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First order transition for the branching random walk at the critical parameter

Considering a critical branching random walk on the real line. From a study of the law of the trajectory of a particle chosen under the polymer measure, we establish a first order transition for the partition function at the critical parameter. This result is strongly related to a recent paper of A\"id\'ekon and Shi in which they solved the problem of the normalisation of the partition function in the critical regime.


Introduction
We consider a real-valued branching random walk : Initially, a single particle sits at the origin. Its children together with their displacements, form a point process Θ on R and the first generation of the branching random walk. These children have children of their own which form the second generation, and behave -relatively to their respective positions at birth-like independent copies of the same point process Θ. And so on.
Let T be the genealogical tree of the particles in the branching random walk. Plainly, T is a Galton-Watson tree. We write |z| = n if a particle z is in the n-th generation, and denote its position by V (z). The collection of positions (V (z), z ∈ T) is our branching random walk.
Following [1], assume throughout the paper the following conditions The branching random walk is then said to be in the boundary case (Biggins and Kyprianou [9]). We refer to [17] for detailed discussions on the nature of the assumption (1.1) and (1.2)).
Let Φ(t) := log E which can be viewed as the normalized partition function of a directed polymer on trees, see the forthcoming (1.12). In the literature W 1,n is called the critical additive martingale associated with the branching random walk. For notational simplification, we write W 1,n = W n for any n ≥ 0 [W 0 := 1]. Under (1.1), T is infinite with positive probability. Moreover the results stated here make only a trivial sense if the system dies out, so it is convenient to introduce : P * (·) := P (· | T is infinite) .
According to the terminology in the study of polymers (see e.g [13]), we say that the region β > 1 is the strong disorder regime, β ≤ 1 the weak disorder regime and β = 1 the critical case.
We are interested here in the regularity of β → W β at β = 1. Biggins [7] proved that the martingale (W n,β ) β∈C converges uniformly on any compact subset of a set Λ * ⊂ C almost surely and in mean. As a by-product, he obtained the analyticity of W β on (0, 1). We shall show that there is a first order transition at β = 1. In order to state our main result, we need to assume that there exist 1 4 > ǫ 0 > 0 and ǫ 0 2 > δ − > 0 such that  max{0, V (x)}e −V (x) , Biggins and Kyprianou, [8], have shown that under the condition E(X(max(0, log X) 2 ) < ∞, E(X max(0, logX)) < ∞, (1. 7) there exists a random variable D ∞ positive on the set of non-extinction such that (1.8) lim n→∞ D n = D ∞ , P * a.s.
Our first result in this paper is the following theorem : Theorem 1.1 Assume (1.1), (1.2) and (1.5). We have : where the convergence holds in P * probability.
Theorem 1.1 relies on a study of the polymer measure at the critical point which will be our second result in this paper. Following Derrida and Spohn [14], we associate each vertex x ∈ T to [∅, x] the unique shortest path relating x to the root ∅ and x i (for 0 ≤ i ≤ |x|) the vertex on [∅, x] such that |x i | = i. The trajectory of x ∈ T corresponds to the ancestor's positions of x, i.e the vector V (x 1 ), ..., V (x |x| ) , whereas (V s (x)) s∈[0,1] designates the linear interpolation of the trajectory of x ∈ T and is defined by Then for each parameter β > 0 and n ∈ N, we define the polymer measure µ (β) n on the disordered tree T n by We will study the law of the trajectory of a particle chosen under the polymer measure. So let (C, ||.|| ∞ ) be the set of continuous function on [0, 1] endowed with the sup-norm ||.|| ∞ and for any A ∈ B (the σ-algebra generated by the open sets of (C, ||.|| ∞ )) define The model of directed polymer on a disordered tree corresponds in some sense to a mean field limit when the dimension goes to infinity, see [14]. This approximation is can be understood in large lattice dimension : indeed, as d increases, two independent paths V 1 and V 2 on the lattice have smaller probability to ever meet in the future. The models on the d-dimensional lattice and on tree with branching number b are asymptotically alike when b = 2d → ∞.
Many authors have already worked on this subject introduced in 1988 by Derrida and Spohn [14]. Recently Mörters and Ortgiese [21] studied the phase transition arising from the presence of a random disorder. Hu and Shi in [15] showed that the derivative martingale appears naturally in the rate convergence of W n → 0. Furthermore, Aïdékon and Shi [1] This result, used repeatedly in our paper, solved the problem of the normalisation of the partition function in the critical regime. Moreover we use the powerful method developed in [1] and establish our second result : where (R s ) s∈[0,1] is a Brownian meander.
This convergence represents an important step in the proof of Theorem 1.1 but it may also have an independent interest. For example we mention an interesting paper by Alberts and Ortgiese [4] who also study a phase transition at the critical case. Theorem 1.2 would yield their Theorem 1.2. Theorem 1.2 gives also an interesting consequence on the "overlap" of the branching random walk which is introduced in [14]. For |u|, |v| = n we define the overlap by Similarly for |u| = |v| = n we can introduce the fraction of time in which the two paths (V (u 1 ), ..., V (u n ))) and (V (v 1 ), ..., V (v n ))) are identical, i.ẽ Clearly, 0 < Q u,v ≤Q u,v ≤ 1.
Finally we stress that Theorem 1.2 is not true P * almost surely. Indeed let us introduce According to Lemma 6.3 in [1] and Theorem 1.5 in [15], there exist C, N > 0 large and c > 0 small, such that for any n > N, P * (A n ) ≥ c. Clearly for any ǫ > 0, n ≥ ǫ −2 , on the set A n , we have µ which implies that (1.14) can not hold P * almost surely.
Similarly, (1.15) can not be strengthened in P * almost sure convergence.
The rest of the paper is organized as follows. In Section 2 we present some preliminaries on branching random walks. Section 3 is devoted to the proof of the Theorem 1.2 and Corollary 1.3. Finally, in Section 4 we prove the Theorem 1.1.

