Adrien Joseph
email: .adrien.joseph@etu.upmc.fr

A phase transition for the heights of a fragmentation tree

Keywords: Occupancy scheme, multiplicative cascade, random split tree, homogeneous fragmentation, branching random walk

We provide information about the asymptotic regimes for a homogeneous fragmentation of a finite set. We establish a phase transition for the asymptotic behaviours of the shattering times, defined as the first instants when all the blocks of the partition process have cardinality less than a fixed integer. Our results may be applied to the study of certain random split trees.

Introduction

The occupancy scheme is a simple urn model which often occurs in probability, statistics, combinatorics and computer science. We can cite for example species sampling [START_REF] Bunge | Estimating the number of species : a review[END_REF][START_REF] He | Estimating species abundance from occurrence[END_REF], analysis of algorithms [START_REF] Gardy | Occupancy urn models in the analysis of algorithms[END_REF], learning theory [START_REF] Boucheron | An urn model from learning theory[END_REF], etc. The books by Johnson and Kotz [START_REF] Johnson | Urn Models and Their Application. An Approach to Modern Discrete Probability Theory[END_REF] and by Kolchin et al. [START_REF] Kolchin | Random Allocations[END_REF] are standard references. The present work is partly motivated by the study of fragmentation trees.

Let us recall the occupancy scheme. Let I be a countable set and p = (p i : i ∈ I) be a probability measure on I. The occupancy scheme on (I, p) is described as follows. For all i ∈ I such that p i = 0, one places a box at i. One then throws successively and independently n balls in the boxes by assuming that each ball has probability p i of falling into the box located at i. We may be interested in the number of boxes containing exactly j balls, or in the number of occupied boxes, etc.

We consider here a variant of the occupancy scheme which corresponds to a nested family of boxes. We introduce the infinite genealogical tree U :

U := ∞ k=0 N k ,
with N = {1, 2, . . . } and the convention N 0 = {∅}. The elements of U are called individuals, and for every integer k ∈ Z + , the k-th generation of U is formed by the individuals i ∈ N k (we write |i| = k). For every individual i = (i 1 , . . . , i k ) of U and j ∈ N, the individual ij = (i 1 , . . . , i k , j) is called the j-th child of i and i is the parent of ij. We suppose a real number p i ∈ [0, 1] is assigned to each individual i, such that p ∅ = 1 and j∈N p ij = p i for all i ∈ U. Note that for all k, p(k) := (p i : |i| = k) is a probability measure on N k . We can couple the occupancy schemes on (N k , p(k)) as follows. Each individual i should be viewed as a box, such that the box ij is contained into the box i for every i ∈ U and j ∈ N such that p ij = 0. Initially the n balls are thrown in the box located at ∅. Then one places successively and independently the n balls in the boxes of the first generation by assuming that each ball has probability p i of falling into the box situated at i ∈ N. Likewise, by iteration, a ball located in the box i is placed independently of the others into the sub-box ij with probability p ij /p i . We denote by H n,j the first generation at which all the boxes have less than j ≥ 2 balls when n have been thrown (H n,j is called a height) and by G n,j the first level at which there exists a box containing less than j ≥ 1 balls (G n,j is called a saturation level ). Our aim is to study the asymptotic behaviours of H n,j and G n,j as n tends to infinity when we consider a certain randomized version of (p i , i ∈ U).

More precisely, in the present work, we shall assume that we are given a random probability measure ρ = (ρ 1 , ρ 2 , . . . ) on N. We assign to each individual i an independent copy ρ(i) of ρ. The real numbers p i , i ∈ U, are defined by induction : p ∅ := 1 and p ij := p i ρ j (i), for all i ∈ U and j ∈ N. It means that ρ(i) describes how the mass of the individual i is splitted to its chidren. This model is called the occupancy scheme of multiplicative cascades. We put emphasis on the fact that there are two levels of randomness in our model, namely the arrangement of boxes and the way one throws balls.

In the particular case when ρ is supported by a finite number of integers, in the sense that #{j : ρ j > 0} ≤ b a.s. for some integer b ≥ 2, the height H n,j has a natural interpretation in terms of a special class of random split trees which have been considered e.g. by Devoye [START_REF] Devroye | Universal limit laws for depths in random trees[END_REF]. Specifically, imagine that each box has a rupture threshold of j, in the sense that when a ball falls into some box i already containing j -1 balls, then this box is removed and the j balls are shared out amongst the children of i according to the random probability ρ(i) (i.e. conditionally on ρ(i), each ball is put in the box ij with probability ρ j (i), independently of the other balls). This procedure yields a random tree where all balls are stored at leaves, and H n,j is the height of this tree when n balls have been thrown.

We further point out that the height H n,j also arises as a natural shattering time in homogeneous fragmentation chains, a class of partition-valued Markov chains. More precisely, the shattering time is defined as the first instant when all the blocks of the partition process have cardinality less than j. See [START_REF] Bertoin | Random Fragmentation and Coagulation Processes[END_REF] for background and Section 3.2 in [START_REF] Bertoin | Asymptotic regimes for the occupancy scheme of multiplicative cascades[END_REF] for a description which is closely related to the present work. In a different direction, we mention the work of Haas et al. [START_REF] Haas | Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models[END_REF] who associate another random tree to homogeneous fragmentation processes.

We shall show the following result : there exist an integer j * ∈ {2, 3, . . . } ∪ {∞} and a sequence of positive real numbers

C 2 > • • • > C j * = C j * +1 = • • • =: C * such that for every integer j ≥ 2, H n,j ∼ n→∞ C j ln n a.s.
It should not be surprising that the heights H n,j have logarithmic asymptotics, as a large class of trees have such a behaviour, like random split trees. On the other hand, it is remarkable that there exists a critical parameter j * ∈ {2, 3, . . . } ∪ {∞} at which a phase transition occurs (provided that 2 < j * < ∞). It is a peculiar behaviour for a random split tree. Devroye indeed proved in [START_REF] Devroye | Universal limit laws for depths in random trees[END_REF] that for a large class of random split trees such that their internal nodes contain at least one ball1 , then all the heights, regardless of the value of the rupture threshold j, have the same asymptotic behaviour (in probability). In particular, no phase transition occurs. It may also be interesting to consider the situation where the parameter j depends on the number of balls n. In the case of power functions, we shall prove that under some technical conditions that will be detailed below,

H n,n α ∼ n→∞ (1 -α)C * ln n a.s.,
for all α ∈ (0, 1), so that there is no phase transition in the asymptotics of H n,n α .

Concerning the asymptotics of the saturation level G n,j , we shall prove the following result : under some technical conditions that will be detailed below, there exists a constant C * < C * such that for every positive integer j,

G n,j ∼ n→∞ C * ln n a.s.
In particular, there is no phase transition in the asymptotics of G n,j . We shall also show that under some further technical conditions, for all α ∈ (0, 1),

G n,n α ∼ n→∞ (1 -α)C * ln n a.s.,
so that there is also no phase transition in the asymptotic behaviours of G n,n α .

Our approach essentially relies on the theory of branching random walks, and more precisely on their large deviations behaviours whose descriptions are due to Biggins [START_REF] Biggins | Uniform convergence of martingales in the branching random walk[END_REF]. The construction of the boxes given by the multiplicative cascades indeed enables us to define a branching random walk giving the sizes of the boxes at each generation. We shall see that the critical parameter j * and the real numbers C 2 , . . . , C j * and C * are described by that branching random walk. Another key technique is Poissonization; instead of throwing exactly n balls, one throws P n balls, where P n is a Poisson variable with parameter n which is independent of (ρ(i), i ∈ U). For every integer k, conditionally on the sizes of the boxes of the k-th generation, the numbers of balls per box of the k-th generation are thus independent Poisson variables.

The results will be stated in Section 2. We shall see the main techniques in Section 3. Section 4 will be devoted to the study of the heights. We shall first turn our attention to the upper bound. Due to the phase transition, we shall give two different proofs to show the lower bound of H n,j , depending on whether the integer j is less than the critical parameter j * or not. The results on the saturation levels will be proved in Section 5. Finally, in Section 6, we shall explain heuristically why a phase transition may occur in the asymptotics of H n,j , but not in those of H n,n α , G n,j and G n,n α .

Formulation of the main results

Recall that ρ is a random probability measure on N. We denote its law by ν. If we denote by Prob N the space of probability measures on N, ν is a probability measure on Prob N , called the splitting law. We assume that ν is not geometric2 , in the sense that there is no real number r ∈ (0, 1) such that with probability one, all the masses of the atoms of ρ belong to {r n , n ∈ Z + }. In particular, the degenerate case when ρ is a Dirac point mass a.s. is therefore excluded.

As explained in the introduction, we consider a family (ρ(i), i ∈ U) of independent copies of ρ labeled by the individuals of the genealogical tree U. The multiplicative cascade construction defines for each generation k a probability measure (p i , |i| = k) on N k . Taking logarithm of masses, we may encode the latter by the following random point measure on

R + Z (k) (dy) := i:|i|=k δ -ln p i (dy),
where δ z stands for the Dirac point mass at z. Note that if p i = 0, i.e. if there is no box at i, the individual i is omitted in the sum defining Z (k) . Likewise, in the sequel, we shall always consider individuals which have a positive mass. It follows immediately from the structure of the multiplicative cascades that (Z (k) , k ∈ Z + ) is a branching random walk, in the sense that for every integers k, k ′ ≥ 0, Z (k+k ′ ) is obtained from Z (k) by replacing each atom z of Z (k) by a family {z + y, y ∈ Y z }, where each Y z is an independent copy of the family of atoms of Z (k ′ ) . Let us introduce quantities defined via the splitting law ν. First, we define the Laplace transform of the intensity measure Z (1) by L(θ) := E Z (1) , e -θ• for θ ∈ R. We can also write

L(θ) = E j∈N ρ θ j = Prob N j∈N p θ j ν(dp)
with the convention that p θ = 0 when p = 0 even when θ ≤ 0. Because ρ is not a Dirac point mass a.s., L(0) > 1. The function L : R → (0, ∞] is decreasing with L(1) = 1. We define

θ := inf{θ ∈ R : L(θ) < ∞},
so that L(θ) < ∞ when θ > θ. Note that θ may be negative. We can show from Hölder's inequality that ln L is a convex function, which implies the convexity of L and that the function

ϕ : θ → ln L(θ) -θ L ′ (θ) L(θ)
is increasing on (θ, 0) and decreasing on (0, ∞). As L(1) = 1 and L decreases, we have ϕ(1) = -L ′ (1) > 0, and thus the set of θ ∈ (θ, ∞) such that ϕ(θ) > 0 is a non-empty open interval (θ * , θ * ) :

θ * := inf{θ > θ : ϕ(θ) > 0} and θ * := sup{θ > θ : ϕ(θ) > 0}.
Note that θ * > 1 and θ * < 0 if θ < 0. The convexity of ln L implies that -L/L ′ is an increasing function. We define

C * := lim θ↓θ * ↓ - L(θ) L ′ (θ) and C * := lim θ↑θ * ↑ - L(θ) L ′ (θ) .
Inspired by the article of Hu and Shi in [START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF] (see Lemma 4 below), we shall sometimes need the following assumption : there exists δ > 0 such that L(-δ) < ∞ and

Prob N i∈N 1 p i >0 1+δ ν(dp) < ∞. (1)
To study the asymptotics of the saturation levels, we shall sometimes need the following hypothesis :

-∞ < θ * < 0 and ϕ(θ * ) = 0.

(

) 2
We can now state the results that we shall prove. Concerning the asymptotic behaviours of the heights H n,j , we have the following results which complete and improve Proposition 2 in [START_REF] Bertoin | Asymptotic regimes for the occupancy scheme of multiplicative cascades[END_REF].

Theorem 1 Let j ≥ 2 be an integer, set

C j := -j/ ln L(j) if j < θ * , C * if j ≥ θ * . Then H n,j ∼ C j ln n a.s.
More precisely, H n,j ≤ C j ln n + O(ln ln n) a.s., [START_REF] Bertoin | The asymptotic behavior of fragmentation processes[END_REF] and ( 3) is an equality if j < θ * or if [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] holds.

We see that there is a phase transition in the asymptotic behaviour of H n,j at the integer ⌈θ * ⌉ when 2 < θ * < ∞. We point out that it may happen that θ * ≤ 2 or θ * = ∞, in which case there is no phase transition. For instance, one can show that if ρ = (ρ 1 , ρ 2 , . . . ) with ρ 1 = 1 -0.75U, ρ j = 0.05U for all j ∈ 2, . . . , 16 and ρ j = 0 for all j ≥ 17, where U is uniformly distributed on (0, 1), then θ * < 1.99. On the other hand, for every α ∈ [1/2, 1), if ρ is equal to (1/2, 1/2, 0, . . . ) with probability α and to (1/3, 1/3, 1/3, 0, . . . ) with probability 1α, then θ * = ∞.

