END_REF],

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 0 < b < b 0 lim n→+∞ 1 a n log P min |z|=n V (z) ≤ ba n = b -b 0 . (2.5)
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

P min |z|=n V (z) > µa n ≤ λ n + o(λ n ), (2.6)
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 .

Proof.

Clearly for

z 1 < z, V (z) ≤ V (z 1 ) + Ṽ (z 1 , z) where Ṽ (z 1 , z) = max z 1 <x≤z V (x) -V (z 1
). In the sequel, writing Ṽ (z 1 , z) implies that z 1 < z implicitly. For 0 < η < 1 and v n := ηb 0 an /α

P min |z|=n V (z) > 2ηb 0 a n ≤ P min |z 1 |=vn min |z|=n Ṽ (z 1 , z) > 2ηb 0 a n -max |z 1 |=vn V (z 1 ) .
Using that max |z 1 |=vn V (z 1 ) ≤ αv n by ellipticity and for A n := Z vn ≥ e ηψ(0)vn

P min |z|=n V (z) > 2ηb 0 a n ≤ P min |z 1 |=vn min |z|=n Ṽ (z 1 , z) > ηb 0 a n ≤ P min |z 1 |=vn min |z|=n Ṽ (z 1 , z) > ηb 0 a n , A n + P ( Ān ).
Theorem A.2 tells that if q 0 + q 1 > 0, there exists ν > 0 such that P ( Ān ) ≤ e -ν(1-η)ψ(0)vn , otherwise there exists β ′ > 0 such that log P ( Ān ) ∼ -e β ′ (1-η)ψ(0)vn . Stationarity gives that min |z|=n Ṽ (z 1 , z) and min |z|=n-vn V (z) have the same law, and independence of the sub-branching processes rooted at generation v n together with (2.5) imply P min

n := Φ(n)(1-ε)/γ and such that ∀z ′ ∈ C ℓ (z), V (z ′ ) ≤ Φ(n).
In other words for |z| < ℓ, let Z z ℓ the number of descendants of z at generation ℓ, we prove

lim n→+∞ P   |z|=ℓ-wn #{z ′ ∈ C ℓ (z), V (z ′ ) ≤ Φ(n)} = Z z ℓ   = 1, (2.7)
which implies according Theorem A.2 that

lim n→+∞ P R Φ(n) (ℓ) ≥ e ψ(0)wn(1-ε) = 1. Let B := |z|=ℓ-wn {V (z) ≤ y n , R z Φ(n)-yn (ℓ) = Z z ℓ } where y n := µℓ 1/3 P (B) ≥ P   {R yn (ℓ -w n ) ≥ 1} |z|=ℓ-wn,V (z)≤yn R z Φ(n)-yn (ℓ) = Z z ℓ   = k≥1 P (R yn (ℓ -w n ) = k) P   |z|=ℓ-wn,V (z)≤yn R z Φ(n)-yn (ℓ) = Z z ℓ R yn (ℓ -w n ) = k   .
Let us denote z 1 , . . . , z k , . . . the ordered points at generation ℓw n satisfying V (z i ) ≤ y n .

Conditionally on {R yn (ℓw n ) = k}, z 1 exists and

R z 1 Φ(n)-yn (ℓ) = Z z 1 ℓ ⊂ |z|=ℓ-wn,V (z)≤yn R z Φ(n)-yn (ℓ) = Z z ℓ .
Furthermore, by stationarity R z 1 Φ(n)-yn (ℓ) and R Φ(n)-yn (w n ) have the same law, so

P (B) ≥ P R Φ(n)-yn (w n ) = Z wn k≥1 P (R yn (ℓ -w n ) = k) ≥ P max |z|=wn V (z) ≤ Φ(n) -y n P min |z|=ℓ-wn V (z) ≤ y n As y n = o(Φ(n))
, the first probability tends to one thanks to a result of Mac-Diarmid [START_REF] Mcdiarmid | Minimal position in a branching random walk[END_REF] (see also [START_REF] Andreoletti | The number of generations entirely visited for recurrent random walks on random environment[END_REF] 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 -(nA - n + B - n ) ≤ E[K n (ℓ)] ≤ C + (nA + n + B + n )
where

A + n := E e S ℓ -Sℓ 1 ℓ i=1 e S i >c -n , B + n := E e S ℓ 1 Sℓ ≤log(c + n) , A - n := E e S ℓ ℓ i=1 e S i 1 Sℓ >log(c + n) and B - n := E e S ℓ 1 ℓ i=1 e S i ≤c -n ,
C -and c -(respectively C + and c + ) are positive constants that may decrease (respectively increase) from line to line.

Proof.

Markov property gives E

E [K n (ℓ)] = |z|=ℓ (1 -e n log(1-pz) ), with p z := P E φ (T z < T φ ).
Obviously on

{np z ≥ 1}, 1 -e -1 ≤ 1 -e n log(1-pz) ≤ 1. As for x ∈ [0; 1[, -x(1 + x /2) ≤ log(1 -x) ≤ -x and x(1 -x /2) ≤ 1 -e -x ≤ x, on {np z < 1} -3p z /2 ≤ log(1 -p z ) ≤ -p z and np z 4 ≤ 1 -e n log(1-pz) ≤ 3np z 2 , then C -(np z 1 npz<1 + 1 npz≥1 ) ≤ 1 -e n log(1-pz) ≤ C + (np z 1 npz<1 + 1 npz≥1 ).
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) F ℓ-j e (εj∧dn)g((εj∧dn)/j)

B - n ≤ E |z|=ℓ 1 npz≥1 ≤ B + n .

Similar arguments show C

-A - n ≤ E |z|=ℓ p z 1 npz<1 ≤ C + A + n .
d n ≍ 1 log n e (log n)g(log n/ℓ) , (3.2)
where for any x, g(x) := f (x) -1. For d n > εj, a Markov inequality gives

ℓ j= dn /α L j F ℓ-j ≤ C + e -εdn/2 ℓ -3/2 ≤ C + e -ε log n/2 ℓ -3/2 , (3.3)
so as ζ > 0, considering (3.2)

ℓ j= dn /α L j F ℓ-j ≍ e (log n)g(log n/ℓ) log n . ( 3
(1-ε)ℓ j= dn /α L j F ℓ-j ≤ C + (εℓ) -3/2 (1-ε)ℓ j= dn /α P (S j > 0) ≤ C + ℓ -3/2 (1-ε)ℓ j= dn /α j -1/2 ≤ C + ℓ -1 , also ℓ j=(1-ε)ℓ L j F ℓ-j ≤ C + ℓ j=(1-ε)ℓ F ℓ-j P (S j > 0) ≤ C + ℓ j=(1-ε)ℓ (ℓ -j + 1) -3/2 j -1/2 ≤ C + ℓ -1/2 . (3.5)
For any j ≥ (1ε)ℓ, d n ≤ Bj 1/2 for all B > 0, and let