Preliminaries
This section collects some preliminary results on the branching random walk (change of probabilities, an associated one-dimensional random walk), and it entirely comes from Aïdékon and Shi [1].

The many-to-one Lemma
Let (V (x)) be a branching random walk satisfying (1.1) and (1.2). Let (S n ) n≥0 a random walk such that the law of (S 1 ) is given by The condition (1.1) and (1.2) implies that (S n ) is a mean zero random walk and E(S 2 1 ) = σ 2 < ∞. From a simple induction it stems that for any n ≥ 0 and g : R n → R + measurable we have : Equality (2.2) forms the so called many-to-one Lemma which plays a fundamental role in many computations of expectations. The presence of the random walk (S i ) is explained in Lyons, Pemantle, Peres [20], Lyons [19] and Biggins, Kyprianou [8].

The renewal function associated with a one-dimensional random walk
Associated to (S n ), which is a centered random walk real-valued with σ 2 = E[S 2 1 ] ∈ (0, ∞), let h 0 be its renewal function defined by In the following this function will play an important role, so we collect here some facts about h 0 . For any u ≥ 0, h 0 satisfies If we write (2.5) S n := min j≤n S j , n ≥ 0, it is known that there exists c 0 > 0 and θ > 0 such that As a consequence there exists constants c, C > 0 such that As in [1], we will need the following uniform version of (2.6) : as n → ∞, Finally we mention the inequality due to [1] : there exists c > 0 such that for u > 0, a ≥ 0, b ≥ 0 and n ≥ 1,

A spine conditioned to stay positive
Let (V (x)) be a branching random walk satisfying (1.1) and (1.2), let (F n ) be the sigmaalgebra generated by the branching random walk in the first n generations. Since Lyons [19], the spinal decomposition is a widespread technique to study the branching random walk. Usually we introduce the martingale W n := |z|=n e −V (z) to define the probability Q satisfying for any n ∈ N, Q| Fn := W n .P| Fn . Then we obtain a description with a spine of the branching random walk under Q , moreover this spine behaves like a centered random walk. Here we will need a slightly different decomposition, we will work with a spine whose the law is as a random walk conditioned to stay positive. V (y). Then for α ≥ 0 and u ≥ −α let h α (u) := h 0 (u + α). Finally we define the processes From (2.4) and the branching property stem that for any α ≥ 0, (D (α) n , n ≥ 0) is a nonnegative martingale with respect to F n (see Biggins Kyprianou [8] or [1] for a proof). So associated with D (α) n we introduce the new probability measure Q (α) which satisfies for any n, Now we will give a representation with spine of the branching random walk under Q (α) . A justification of this representation can be founded in [1]. Recall that the point process which governs the law at the first generation of (V (x), |x| = 1) is distributed under P as the point process Θ. For any u ≥ −α, the new probability Q (α) makes appear the point procesŝ Θ (α) u whose distribution is the law of (u + V (w), |x| = 1) under Q (u+α) . Then the branching random walk under Q (α) is governed by the followings rules : -w n gives birth to particles distributed according to Θ -Subtrees rooted at all other brother particles are independent branching random walks under P.
See below three facts which we will use continuously : (ii) For any n and any vertex x with |x| = n, we have , is distributed as a Markov chain with transition probabilities given by where p(u, dv) := P(S 1 + u ∈ dv) is the transition probability of (S n ). In the sense of Doob's h-transform, (V (w (α) n )) n∈N has the law of the random walk (S n ) n≥0 conditioned to stay in [−α, ∞]. A convey way to represent this processes is the following identity : for any n ≥ 1 and any measurable function g : Convention : Throughout the paper, c, c ′ , c ′′ denote generic constants which may change from paragraph to paragraph, but are independent of n.

