Itai Benjamini

Hugo Duminil-Copin

Gady Kozma

Cyrille Lucas

Internal diffusion-limited aggregation with uniform starting points

Keywords: Growth model, Random walk, IDLA, Harmonic measure. Mots-Clés : Modèle de croissance, Marche aléatoire, IDLA, mesure harmonique

We study internal diffusion-limited aggregation with uniform starting points on Z d . In this model, each new particle starts from a vertex chosen uniformly at random on the existing aggregate. We prove that the limiting shape of the aggregate is a Euclidean ball.

Nous étudions le modèle d'agrégation limitée par diffusion interne avec points de départ uniformes sur Z d . Dans ce modèle, chaque nouvelle particule est ajoutée à un point choisi uniformément au hasard parmi ceux de l'agrégat existant. Nous prouvons que l'agrégat normalisé admet comme forme limite la boule euclidienne.

Introduction 1.Historical introduction and motivation

Internal diffusion-limited aggregation (IDLA) was introduced by Diaconis and Fulton in [START_REF] Diaconis | A growth model, a game, an algebra, Lagrange inversion, and characteristic classes[END_REF], and gives a protocol for recursively building a random aggregate of particles. At each step, the first vertex visited outside the current aggregate by a random walk started at the origin is added to the aggregate. In a number of settings, this model is known to have a deterministic limit-shape, meaning that a random aggregate with a large number of particles has a typical shape. On Z d , Lawler, Bramson and Griffeath [START_REF] Lawler | Internal diffusion limited aggregation[END_REF] were the first to identify this limit-shape, in the case of simple random walks, as the Euclidean ball. Their result was later sharpened by Lawler [START_REF] Gregory | Subdiffusive fluctuations for internal diffusion limited aggregation[END_REF], and was recently drastically improved with the simultaneous works of Asselah and Gaudillère [START_REF] Asselah | From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models[END_REF][START_REF] Asselah | Sublogarithmic fluctuations for internal DLA[END_REF] and Jerison, Levine and Sheffield [START_REF] Jerison | Internal DLA in higher dimensions[END_REF][START_REF] Jerison | Logarithmic fluctuations for internal DLA[END_REF][START_REF] Jerison | Internal DLA and the Gaussian free field[END_REF], where logarithmic bounds are proved for fluctuations of the boundary.

The IDLA model has been extended in several contexts including drifted random walks [START_REF] Lucas | The limiting shape for drifted internal diffusion limited aggregation is a true heat ball[END_REF], Cayley graphs of finitely generated groups [START_REF] Blachère | Internal diffusion limited aggregation on discrete groups of polynomial growth[END_REF][START_REF] Blachère | Internal diffusion limited aggregation on discrete groups having exponential growth[END_REF][START_REF] Duminil-Copin | Internal Diffusion Limited Aggregation on groups of polynomial growth[END_REF][START_REF] Huss | Internal diffusion-limited aggregation on non-amenable graphs[END_REF] and random environments [START_REF] Duminil-Copin | Containing Internal Diffusion Limited Aggregation[END_REF][START_REF] Shellef | Idla on the supercritical percolation cluster[END_REF].

Another interesting growth model is provided by Richardson's model [START_REF] Richardson | Random growth in a tesselation[END_REF], which is defined as follows. At time 0, only the origin is occupied. A vacant site becomes occupied at an exponential time with a rate proportional to the number of occupied neighbours, and once occupied a site remains occupied. The set of vertices occupied by time t is the ball of radius t centered at the origin in first passage percolation with exponential clocks (see [START_REF] Kesten | Aspects of First Passage Percolation[END_REF]). Eden [START_REF] Eden | A two-dimensional growth process[END_REF] first asked about the shape of this process on Euclidean lattices and Richardson proved that a limiting shape exists. It is believed that the convex centrally symmetric limiting shape is not a Euclidean ball. This was established by Kesten in high dimensions (unpublished, but see [START_REF] Couronné | Construction of a short path in highdimensional first passage percolation[END_REF]) together with the fact that the boundary has t 1/3 fluctuations, a long standing conjecture.

Internal diffusion-limited aggregation with uniform starting points (from here on shortened to uIDLA) is a growth model interpolating between standard internal diffusionlimited aggregation and Richardson's model. In uIDLA, particles are born uniformly on the shape and relocate to the outer boundary according to harmonic measure seen from the site they appeared at. While usual IDLA approaches rely on estimating the number of visits to a given point by particles starting from the origin, either directly or as the solution to a discrete partial differential equation, the study of uIDLA is more difficult because of the self-dependence involved in the construction.