A > B ℓ j=(1-ε)ℓ L j F ℓ-j ≥ ℓ j=(1-ε)ℓ F ℓ-j P (Bj 1/2 < S j ≤ Aj 1/2 , S j > 0) = ℓ j=(1-ε)ℓ F ℓ-j Aj 1/2 -1 k=Bj 1/2 P (S j ∈ (k, k + 1], S j > 0) (3.6)
then Lemma B.6 yields

Aj 1/2 -1 k=Bj 1/2 P (S j ∈ [k, k + 1), S j > 0) ≍ Aj 1/2 k=Bj 1/2 kj -3/2 e -k 2 /(2σ 2 j) ≥ C -j -1/2 . Inserting this in (3.6) give ℓ j=(1-ε)ℓ L j F ℓ-j ≥ C -ℓ -1/2
, this together with (3.5) and the above inequality implies

ℓ j= dn /α L j F ℓ-j 1 dn≤εj ≍ ℓ -1/2 ≍ ℓ -1/2 e (log n)g(log n/ℓ) . (3.7)
Collecting (3.1), (3.4) and (3.7) yields

A + n ≤ E e S ℓ -Sℓ 1 Sℓ >dn ≍ e (log n)g(log n/ℓ) (log n) -1 1 0<ζ<1 + ℓ -1/2 1 1≤ζ<2 . • For A - n , with B ℓ := { ℓ i=1 e S i ≤ ℓ ε e Sℓ } and b n := log(c + n) ℓ ε A - n ≥ E e S ℓ -Sℓ 1 Sℓ >bn,B ℓ = E e S ℓ -Sℓ 1 Sℓ >bn -1 Sℓ >bn,B ℓ := Γ 1 -Γ 2 . (3.8)
Γ 1 can be treated as E e S ℓ -Sℓ 1 Sℓ >dn so

Γ 1 ≍ e (log n)g(log n/ℓ) (log n) -1 1 0<ζ<1 + ℓ -1/2 1 1≤ζ<2 . (3.9)
Recalling that S j = {S j = Sℓ , Sj-1 < S j }, Γ 2 = ℓ j=1 E e S ℓ -S j 1 S j >bn,B ℓ ,S j . Note that on S j , B ℓ = {Y 1 (j) + Y 2 (j) > ℓ ε } where Y 1 (j) := j i=1 e S i -S j and Y 2 (j) := ℓ i=j+1 e S i -S j . As for 0

< δ < 1/2, B ℓ ⊂ {Y 1 (j) > δℓ ε } ∪ {Y 2 (j) > δℓ ε } Γ 2 ≤ ℓ j=bn/α E e S ℓ -S j 1 S j >bn,S j 1 Y 1 (j)>δℓ ε + 1 Y 2 (j)>δℓ ε =: ℓ j=bn/α (Π j + Ω j ).
(3.10) For ℓ j=bn/α Ω j , conditioning by σ(S k , 0 ≤ k ≤ j)

Ω j = P (H j )E e S ℓ-j 1 Sℓ-j ≤0,Y + (ℓ-j)>δℓ ε
where Y ± (k) := k i=1 e ±S i and H m := {S m > b n , S m > 0}. Moreover, using (B.1) and the fact that, on { Sℓ-j ≤ 0}, {Y + (ℓj) > δℓ ε } = ∅ for j ≤ ℓδℓ ε E e S ℓ-j 1 Sℓ-j ≤0,Y + (ℓ-j)>δℓ ε ≤ C + (ℓj + 1) -3/2 1 ℓ-j≥δℓ ε , and P (H j ) can be treated like L j . Then, for n large enough

ℓ j=bn/α Ω j ≤ e (log n)g(log n/ℓ) δℓ ε (log n) -1 1 0<ζ<1 + ℓ -1/2 1 1≤ζ<2 . (3.11) For ℓ j=bn/α Π j , we have Π j = P (H j , Y -(j) > δℓ ε ) F ℓ-j . Let R := sup{0 < k ≤ j, S k ≤ γ log ℓ} with γ > 1, τ + x := inf{k > 0, S k ≥ x} for x ∈ R and t n = (log ℓ) 2 , then Π j = (P H j , Y -(j) > δℓ ε , R ≤ τ + tn + P H j , Y -(j) > δℓ ε , R > τ + tn )F ℓ-j =: Γ 3 + Γ 4 .
Upper bound for Γ 3 , let τ -

x := inf{k > 0, S k ≤ x} for x ∈ R. Notice that on {R ≤ τ + tn }, (a) First case (3) (b) Second case (4)
Figure 2: Two cases P z+tn (H j-k ) ≤ C + t n b n -1 e (εj∧bn)g((εj∧bn)/j) .

j k=τ + tn +1 e -S k ≤ ℓ 1-γ implying that {Y -(j) > δℓ ε } ⊂ {Y -(τ + tn ) > δℓ ε /2}. Thus, using strong Markov property P H j , Y -(j) > δℓ ε , R ≤ τ + tn ≤ P H j , Y -(τ + tn ) > δℓ ε /2, R ≤ τ + tn ≤ j k=tn/α P H j , Y -(τ + tn ) > δℓ ε /2, τ + tn = k ≤ j k=tn/α sup z∈[0,α] P z+tn (H j-k ) P (Y -(τ + tn ) > δℓ ε /2, τ - 0 > τ + tn = k). ( 3
So Γ 3 ≤ C + t n b -1 n ℓ -ε (ℓ -j + 1) -3
/2 e (εj∧bn)g((εj∧bn)/j) . Case ζ ≥ 1, here the following upper bound for (3.12) is useful

P (Y -(τ + tn ) > δℓ ε /2, τ - 0 > τ + tn ) sup tn/α≤k≤j/2 sup z∈[0,α] P z+tn (H j-k ) + P (τ - 0 ∧ τ + tn ≥ j/2) ≤ P (Y -(τ + tn ) > δℓ ε /2, τ - 0 > τ + tn )P α+tn (S j/2 > 0) + 2E[τ - 0 ∧ τ + tn ]/j ≤ C + t n 1 ℓ ε j 1/2 + 1 j ,
with Lemmata B.1 and B.4. So

Γ 3 ≤ C + t n (ℓ -j + 1) 3/2 1 ℓ ε j 1/2 + 1 j .
Upper bound for Γ 4 , first note that on {R > τ + tn }, the following hitting times τ -