In this direction, we also point out that the critical parameter θ * is finite whenever

max j∈N ρ j ∞ = 1,
where ρ = (ρ j , j ∈ N) denotes a random probability measure on N with law ν. Indeed, it is easily seen that lim θ→∞

L(θ) 1/θ = max j∈N ρ j ∞ .
If the right-hand side equals 1, then g : θ →ln L(θ)/θ has limit 0 at infinity. Now g is derivable and g(1) = 0, so there exists

θ 0 ∈ (1, ∞) such that g ′ (θ 0 ) = 0. As g ′ (θ) = θ -2 ϕ(θ), we conclude that θ 0 = θ * < ∞.
We may also be interested in the asymptotics of H n,n α , where α ∈ (0, 1). We shall prove the following result.

Proposition 1 Suppose θ * < ∞. Let α ∈ (0, 1). Then H n,n α ∼ (1 -α)C * ln n a.s. Furthermore, H n,n α ≤ (1 -α)C * ln n + O(ln ln n) a.s., (4)
and ( 4) is an equality whenever (1) holds.

Remark 1

The final assertions in Theorem 1 and Proposition 1 rely on the work of Hu and Shi [START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF]. McDiarmid's setting in [START_REF] Mcdiarmid | Minimal positions in a branching random walk[END_REF] can however be considered; we can prove that the results stated in Theorem 1 and in Proposition 1 still hold if the assumption (1) is replaced by the following :

Prob N i∈N 1 p i >0 2 ν(dp) < ∞. (5)
For instance, suppose that ρ = (ρ 1 , 1ρ 1 , 0, 0, . . . ), where ρ 1 is a random variable with density 1 0<x<e -1 x -1 ln -2 xdx. Then θ = 0, so (1) does not hold. Nonetheless, ρ is supported by two integers a.s., so (5) holds. As a result, for all j ≥ 2, H n,j = C j ln n + O(ln ln n) a.s. Furthermore, as max j∈N ρ j ∞ = 1, the discussion below Theorem 1 ensures that θ * is finite, so for all α ∈ (0, 1),

H n,n α = (1 -α)C * ln n + O(ln ln n) a.s.
For the sake of simplicity, we shall show Theorem 1 and Proposition 1 only in the setting of [START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF] (see Proposition 4 below). The general proof can however be easily carried out.

Concerning the asymptotic behaviours of the saturation levels G n,j , we shall prove the following theorem.

Theorem 2 Let j ≥ 1 be an integer.

• If θ * = -∞, then G n,j ∼ C * ln n a.s.
• Suppose that (2) holds. Then

G n,j ∼ C * ln n a.s. More precisely, G n,j ≥ C * ln n + O(ln ln n) a.s., (6)
and ( 6) is an equality if (1) holds.

A sufficient condition to guarantee that -∞ < θ * < 0 and ϕ(θ * ) = 0 is :

θ > -∞ and lim θ↓θ ↑ L(θ) = ∞.
Indeed, suppose this condition fulfilled. Imagine that for all θ < 0, ϕ(θ

) > 0. Let θ 0 ∈ (θ, 0). As θ → ln L(θ)/θ is decreasing (its derivative is -θ -2 ϕ(θ) < 0), we have for all θ < θ 0 : ln L(θ) ≤ θ ln L(θ 0 ) θ 0 ≤ θ ln L(θ 0 ) θ 0 , which contradicts lim θ→θ L(θ) = ∞.
We shall also show the following proposition.

Proposition 2 Suppose that (2) holds. Let α ∈ (0, 1). Then

G n,n α ∼ (1 -α)C * ln n a.s. Moreover, G n,n α ≥ (1 -α)C * ln n + O(ln ln n) a.s., (7)
and ( 7) is an equality if (1) holds.

We now conclude this section by discussing an illustrative example. Consider the case ρ = (U, 1 -U, 0, 0, . . . ), where U is uniformly distributed on [0, 1]. Then L(θ) = 2/(θ + 1), so θ = -1 and (1) holds. The discussions below the theorems yield : θ * < ∞ and (2) hold. Hence, all our results may be applied. Easy calculations yield ϕ(θ) = ln 2-ln(θ+1)+θ/(θ+1) and ⌈θ * ⌉ = 4, so that there is a phase transition. We can show that C 2 = 2/ ln(3/2) ≈ 4, 93260..., C 3 = 3/ ln 2 ≈ 4, 32808... and that C * > C * are the solutions of the equation

ln 2 c + c -1 c = 0,
i.e. C * ≈ 4, 31107... and C * ≈ 0, 37336... As ρ is supported by two integers, our model may be interpreted in terms of random split trees; the procedure described in the introduction yields a random tree where all balls are stored at leaves. We denote by T n,j the tree obtained when n balls have been thrown and when the boxes have a rupture threshold of j. We now define another random split tree Tn,j also related to our model. We imagine that each box has a rupture threshold of j, but when a ball falls into some box i already containing j -1 balls, it remains in that box and the j -1 other balls are shared out amongst the two children of i independently and with probability (1/2, 1/2). No other ball is then allowed to be stored at the box i : when a ball falls into i, it is placed into one of the two children of i with probability (1/2, 1/2), and one has to consider whether the ball stays at that box or not. We denote by Tn,j the tree obtained when n balls have been thrown. Note that each internal node of Tn,j has exactly one ball, whereas its leaves have less than j balls. We see that the height Hn,j of the tree Tn,j is less than or equal to the height H n,j of the tree T n,j .

It is remarkable that Tn,2 has the law of the random binary search tree. Robson [START_REF] Robson | The height of binary search trees[END_REF] was the first to be interested in the height of the random binary search tree. In [START_REF] Pittel | On growing random binary trees[END_REF], Pittel studied its height and its saturation level. Devroye proved in [START_REF] Devroye | A note on the height of binary search trees[END_REF][START_REF] Devroye | Branching processes in the analysis of the heights of trees[END_REF] that the saturation level is asymptotically equivalent to C * ln n in probability as in our model, but that the height Hn,2 is equivalent to C * ln n in probability, so that Hn,2 is not equivalent to H n,2 . Regarding the random binary search tree as a random split tree, he proved in [START_REF] Devroye | Universal limit laws for depths in random trees[END_REF] that all the heights Hn,j (regardless of the value of j) of the random binary search tree are asymptotically equivalent to C * ln n in probability. In particular, there is no phase transition, contrary to our case. There is of course no contradiction, as the random binary search tree does not correspond to a model treated in this work. Indeed, in terms of random split trees, internal nodes of the binary search tree retain exactly one ball whereas all the balls are stored at leaves in our case.

Preliminaries

Some results on branching random walks

In this section, we recall some results on branching random walks that will be used in the proofs.

We begin by stating a key result obtained by Biggins in [START_REF] Biggins | Uniform convergence of martingales in the branching random walk[END_REF]. For every θ > θ, we introduce

W (k) (θ) := L(θ) -k Z (k) , e -θ• = L(θ) -k i:|i|=k p θ i .
For every k ∈ Z + ∪ {∞}, we denote by F k the σ-algebra generated by (p i , |i| ≤ k).

Lemma 1 For every θ > θ, (W (k) (θ), k ∈ Z + ) is a martingale with respect to the filtration

(F k ) k∈Z + . Moreover, if θ ∈ (θ * , θ * ), it is bounded in L γ (P)
for some γ > 1 and therefore uniformly integrable, and its terminal value

W (θ) := lim k→∞ W (k) (θ)
is positive a.s.

Applying Corollary 4 in [START_REF] Biggins | Uniform convergence of martingales in the branching random walk[END_REF], we can obtain a precise estimate of the number of boxes at generation k with size of order exp(k L ′ (θ)/ L(θ)). Specifically Lemma 2 For all real numbers a and b such that a < b and for all θ ∈ (θ * , θ * ), we have with probability one that

lim k→∞ √ ke -kϕ(θ) # i ∈ N k : exp (-b + kL ′ (θ)/ L(θ)) ≤ p i ≤ exp (-a + kL ′ (θ)/ L(θ)) = 1 √ 2π e θb -e θa θ L ′′ (θ) L(θ) - L ′ (θ) L(θ)
We next turn our interest to the asymptotic behaviours of extreme sizes of boxes at generation k : p(k) := inf {p i : p i > 0, |i| = k} and p(k) := sup {p i : |i| = k} .

We have the general following results :

Lemma 3 If θ * < ∞, then lim k→∞ (-ln p(k) -k/C * ) = ∞ and lim k→∞ - ln p(k) k = 1 C * a.s.
Likewise, if (2) holds, then

lim k→∞ ln p(k) + k/C * = ∞ and lim k→∞ - ln p(k) k = 1 C * a.s.
To have precise estimates of the behaviours of the heights and of the saturation levels, we shall sometimes need sharper results of the asymptotics of p(k) and p(k). The following result, proved by Hu and Shi in [START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF], will be very useful.

Lemma 4 We assume that (1) holds.

• If θ * < ∞, then lim sup k→∞ -ln p(k) -k/C * ln k = 3 2θ * a.s. • If (2) holds, then lim sup k→∞ ln p(k) + k/C * ln k = - 3 2θ * a.s.
Addario-Berry and Reed in [START_REF] Addario-Berry | Minima in branching random walks[END_REF] also studied the minima in branching random walks, but they require a stronger condition than (1). Their assumption is however fulfilled for random split trees.

Poissonization

Let us present the methods used in the proofs. We have to consider the number of balls belonging to each box at each generation. Now, conditionally on F k , when n balls have been thrown, the number of balls in the box i, |i| = k, follows a binomial law of parameter np i . Furthermore, these random variables are not independent. A classical idea to circumvent those difficulties (see for instance Gnedin et al. in [START_REF] Gnedin | Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws[END_REF] or Holst in [START_REF] Holst | On birthday, collectors', occupancy and other classical urn problems[END_REF]) is to consider a randomized version of the total number of balls : instead of throwing initially n balls, one throws P n balls, where P n is a Poisson variable with parameter n which is independent of (ρ(i), i ∈ U).

More precisely, we suppose we are given a standard Poisson process (P x ) x≥0 independent of F ∞ . For every individual i ∈ U and for every x ∈ (0, ∞), we denote by C(i; x) the number of balls at i when the first P x balls have been thrown. For all x, y ∈ (0, ∞) and k ∈ Z + , we denote by N x,y (k) the number of boxes at generation k containing at least y balls when the first P x balls have been thrown :

N x,y (k) := #{i ∈ N k : C(i; x) ≥ y}.
Conditionally on F k , the random variables (C(i; n)) |i|=k are independent Poisson variables with parameters np i . That is why in our proofs, we shall first focus on N n,j (k). We shall then show that N n,j (k) is close to the number N n,j (k) of boxes at generation k containing at least j balls when exactly n balls have been thrown. Similarly, we denote by M x,y (k) the number of boxes at generation k containing less than y balls when the first P x balls have been thrown :

M x,y (k) := #{i ∈ N k : C(i; x) < y}
and by M n,j the number of boxes at generation k containing less than j balls when n balls have been thrown. We now prove two estimates using the technique of Poissonization that will then be used in the sequel.

P (P x ≥ j) j p x -p ≤ c(p), (8)
and

sup j∈N sup x>0 P (P x < j) j -p x p ≤ d(p). (9)
Proof : We begin by showing [START_REF] Boucheron | An urn model from learning theory[END_REF]. Let j ≥ p be an integer. Let x > 0. By Markov's inequality, P (P x ≥ j) = P (P p x ≥ j p ) ≤ j -p E [P p x 1 Px≥j ] . Therefore we only have to bound from above

x -p E [P p x 1 Px≥j ] = x -p ∞ k=j k p e -x x k k! .
As Γ(kp + 1)k p /k! → 1 as k tends to infinity, there exists a finite constant c(p) ≥ 1 such that for all k > p -1,

k p k! ≤ c(p) Γ(k -p + 1)
.

As j ≥ p > p -1, we thus have

x -p ∞ k=j k p e -x x k k! ≤ c(p)e -x ∞ k=j x k-p Γ(k -p + 1) = c(p)e -x ∞ k=0 x k+u Γ(k + u + 1)
,

where u := jp ≥ 0. Applying the formulae 6.5.1, 6.5.4 and 6.5.29 in [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], we get that

∞ k=0 x k+u Γ(k + u + 1) ≤ e x ,
which proves (8).