Step A)
For any metric spaces E and F we denote C b (E, F ) := {f : E → F, continuous and bounded}. Let W + the law of the Brownian meander, and (E * (µ n ⊗µ n (·))) n≥0 the sequence of probability measure on First we shall prove that (3.1) implies : In order to obtain (3.3), recall from [10] that for any continuous process the convergence of the finite-dimensional laws and the relative compactness imply the weak convergence. We will obtain the relative compactness via the following criteria (see [23]) : A sequence (P n ) of probability measures on C 2 is weakly relatively compact if and only if the following two conditions hold : i) for every ǫ > 0, there eixst a number A and an integer n 0 such that ii) for every η, ǫ > 0, there exists a number δ and an integer n 0 Proof of the relative compactness of (E * (µ n ⊗ µ n (·))). The first condition is trivially satisfied. For the second we need to control E * 1 (1.13) and (1.4) in [3]. Thus for any ǫ > 0 there exist A, α > 0 large enough such that for any n ∈ N * , we have : Now using the Many-to-one Lemma we have [16] pp 615), we deduce that for any η, ǫ > 0 there exist δ > 0, N ∈ N such that for any n ≥ N, E * (3.4) ≤ 3ǫ. It ends the proof of the relative compactness of (E * (µ n ⊗ µ n (·))).
Now we have to prove that the finite dimensional distributions of E * (µ n ⊗ µ n (·)) converge to those of According to (3.1), for any t The left hand terms of (3.6) and (3.7) are bounded, then by taking the expectation of the product of (3.6) and (3.7) we get This equality is sufficient to affirm that the law of (V (u, t), (V (v, t)) under E * (µ n ⊗ µ n (·)) converges to this one (R t , R ′ t ) under P, which implies (3.5).
To conclude step A) it remains to show that (3.3) implies Theorem 1.2 and Corollary 1.3. For any F ∈ C b (C, R) let F 2 * ∈ C b (C 2 , R) be the function defined by : Then according to (3.3), for any which gives Corollary 1.3.
So we can turn now to the proof of (3.1).

3.2
Step B) : proof of (3.1) Let us also introduce for any α > 0, Following [1], to prove (3.1), we firstly show the following result on Q (α) : for any α ≥ 0, Equality (3.9) represents the exact analogue of (3.1) under Q (α) . Because of the relation (2.10) we will see at the end of this section how to obtain (3.1) from (3.9) by letting α goes to infinity. Working under Q (α) presents the following advantage : under Q (α) the spine remains above a barrier positioned at −α, then the random variables (W n ) are much more concentrated around their mean than W n and D n under P.
Let us start the Proof of (3.9). As in [1] we need to rewrite W x. For any i ≥ 1, let stands for the set of "brothers" of w (α) i ). Accordingly, Let k n < n be an integer such that k n → ∞ (n → ∞). We write |x|=n,x≥y Let also E n, Under Q (α) , (V (w (α) n )) n≥0 has the law of a centered random walk conditioned to stay positive. Moreover it is well known that a such process "tends to infinity" when n goes to infinity. Then keeping this fact in minds we are inclined to affirm that W  Let (k n ) n∈N * := (⌊(log n) 30 ⌋) n∈N * . Fix ǫ, α > 0, t ∈ [0, 1] d and F ∈ C b (R d , R). As ||F || < ∞, for any event A, the expectation in (3.9) ( :=E n ) 2 , thus the expectation in the right hand term of (3.16) is smaller than : where we have recognize in (3.17) the expression of Q (α) w (α) n = u|F n as described in (2.11). Finally there exists n 0 > 0 such that for any n ≥ n 0 , and (3.19) . (3.20) In the following we shall prove that |I n − L n | → n→∞ 0. To achieve this goal we also introduce and (3.21) , (3.22) in order to prove that :