γ log ℓ := inf{k ∈ τ + tn , j , S k ≤ γ log ℓ} = τ + tn + θ τ + tn • τ - γ log ℓ , and τ + tn := inf{k ∈ τ - γ log ℓ , j , S k ≥ t n } = τ - γ log ℓ + θ τ - γ log ℓ • τ +
tn (where θ is the shift operator) exist. With these notations according to Lemma B.1

P (τ - 0 > τ + tn ) = P (τ + tn < τ + tn + θ τ + tn • τ - γ log ℓ < τ - γ log ℓ + θ τ - γ log ℓ • τ + tn < τ - 0 ) ≤ P (τ - 0 > τ + tn ) sup z∈[0,α] P γ log ℓ-z (τ - 0 > τ + tn ) ≤ C + γ log ℓ (t n ) 2 .
Again at this point we distinguish the cases ζ < 1 or ζ ≥ 1. When ζ < 1 the above inequality yields

P H j , R > τ + tn ≤ sup 2tn/α≤k≤j sup z∈[0,α] P z+tn (H j-k ) P (τ - 0 > τ + tn ) ≤ C + γ log ℓ (t n ) 2 sup 2tn/α≤k≤j sup z∈[0,α]
P z+tn (H j-k ) .

Finally using that

P H j , Y -(j) > δℓ ε , R > τ + tn ≤ P H j , R > τ + tn , Lemma B.

and (B.1)

Γ 4 ≤ C + e (εj∧bn)g((εj∧bn)/j) b n (ℓj + 1) 3/2 log ℓ .

When ζ ≥ 1, we have

P H j , Y -(j) > δℓ ε , R > τ + tn ≤ P H j , j/2 ≥ R > τ + tn + P H j , R > τ + tn ∨ j/2 .
Moreover P H j , j/2 ≥ R > τ + tn ≤ P τ + tn < τ - 0 sup -α≤x≤α P γ log ℓ+x (S j/2 > 0) and P H j , R > τ + tn ∨ j/2 ≤ P (S j/2 > 0) sup -α≤x≤α P γ log ℓ+x (τ - 0 > τ + bn ). So using Lemma B.1

Γ 4 ≤ C + j 1/2 (log ℓ)(ℓ -j + 1) 3/2 .
Collecting (3.11) and Γ 3 , Γ 4 with (3.10)

Γ 2 ≤ C + e bng(bn/ℓ) b n log ℓ ∼ C + e (log n)g(log n/ℓ) (log n) log ℓ , if ζ < 1 and Γ 2 ≤ C + ℓ 1/2 log ℓ ≤ C + e (log n)g(log n/ℓ) ℓ 1/2 log ℓ , if ζ ≥ 1. Γ 2 is therefore negligible compared to Γ 1 (see (3.9)) and (3.8) implies that A - n ≥ C + e (log n)g((log n) -ζ ) (log n) -1-ε 1 0<ζ<1 + ℓ -1/2-ε 1 1≤ζ<2
. This with Lemma 3.1 finish the proof of (1.7).

3.2 From R Φ(n) (ℓ) to K n (ℓ) and M n (ℓ) (proof of (1.6) and (1.9))

Let Φ 1 (n) := (1 -2ε) log n, we need the following Lemma 3.1 Let A := {|z| = ℓ, V (z) ≤ Φ 1 (n)}, lim n→+∞ P min z∈A L(z, T n 1-ε φ ) ≥ 1 = 1, which implies lim n→+∞ P(K n 1-ε (ℓ) ≥ R Φ 1 (n) (ℓ)) = 1.

Proof.

Applying Corollary C.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

P E (∪ z∈A {L(z, T n 1-ε φ ) = 0}) ≤ |A|e -c -n ε ℓ -1 ≤ R Φ 1 (n) (ℓ)e -c -n ε/2 Using (1.13), E[R Φ 1 (n) (ℓ)] ≤ e Φ 1 (
lim n→+∞ P max |z|=ℓ-log n/γ min y∈C ℓ (z) L(y, T n 1-ε φ ) ≥ 1 = 1 (3.13)
To obtain the lower bound in (1.9) we finally use the following result that can be deduced from [START_REF] Faraud | Almost sure convergence for stochastically biased random walks on trees[END_REF] (see [START_REF] Andreoletti | The number of generations entirely visited for recurrent random walks on random environment[END_REF] Lemma 3.2 and what follows for details)

∀δ > 0, lim n→+∞ P(L(φ, n) ≥ n 1-δ ) = 1. (3.14)
For the lower bound in (1.6), we use Lemma 3.1, (3.14) and finally the lower bound in (1.13).

For the upper bound in (1.6), denote u n := C log log n ∨ (log n) 1-ζ , where C > 0. As n ≤ T n φ , by Markov inequality and (1.7), P (log

M n (ℓ) ≥ log n -u n ) ≤ P(K n (ℓ) ≥ ne -un ) ≤ C + e un e -(log n) 1-ζ /2σ 2 ℓ -1/2
which gives the upper bound adjusting C properly.

Visited points along the GW

In this paragraph we study the manner the random walk visits the tree.

Visits of clusters at deterministic cuts (proof of (1.11))

Recall that a cluster initiated at z with end generation m is the set

C m (z) = {u > z, |u| = m}. Also take k n = Φ(n) k , r n = Φ(n) r , s n = Φ(n) s with s > 0 and h n sequences such that h n = ℓ -k n r n k n -1 , k n (αr n + s n ) -s n ≤ Φ(n)(1 -2ε), (4.1)
where α := | log ε 0 | (see (A.1) for details).

We define recursively clusters at generations ir n + (i -1)h n for all 1 ≤ i ≤ k n in the following way (see also Figure 3): the iteration starts with ž0 = φ and

∀z i ∈ C irn+(i-1)hn (ž i-1 ), ži = inf {u > z i , |u| = i(r n + h n ), V (u) ≤ i(αr n + s n )}.
The individuals of these clusters form a subtree of the GW, moreover for all z of this subtree at generation ℓ, V (z) ≤ Φ(n)(1 -2ε). For a fixed i ∈ 1; k n , C i denotes, among the previously defined clusters, the ones rooted at generation (i -1)(r n + h n ) and with end points at generation ir n + (i -1)h n , in other words C irn+(i-1)hn (.). We first give an upper bound for the probability that for all i ≤ k n every clusters in C i are fully visited before

T n 1-ε φ P E   kn i=1 D∈C i z∈D L(z, T n 1-ε φ ) = 0   ≤ kn i=1 D∈C i z∈D P E L(z, T n 1-ε φ ) = 0 .
In the previous formula kn i=1 D∈C i is an abuse of notation as the sets of clusters are defined recursively. With a similar reasoning as the one for Corollary C.1 and the ellipticity condition for the number of descendants