To show (9), we write by Markov's inequality :

P(P x < j) = P (P x + 1) -p ≥ j -p ≤ j p E (P x + 1) -p = j p ∞ k=0 (k + 1) -p e -x x k k! .
As Γ(k + p + 1)(k + 1) -p /k! → 1 as k tends to infinity, there exists a finite constant d(p) ≥ 1 such that for all k ∈ Z + , Γ(k + p + 1)(k + 1) -p /k! ≤ d(p). We thus have

P(P x < j) ≤ d(p)j p x -p e -x ∞ k=0
x k+p Γ(k + p + 1) .

We conclude as before.

Remark 2 We stress that in [START_REF] Boucheron | An urn model from learning theory[END_REF], the integer j cannot be less than p. This restriction lies in the heart of the existence of the phase transition in the asymptotics of H n,j (see Proposition 3 below, and more precisely Lemma 6). On the other hand, there is no restriction in (9); no phase transition appears in the asymptotics of G n,j .

Study of the heights

In this section, we prove Theorem 1 and Proposition 1. We are first interested in the upper bound. We shall then focus on the lower bound. We stress that the phase transition is glimpsed at the beginning of the study of the upper bound (see Remark 3 below). It is however proved in the paragraph dealing with the lower bound.

Upper bound

Equation (8) will enable us to have a uniform upper bound of N n,j (k) independent of the sizes of the boxes that will eventually lead to the inequalities (3) and (4).

Proposition 3 Let j ≥ 1 be an integer and α ∈ [0, 1) such that (j, α) = (1, 0). Then

lim sup n→∞ H n,jn α -(1 -α) θ -ln L(θ) ln n ln ln n ≤ 1 -ln L(θ) a.s.
for every θ > 1 if α > 0, and for every θ ∈ (1, j] if α = 0.

Remark 3 Suppose that α = 0. We deduce from Proposition 3 that :

H n,j ≤ min θ∈(1.j] h(θ) ln n + O(ln ln n) a.s.,
where h is the function θ → -θ/ ln L(θ). Now, h is derivable and h ′ (θ) = -ϕ(θ) ln -2 L(θ), so h is decreasing on (1, θ * ] and increasing on [θ * , ∞). The minimum of h is therefore

• -j/ ln L(j) if j < θ * , • -θ * / ln L(θ * ) = C * if j ≥ θ * .
The phase transition is shown. It will be proved in the study of the lower bound.

Recall that H n,jn α ≤ k if and only if at genetation k, every box contains less than jn α balls when n balls have been thrown. Because we want to show that H n,jn α is bounded

from above by (1 -α) θ -ln L(θ) + ε ln(n), we take k ≈ (1 -α) θ -ln L(θ) + ε ln(n).
In other words, one initially throws n ≈ exp k 1 1-α ln L(θ) θ

ε balls and we show that every box of the k-th generation contains less than jn α balls.

Let u and u ′ be two real numbers such that

u > 1 (1 -α)θ and u < u ′ < 1 α u - 1 θ ,
where for α = 0, the second condition reduces to u ′ > u. Define for every k ≥ 1

x k := k -u exp k 1 1 -α -ln L(θ) θ and φ k := k -u ′ exp k 1 1 -α -ln L(θ) θ .
Informally, x k corresponds to the number of balls thrown when we consider boxes at generation k. Note that

x k ≈ exp k 1 1-α -ln L(θ) θ -ε .
As mentionned in the preliminaries, the argument relies on Poissonization. We are first interested in N x,y (k) and we shall see how to depoissonize.

Lemma 6

For almost all ω, there exists k 0 (ω) such that

N x k ,jφ α k (k) = 0, for all k ≥ k 0 (ω).
Proof : Let x > 0 and y ≥ θ. We calculate E [N x,y (k)|F k ]. We write

N x,y (k) = i:|i|=k 1 C(i;x)≥y = i:|i|=k 1 C(i;x)≥⌈y⌉ .
Conditionally on F k , (C(i; x)) |i|=k are Poisson variables with parameters xp i , so

E [N x,y (k)|F k ] = i:|i|=k P(P xp i ≥ ⌈y⌉).
As ⌈y⌉ ≥ θ, [START_REF] Boucheron | An urn model from learning theory[END_REF] ensures that

E [N x,y (k)|F k ] ≤ i:|i|=k c(θ)⌈y⌉ -θ (xp i ) θ (see Remark 2). Now, as y ≥ θ, ⌈y⌉ ≥ y -1 y y ≥ θ -1 θ y.
In the notations of Lemma 1, we have :

E [N x,y (k)|F k ] ≤ c ′ (θ)y -θ x θ L(θ) k W (k) (θ),
where c ′ (θ) := c(θ)(1 -1/θ) -θ . We finally get

E [N x,y (k)] ≤ c ′ (θ)y -θ x θ L(θ) k .
Taking x = x k and y = jφ α k , we get for all k sufficiently large so that jφ α k ≥ θ (note that if α = 0, then for all k, jφ α k ≥ θ) :

E N x k ,jφ α k (k) ≤ c ′ (θ)j -θ k θ(u ′ α-u) . Now θ(u ′ α -u) < -1. As a consequence E k∈N 1 N x k ,jφ α k (k)≥1 ≤ E k∈N N x k ,jφ α k (k) < ∞.
In particular, there is an a.s. finite number of integers k such that

N x k ,jφ α k (k) ≥ 1.
We now show how to have information on N n,jn α (k) itself.

Proof of Proposition 3 : Let E k denote the event {P x k ≤ φ k+1 }. Because u ′ > u, φ k+1 /x k tends to 0 as k tends to infinity, so there exists an integer k 1 such that for all k ≥ k 1 , φ k+1 ≤ x k /2. Consequently,

P (E k ) ≤ P (P x k ≤ x k /2) = P (P x k -x k ≤ -x k /2) ≤ P (|P x k -x k | ≥ x k /2) .
The variance of the Poisson variable P x k being x k , we get by Chebichev's inequality:

P (E k ) ≤ 4x -1 k for all k ≥ k 1 . As k∈N x -1
k < ∞, the Borel-Cantelli lemma ensures that for almost all ω, there exists k 2 (ω) such that P x k > ⌊φ k+1 ⌋ for all k ≥ k 2 (ω).

Applying Lemma 6, we deduce that, if we define the event Ω 0 by Ω 0 := ω : there exists k 3 (ω) such that N ⌊φ k+1 ⌋,jφ α k (k) = 0 for all k ≥ k 3 (ω) , then P(Ω 0 ) = 1. Notice that there exists an integer k 4 such that for all k ≥ k 4 ,

k -u ′ ≥ exp -k 1 2(1 -α) -ln L(θ) θ .
There exists a rank k 5 from which the sequence (φ k ) k≥k 5 is increasing. Furthermore, that sequence tends to infinity. Let ω ∈ Ω 0 . Let n be an integer greater than φ k 5 large enough so that the unique k ≥ k 5 satisfying φ k ≤ n < φ k+1 is greater than k 3 (ω) and k 4 . Because

N ⌊φ k+1 ⌋,jφ α k (k) = 0 and n ≤ ⌊φ k+1 ⌋, N n,jφ α k (k) = 0. Now, jn α ≥ jφ α k , so N n,jn α (k) = 0 and H n,jn α ≤ k. Moreover, as n ≥ φ k , k ≤ (1 -α) θ -ln L(θ) ln n + u ′ (1 -α) θ -ln L(θ) ln k.
Further, as k ≥ k 4 , we have :

exp k 1 2(1 -α) -ln L(θ) θ ≤ k -u ′ exp k 1 1 -α -ln L(θ) θ ≤ n, so k ≤ 2(1 -α) θ -ln L(θ) ln n. Therefore H n,jn α ≤ k ≤ (1 -α) θ -ln L(θ) ln n + u ′ (1 -α) θ -ln L(θ) ln 2(1 -α) θ -ln L(θ) ln n .
We conclude that lim sup

n→∞ H n,jn α -(1 -α) θ -ln L(θ) ln n ln ln n ≤ u ′ (1 -α) θ -ln L(θ)
a.s.

The fact that u ′ can be chosen arbitrarily close to 1 (1-α)θ completes the proof.

Lower bound

We now study the lower bound. Combined with the upper bound supplied by Proposition 3, Theorem 1 and Proposition 1 will be proved.

4.2.1

The case j ≥ θ * or α ∈ (0, 1) and θ * < ∞

In this section, we prove that H n,j ≥ C * ln n + O(ln ln n) a.s. if j ≥ θ * and (1) holds. We also prove that Let γ, γ ′ and γ ′′ be three real numbers such that

H n,n α ≥ (1 -α)C * ln n + O(ln ln n) a.s. if θ * < ∞
γ > 3 2θ * , γ ′ > γ 1 -α and γ ′ < γ ′′ < γ ′ -γ α .
Note that if α = 0, the third condition is simply γ ′′ > γ ′ . Define for all k ∈ Z +

x k := k γ ′ exp k 1 (1 -α)C * and φ k := k γ ′′ exp k 1 (1 -α)C * .
We first show that a.s, N x k ,jφ α k (k) ≥ 1 for all integers k sufficiently large. To do so, we simply consider the largest box.

P(A k ∩ B k ) ≤ P(B k |A k ) = P(P x k p(k) < jφ α k
). Applying (9), we get :

P(A k ∩ B k ) ≤ d(p)j p k -2 ,
where p := 2/(γ ′γαγ ′′ ) > 0, and hence

P(A k ∩ B k ) < ∞.
By the Borel-Cantelli lemma, we deduce that a.s., for all k sufficiently large, p(k) ≥ p(k) or the box b(k), which is contained in the box b(k), has at least jφ α k balls when the first P x k balls have been thrown. Now, by Lemma 4, we know that a.s., for all integers k sufficiently large, p(k) < p(k). Lemma 7 is therefore proved.

We now deduce from Lemma 7 the lower bound of H n,jn α .

Proof of Proposition 4 :

The same calculations performed at the beginning of the proof of Proposition 3 show that for almost all ω, there exists k 1 (ω) such that P x k < ⌈φ k-1 ⌉ for all k ≥ k 1 (ω). Applying Lemma 7, we deduce that, if we define the event Ω 0 by Ω 0 := ω : there exists k 2 (ω) such that N ⌈φ k-1 ⌉,jφ α k (k) ≥ 1 for all k ≥ k 2 (ω) , then P(Ω 0 ) = 1. Let ω ∈ Ω 0 . The sequence (φ k ) is increasing and tends to infinity. Let n be an integer large enough so that the unique integer

k satisfying φ k-1 < n ≤ φ k is greater than k 2 (ω). Because N ⌈φ k-1 ⌉,jφ α k (k) ≥ 1 and ⌈φ k-1 ⌉ ≤ n, N n,jφ α k (k) ≥ 1. Now, jn α ≤ jφ α k , so N n,jn α (k) ≥ 1 and H n,jn α > k. Further, as n ≤ φ k , k ≥ (1 -α)C * ln n -γ ′′ (1 -α)C * ln k. As exp (k -1) 1 (1-α)C * ≤ φ k-1 ≤ n, we have : k ≤ (1 -α)C * ln n + 1. Thus H n,jn α > k ≥ (1 -α)C * ln n -γ ′′ (1 -α)C * ln ((1 -α)C * ln n + 1) and lim inf n→∞ H n,jn α -(1 -α)C * ln n ln ln n ≥ -γ ′′ (1 -α)C * a.s.
The fact that γ ′′ can be chosen arbitrarily close to

4.2.2

The case 2 ≤ j < θ * and α = 0

We now prove that H n,j ≥ C j ln n + O(ln ln n) a.s. if j < θ * . We shall notice that contrary to the case j ≥ θ * , the number of boxes matters more than their sizes.

Proposition 5 Let j < θ * be an integer greater than 1. Then

lim inf n→∞ H n,j - j -ln L(j) ln n ln ln n ≥ 1 2 ln L(j)
a.s.

The main difference with the case j ≥ θ * is that we cannot consider the largest box any more. If θ * < ∞, one can even show by performing the usual calculations that with probability one, for all k ≥ C * ln n + o(ln n), the largest box of the k-th generation contains no ball when n balls have been initially thrown. We shall rather focus on some other boxes which are smaller but sufficiently numerous so that it is very unlikely that all of them contain less than j balls when n balls have been thrown. As we want to prove that H n,j is bounded from below by j ln L(j)ε ln n, we consider the situation at the k-th generation when one initially throws approximatively exp (k (-ln L(j)/j + ε)) balls.