Another related model is excited to the center. In this model a single particle walks around the lattice Z d doing simple random walk, except when it arrives at a vertex it never visited before ("a new vertex"), in which case it gets a drift towards the point 0. To compare excited to the center to the models described so far, think about standard IDLA as a single particle which, upon reaching a new vertex, is teleported to 0; and about uIDLA as a single particle which, upon reaching a new vertex, is teleported to a random location in the visited area. Very little is known about random walk excited to the center -there is an unpublished result showing that it is recurrent in all dimensions, but the shape of visited vertices is very far from being understood. Simulations and some heuristics indicate that at time t the set of visited vertices should be a ball with radius approximately t 1/(d+1) .

For uIDLA, we show that the limiting shape is a Euclidean ball, hence showing a behaviour close to the standard IDLA behaviour. Yet, the boundary fluctuations are expected to be slightly stronger than that of standard IDLA. This is not surprising, since part of the growth is due to particles emerging near the boundary thus behaving very roughly like the Richardson model. This suggests that the local regularity will be determined by some competition between particles born locally à la Richardson and particles arriving from far away as in standard IDLA. Furthermore, simulations like the one we present below seem to indicate a mesoscopic shift in the center of mass of the cluster, which occurs in a random direction. This paper deals with the limiting shape and not the fluctuations.

Definition of the model and statement of the main theorem

We consider the lattice

Z d with d ≥ 1. Let S ⊆ Z d be a finite subset of Z d .
In order to define both standard and uniform starting point IDLA, first define the action of adding a particle to an existing aggregate S. Let ξ = (ξ(0), ξ(1), . . .) be a random walk on Z d and let t S be the first time this walk is not in S. By random walk we mean the simple random walk choosing one of its 2d neighbours uniformly and independently at random at each step. Define Add[ξ, S] := S ∪ {ξ(t S )}. Standard IDLA Fix an integer n ≥ 0. Let DA n be the aggregate with n particles started at 0, constructed inductively as follows: DA 0 = ∅ and

DA n+1 := Add[ξ 0 n , DA n ]
where ξ 0 n is a random walk starting at 0 which is independent from ξ 0 0 , . . . , ξ 0 n-1 . This process is referred to as IDLA.

Note that the equivalent initialisation DA 1 = {0} is sometimes used.

IDLA with uniform starting point Fix an integer n ≥ 0. Let A n be the uniform starting point aggregate with n particles constructed inductively as follows: A 1 = {0} and

A n+1 := Add[ξ Xn n , A n ]
where X n is a point chosen uniformly on A n , and ξ Xn n is a random walk starting at X n and independent of ξ X 0 0 , . . . , ξ

X n-1 n-1 . This process is referred to as uIDLA. Let | • | be the Euclidean distance in R d . For n > 0, let B[n] := {y ∈ Z d : |y| ≤ n} and b n := |B[n]|. Theorem 1.1. Let d ≥ 2.
There exists positive constants c 1 , c 2 , C 1 and C 2 depending only on the dimension, such that almost surely,

B n(1 -C 1 n -c 1 ) ⊆ A bn ⊆ B n(1 + C 2 n -c 2 )
for n large enough.

Remarks In dimension 1, the uIDLA aggregate with n points is a set of consecutive integers of length n, therefore it is entirely determined by the position of its middle point (called M n ). It is clear, either from a quick computation using the gambler's ruin or from a symmetry argument, that the probability for the cluster to grow on either of the two sides is exactly 1/2. Therefore the process M n is exactly a simple random walk on integers and half-integers, and the behaviour of the cluster is obvious, with a law of large numbers and CLT fluctuations.

In dimensions bigger than 2 we expect much smaller fluctuations, and our theorem is not satisfactory in this regard. We have chosen not to optimize the n -c , mainly in order to help alleviate notations, but also because we do not hope to capture the true order of the error term with our method.

Structure of the paper

The first section contains five lemmas. They provide useful information on comparing uIDLA to IDLA. As they are of interest on their own, we isolate them from the proof of the theorem.