P E   kn i=1 D∈C i z∈D L(z, T n 1-ε φ ) = 0   ≤ exp(k n r n log N 0 -c -n 1-ε e -Φ(n)(1-2ε) /ℓ). (4.2)
We now prove the existence of such clusters, this implies new constraints on k, r and s in addition to (4.1). First, ellipticity conditions imply that for any site z > y, V (z

) -V (y) ≤ α(|z| -|y|) a.s. Thus, for all a ∈ N and b > 0, {∃ž > z, |ž| = |z| + a, Ṽ (z, ž) ≤ b} is a.s. contained in {V (z) -V (y) ≤ α(|z| -|y|), ∃ž > z, |ž| = |z| + a, Ṽ (y, ž) ≤ α(|z| -|y|) + b} ( Ṽ is defined in the proof of (2.6)).
Then, with our slight abuse of notation, a.s. B := {the clusters C . (.) exist} contains

kn i=1    z i ∈C irn+(i-1)hn (ž i-1 ) ∃ž i > z i , |ž i | = i(r n + h n ), Ṽ (z i , ži ) ≤ s n    .
P (B) ≥ P (∃|z| = h n , V (z) ≤ s n ) N knrn 0 = P min |z|=hn V (z) ≤ s n N knrn 0 . Assuming s ≥ (1+ζ -k)/3, Lemma 2.3 yields P (B) ≥ (1-λ hn ) N knrn 0 ∼ exp -N knrn 0
λ hn . To choose k, r and s, we have to take into account the last constraint in (4.1), s ≤ 1k and that if q 0 + q 1 = 0, k + r < s. We distinguish two cases

• if 0 < ζ ≤ 1, let 0 < δ < ζ/2, take s = (1 + ζ)/2 -δ, k = δ/

Thus in both cases

P (B) ≥ 1 -C + e -c 1 sn N knrn 0 -→ n→+∞ 1. (4.3)
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 lim n→+∞ P   kn i=1 D∈C i z∈D L(z, T n 1-ε φ ) ≥ 1 , B   = 1.
According to (3.14), as P(n ≥ T n 1-ε φ

) tends to one we finally obtain

lim n→+∞ P   kn i=1 D∈C i z∈D {L(z, n) ≥ 1} , B   = 1.
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)

P ∃i ∈ 2, k n , |C i | ≤ e ψ(0)(i-1)rn /2 ≤ P ∃i ∈ 2, k n , |C i | ≤ e ψ(0)(i-1)rn /2 , B + P (B) ≤ kn i=2
P Z (i-1)rn ≤ e ψ(0)(i-1)rn /2 + P (B)

≤ e -ψ(0)νrn/4 + C + e -c 1 sn N knrn 0 .

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, ≤ 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 .

P ∃i ∈ 2, k n , D(C i ) ≤ e ψ(
Consequently P ∃i ∈ 2, k n , D(C i ) ≤ e ψ(0)hn/2 ≤ e 3ψ(0)knrn-νψ(0)hn/2 + 2e -ψ(0)rn + C + e -c 1 sn N knrn 0 , moreover as h n ∼ (log n) 1+ζ-k , k + r < s < 1 and k < ζ we obtain the result.

Proof of (1.10)

Let m = εℓ 1/3 , δ > 0, define B the set of points z ′ such that for all |z| = m, z

′ := inf {u > z, |u| = ℓ, V (u) ≤ Φ(n)(1 -δ)}. Corollary C.1 gives P E z ′ ∈B L(z ′ , T n 1-ε φ ) = 0 ≤ |B|e -c -n 1-ε e -Φ(n)(1-δ) /ℓ ≤ Z m e -c -n 1-ε e -Φ(n)(1-δ) /ℓ .
As Φ(n) ≤ log n + o(log n) and E[Z m ] = e ψ(0)m , taking δ = 2ε

P z ′ ∈B L(z ′ , T n 1-ε φ ) = 0 ≤ e -c -n ε/2 . (4.4)
We now prove that lim n→+∞ P (|B| = Z m ) = 1. From [START_REF] Mcdiarmid | Minimal position in a branching random walk[END_REF] (see also [START_REF] Andreoletti | The number of generations entirely visited for recurrent random walks on random environment[END_REF] Lemma 2.1), lim n→+∞ P (max |z|=m V (z) ≤ 2γm) = 1, so as for n large enough ℓ 1/3 /Φ(n) ≤ δ with the same arguments used in the proof of Lemma 2.3

P (|B| < Z m ) ≤ P   |z|=m ∀z ′ > z, |z ′ | = ℓ, V (z ′ ) > (1 -δ)Φ(n)   ≤ 1 -1 -P min |z|=ℓ-m V (z) > (1 -2δ)Φ(n) e 2ψ(0)m + e -ψ(0)m ≤ 2P min |z|=ℓ-m V (z) > (1 -4ε)Φ(n) e 2ψ(0)m + e -ψ(0)m .
To finish we put ourself in the case q 0 + q 1 > 0 (the other case is treated similarly), using Lemma 2.3

P min |z|=ℓ-m V (z) > (1 -4ε)Φ(n) ≤ e -c 1 ℓ 1/3 .
We can now choose ε small enough and obtain,

P (|B| = Z m ) ≥ 1 -e -c 1 ℓ 1/3 /2 . Moving back to (4.4) P(∀z ∈ B, L(z, T n 1-ε φ ) ≥ 1, |B| = Z m ) ≥ 1 -o(1)
. Finally to obtain (1.10) we apply (3.14).

E   |x|=n e -V (x) F (V (x i ), 1 ≤ i ≤ n)   = E[F (S i , 1 ≤ i ≤ n)] (A.1)
where (S i -S i-1 ) i≥1 are i.i.d. random variables, and the law of S 1 is determined by

E[f (S 1 )] = E   |x|=1 e -V (x) f (V (x))   , (A.2)
for any measurable function f : R → [0, +∞). A proof can be found in [START_REF] Biggins | Senata-heyde norming in the branching random walk[END_REF], see also [START_REF] Shi | Random walks and trees[END_REF]. We have the following identities

ψ(t) = log E[e (1-t)S 1 ], ψ ′ (t) = - E[S 1 e (1-t)S 1 ] E[e (1-t)S 1 ] . In particular, E[N ] = e ψ(0) = E[e S 1 ] and the hypothesis ψ ′ (1) = 0 equates to E[S 1 ] = 0. Remark A.1 Let α := | log ε 0 |, we have P(|S 1 | ≤ α) = 1. Indeed taking f (x) = 1 |x|≤α and using Biggins-Kiprianou, P(|S 1 | ≤ α) = E[f (S 1 )] = e ψ(1) = 1.