The boxes that will play a key role are those appearing in Lemma 2 (recall that j < θ * ) : with probability one,

lim k→∞ √ ke -kϕ(j) # i ∈ N k : s(k) ≤ p i ≤ 2s(k) = Q(j),
where

s(k) := exp -k -L ′ (j) L(j)
and

Q(j) := 1 √ 2π 1 -2 -j j L ′′ (j) L(j) - L ′ (j) L(j) 2 -1/2 W (j).
Notice that, by Lemma 1, as j < θ * , Q(j) > 0 a.s. Define

ν(k) := Q(j)k -1/2 e kϕ(j) /2 .
Then (recall that ϕ(j) > 0), for almost all ω, there exists k 0 (ω) such that

# i ∈ N k : p i ≥ s(k) ≥ ν(k) ≥ 1, for all k ≥ k 0 (ω). ( 10
)
We can now prove the following result.

Lemma 8 Define for all

k ∈ Z + x k := k u exp k -ln L(j) j , where u > 1 2j
.

Then for almost all ω, there exists k 1 (ω) such that

N x k ,j (k) ≥ 1, for all k ≥ k 1 (ω).
Proof : As N k is countable, we can order the boxes of the k-th generation in the decreasing order. Denote by G k the family of boxes having a size at least s(k). We create ν Conditionally on F ∞ and A k , the boxes b l (k) have size s(k) so the numbers of balls contained in b l (k), l ≤ ν(k), when P x k balls have been thrown are independent Poisson variables with parameters x k s(k). Therefore

(k) new boxes : if #G k ≥ ν(k),
P(A k ∩ B k |F ∞ ) ≤ P(P x k s(k) < j) ν(k) . Thus ln P(A k ∩ B k |F ∞ ) ≤ Q(j) 2 k -1/2 e kϕ(j) ln P(P x k s(k) < j).
It can be easily seen that x k s(k) tends to 0, and that, as a result, ln

P(P x k s(k) < j) ∼ -x j k s(k) j /j!.
Thus there exists a constant c > 0 such that for all k ∈ Z + , ln P(P x k s(k) < j) ≤ -2cx j k s(k) j .

Finally, we get

P(A k ∩ B k |F ∞ ) ≤ exp -cQ(j)k -1/2 e kϕ(k) x j k s(k) j = exp -cQ(j)k ju-1/2 .
As u > 1 2j and Q(j) > 0 a.s., we get [START_REF] Chauvin | Martingales and profile of binary search trees[END_REF], this proves that a.s., for all integers k sufficiently large, there exists an imaginary box containing at least j balls when P x k have been thrown. As every imaginary box is contained in a real box, Lemma 8 is proved.

E [ 1 A k ∩B k |F ∞ ] < ∞ a.s., so 1 A k ∩B k < ∞ a.s. Combined with
We now deduce from Lemma 8 the lower bound of H n,j .

Proof of Proposition 5 : Let u and u ′ be two real numbers such that 1 2j < u < u ′ . Define x k := k u exp (-k ln L(j)/j) and φ k := k u ′ exp (-k ln L(j)/j). One can show that for almost all ω, there exists k 2 (ω) such that P x k < ⌈φ k-1 ⌉ for all k ≥ k 2 (ω). Applying Lemma 8, we deduce that if we define the event Ω 0 by Ω 0 := ω : there exists k 3 (ω) such that N ⌈φ k-1 ⌉,j (k) ≥ 1 for all k ≥ k 3 (ω) , then P(Ω 0 ) = 1. The sequence (φ k ) is increasing and tends to infinity. Let ω ∈ Ω 0 . Let n be an integer large enough so that the unique integer

k satisfying φ k-1 < n ≤ φ k is greater than k 3 (ω). Because N ⌈φ k-1 ⌉,j (k) ≥ 1 and n ≥ ⌈φ k-1 ⌉, N n,j (k) ≥ 1, so H n,j > k. Moreover, as n ≤ φ k , k ≥ j -ln L(j) ln n -u ′ j -ln L(j) ln k.
As n ≥ φ k-1 ≥ exp (-(k -1) ln L(j)/j), we have : k ≤ j ln L(j) ln n + 1. Therefore

H n,j > k ≥ j -ln L(j) ln n -u ′ j -ln L(j) ln j -ln L(j) ln n + 1 .
We conclude that

lim inf n→∞ H n,j - j -ln L(j) ln n ln ln(n) ≥ -u ′ j -ln L(j)
a.s.

The fact that u ′ can be chosen arbitrarily close to 1 2j completes the proof.

Remark 4 Theorem 1 states that H n,j ∼ C j ln n a.s. This asymptotic behaviour was proved in the case θ * = ∞ (we even showed that H n,j = C j ln n + O(ln ln n) a.s.). If θ * < ∞, then ln p(k)/k tends to 1/C * a.s. (see Lemma 3). We deduce from an argument similar to that in Proposition 4 that H n,j ∼ C j ln n a.s. Finally, Theorem 1 and Proposition 1 have been proved.

Study of the saturation levels

In this section, we prove Theorem 2 and Proposition 2. We shall first assume that (2) holds.

We shall then study the case θ * = -∞ to complete the proof of Theorem 2.

5.1

The case -∞ < θ * < 0 and ϕ(θ * ) = 0

Throughout this section, we suppose that (2) holds. In particular, C * = -θ * / ln L(θ * ) = -L(θ * )/ L ′ (θ * ). We are first interested in the lower bound. We shall then focus on the upper bound. We are inspired by the techniques developed in the previous section.

Lower bound

We show the inequalities ( 6) and ( 7) when (2) holds. Equation ( 9) will be the key tool.

Proposition 6 Suppose that (2) holds. Let j ≥ 1 be an integer and α ∈ [0, 1). Then

lim inf n→∞ G n,jn α -(1 -α)C * ln(n) ln ln(n) ≥ - 1 ln L(θ * ) a.s.
Recall that G n,jn α > k if and only if at generation k, every box contains at least jn α balls when n balls have been thrown. Because we want to show that G n,jn α is bounded from below by ((

1 -α)C * -ε) ln(n), we take k ≈ ((1 -α)C * -ε) ln(n), i.e. n ≈ exp k 1 (1-α)C * + ε .
Proof : Let u and u ′ be two real numbers such that

u > - 1 (1 -α)θ * and u < u ′ < 1 α u + 1 θ * ,
where the second condition reduces to u ′ > u for α = 0. Define for all k ∈ N :

x k := k u exp k 1 (1 -α)C * and φ k := k u ′ exp k 1 (1 -α)C * .
Proceeding as usual, let us prove the following key result : for almost all ω, there exists

k 0 (ω) such that M x k ,jφ α k (k) = 0, for all k ≥ k 0 (ω). (11)
Let x and y be two real numbers such that y ≥ j and k ∈ N. We calculate

E [M x,y (k)|F k ]. We write M x,y (k) = i:|i|=k 1 C(i;x)<y = i:|i|=k 1 C(i;x)<⌈y⌉
Now, conditionally on F k , (C(i; x)) |i|=k are Poisson variables with parameters xp i , so

E [M x,y (k)|F k ] = i:|i|=k P(P xp i < ⌈y⌉).
Applying (9), we get

E [M x,y (k)|F k ] ≤ d(-θ * )⌈y⌉ -θ * i:|i|=k (xp i ) θ * . Now, as y ≥ j, ⌈y⌉ ≤ y + 1 y y ≤ j + 1 j y.
In the notations of Lemma 1, we have :

E [M x,y (k)|F k ] ≤ d ′ y -θ * x θ * L(θ * ) k W (k) (θ * ),
where d ′ := d(-θ * )(1 + 1/j) -θ * . We finally get

E [M x,y (k)] ≤ d ′ y -θ * x θ * L(θ * ) k .
For x = x k and y = jφ α k ≥ j, we obtain for every integer k ∈ N :

E M x k ,jφ α k (k) ≤ d ′ j -θ * k θ * (u-u ′ α) . Now θ * (u -u ′ α) < -1. As a consequence E   k∈Z + 1 M x k ,jφ α k (k)≥1   ≤ E   k∈Z + M x k ,jφ α k (k)   < ∞.
In particular, there is an a.s. finite number of integers k such that M x k ,jφ α k (k) ≥ 1, which proves [START_REF] Devroye | A note on the height of binary search trees[END_REF].

Performing the same calculations as at the end of the proof of Proposition 4, one finally gets :

lim inf n→∞ G n,jn α -(1 -α)C * ln n ln ln n ≥ -u ′ (1 -α)C * a.s.
The fact that u ′ can be chose arbitrarily close to -1 (1-α)θ * completes the proof.

Upper bound

In this section, we are interested in the upper bound of the saturation levels G n,j . We prove that the inequalities ( 6) and ( 7) are in fact equalities whenever (1) and ( 2) hold by studying the smallest box, regardless of the value of j; there is no phase transition.

Proposition 7 Suppose that ( 1) and ( 2) hold. Let j ≥ 1 be an integer and α ∈ (0, 1). a.s.

In order to prove both inequalities simultaneously, we may suppose that α ∈ [0, 1), the case G n,j corresponding to G n,jn α with α = 0. By definition, G n,jn α ≤ k if and only if at generation k, there exists a box containing less than jn α balls when n balls have been thrown. We shall see that it suffices to consider the smallest box. As we intend to show that G n,jn α is bounded from above by

((1 -α)C * + ε) ln n, we take k ≈ ((1 -α)C * + ε) ln n, i.e. n ≈ exp k 1 (1-α)C * -ε .
Proof : Let γ, γ ′ and γ ′′ be three real numbers such that :

• γ > -3 2θ * , γ ′ > γ + 1 j and γ ′′ > γ ′ if α = 0, • γ > -3 2θ * , γ ′ > γ 1-α and γ ′ < γ ′′ < γ ′ -γ α if α > 0. Define x k := k -γ ′ exp k 1 (1 -α)C * and φ k := k -γ ′′ exp k 1 (1 -α)C * .
Let us prove the following result : for almost all ω, there exists k 0 (ω) such that The box b(k) has thus size p(k) ∧ 1, and whenever p(k) > p(k), it contains the smallest box.

M x k ,jφ α k (k) ≥ 1, for all k ≥ k 0 (ω). (12)
From Lemma 4, we know that a.s., for all integers k sufficiently large, p(k) > p(k). To prove Lemma 12, all we have to do is therefore to prove that P(lim sup A k ) = 0, where the event A k is defined by A k := {b(k) contains at least jφ α k balls when the first P x k balls have been thrown}.

By the Borel-Cantelli lemma, it suffices to show that P(A k ) < ∞. Let k be an integer sufficiently large so that p(k) < 1. The imaginary box has then size p(k), so we have :

P(A k ) = P P x k p(k) ≥ jφ α k
We would like to apply [START_REF] Boucheron | An urn model from learning theory[END_REF]. To do so, let p := j if α = 0 and p := 2/(γ ′γαγ ′′ ) > 0 if α > 0. Note that for all integers k sufficiently large, jφ α k ≥ p. Equation ( 8) then ensures that :

P(A k ) ≤ c(p)j -p k p(αγ ′′ +γ-γ ′ ) .
As p(αγ ′′ + γγ ′ ) < -1, P(A k ) < ∞, which proves [START_REF] Devroye | Branching processes in the analysis of the heights of trees[END_REF]. Performing the same calculations as at the end of the proof of Proposition 3, one finally gets :

lim sup n→∞ G n,jn α -(1 -α)C * ln n ln ln n ≤ γ ′′ (1 -α)C * a.s.
The fact that γ ′′ can be chosen arbitrarily close either to

-3 2θ * + 1 j if α = 0 or to -3 2(1-α)θ * if α > 0 completes the proof.
Proposition 2 and a part of Theorem 2 have been proved. We now turn our attention to the other part of Theorem 2.

The case θ * = -∞

In this section, we prove that if θ * = -∞, then G n,j ∼ C * ln n a.s. We begin by showing the following lemma :

Lemma 9 If θ * = -∞, then -θ/ ln L(θ) tends to C * as θ tends to -∞.