Combining (3.25) with (3.24) we get (i).
(ii) Proof of lim n→∞ |J n − K n | = 0 : By the triangular inequality observe that |J n − K n | is smaller Recalling (3.14), on E n we have |W  , on E n .
Finally it stems that |J n − K n | is smaller which concludes the proof of (ii).
Moreover, by the branching property, conditionally at V (w is independent of G kn , so the term in (3.27) is equal to According to [1] (see (4.9) p21) we know that Furthermore by the Cauchy-Schwartz inequality, for any u ∈ [k 1 3 n , k n ], n , k n ] we have : Recalling that k n = o(n 1 2 ) and using (2.6) and (2.9), we get that for any n ∈ N large enough and any u ∈ [k 1 3 n , k n ], Finally by combining (3.29) and (3.28) we obtain that lim n→∞ |K n − L n | = 0.
It remains to prove that L n → 0 when n goes to infinity. By using the Markov property (assuming n large enough such that k n ≤ min i∈ [1,d] t i n), we get that where we have used (3.28) in the last inequality. Recalling the definition of V t in (1.10) using the Markov property at time k n , then (2.13) we can affirm that for any u ∈ [k 1 3 n , k n ], the expectation in (3.30) is equal to which we rewrite (according to (2.6)), As a straight-forward consequence of the Lemma 5.1, in the Appendix, we have that : uniformly in u ∈ [k 1 3 n , k n ], as n → ∞, Then by combining (3.34), (3.30) and assertion (i), (ii), (iii) we obtain (3.9) We turn now to the Proof of (3.1). Let ǫ > 0. From Theorem 1.1 [3], we know that From (2.6) there exists M = M(ǫ) > 0 such that Now we fix α = α(ǫ) := k + M. Since h α (u) = h 0 (u + α), we have for all vertices x We deduce that on Ω k , for any n ∈ N * , Furthermore P * a.s, D n → n→∞ D ∞ > 0, then let η(ǫ), N(ǫ) such that for any n ≥ N, Gathering all these facts we finally deduce that for any n ≥ N(ǫ), which is smaller that 3ǫ for n large enough according to (3.9). This last inequality ends the proof of (3.1).

An extension of Theorem 1.2
As we will see in the next section, to prove Theorem 1.1 we will need a slightly extension of Theorem 1.2. Formally, it corresponds to the case where F C (f ) = e Cf (1) , f ∈ C, C > 0 : Proposition 3.2 Under (1.1), (1.2) and (1.5), for any C > 0, the following equality is true in P * probability, where R 1 denotes a Brownian meander at time 1.
Proof of Proposition 3.2. From Theorem 1.2 we can affirm that √ n ≤p} = E e σCR 1 , in P * probability.
So in order to prove (3.39) it remains to show that : for any ǫ > 0 as n → ∞ then p → ∞, Let ǫ > 0. We choose k = k(ǫ), α = α(ǫ) as in the proof of (3.1). From (1.13) we recall that lim Recalling also the definition of Ω k we deduce that for A and n large enough, we have where in the last inequality we have used the identity (2.2). By Lemma 5.2 (Appendix) this is smaller than cA ǫ e − p 4 + 2ǫ. Finally we conclude that which ends the proof of Proposition 3.2.
In order to prove the Lemma 4.1, a first step consists to show the following assertion : ( ) There exists c > 0 and α 0 < 1 such that : E W 1+ α 2 β ≤ c for any 0 < α < α 0 .
Proof of ( ). We recall that under the condition (1.6) there exists ǫ 0 , δ − > 0 such that sup The proof of ( ) is similar to this one of Lemma 3 in [6]. Let us introduce the probability measure Q β defined by (4.7) Q β := W β .P.

Appendix
Recall that (S n ) n≥0 is a centred random walk with E(S 2 1 ) := σ 2 < ∞. For any t ∈ [0, 1], let S tn := S ⌊tn⌋ . Recall also the definition of ∆S and Proof of (5.1). To prove (5.1) we can suppose without lost of generality that d = 1. Let t ∈ (0, 1], recalling (3.33) observe there exists c > 0 such that for any n ∈ N large enough we have It is clear that Thus we only need to prove that uniformly in x ∈ [0, k n ], According to the Markov property at time nt − k n we have which gives (5.4).
Proof of (5.2).Let t ∈ (0, 1] d and F : [0, ∞) d → R a continuous function bounded by M > 0. For any (x, y) ∈ [0, b n ] × B(0, b n ) we denote E x (5.1) (y) the expectation in (5.1). According to [2] (p11) we have : for any i ∈ [1, d], ǫ > 0, there exists A(ǫ) large enough such that Thus we can suppose that F is a continuous function with compact support. By approximation, we can also assume that F is Lipschitz. Let (m n ) n≥0 be a sequence of integers such that n mn and mn k 2 n go to infinity. Decomposing E x (5.1) (y) according to the time j such that S j = S n gives : The amount in (5.6) is negligible, indeed according to (2.9) we have Let ǫ > 0. For n large enough and k ≤ m n , we have from (2.6), |P(S n−j ≥ 0)) − θ √ n | ≤ ǫ √ n . Combined with (5.11) and (5.12), for n large enough and any z ∈ B(0, k n ), j ≤ m n , this gives a n (z, j) − P(S n−j ≥ 0)E(F (R t )) ≤ c ǫ √ n .
for n greater than some n 1 , j ≤ m n , z ∈ B(0, k n ). We use this inequality for every k = 0..., m n and we obtain mn j=0 E x a n ((S j + y Together with (5.8) and (5.9), it yields that there exists n 0 ≥ 0 such that for any n ≥ n 0 , x ∈ [0, k n ], y ∈ B(0, k n ) we have which yields (5.2).
The following lemma is a consequence of [12] (pp 8).