A.2 Left tail of Z n

Recall that the positive martingale (W n ) n≥0 := ( Zn /e ψ(0)n ) n≥0 converges a.s. to a non degenerate limit W (see for instance [START_REF] Shi | Random walks and trees[END_REF]). Moreover W has a positive continuous density function denoted w. Bingham [START_REF] Bingham | On the limit of a supercritical branching process[END_REF] shows that for the Schröder case (q 0 + q 1 > 0), there exists 0 < ν < 1 such that for small x, w(x) ∼ x ν-1 and for the Böttcher case (q 0 + q 1 = 0) there exists β ∈ (0, 1) such that when x → 0, log w(x) ∼ -x -β 1-β . The results of [START_REF] Athreya | Branching processes[END_REF] and then [START_REF] Fleischmann | Lower deviation probabilities for supercritical galton-watson processes[END_REF] (Theorems 4 and 5) and [START_REF] Fleischmann | On the left tail asymptotics for the limit law of a supercritical galton-watson processes in the bötcher case[END_REF] (Theorem 7) lead to Theorem A.2 Let 0 < κ < 1 then P (Z n ≤ e κψ(0)n ) ∼ e -νψ(0)(1-κ)n in the Schröder case, and log P (Z n ≤ e κψ(0)n ) ∼ log w e ψ(0)(κ-1)n in the Böttcher case.

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 [START_REF] Aidékon | Tail asymptotics for the total progeny of the critical killed branching random walk[END_REF] and [START_REF] Spitzer | Principle of Random Walks[END_REF].

Lemma B.1 For all x ∈ [0, y] and m large enough

P x (τ + y < τ - 0 ) ≍ x + 1 y + 1 , E[τ + y ∧ τ - 0 ] ≍ y and P x (τ - 0 > m) ≍ x + 1 √ m .
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

P Y -(τ + a ) > M, τ + a < τ - 0 ≤ C +/M .
Proof.

According to [START_REF] Aidékon | The precise tail behavior of the total progeny of a killed branching random walk[END_REF] p.19, there exists C + > 1 such that for all 0 ≤ a ≤ L ≤ 1,

E Y -(τ - 0 ∧ τ + a ) ≤ C + a+1 /a. As P Y -(τ + a ) > M, τ + a < τ - 0 ≤ P Y -(τ - 0 ∧ τ + a )
> M , we conclude using the Markov inequality.

Lemma B.3 For any

m ≥ 1, E[e Sm 1 τ - 0 >m ] ≍ (m + 1) -3/2 . (B.1)
Proof.

The upper bound can be found in [START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF] p.44, the lower bound can be obtained as follows:

E[e Sm 1 τ - 0 >m ] ≤ αm k=0 e -k P (S m ∈ [-k, -k + 1], Sm < 0) = αm k=0 ke -k m -3/2 ≍ (m + 1) -3/2 .
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

P Y -(τ + a ) > M, τ + a < τ - 0 ≤ C +/M .
The following Lemma may be found in the literature, however as we can prove it easily for our case we present a short proof. Proof.

(B.4) is F. Caravenna [START_REF] Caravenna | A local limit theorem for random walks conditioned to stay positive[END_REF] result and (B.5) can be obtained with [START_REF] Petrov | Sums of Independent Random Variables[END_REF] 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 :

P E x ′ x (T x < T x ′ ) = e V (x ′ x )
z∈ x ′ ,x e V (z) , (C.1)

P E ← x (T x ′ < T x ) = e V (x)
z∈ x ′ ,x e V (z) .

(C.2)

where x ′ x is the only children of x ′ in x ′ , x .

The result is classical (see for instance [START_REF] Andreoletti | The number of generations entirely visited for recurrent random walks on random environment[END_REF]) and a useful direct consequence of this latter is the following Proof.

Obviously (C.4) is a consequence of (C.3). Thanks to formula (C.1), for z ∈ A, P E φ (T z < T φ ) ≥ C -e -V (z) /ℓ. Then using the strong Markov property and the recurrence of X, for n large enough P E φ T n κ φ < T z = (1 -P E φ (T z < T φ )) n κ ≤ exp e -c 7 n κ V (z) /ℓ .

  1 , z) > ηb 0 a n , A n ≤ P min |z|=n-vn V (z) > ηb 0 a n e ηψ(0)vn ≤ 1e -(b 0 (1-η)+o(1))an e ηψ(0)vn , we conclude choosing η sufficiently close to 1 to get (1η) < η 2 ψ(0)/α.

Figure 1 :

 1 Figure 1: One large cluster

. 4 )

 4 When 1 ≤ ζ < 2 , first Lemma B.1 and (B.1) give

Figure 3 :

 3 Figure 3: Clusters at regular cuts

2 ,

 2 and r = (1ζ)/2 + δ /2, • if 1 < ζ < 2, let 0 < δ < (2ζ)/3, take s = (1 + ζ)/3, k = δ and r = (1 + ζ -4δ)/3.

  0)hn/2 ≤ P (B) +kn i=2 P D(C i ) ≤ e ψ(0)hn/2 , B {|C i(rn+hn) (z)| ≤ e ψ(0)hn/2 } ∩ B   .As conditionally on B, |C i(rn+hn) (z)| and Z hn are equal in law, onD i := {|C i | ≤ e 2ψ(0)(i-1)rn }, |D| |C i | ≤ N rn 0 e 2ψ(0)(i-1)rn so Theorem A.2 yields P D(C i ) ≤ e ψ(0)hn /2 , B ≤ P   D∈C i-1 z∈D {|C i(rn+hn) (z)| ≤ e ψ(0)hn/2 } ∩ B ∩ D i   + P (D i )

Figure 4 :

 4 Figure 4: Distant visited sites

Lemma B. 5 √P. 4 )