Proof : The condition θ * = -∞ means that ϕ(θ) > 0 for all θ < 0. Let ψ = ln L. Recall that C * = lim θ→-∞ -1/ψ ′ (θ). As ψ is convex decreasing, it is known that -ψ(θ)/θ tends to l ∈ (0, ∞] as θ tends to -∞. We distinguish two cases. Either l is finite, then one can easily show that θ → ψ(θ) + lθ is increasing. As a result,

ψ ′ ≥ -l. If l = 1/C * , then ψ ′ (θ)
-l as θ tends to -∞ , so there exists ε > 0 such that ψ ′ ≥ εl. Let θ < 0. Then 0 θ ψ ′ ≥ (εl)(-θ), i.e. ψ(θ) + lθ ≤ ψ(0) + εθ. Dividing by -θ and taking the limit, we get ε ≤ 0, which is absurd. We deduce that l = 1/C * , which means that -θ/ ln L(θ) → C * as θ tends to -∞.

Or l is infinite. As for all θ < 0, ϕ(θ) > 0, the function θ ∈ (-∞, 0) → θ/ψ(θ) -1/ψ ′ (θ) is increasing. It is also positive. If it does not tend to 0 at -∞, then it is bounded from below by some ε > 0. Multiplying by -ψ ′ (θ) > 0, we get : 1 ≥ 1θψ ′ (θ)/ψ(θ) ≥ -εψ ′ (θ).

Consequently ψ ′ (θ) ≥ -1/ε and ψ(0)ψ(θ) ≥ θ/ε. Dividing by θ < 0, we obtain : -ψ(θ)/θ ≤ 1/εψ(0)/θ. Taking the limit, we have ∞ ≤ 1/ε, which is absurd. Finally, θ/ψ(θ) -1/ψ ′ (θ) tends to 0 as θ tends to -∞. Now, by definition of C * , ψ ′ (θ) tends to -1/C * . As a result, -θ/ψ(θ) tends to C * as θ tends to -∞.

Lower bound

We show that G n,j ≥ C * ln n + o(ln n) a.s. Equation ( 9) will be very useful.

Proposition 8 Suppose that θ * = -∞. Let j ≥ 1 be an integer. Then

lim inf n→∞ G n,j ln n ≥ C * a.s.
Proof : We follow the usual strategy. Let θ < 0 and u >ln L(θ)/θ. Define for all k ∈ Z + , x k := e ku . Applying [START_REF] Bunge | Estimating the number of species : a review[END_REF], we can show that

E[M x k ,j (k)|F k ] ≤ d(-θ)j -θ x θ k L(θ) k W (k) (θ), so E[M x k ,j (k)] ≤ d(-θ)j -θ exp {kθ(u + ln L(θ)/θ)} . As θ(u + ln L(θ)/θ) < 0, E [ M x k ,j (k)
] < ∞, so a.s., for all integers k sufficiently large, M x k ,j (k) = 0. The usual calculations yield :

lim inf n→∞ G n,j ln n ≥ -θ ln L(θ)
a.s.

Lemma 9 enables us to complete the proof.

Upper bound

We finally prove that G n,j ≤ C * ln n + o(ln n) a.s. As θ * = -∞, Lemma 3 cannot be applied (we do not know how the size of the smallest box behaves). Surprisingly, we are inspired by the proof of Proposition 5.

Proposition 9 Suppose that θ * = -∞. Let j ≥ 1 be an integer. Then

lim sup n→∞ G n,j ln n ≤ C * a.s.
Proof : We may suppose that j = 1. Let θ < 0. Here, the boxes of the k-th generation that will play a key role are those having a size approximatively s(k), where

s(k) := exp -k -L ′ (θ) L(θ) .
Applying Lemma 2, we have with probability one :

lim k→∞ √ ke -kϕ(θ) # i ∈ N k : s(k)/2 ≤ p i ≤ s(k) = Q(θ),
where

Q(θ) := 1 √ 2π 1 -2 θ -θ L ′′ (θ) L(θ) - L ′ (θ) L(θ) 2 -1/2 W (θ).
Notice that, by Lemma 1, Q(θ) > 0 a.s. Define

ν(k) := Q(θ)k -1/2 e kϕ(θ) /2 .
Then (recall that ϕ(θ) > 0), for almost all ω, there exists k 0 (ω) such that

# i ∈ N k : p i ≤ s(k) ≥ ν(k) ≥ 1, for all k ≥ k 0 (ω). (13)
Let u < -L ′ (θ)/ L(θ). Define for all k ∈ Z + , x k := e ku . Let us prove that for almost all ω, there exists k 1 (ω) such that

M x k ,1 (k) ≥ 1, for all k ≥ k 1 (ω). ( 14
)
Denote by G k the family of boxes having a size at most s(k). We define ν Conditionally on F ∞ and A k , the boxes b l (k) have size s(k) so the numbers of balls contained in b l (k), l ≤ ν(k), when P x k balls have been thrown are independent Poisson variables with parameters x k s(k). Therefore [START_REF] Devroye | Branching processes and their applications in the analysis of tree structures and tree algorithms[END_REF], this proves that a.s., for all integers k sufficiently large, there exists an empty imaginary box when P x k have been thrown. As every imaginary box contains a real box, ( 14) is proved.

(k) new boxes : if #G k ≥ ν(k), we create ν(k) imaginary boxes b 1 (k), . . . , b ν(k) (k) of
P(A k ∩ B k |F ∞ ) ≤ P(P x k s(k) ≥ 1) ν(k) . Thus ln P(A k ∩ B k |F ∞ ) ≤ Q(θ) 2 k -1/2 e kϕ(θ) ln P(P x k s(k) ≥ 1). Now P(P x k s(k) ≥ 1) ≤ x k s(k), so P(A k ∩ B k |F ∞ ) ≤ exp u + L ′ (θ) L(θ) Q(θ) 2 k 1/2 e kϕ(θ) . As u + L ′ (θ)/ L(θ) < 0, ϕ(θ) > 0 and Q(θ) > 0 a.s., we get E [ 1 A k ∩B k |F ∞ ] < ∞ a.s., so 1 A k ∩B k < ∞ a.s. Combined with
Proceeding as usual, we deduce that for all θ < 0,

lim sup n→∞ G n,1 ln n ≤ L(θ) -L ′ (θ) a.s.
We get the result by letting θ tend to -∞.

Finally, Theorem 2 has been proved.

An explanation for the phase transition

In this section, we explain heuristically why a phase transition may occur in the asymptotics of H n,j but not in those of H n,n α , G n,j and G n,n α . To do so, we rephrase the setting in terms of interval-fragmentations. One can easily construct a family (F (k), k ∈ Z + ) of random open subsets of (0, 1) with F (0) = (0, 1), which is nested in the sense that F (k ′ ) ⊆ F (k) for all k ≤ k ′ , and such that the set of the lengths of the interval components of F (k) is {p i : |i| = k} for every integer k. The boxes correspond to the interval components and the balls to a sequence (U i ) i∈N of independent random variables uniformly distributed on (0, 1) and independent of the fragmentation F . The height H n,j corresponds to the least integer k such that every interval component of F (k) contains less than j elements of the set {U 1 , . . . , U n }, and the saturation level G n,j to the least integer k such that there exists an interval component of F (k) containing less than j elements of the set {U 1 , . . . , U n }. Roughly speaking, the height H n,j depends crucially on the minimal length m n,j of the intervals [ Ûi , Ûi+j-1 ] for 1 ≤ i ≤ nj + 1, where 0 < Û1 < • • • < Ûn < 1 are the ordered statistics of the family (U 1 , . . . , U n ), whereas the saturation level G n,j is related to the maximal length m n,j of the intervals [ Ûi , Ûi+j ] for 0 ≤ i ≤ nj + 1, where Û0 := 0 and Ûn+1 := 1. Indeed, the height H n,j is the first time k when no cluster [ Ûi , Ûi+j-1 ], 1 ≤ i ≤ nj + 1, of size j (in particular the smallest one) is included in an interval component of F (k), and the saturation level G n,j is the first time k when there exists a cluster [ Ûi , Ûi+j ], 0 ≤ i ≤ nj + 1, of size j + 1 (possibly the largest one) containing an interval component of F (k).

It is easy to show from Lemma 3 that if we used equidistributed points {j/(n + 1) : 1 ≤ j ≤ n} instead of i.i.d. uniform points, then no phase transition would occur. More precisely, all the heights H n,j would be equivalent to C * ln n a.s. Further, the heights H n,n α would be equivalent to (1α)C * ln n a.s. and the saturation levels G n,jn α to (1α)C * ln n a.s.

We first explain why the clusters of size n α behave as if the points U i , 1 ≤ i ≤ n, were equidistributed on (0, 1). It will follow that no phase transition occurs in the asymptotics of H n,n α and G n,n α . To do so, let us first prove that m n,n α ≥ n -1+α+o(1) a.s. Let ε > 0. We have :

P(m n,n α ≤ n -1+α-2ε ) ≤ nP( Ûn α -Û1 ≤ n -1+α-2ε ).

Let (e i ) i∈N be a sequence of independent exponential variables with parameters 1. Define γ k := 1≤i≤k e i , for all k ≥ 1. Then we have :

( Û1 , . . . , Ûn )

= (γ 1 /γ n+1 , . . . , γ n /γ n+1 ), [START_REF] Drmota | Stochastic analysis of tree-like data structures[END_REF] so

P( Ûn α -Û1 ≤ n -1+α-2ε ) = P γ n α -γ 1 γ n+1 ≤ n -1+α-2ε
≤ P γ n αγ 1 < n 1+ε n -1+α-2ε + P γ n+1 ≥ n 1+ε = P (P n α-ε ≥ n α ) + P (P n 1+ε ≤ n + 1) since the sequence (γ i ) i∈N has the same law as the ordered sequence of points of a standard Poisson process. Applying Lemma 5 with p = 3/ε, we have for every integer n sufficiently large : P (P n α-ε ≥ n α ) + P (P n 1+ε ≤ n + 1) ≤ (c(p) + 2 p d(p)) n -3 .

We deduce that that for every integer n sufficiently large, we have :

P(m n,n α ≤ n -1+α-2ε ) ≤ (c(p) + 2 p d(p)) n -2 .
Applying the Borel-Cantelli lemma and letting ε tend to 0, we easily conclude that m n,n α ≥ n -1+α+o(1) a.s. Similar calculations yield : m n,n α ≤ n -1+α+o(1) a.s. We conclude that the clusters of size n α essentially behave as if the points U i , 1 ≤ i ≤ n, were equidistributed on (0, 1), which explains why no phase transition occur in the asymptotics of H n,n α and G n,n α . On the contrary, the clusters of size j behave very differently from the equidistribution case. For instance, thanks to the identification [START_REF] Drmota | Stochastic analysis of tree-like data structures[END_REF], we may apply well-known results on extreme values (see e.g. [START_REF] Resnick | Extreme Values, Regular Variation, and Point Processes[END_REF]). One finds that m n,j ≈ n -j/(j-1) , so that m n,j is much smaller than j/n, even at a logarithmic scale. Furthermore, the larger the integer j is, the closer these two quantities are, so that H n,j may eventually be equivalent to C * ln n a.s. if j is sufficiently large. On the other hand, if the integer j is too small, then H n,j may be greater than (1+ε)C * ln n+o(ln n) a.s. for some ε > 0 : one has to wait for a long time before the smallest clusters of size j are no longer included in any interval component of the fragmentation; in that case, there is a phase transition.

Concerning the saturation level G n,j , one gets m n,j ≈ ln n/n, which differs from the equidistribution case only up to a logarithmic factor, and the impact of the latter is asymptotically negligible. It explains why no phase transition occurs in the asymptotics of G n,j .

Lemma 5

 5 Let p ∈ (0, ∞). There exist two finite constants c(p) and d(p) such that sup j≥p j∈N sup x>0

Lemma 7

 7 For almost all ω, there exists k 0 (ω) such thatN x k ,jφ α k (k) ≥ 1,for all k ≥ k 0 (ω). Proof : For every generation k ∈ N, we consider an imaginary box b(k) contained in the largest box b(k) of size p(k) such that a ball fallen in b(k) is thrown in b(k) with probability p(k) p(k) ∧ 1, where p(k) := k -γ e -k/C * . Informally, the box b(k) has size p(k) ∧ p(k). We denote by A k the event {p(k) < p(k)} and by B k the event defined by B k := {the box b(k) contains less than jφ α k balls when the first P x k have been thrown}. Conditionally on A k , the box b(k) has size p(k) so the number of balls contained in b(k) when P x k balls have been thrown is a Poisson variable with parameter x k p(k). As a result

3 2( 1

 31 -α)θ * completes the proof.

  consider the first ν(k) boxes belonging to the family G k , denoted by b 1 (k), . . . , b ν(k) (k). For all l ≤ ν(k), we place an imaginary box b l (k) inside the box b l (k) such that a ball fallen in b l (k) is thrown in b l (k) with probability s(k)/s l (k), where s l (k) is the size of b l (k). In particular, every imaginary box has size s(k). If #G k < ν(k), we denote by b 1 (k), . . . , b ν(k) (k) the first ν(k) boxes. Introduce the events A k := {#G k ≥ ν(k)} and B k := {∀l ≤ ν(k), b l (k) contains less than j balls when the first P x k have been thrown} .

  Define p(k) := k γ e -k/C * . We consider an imaginary box b(k) at generation k such that 1. if 1 ≤ p(k) : every ball thrown is placed in b(k).

2 .

 2 if p(k) < p(k) < 1, where p(k) is the size of the smallest box b(k) : every ball fallen in b(k) is also placed in the imaginary box and every other ball is placed in b(k) with probability (p(k)p(k))/(1p(k)). Hence, the imaginary box has size p(k).

3 .

 3 if p(k) ≤ p(k) : every ball fallen in the smallest box is placed in the imaginary box with probability p(k)/p(k), and no other ball is placed in b(k).

  size s(k) containing the first ν(k) real boxes belonging to the familyG k . If #G k < ν(k), we denote by b 1 (k), . . . , b ν(k) (k) the first ν(k) boxes of the k-th generation. Introduce the event A k := {#G k ≥ ν(k)} and B k := {∀l ≤ ν(k), b l (k)contains at least one ball when the first P x k have been thrown} .

We stress that this assumption is crucial in the proof of Theorem 1 in[START_REF] Devroye | Universal limit laws for depths in random trees[END_REF] and seems to have been overlooked in the statement.

Working with a geometric splitting law would induce a phenomenon of periodicity which we shall not discuss here for simplicity. However results similar to those proven in this work can be established by the same techniques for geometric splitting laws.

-1/2 W (θ).

Acknowledgements. I would like to express my gratitude to my thesis director Jean Bertoin who suggested this subject to me. I sincerely thank him for his invaluable advice and for his careful reading.


Xhek Turkeshi

Lorenzo Piroli

Marco Schiró

Enhanced entanglement negativity in boundary driven monitored fermionic chains

We investigate entanglement dynamics in continuously monitored open quantum systems featuring current-carrying non-equilibrium states. We focus on a prototypical one-dimensional model of boundary-driven non-interacting fermions with monitoring of the local density, whose average Lindblad dynamics features a well-studied ballistic to diffusive crossover in transport. Here we analyze the dynamics of the fermionic negativity, mutual information, and purity along different quantum trajectories. We show that monitoring this boundary-driven system enhances its entanglement negativity at long times, which otherwise decays to zero in absence of measurements. This result is in contrast with the case of unitary evolution where monitoring suppresses entanglement production. For small values of γ, the stationary-state negativity shows a logarithmic scaling with system size, transitioning to an area-law scaling as γ is increased beyond a critical value. Similar critical behavior is found in the mutual information, while the late-time purity shows no apparent signature of a transition, being O(1) for all values of γ. Our work unveils the double role of weak monitoring in current-driven open quantum systems, simultaneously damping transport and enhancing entanglement.

I. INTRODUCTION

Understanding entanglement dynamics in many-body systems is a fundamental challenge that bridges condensed matter to quantum information theory and helps us to characterize the wide spectrum of possible dynamical phases of matter. While generic isolated systems with local interactions exhibit universal features of entanglement growth [1,2], different scenarios for entanglement behavior arise in the presence of ergodicity breaking, for instance due to many-body localization [3], kinetic constraints [4][5][6][7], long-range interactions [8][9][10][11][12][13] or integrability [14][15][16][17][18][19][20].

Entanglement is commonly believed to be destroyed by bulk coupling to a noisy environment [START_REF] Breuer | The theory of open quantum systems[END_REF]. This is usually the case for open-system dynamics which are described by a Lindbladian master equation and in which the environment is dealt with as a black box. A different behavior emerges by considering an open-system dynamics induced by monitoring the system [START_REF] Plenio | [END_REF][START_REF] Wiseman | Quantum Measurement and Control[END_REF]: in this setting, the environment is given by the measurement apparatus, which allows us to gain information on its state. The renewed interest in this type of dynamics has been largely motivated by the progress in quantum optics experiments, allowing us to manipulate and probe quantum systems to an unprecedented degree of control [START_REF] Basché | [END_REF][25][26][27][28].

The possibility of monitoring the many-body dynamics has already proven its potential to realize new nonequilibrium phases, as explicitly shown in the simplest case where local unitary evolution is interspersed by local measurements [29][30][31][32][33][34][35][36]. In this setting, extensive theoretical research has provided strong evidence that generic systems described by random unitary circuits undergo a new type of measurement induced phase transition (MIPT), characterized by a change in the scaling of the subsystems entanglement entropy, from volume-law to area-law. On the other hand, certain classes of noninteracting systems under different types of monitoring protocols have been shown to display MIPT between a phase with sub-extensive entanglement growth and an area law [63][64][65][66][67][68][69].

In this work, we put forward a different setting where monitoring might be expected to give rise to novel nonequilibrium entanglement behavior. Namely, we focus on current-driven open many-body quantum systems featuring non-thermal non-equilibrium stationary states (NESS). For concreteness, we consider the case of a system with a local U(1) charge, coupled at its ends to two reservoirs at different chemical potentials -a prototypical framework to investigate quantum transport [70,71]. Accordingly, the dynamics under study is characterized by three ingredients: unitary evolution, boundary driving and monitoring of the U(1) charge.

Current-driven setups are known to give rise to a nontrivial interplay between transport and entanglement. For instance, current-carrying states of non-interacting diffusive fermions can sustain extensive entanglement [72,73], in stark contrast to the universal area-law scaling characterizing thermal phases [74,75]. In the presence of monitoring, one may ask how trajectory-resolved features of entanglement depend on its rate. This question is particularly natural in light of the apparent competition between different effects: on the one hand, the driving forces a particle flow through the system while, on the other, large monitoring tends to pin it to an eigenstate of particle-density operators.

We will address such a question by studying a simple but prototypical model of quantum transport: a one-dimensional chain of non-interacting fermions, where particles are injected and extracted at the two ends, respectively. In addition we will consider continuously monitoring the fermionic particle number, according to the so-called quantum-state-diffusion (QSD) protocol [76][77][78], (see Fig 1).

This model was introduced in Ref. [79] where its transport properties were analyzed within the limit of infinite monitoring rate. We also note that, in the absence of boundary driving terms, it coincides with that studied in Refs. [63,65,80,81]. In Ref. [63,65], in particular, it was shown that the system undergoes a MIPT between two phases, with logarithmic and area-law scaling of the entanglement entropy [START_REF]The absence of a volume-law phase is consistent with the non-interacting nature of the model, limiting its ability to scramble quantum information[END_REF]. While our analysis builds upon these works, the boundary driving introduces crucial differences. Most prominently, because the particle reservoirs are treated as an inaccessible environment, the state of the system along each quantum trajectory is a mixed state. Accordingly, in order to quantify the corresponding entanglement, we need to rely on the socalled negativity [START_REF] Eisert | [END_REF][84][85][86][87][88][89][90], since the entanglement entropy is known not to be a genuine measure of entanglement for mixed states [START_REF] Nielsen | Quantum computation and quantum information[END_REF].

In our model, the evolution of the average density matrix is described by a well studied Lindbladian master equation, with boundary driving and bulk dephasing [START_REF] Žnidarič | [END_REF][93][94][95][96][97]. At late time, any non-zero rate of monitoring leads to diffusive behavior, thus damping the ballistic transport characterizing the isolated system. On the contrary, we show that monitoring can enhance entanglement. Specifically, we find that the negativity displays a logarithmic growth for small values of γ, finally transitioning to an area-law scaling as γ is increased beyond a critical value.

Before leaving this section, we note that two recent works also explored the role of a dissipative environment for a system that is simultaneously monitored [98,99], although from a different point of view. Ref. [99] considered a fermionic chain with a coupling to a dephasing bath at each bulk site in the system. Ref. [98], instead, focused on quantum circuit models with boundary dephasing, but featuring no local conservation law and therefore no current driving.

The rest of this paper is organized as follows. In Sec. II we introduce the model and the protocol we consider. In Sec. III we study the purity and entanglement in the late-time regime, providing evidence of a transition in the scaling of the negativity. Our conclusions are consigned to Sec. IV. Finally, the most technical aspects of our work are reported in several appendices.

II. MONITORED DRIVEN FERMIONS

We begin by describing in detail the model studied in this work (cf. Fig. 1). We consider a one-dimensional chain of spinless fermions, governed by the Hamiltonian

H = - L-1 j=1 c † j c j+1 + c † j+1 c j , ( 1
)
where L is the system size, while c j , c † j are canonical fermionic operators. The system is coupled at its boundaries to particle reservoirs at different chemical potentials, and subject to continuous monitoring in the bulk. Within the so-called QSD protocol [76][77][78], the evolution of the system density matrix ρ is captured by the stochastic master equation (SME)

dρ ξ = dtL[ρ ξ ] + L m=1 dΞ m [ρ], (2)
L[•] = -i[H, •] + D bnd [•] + D bulk [•]. (3)
The first term in Eq. ( 2) describes the deterministic part of the evolution and includes several contributions, explicitly reported in Eq. ( 3). The first one is a coherent term encoding the bulk unitary dynamics driven by the Hamiltonian (1). The second one corresponds to the boundary driving and reads

D bnd [•] = D L bnd [•] + D R bnd [•], with D L bnd [•] = Γ L 1 + µ 2 2c † 1 • c 1 -{c 1 c † 1 , •} + 1 -µ 2 2c 1 • c † 1 -{c † 1 c 1 , •} , (4)
and

D R bnd [•] = Γ R 1 -µ 2 2c † L • c L -{c L c † L , •} + 1 + µ 2 2c L • c † L -{c † L c L , •} . (5)
These two terms describe injection/depletion of particles at the two edges of the chain, respectively with rates

Γ L (1 ± µ)/2 and Γ R (1 ∓ µ)/2.
In the following we consider equal overall scales Γ L = Γ R = Γ and unless stated otherwise set µ = 1, corresponding to the maximum difference in driving potential i.e. pure injection/depletion on the left/right edge. We will discuss the dependence on µ in Sec. III B. Finally, the last term in (3) encodes the deterministic back-action due to bulk monitoring of the local particle density n i = c † i c i , which takes the form of a dephasing dissipator

D bulk [•] = - γ 2 L j=1 [n i , [n i , •]] . (6)
Monitoring is also responsible for the stochastic feedback term in (2), which is defined as

dΞ m [•] ≡ √ γdξ m t {n m -n m ξ t , •} . (7)
It is characterized by independent Brownian processes, with Îto differentials dξ i t such that dξ i t dξ j t = δ(t-t )δ i,j dt. In Eq. ( 7) we introduced the notation A ξ t ≡ tr(ρ ξ A). We emphasize the difference of our setting with respect to Ref. [99], where an additional bulk dephasing channel was added on top of the monitoring process.

The solution to the SME ( 3) is a conditional density matrix ρ ξ which encodes complete information on the monitored quantum system. For a given functional of the density matrix, F[•], we may define its statistical distribution as

P (F ) ≡ [dξ]P (ξ)δ(F[ρ ξ ] -F ). (8)
A natural class of functionals is the expectation value of an observable O, that is F O [ρ] ≡ tr(ρO). In this case, the average over the trajectories

O ≡ dOP (O)O, (9)
coincides with the trace over the average density matrix tr(Oρ), where

ρ = [dξ]P (ξ)ρ ξ . (10)
It is easy to show that ρ satisfies

d dt ρ = L[ρ] . (11)
This Lindbladian equation has been extensively studied in the literature [START_REF] Žnidarič | [END_REF][93][94][95][96][97], and the structures of its late-time NESS has been worked out analytically [START_REF] Žnidarič | [END_REF].

For completeness, we review its main properties in Appendix B.

When considering non-linear functionals of the density matrix, the average behavior can not be computed from the averaged density matrix. This is the case, in particular, for the purity and the entanglement negativity, discussed in Sec. III. In general, in order to study the dynamics of these quantities, one needs to solve the SME and sample over different trajectories.

Crucially, in the non-interacting model ( 2), the dynamics along each trajectory can be computed efficiently starting from a Gaussian initial state [100]. Indeed, a fermionic Gaussian state evolved under the SME (4) remains Gaussian [100], so that, along each given quantum trajectory, the state of the system is completely characterized by its covariance matrix.