 54 Let m > 1, assume that b = b(m) ≥ σ 2 √ m log m, with lim m→+∞ b/m = 0, and a = a(m) > 0 is such that lim m→+∞ a/ √ m = 0, then for all m large enough P a (S m > b, S m > 0) ≤ C + a b e b•g(b/m) , (B.2)with g(x) = f (x) -1 = -x 2σ 2 + x 2 λ(x). For all ε > 0 and r > εm.P a (S m ≥ r, S m > 0) ≤ C + a √ m e -sr+mψ(1-s) . (B.3) Proof.For (B.2), let ω a positive function of b and m, such that ω ≤ 2 √ m/b and that we choose later, write P a (S m > b, S m > 0) asP a S m > b, S m > 0, τ + √ m ≤ ωm + P a S m > b, S m > 0, τ + √ m > ωm =: P 3 + P 4 .Strong Markov property and homogeneity give:P a S m > b, S m > 0, τ + √ m = j ≤ P a τ - 0 > τ + √ m = j sup 0≤x≤α P √ m+x S m-j > b, S m-j > 0 ≤ P a τ - 0 > τ + √ m = j P S m-j > b -√ mα implying with Lemma B.1: m-j > b -√ mα ,A classical result of moderate deviations (see for instance[START_REF] Petrov | Sums of Independent Random Variables[END_REF], Chapter VIII, Theorem 1) impliesP S m(1-ω) > b -√ mα ≤ C + m(1ω) m)g((b-√ m)/(m(1-ω)))We now choose ω in such a way that (b-√ m)g((b -√ m)/(m(1ω)))bg(b/m) = O(1), ω is actually a sum which first two terms are ω = 2m 1/2 /bλ(b/m)/ √ m + .... So for any n large enoughP S m(1-ω) > b -√ mα ≤ C + √ m b e b•g(b/m) .With similar computations this upper bound is still true forP (S m-j > b -√ mα) for m(1ε) ≤ j ≤ m, so P 4 ≤ C + a b e bg(b/m) . In the same way a τ - 0 > τ + √ m = j sup 0≤x≤α P √ m+x S m-j > b, S m-j > 0 e -m/(2σ 2 j) (mj) 1/2 b -√ mα e (b- √ m-α)g((b-√ m-α)/(m-j)) ≤ C + (ωm) 1/2 b e -b/(2σ 2 m 1/2 ) e b•g(b/m) = o e b•g(b/m) . b .which finish the proof. (B.3) can be proved in a similar way with classical large deviation estimates.The following Lemma states the local behavior of sums of i.i.d. random variables, recall that S 1 is non-lattice. Lemma B.6 Let ε > 0 small and A > 0 large. For all m large enough, for all 1 ≤ r ≤ A √ m P (S m ∈ (r, r + 1], S m > 0) = r m 3/2 e -r 2 /(2σ 2 m) + o(m -3/2 ). (BFor all A √ m ≤ r ≤ εm P (S m ∈ (r, r + 1], S m > 0) ≍ 1 m e rg(r/m) . (B.5) see Lemma B.5 for the definition of g.

Corollary C. 1

 1 Let A ⊂ {z ∈ T, |z| = ℓ} and κ > 0, there exists a positive constant c 7 such thatP E φ (T n κ φ < T z ) ≤ exp -c 7 n κ e V (z) /ℓ , ∀z ∈ A, (C.3) P E z∈A {L(z, T n κ φ ) = 0} ≤ |A| exp -c 7 n κ e -max z∈A V (z) /ℓ (C.4)

  where, by definition, inf D j+2sup 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

	Theorem 1.2 For 0 < ζ < 2 and ε > 0 recalling that ℓ = (log n) 1+ζ
	lim n→+∞	P	max |z|=ℓ-log n/γ	min y∈C ℓ (z)	L(y, n) ≥ 1 = 1,	(1.9)
	lim n→+∞	P	min z∈C εℓ 1/3 (φ)	max y>z,|y|=ℓ	L(y, n) ≥ 1 = 1.	(1.10)

  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.

• For A + n , first note that { ℓ i=1 e S i > c -n} ⊂ Sℓ > d n , with d n = log(c -n/ℓ).

Recalling the arguments given in (2.1), A + n is bounded from above by

E e S ℓ -Sℓ 1 Sℓ >dn = ℓ j= dn /α P S j > d n , S j > 0 E e S ℓ-j 1 Sℓ-j ≤0 =: ℓ j= dn /α L j F ℓ-j . (3.1)

Like in the proof of Lemma 2.1, we distinguish cases 0

< ζ < 1 and 1 ≤ ζ < 2. When 0 < ζ < 1 then d n > A √ j

for any A > 0, so applying Lemma B.6 like for (2.2) we obtain for d n ≤ εj ℓ j= dn /α L j F ℓ-j ≍ ℓ j= dn /α


E Brézin
email: brezin@lpt.ens.fr

Wilson's renormalization group : a paradigmatic shift

come     Wilson's renormalization group : a paradigmatic shift

1 Why was it so difficult to accept Wilson's viewpoint?

In 1971-72 I was on leave from Saclay, employed by the theory group at the physics department of Princeton University. My background at the time was mainly field theory. Just before this I had been working with C. Itzykson on the use of classical approximations in QED, dealing with problems such as pair creation by an oscillating electromagnetic field , but I was aware that the understanding of the intriguing features of scaling laws and universality near a critical point was a difficult non-perturbative problem. Ken Wilson was invited in the fall of 1971 to give a talk on his recently published articles "Renormalization group and Kadanoff scaling picture [1] , Phase space cell analysis of critical behavior [2]" but Ken said that he could not explain his ideas in one talk. The University (thanks to David Gross I think) had the good idea of inviting him to talk as much as he needed and Ken ended up giving 15 lectures in the spring term of 1972, which resulted in the well-known 1974 Physics Reports " The renormalization group and the ǫ-expansion " by Wilson and Kogut (who had been taking notes throughout the lectures), one of the most influential articles of the last decades [START_REF] Wilson | The renormalization group and the ǫexpansion[END_REF].

Wilson's ideas were so different from the standard views at the time, that they were not easy to accept. His style was also completely different : for him field theory was not a set of abstract axioms but a tool to perform practical calculations even at the expense of oversimplified models whose relationship with reality was tenuous. Let us try to understand why Ken's ideas, nowadays so inbred in the training of any young physicist that they seem nearly obvious, were so difficult to accept. Let us first review the conventional wisdom at the time.

• Renormalization was a technique to remove perturbatively the ultraviolet singularities for those theories which were known to be renormalizable, i.e. theories for which this removal could be done at the expense of a finite number of parameters.

• Why should one consider such theories? Presumably because they were the only ones for which practical calculations are conceivable. For instance in QED, the minimal replacement p → p -eA provides such a theory but gauge invariance alone would allow for more couplings, such as a spin current coupled to the field strength, which would ruin the renormalizability.

• As a result one obtained with renormalized QED a theory which, a priori, was potentially valid at all distances from astronomical scales down to vanishingly small ones.

• The renormalization group in its conventional formulation was applicable only to renormalizable theories ; it allowed one to understand leading logarithms, at a given order in perturbation theory, from lower orders.

So what is it which was surprising in Wilson's approach?

• Critical points such as liquid-vapor, Curie points in magnets, Ising models, etc., are a priori classical problems in statistical mechanics. Why would a quantum field theory be relevant?

• Why renormalization in a theory in which there is a momentum cutoff Λ = a -1 , in which a is a physical short distance such as a lattice spacing, or a typical interatomic distance, and there is no reason to let a go to zero?

• The momentum space reduction of the number of degrees of freedom that Wilson used introduced singularities in the Hamiltonian generated by the flow. It was not quite clear that the procedure could be systematized.