If the initial state is such that

c † i c † j t=0 = 0 , (12)
it is easy to see that this remains true at later times, and the state of the system is completely encoded into the matrix

(C t ξ ) m,n ≡ c † m c n ξ t . (13)
Accordingly, the full many-body evolution, that is a trajectory in the 2 2L -dimensional space of fermionic density matrices, can be replaced by the evolution of the L × L covariance matrix (13). The explicit stochastic equation satisfied by the covariance matrix is reported in Appendix A, where we also provide details on our numerical solution.

III. ENTANGLEMENT NEGATIVITY AND PURIFICATION

In this section, we present our results for the dynamics of entanglement, as quantified by the so-called fermionic negativity. We will also discuss the behavior of mutual information and study the purity of the system.

A. Fermionic negativity

As already mentioned, because the state of the system along each quantum trajectory is mixed, the von Neumann entropy is not a good measure of quantum entanglement [START_REF] Nielsen | Quantum computation and quantum information[END_REF]. Here, we focus on the fermionic negativity [86], which has been recently proposed as a measure of mixed-state entanglement alternative to the logarithmic negativity [START_REF] Eisert | [END_REF][84][85]. Contrary to the latter, the fermionic negativity can be computed efficiently for fermionic Gaussian states [86], while it is also a genuine entanglement monotone [87]. In the context of Lindbladian dynamics of non-interacting fermionic chains, the fermionic negativity has been extensively investigated [101,102], and its behavior has been understood based on a semi-classical quasi-particle picture. Here, we show that qualitative differences arise in the presence of monitoring.

The fermionic negativity is defined based on the concept of partial time reversal [86], as we now briefly summarize. Let us introduce the Majorana operators

ψ 2k-1 = c k + c † k and ψ 2k = i(c k -c † k ).
Given a bipartition A ∪ B, and denoting by a j and b j the Majoranas acting respectively on A and B, the system density matrix is given by

ρ = k1+k2 even k1,k2 ρ q1,...,q k 2 p1,...,p k 1 k1 i=1 a pi k2 j=1 b pj . ( 14
)
Introducing the partial time reversal on the subsystem Entanglement negativity as a function of time. We plot two sets of curves corresponding to zero (γ = 0.0) and weak (γ = 0.5) monitoring. In the two cases, we consider chains of size L = 32÷96 with Γ = 1 and µ = 1. At late times, the negativity vanishes in non-monitored systems, while it approaches a constant increasing with L in the presence of weak monitoring.

A ρ R A = k1+k2 even k1,k2 ρ q1,...,q k 2 p1,...,p k 1 i k1 k1 i=1 a pi k2 j=1 b pj , (15)
the fermionic negativity is defined as

E(ρ; A) = ln tr ρ R A (ρ R A ) † . (16)
It is a measure of the entanglement of the state ρ shared between the regions A and B. Importantly, if the state ρ is Gaussian, so is ρ R A and, in this case, the entanglement negativity can be obtained with polynomial computational resources. For completeness, we detail the procedure to compute it in Appendix C, while here we only report the final results of our analysis. We focus on the partition given in Fig. 1, with |A| = L/2 and begin with the real-time evolution of the average negativity

E(t) ≡ E ξ [E(ρ ξ ; A)] , (17)
from the initial state

|Ψ 0 = c † 1 c † 3 • • • c † L-1 |0 , ( 18
)
where |0 is the vacuum. |Ψ 0 has no entanglement, is Gaussian and satisfies (12). Therefore, we can apply the numerical scheme described in Appendix A. We note that the late-time stationary state does not depend on the choice of the initial state, as we explicitly verified and discuss in Appendix E; there we also detail the parameters used for the numerical simulations. An example of our numerical data is shown in Fig. 2. In the absence of monitoring, we find that the negativity grows linearly in time, displaying an "entanglement barrier". Namely, after a time proportional to L, E(t) reaches a maximum value increasing linearly with the system size L. Within this time frame, we verified that there is a ballistic data collapse according to the scaling function ε(t) = E(t/L, /L)/L. At later times, the fermionic negativity starts decaying, as the quantum correlations are washed out by the boundary coupling with the dissipative environment. Our numerical data are compatible with a zero negativity in the infinite-time limit, for all system sizes. The evolution is qualitatively different for γ > 0. For weak monitoring, corresponding to γ = 0.5 in Fig. 2, the entanglement negativity E grows logarithmically in time, saturating to a non-zero value. Contrary to the case γ = 0, the fermionic negativity does not show an entanglement barrier: this is in line with the expectation that monitored non-interacting fermionic dynamics can not sustain extensive entanglement, due to the absence of scrambling [40,63,80,103]. On the other hand, it is evident from Fig. 2 that, at late times, the negativity for γ > 0 is larger than in the case γ = 0: namely, monitoring a boundary-driven system causes an enhancement of steady-state entanglement, in stark contrast to what happens in monitored isolated-system evolution [29][30][31][32][33][34][35] where entanglement production is suppressed by measurement, with respect to the unitary dynamics.

As expected, we find that the late-time negativity depends on the monitoring strength. For small γ, we see that the scaling of the latter is consistent with a logarithmic growth in L. Instead, for γ sufficiently large, the late-time stationary value of the negativity appears to be convincingly independent of L, despite being generically non-zero. In order to be quantitative, we analyze the system-size scaling of the stationary negativity E ∞ as a function of L, for L ≤ 192. Our results are reported in Fig. 3, from which we see evidence of a transition at a critical value, γ = γ c separating a logarithmic from an area-law scaling. In the small-γ regime, we have per-formed a fit of the negativity against the formula

E = ceff (γ) 3 ln L + e 0 (γ) , (19)
obtaining an estimate for the effective central charge ceff (γ) [104] and the constant e 0 (γ). Both parameters are found to continuously vary with γ, as shown in inset of Fig. 3. For γ 0.8, our fitting procedure gives us ceff (γ) = 0, which allows us to identify γ c 0.8 as the critical value separating the two phases.

We have verified that our estimate for γ c does not depend on the strength of the boundary coupling Γ. In addition, it is consistent with the critical value found in Ref. [63] characterizing the MIPT in the isolated noninteracting fermionic chain (corresponding to Γ = 0). Therefore, the critical behavior of the bipartite entanglement in the stationary state appears to be dominated by the physics in the bulk, i.e. by the competition between unitary hopping and monitoring. We note however that boundary driving can influence bulk properties such as the density profile, cf. Appendix B.

We expect that this transition in the entanglement negativity can be understood based on an approach similar to the one developed in Ref. [65], which introduced an effective n-replica Keldysh field theory capturing the average of the n-th moment of the quantum trajectories, followed by a bosonization of the bulk monitored problem. In this framework the boundary driving should appear as a local non-linearity for the effective replica field theory, not modifying the nature of the bulk transition. In turn, this would imply that the fermionic negativity undergoes a Berezinskii-Kosterlitz-Thouless (BKT) transition, as established in Refs. [63,65] for the entanglement entropy of the isolated system.

In order to substantiate further this claim, it would be useful to provide numerical results for larger system sizes, especially given the expected large finite-size effects characterizing the BKT transition. Unfortunately, we are not able to simulate systems of the same sizes studied in Ref. [63]. In our setting, the main limitation comes from the fact that we need to follow the evolution up to very large times, in order to reach the stationary regime. This can be understood at the level of the average Lindbladian dynamics, cf. Appendix B. Indeed, because the latter displays diffusive behavior [START_REF] Žnidarič | [END_REF], one has that the stationary regime is approached at times t ∼ L 2 /D, where D = (γ + 1/γ) is the diffusion constant [93]. Therefore, in our simulations we need to follow the dynamics up to times that scale quadratically in L, limiting the system sizes which can be analyzed.

Together with the fermionic negativity, we have also studied the Rényi mutual information associated with the bipartition displayed in Fig. 1. It is defined as where

I (α) A:B (t) := S (α) A (t) + S (α) B (t) -S (α) AB (t) , (20)
S (α) S (t) = E ξ [S S [ρ ξ ]] (21)
S S [ρ ξ ] = 1 1 -α [ln tr ρ S ξ (t) α ] , (22)
is the Rényi entropy and ρ S ξ (t) is the density matrix reduced to the subsystem S. It is a measure of both classical and quantum correlations [74] which can be non-zero even for non-entangled states. We recall that the Rényi entropy of a subsystem for a Gaussian state can be efficiently calculated [105,106]. Given the reduced correlation matrix C S i,j = C i,j for i, j ∈ S, the Rényi entropy is given by

S S [ρ ξ ] = 1 1 -α tr ln (C S ξ ) α + (1 1 -C S ξ ) α . (23)
For unitary quench dynamics in non-interacting fermionic chains, it has been shown [19] that, in a scaling limit of large system sizes and times, the fermionic negativity of a bipartition is proportional to the Rényi-1/2 mutual information, i.e.

E = 1 2 I (1/2) A:B . (24)
Recently, this identification has been extended to arbitrary unitary quantum-circuit dynamics up to times linear in the subsystem sizes [107]. On the other hand, Eq. ( 24) does not generally hold for non-unitary evolution, as shown for non-interacting fermionic chains with dephasing noise [102]. Motivated by these discussions, we have probed the validity of Eq. ( 24) in the presence of monitoring. An example of our results is shown in Fig. 4. Interestingly, despite the dynamics being non-unitary, we found that Eq. ( 24) is exactly verified up to times proportional to the system size, both for γ = 0 and γ > 0. At later times, the two deviate from one another, but still remain numerically close and display the same qualitative behavior. In particular, the mutual information shows the same phase transition of the fermionic negativity. We note that a similar relation between the two quantities was also found in Ref. [98] studying monitored quantum circuits with boundary dephasing.

B. The purity

Finally, we study the dynamics of the purity of the total system, i.e.

P(t) = E ξ [tr[ρ 2 ξ ]] . (25)
For a single quantum trajectory, it is related to the Rényi-2 entropy via P[ρ ξ ] = e -S (2) [ρ ξ ] . This observation allows us to compute it efficiently, cf. Eq. ( 23).

For non-interacting fermionic systems, a mixed state subject to monitored unitary dynamics purifies, i.e. becomes a pure state, in a time that is polynomial in the system size [103]. However, this is in general not true in the presence of an incoherent coupling to the environment, since the latter tends to increase the system entropy, competing with the effect of monitoring. In our setting, a natural question then pertains to the scaling of the purity in the late-time stationary regime, and whether the latter is able to diagnose a transition as a function of γ. A similar question was also addressed recently in Ref. [99].

We have found that, contrary to the fermionic negativity, the purity has a strong dependence on µ, modeling the driving potential difference, cf. Eqs. ( 4) and (5). We begin by discussing our results in the simplest case µ = 1. Our numerical data for the late-time limit of the purity for different values of γ are displayed in Fig. 5. The plots indicate that the purity remains close to 1 for large L, so that its logarithm is O(1) as L → ∞. For small γ, the scaling of P shows a non-monotonic behavior, with an asymptotic growth towards 1. This trend is manifest well below the critical value γ c ∼ 0.8 and is visible, within the accessible system sizes, at least down to γ ∼ 0.3.

These numerical results suggest that the purity scaling does not display a transition as a function of γ, and in particular it is never vanishing as L → ∞. Although we were not able to prove this rigorously, in Appendix D we provide a simple heuristic argument to justify it. In essence, the idea is that, as L → ∞, the particle densities in the left-most and right-most sites are close to 1 and 0, respectively. This follows from the knowledge of the average density profile in the Lindbladian steady state, as discussed in Appendix B. Accordingly, the injection and depletion of particles at the ends of the chain is suppressed, damping the rate of entropy growth due to the boundary Lindbladian. Our argument is completed by combining this picture with a lower bound on the purification rate for non-interacting fermionic systems [103].

This discussion also suggests a non-trivial dependence of the purity from µ, because it changes the particle densities at the boundary sites. We have verified this numerically, as shown in Fig. 6, where we report data for the purity and negativity for different values of µ and γ. First, we see that the qualitative behavior of the negativity is independent from the boundary parameters, confirming the picture established in the previous section. On the other hand, the scaling of the purity is more complicated. For µ = 0.5 it appears to be vanishing for weak monitoring (γ = 0.6), while remaining approximately constant for γ 0.9. Our numerics suggest a γ-dependent station-ary value for µ < 1 with a finite stationary mixedness; however, the limited accessible system sizes are not conclusive to rule out that these are finite-size effects, and that for µ < 1 the purity vanishes for L → ∞.