• Irrelevance of most couplings was of course a fundamental piece of the theory but it looked a priori purely dimensional. For instance if one adds to a g 4 ϕ 4 theory in four dimensions a coupling (g 6 /Λ 2 )ϕ 6 one obtains a flow equation for the dimensionless g 6 which concludes at its irrelevance as if the 1/Λ 2 sufficed :

Λ ∂ ∂Λ g 6 -2g 6 = β 6 (g 4 , g 6 )
However the inclusion of the g 6 /Λ 2 into a Feynman diagram with only g 4 vertices produces immediately an extra Λ 2 which cancels the previous one. So why is g 6 irrelevant?

• Although it is now physically very clear, it was not easy to understand that the critical surface had codimension two for an ordinary critical point, in others words that there are two and only two relevant operators in the large space of allowed coupling parameters.

• Even more difficult to take was the statement that four-dimensional renormalizable theories such as ϕ 4 or QED, i.e. non asymptotically free theories (an anachronistic name), if extended to all energy scales could only be free fields. How could one believe this, given the extraordinary agreement of QED with experiment?

I must admit that my initial difficulties at grasping the views exposed in Ken's lectures led me, in a friendly collaboration with David Wallace, to check their consistency. Amazed to see that the theory defeated our skepticism, we ended up working out with Ken the critical equation of state [START_REF] Brezin | Feynman-graph expansion for the equation of state near a critical point[END_REF].

Wilson's renormalization group

.

It would take hundreds of pages to list all that we have learnt from Wilson's RG applied to critical exponents. The basic questions of scaling and universality were immediately made transparent, all the critical indices could be expressed in terms of two of them (plus a leading irrelevant operator for corrections to scaling, plus cross-over exponents). Wilson insisted that RG was a computational tool even in the absence of small parameters, but he provided also new parameters. Together with Michael Fisher, he introduced the famous epsilon-expansion [START_REF] Wilson | Critical exponents in 3.99 dimensions[END_REF] ; he was also the first to realize that critical properties could also be computed in a 1/n expansion (he pointed out that in a ( ϕ 2 ) 2 theory non bubble diagrams were subleading in the large n-limit), he contributed to Monte-Carlo RG, etc. I would like here to insist simply on the conceptual revolutions brought by Wilson's RG:

• Field theory applied to the real world had to be considered as a long distance limit of yet unknown physics, even if the distances at which field theory is applied in particle physics go down to less than 10 -6 femtometers.

• Renormalizability emerged as a consequence of considering this long distance limit, and not because the theory should be applicable down to vanishingly small distances.

• Non asymptotically free theories such as ϕ 4 , QED or the Weinberg-Salam model are merely effective theories. If one tried to push them down to vanishingly small distances they would be forced to be only free field theories. These theories carry their own limitation, without any indication on the underlying physics.

It is interesting to try to track down the maturation of his ideas. He introduced first the operator product expansion in an unpublished Cornell report of 1964 "On products of quantum field operators at short distances". In his Nobel lectures Ken wrote I submitted the paper for publication; the referee suggested that the solution of the Thirring model might illustrate this expansion. Unfortunately, when I checked out the Thirring model, I found that while indeed there was a short distance expansion for the Thirring model, my rules for how the coefficient functions behaved were all wrong, in the strong coupling domain. I put the preprint aside, awaiting resolution of the problem. He computed indeed the anomalous dimensions of the operators in the Thirring model (Operator-product expansions and anomalous dimensions in the Thirring model, [START_REF] Wilson | Operator-product expansions and anomalous dimensions in the Thirring model[END_REF] :

It is shown that ψ has dimension 1/2 + λ 2 /4π 2 1-λ 2 /4π 2 .
) Clearly this had forced Ken to consider non-canonical dimensions of operators and the possibility that this could spoil the use of the OPE. Right after this he published his paper on . "Anomalous dimensions and breakdown of scale invariance in perturbation theory" [START_REF] Wilson | Anomalous dimensions and breakdown of scale invariance in perturbation theory[END_REF]. Other scientists at the time were also realizing that the Gell-Mann and Low RG could be interpreted as asymptotic restoration of scale invariance broken by the regularization of quantum fluctuations, in particular C. Callan ("Broken scale invariance in scalar field theory" [START_REF] Callan | Broken scale invariance in scalar field theory[END_REF] )and K. Symanzik ("Small distance behaviour analysis and Wilson expansions " [START_REF] Symanzik | Small distance behaviour analysis and Wilson expansions[END_REF]).

The application of these ideas to critical phenomena was a natural illustration of Ken's views and he acknowledges in his 1971, Phys. Rev. B papers quoted here-above the role of Kadanoff's block spin picture [START_REF] Kadanoff | Scaling laws for Ising models near Tc[END_REF] and his conversations with Ben Widom and Michael Fisher at Cornell. However Ken clearly was not very fond of the ǫ or 1/n expansions, although he had invented them. He regarded his RG as a tool that is applicable in the absence of any small parameter and thus the pseudo-small ǫ or 1/n were going back to directions that he didn't like. Likewise he regarded the field theoretic method such as the Callan-Symanzik equations as a very special case of a general method. To prove his point the participants of the 1973 Cargèse summer school (which I co-organized) remember that Ken arrived with a brilliant solution of the Kondo problem [START_REF] Wilson | Renormalization group-critical phenomena and Kondo problem[END_REF] (Renormalization groupcritical phenomena and Kondo problem , Rev. Mod. Phys. 47, 773 (1975)) and his Nobel lectures he stated [] the solution of the Kondo problem is the first example where the full renormalization group (as the author conceives it) has been realized : the formal aspects of the fixed points, eigenoperators, and scaling laws will be blended with the practical aspect of numerical approximate calculations of effective interactions to give a quantitative solution [] to a problem that previously had seemed hopeless.

It is also at this Cargèse school that he presented for the first time lattice gauge theories, which turned QCD calculations of mass spectra into a problem of statistical mechanics which remains as the main tool for understanding quantitatively the predictions of QCD.

Quantitative aspects

If there is no doubt forty years later that the RG has led to a complete and extremely beautiful qualtitative understanding of critical fluctuations, succeeding to explain universality, scaling laws and many new phenomena such as the geometry of polymers, percolation, etc, the quantitative aspects are often less clear. Indeed the connection with the three dimensional physical world remains a difficult non-perturbative problem and various numerical strategies have to be devised. How systematic are some of these methods, such as real space RG, living with the size limitations of numerical simulations, has left room for analytical methods and I think that renormalized field theory has played a significant role. Indeed the theory deals with correlations extending to distances much larger than the lattice spacing x ≫ a , a correlation length ξ ≫ a and a finite ratio x/ξ. The effective renormalizable theory, the a → 0 limit, provides exactly all the correlation functions in this limit. There are several strategies.