IV. CONCLUSIONS

We have investigated the entanglement dynamics in a prototypical one-dimensional model of boundary-driven non-interacting fermions in the presence of monitoring. We have shown in this context that the interplay between boundary dissipation and monitoring can enhance entanglement, as quantified by the fermionic negativity, as opposed to the unitary case where monitoring is detrimental to entanglement production. Furthermore, we have provided evidence that the system undergoes a phase transition that manifests itself in the scaling of the late-time entanglement negativity, going from a logarithmic to an area-law scaling. We have also shown that the transition can be diagnosed from the bipartite mutual information, but not from the purity of the whole system. Our results complement recent works studying the effect of a dephasing environment on monitored unitary dynamics [98,99].

Our work raises several questions. First, it would be interesting to substantiate analytically our results on the MIPT of the entanglement negativity. As mentioned, we expect that a possible strategy could be to extend the field-theoretical approach developed in Ref. [65]. Second, a straightforward direction would be to investigate measurement dynamics in transport settings beyond the noninteracting case considered in this work. For instance, a natural question is how our findings are modified in the presence of additional unitary noise, such as in the Quan-tum Symmetric Simple Exclusion Process [108][109][110][111][112][113][114], or of interactions. We believe that an interesting tractable model to study the latter problem is given by random unitary circuits featuring a U (1) conserved charge [115][116][117], with additional incoherent boundary terms implementation charge injection and extraction.

Perhaps, the most interesting direction pertains to the study of transport features beyond the average Lindbladian dynamics. In this respect, a non-trivial task is to define a meaningful notion of transport at the level of individual quantum trajectories, since the monitoring brings about violations of the local charge continuity equation. We leave these questions for future work. for some constant c. Next, we would like to estimate the variation of S proxy after the discrete steps (ii) and (iii) described above, which we denote by ∆S m . On average, the measurements decrease S proxy since they have the tendency of purifying the system. Suppose we replace the finite-depth circuit with a random Gaussian unitary acting on the whole system. In this case, based on the analysis of Ref. [103], and using that we have on average pL measurements, we would obtain the estimate |∆S m | ∼ (S proxy ) 2 /L. For our quantum-circuit model, we expect |∆S m | to be larger than this (measurements purify more), because the unitary dynamics is less scrambling and so less effective in protecting quantum information from the measurements. Therefore, we obtain the lower bound

|∆S m | ≥ c (S proxy ) 2 L . (D5)
Now, after a single application of steps (i), (ii) and (iii), the total change of S proxy is ∆S proxy = ∆S c -|∆S m |.

In the stationary regime, we must have ∆S proxy = 0. Putting together (D4) and (D5), this readily implies Namely, even in the weak monitoring limit, the late-time Rényi-2 entropy does not grow with the system size, consistently with our numerical results.

Figure 1 .

 1 Figure 1. Pictorial representation of the setup. -A chain of fermionic degrees of freedom is subject to unitary dynamics dictated by the Hamiltonian H, weak monitoring of strength γ, and particle injection/depletion of the left/right edge sites. The negativity is computed between the two halves of the system A and B.

Figure 2 .

 2 Figure 2.Entanglement negativity as a function of time. We plot two sets of curves corresponding to zero (γ = 0.0) and weak (γ = 0.5) monitoring. In the two cases, we consider chains of size L = 32÷96 with Γ = 1 and µ = 1. At late times, the negativity vanishes in non-monitored systems, while it approaches a constant increasing with L in the presence of weak monitoring.

Figure 3 .

 3 Figure 3. Large-time limit of the fermionic negativity. We consider chains of size L = 8 ÷ 192 with Γ = 1, µ = 1, and γ = 0.15 ÷ 2. Inset: numerical fit for the effective central charge appearing in Eq. (19).

Figure 4 .

 4 Figure 4. Comparison between the dynamics of Rényi-1/2 mutual information and fermionic negativity, from the initial state (18). We consider chains of size L = 32 ÷ 128 with Γ = 1, µ = 1, and γ = 0.0, γ = 0.25.

Figure 5 .

 5 Figure 5. Large-time limit of the averaged purity. We consider chains of size L = 8 ÷ 192 with Γ = 1, µ = 1, γ = 0.15 ÷ 1.5.

Figure 6 .

 6 Figure 6. Large-time limit of the averaged purity and entanglement negativity, for different choices of the driving potential difference µ. We fix Γ = 1 and γ = 0.6 ÷ 1.5. As detailed in the text, the purity is qualitatively sensible to the values of µ, while the negativity is only quantitatively affected by µ, the scaling being qualitatively the same.

Figure 7 .

 7 Figure 7. Choice of saturation time. -We consider Γ = 1, and show which saturation time (red line) is chosen for γ = 0.5, 2 and L = 16 ÷ 128.

  S 2 proxy ≤ (c/c ) ∼ O(1) . (D6)

Figure 8 .

 8 Figure 8. Robustness against different initial conditions. -As an example, here we consider γ = 0.5 and various system sizes with the infinite temperature initial state (blue) and the initial state (18) (orange). As expected, the dynamics show a different transient regime, but the same stationary state.

ACKNOWLEDGMENTS

LP acknowledges Denis Bernard for interesting discussions and collaborations on related topics. XT is grateful to V. Alba, P. Ruggiero, and V. Vitale for discussions. XT and MS were supported by the ANR grant "NonEQuMat"(ANR-19-CE47-0001). We acknowledge computational resources on the Collége de France IPH cluster.

Appendix A: Equation of motion for the correlation matrix

In this section, we derive the equation of motion for the correlation matrix C t ξ . For readability, in this section, we drop the time label and the trajectory label. Following the prescriptions in Refs. [63-65, 80, 81], the stochastic Schrödinger equation for the correlation matrix is easily derived and reads dC i,j = idt(C i-1,j + C i+1,j -C i,j+1 -C i,j-1 ) -γC i,j dt

Up to a sub-leading Trotterization error, we can consider the equation of motion from two separate contributions: (i) the Hamiltonian and the boundary Lindbladian, (ii) the monitoring contribution.

The former has been considered in a variety of works (see e.g. Refs. [START_REF] Žnidarič | [END_REF]97,118,119]), and is given by the terms in Eq. (A1) which are not proportional to γ or to dξ j,t

The linear operator L and the matrix P are simply read out from the corresponding terms in Eq. (A1), and Eq. (A2) can be integrated with standard means (e.g. Runge-Kutta algorithms) to obtain the infinitesimal solution Ct+dt .

The noisy contribution can be integrated as well, to obtain [80]

where C diag is the diagonal part of C. The transformation Eq. (A3) corresponds to

We note that this transformation preserves the norm only up to terms o(dt), and the errors accumulate during the evolution. Thus, we choose to renormalize the state after each step and consider instead ρ = e m dξ m t (nm-nm )-γdt m (nm-nm) 2 ρe m dξ m t (nm-nm )-γdt m (nm-nm) 2 tr e 2 m dξ m t (nm-nm )-2γdt m (nm-nm) 2 ρ .

(A5)

We can think of this map as a sequence of single-site commuting transformations. The associated map for the correlation matrix can be obtained within the framework of linear fermionic optics [100,103]. The final result is given by the combined action of L commuting channels

, each of which is associated to a given site, and reads

where

Appendix B: Average Lindbladian equation and NESS

In this section we give a brief summary of known results for the average stationary state ρ = E ξ [ρ ξ ]. The density, current density and two-point density functions in the non-equilibrium steady state were computed in Ref. [START_REF] Žnidarič | [END_REF]. The average current

and displays in the large system size limit L 1 a diffusive behavior. The average density, on the other hand, shows a linearly decreasing gradient profile from source (l = 1) to drain (l = L)

with a slope given by the average current j NESS ∼ 1/L. The full dynamics was analyzed in Ref. [93]. Within the Lindbladian framework, large deviations in the statistics of the current were later analyzed in Refs. [95,120].

It is also important to mention that quantum trajectories in this model were analyzed before in Ref. [96], which, however, considered a unitary unraveling of the averaged Lindbladian evolution. Within this framework, and for γ > 0 the entanglement negativity has been estimated in Ref. [101].

Appendix C: Fermionic negativity of Gaussian states

In this section we detail the computation of the fermionic negativity for Gaussian fermionic states. We consider a density matrix ρ AB over a bipartite system A ∪ B, characterized by the 2N × 2N Majorana matrix

To compute Eq. ( 16), the starting point is the block decomposition over

From (15), it follows that

are the covariance matrices associated with ρ R A and [ρ R A ] † . The product of these covariance matrices can be performed following Refs. [121,122]. The resulting density matrix is Gaussian with covariance matrix

It can shown that this matrix is antisymmetric and purely immaginary, and so Hermitian. Furthermore, the product density matrix has a normalization factor given by [121] det

Collecting all the terms, we arrive at the final result

Here we denoted by {ξ j , -ξ j } and {ζ j , -ζ j } the eigenvalues of M * and M , respectively (in both cases they come in pairs of opposite sign, because they are symmetric). So in the above sums only a single element in each pair must be included.

The formula Eq. (C5) holds for any quadratic fermionic system [102]. For the specific instance considered in the main text, and for the initial condition c † a c † b = 0, the computation of the negativity can be simplified. Defining the matrix G m,n = 2C m,n -δ m,n , and given the bipartition A ∪ B, we have

Then, in a similar fashion to the Majorana case, one can compute

and the matrix

The final expression for the negativity for the restricted correlation is given by

(C9) where µ j are the eigenvalues of G * and λ j are the eigenvalues of C.

Appendix D: Late-time scaling of the purity

In this section, we provide an heuristic argument to justify that, for µ = 1, the average of the late-time Rényi-2 entropy is O(1) as L → ∞. First, following Ref. [103], we introduce the quantity

It is not difficult to show that, for a Gaussian state,

Therefore, S proxy (C) and the Rényi-2 entropy S (2) have the same scaling in L, so that S proxy can be considered a proxy for S (2) . Next, we consider a simplified dynamics which we expect to display the same qualitative behavior of the SME (2). Namely, we focus on a discrete, rather than continuous, model where the single time-step consists in three parts: (i) the application of a quantum channel [START_REF] Nielsen | Quantum computation and quantum information[END_REF] acting at the boundary sites, implementing extraction/injection of particles; (ii) a finite-depth quantum-circuit Gaussian dynamics; (iii) a round of random measurements of the local density. For simplicity, we take the measurements to be projective, although our conclusions also hold if they are weak. They are performed at each site with a finite probability p ∈ [0, 1]. In the following, we do not need to specify the quantumcircuit gates, which can be obtained, for instance, by a Trotterization of the Hamiltonian in Eq. ( 2). Finally, for the right quantum channel we choose a Gaussian operation implementing extraction of particles,

where

Here c A , c † A act on an ancillary degree of freedom initialized in the vacuum, |0 , and eventually they are traced over. Note that the symbol ⊗ in (D3) denotes graded tensor product. Analogously, we can define a Gaussian operation implementing injection of particles at the left boundary.

Let us consider the stationary state for µ = 1. We want to estimate the variation of S proxy due to the action of the right quantum channel, denoted by ∆S R . First, we recall that the average density at the right boundary site In this section, we briefly summarize the numerical implementation and give further numerical results.

We have used a Runge-Kutta algorithm of fourth order to solve the combined evolution consisting of boundarydriving and Hamiltonian dynamics and implemented the mapping Eq. (A6) for the noise contribution. For the time-evolution of the quantum trajectories, we considered the average over N = 200 ÷ 1000 trajectories. Furthermore, we have evidence that a self-averaging property occurs at late time: hence we also average the stationary state values over the last T = 5000 ÷ 20000 time-steps in the stationary regime. Throughout this paper, we have chosen dt = 0.05, but we tested, but not shown here, that the protocol gives the same average results for dt = 0.01. Another test we have performed is that the results are qualitatively robust against varying Γ, which have overall set to Γ = 1 in this paper.

In Fig. 7 we illustrate the choice of the stationary time t stat for various γ and various system sizes. Typically, due to the diffusive nature of the average state, the saturation time scales ad O(L 2 /γ), which combined with the O(L 3 ) simulation cost of each time step, results in O(L 5 ) cost.

Independence from the initial conditions.-Initial conditions affect the transient dynamics, but results in the same stationary state. As an example, in Fig. 8 we consider two different initial conditions, the infinite temperature state ρ = 1/2 L and the Néel state (18). focusing on γ = 0.5, Γ = 1. Our numerics show the conditional average of the negativity and of the purity saturates to the same stationary value. We have also checked the independence from the initial conditions by taking random product states and different values of γ (not shown here).