For instance in the "minimal subtraction scheme"

β d (g) = -(4 -d)g + β 4 (g) γ d (g) = γ 4 (g)
and all calculations in dimension d follow from computations in the critical (i.e. massless) theory in 4D. High order calculations in the ǫ or 1/n expansions follow from such techniques. However these expansions do not converge, no matter how small are ǫ or 1/n. Using Lipatov's instanton method [START_REF] Lipatov | Divergence of the perturbation-theory series and pseudo particles[END_REF] one can find the behavior of the perturbative series at order k for large k : for instance the successive terms of the ǫ expansion for the anomalous dimension of the field grow asymptotically like ǫ k k!(-1/3) k k 7/2 (Ising like systems). This understanding led Zinn-Justin and co-workers [START_REF] Le Guillou | Critical exponents from field theory[END_REF], using mathematical techniques such as Borel transform plus conformal maps, to accurate values of the critical exponents.

Parisi proposed to work directly in three dimensions [14], perturbation theory then makes sense only in the massive case, i.e. away from T c . Again using the Lipatov type asymptotic estimate one can obtain at the end a good quantitative agreement with known values [START_REF] Le Guillou | Critical exponents from field theory[END_REF][START_REF] Brezin | Critical exponents and large-order behavior of perturbation theory[END_REF].

The point here is that it seems that one can transform the divergent perturbation series into seemingly convergent algorithms. Initially the impression was that one should better stop at some low order in the ǫ expansion if one did one want to run into absurdities and that was the best the theory could do. But now this says that the accuracy of the theory is not a priori limited : if comparison with experiment required better theoretical estimates, computing one more order in perturbation theory, although long and painful, would still improve the theoretical accuracy.

A crucial test of RG

In 1969 in a remarkable pre-Wilson article, Larkin and Khmel'nitskii [START_REF] Larkin | nitskii[END_REF] made two interesting points 1. in 4D mean field theory is violated by powers of logarithms, and they computed explicitly those logs with a "parquet" approach (i.e. sum of leading logs at each order) which is equivalent to a one-loop RG 2. if one considers an Ising-like magnet with strong dipolar forces, instead of the usual short-range exchange forces, then it is in 3D and not in 4D, that one should observe those logarithmic violations of mean field scaling.

They predicted for instance that the specific heat, which in mean field theory remains finite at T c with a simple step when one crosses the critical temperature, would diverge instead as

C = A ± | log | T -T c T c || 1/3 with A + A - = 1 4
Note that this is a prediction which does not require any approximation scheme ; it is a pure test of RG very similar in spirit to the logarithmic deviations to scaling in deep inelastic electron scattering which provided the early tests of QCD [START_REF] Gross | Applications of the renormalization group to high-energy physics[END_REF]. The experiment was performed in 1975 by G.Ahlers et al. [START_REF] Ahlers | Logarithmic Corrections to the Landau Specific Heat near the Curie Temperature of the Dipolar Ising Ferromagnet LiTbF4[END_REF] and their finding was : Quantitative experimental results for the logarithmic corrections to the Landau specific heat are reported for the dipolar Ising ferromagnet LiT bF 4 near its Curie temperature T c = 2.885K. The power of the leading logarithmic term is found to be 0.34 ± 0.03, and the corresponding amplitude ratio is 0.24 ± 0.01. These results are in agreement with the predicted values of 1/3 and 1/4, respectively. 1

A few remarks concerning RG and conformal invariance

The idea of asymptotic conformal invariance of strong interactions was present at an early stage in the work of Mack, Kastrup, Salam, Symanzik and others [START_REF] Mack | itIntroduction to Conformal Invariant Quantum Field Theory in Two and More Dimensions in Nonperturbative Quantum Field Theory[END_REF]. Wilson himself in his 1970 paper Broken scale invariance and anomalous dimensions [START_REF] Kg Wilson | proceedings of the Midwest Conference on Theoretical Physics[END_REF] refers to this asymptotic conformal symmetry. At the same period at the Landau Institute Polyakov [START_REF] Polyakov | Conformal symmetry of critical fluctuations[END_REF] and Migdal [START_REF]Conformal invariance and bootstrap[END_REF] realized that the same conformal symmetry should hold near a critical point, 1 We realized with Zinn-Justin in 1975 [START_REF] Brezin | Critical behaviour of uniaxial systems with strong dipolar interactions[END_REF] that the correspondance between short range forces in 4D, a ϕ 4 -theory with propagator 1/q 2 , and 3D with strong dipolar forces a ϕ 4theory with propagator 1/q 2 ⊥ + q 2 z /q 2 ⊥ , would not extend beyond one-loop. A two-loop RG would give for the former

C shortrange,d=4 = A ± | log |t|| 1/3 (1 - 25 81 | log | log |t|| | log |t|| )
and for the latter

C dipolar,d=3 = A ± | log |t|| 1/3 (1 - 1 243 (41 + 108 log 4 3 ) | log | log |t|| | log |t|| )
A log-log is so slowly varying that this prediction has remained out of reach for experiments.

and devised "bootstrap" equations to determine the theory. However this approach was temporarily abandoned for a number of reasons

• The self-consistent bootstrap equation for the coupling constant did not contain any small parameter ... until the ǫ-expansion was discovered.

• This self-consistent equation was shown to be equivalent to the fixed point equation of RG [START_REF] Mack | Conformal invariance and short distance behavior in quantum field theory[END_REF] • The full conformal group in generic dimension D has (D + 1)(D + 2)/2 parameters, but if the generators of invariance under rotations, translations, and dilatation are conserved, the remaining D generators are conserved as well. Therefore there was not much interest to go beyond the standard RG.

The point of view changed drastically for D = 2 after the work of Belavin, Polyakov, Zamolodchikov [START_REF] Belavin | I"nfinite conformal symmetry in two-dimensional quantum field theory[END_REF] who pointed out that this was an infinite symmetry. The influence of this article on the last 30 years has been of course considerable. However it is only recently that the use of conformal symmetry plus crossing symmetry and unitarity was revived for three dimensional physics [START_REF] El-Showk | Solving the 3D Ising Model with the Conformal Bootstrap[END_REF]. Using this symmetry and bounds coming from unitarity the authors determine an allowed domain in the plane ∆ σ , ∆ ǫ , the dimensions of the spin and energy operators. For reasons which do not seem yet well understood the 3D Ising exponents lie on this boundary at a well-defined location. The authors also match their method with the ǫ-expansion and the agreement is impressive.

The theory is thus far from finished but without Wilson's RG which solved the problem of critical phenomena, and modified completely the understanding of particle physics, we would still be in the stone age.