The second section of the article deals with the stability properties of the Euclidean ball under the uIDLA process. We first investigate the claim that the process started from a configuration that includes a ball will contain a growing ball with high probability. Then we take the converse and prove that the process started from any configuration inside a ball will stay contained in a slightly bigger growing ball. To prove this statement we examine the cluster together with the genealogical tree describing the starting points of the random walks. Our proof involves a comparison with a First Passage Percolation process on random trees.

In our third section, we bring these elements together for a proof of our theorem. The inner bound is proved first, using a refinement method that relies heavily on our coupling properties. The outer bound is then proved using the genealogical construction from the previous section.

Further notation For every y ∈ Z d , let P y denote the law of a simple random walk on Z d starting from y. For a set S ⊂ Z d we will denote by ∂S the set of vertices in Z d \ S with a neighbour (or more than one) in S.

Comparison lemmas

We start with the following notations which will enable us to state our lemmas more easily. Given vertices x 1 , . . . , x k in Z d , define DA x 1 ,...,x k (S) to be the IDLA aggregate formed by launching additional particles from points x 1 , . . . , x k . Note that x i need not be in the set S. Naturally, for x ∈ S, DA x (S) = S ∪ {x} deterministically. Recall that, classically, the law of the aggregate does not depend on the order in which these particles are added. Therefore, we also define DA X (S) := DA x 1 ,...,x k (S), where X is the multi-set X = {x 1 , . . . , x k }. If the multi-set X is just k repetitions of the origin, we denote for conciseness DA k (S) := DA X (S). Remark that this notation is consistent with our initial definition, in that DA n = DA n (∅).

Similarly, for the uIDLA process, we denote A 1 (S) the result of adding a particle started uniformly on S to the set S. We also denote A k (S) for k ∈ N the result of the recursive process of adding k particles to the aggregate S, where the first one starts uniformly on S, and the j-th particle starts uniformly on A j-1 (S).

We start with the following lemma. It states that the aggregate obtained by launching k particles from arbitrary points in B[n/2] is bigger than the aggregate obtained by launching a smaller yet comparable number of particles from the origin. Lemma 2.1. There exists η > 0 (depending only on the dimension) such that for any multi-set X of cardinality k in B[n/2], DA X (B[n]) stochastically dominates DA κ (B[n]), where κ follows a binomial distribution B(k, η).

Proof. Let x be a point in B[n/2] and A a set containing B[n]. We consider the function evaluating the probability that the random walk starting at x exits A through a point y. We consider the stopping time τ A = inf {t : ξ x (t) / ∈ A} and the function

h y (x) = P x (ξ x (τ A ) = y) .
This function is harmonic in x on B[n -1], hence the Harnack inequality [19, Theorem 6.3.9] implies that there exists η > 0 such that for all

A ⊇ B[n], x ∈ B[n/2] and y / ∈ A, h y (x) ≥ ηh y (0).
This inequality allows to construct a coupling between DA X and DA κ as follows. Let E k and F k be constructed recursively.

Set E 0 = F 0 = B[n]. Index sites of X by {x 1 , . . . , x |X| }. Assume that F k ⊆ E k have been constructed. Construct E k+1 = Add[ξ x k+1 k+1 , E k ]. Consider a killed random walk ξ 0 k+1 coupled with ξ x k+1
k+1 in such a way that:

• ξ 0 k+1 is killed at 0 with probability 1 -η. • if ξ 0 k+1 exit E k through y, so does ξ x k+1 k+1 ,
The existence of this coupling is guaranteed by h y (x) ≥ ηh y (0). After exiting B[n] we couple the walks in the usual way: they walk together until exiting their respective aggregates (since

F k ⊆ E k , the walk on F k would exit first). Construct F k+1 = Add[ξ 0 k+1 , F k ] if the particle is not killed. Note that F k+1 ⊆ E k+1 , since either ξ 0 k+1 exits F k throughout a point of E k ,
or it does through a point not in E k , but in this case the coupling guarantees that the exiting point is in E k+1 . The total number of coupled particles, κ, follows a binomial distribution with parameters (k, η).

The following lemma controls the behaviour of a standard IDLA with M points started with a Euclidean ball B[n] already occupied. It closely follows the spirit of [START_REF] Asselah | From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models[END_REF], but instead of pushing the precision to get the best almost sure bound, we only look at points that are far enough from the edge of the theoretical shape to keep the probability of inclusion exponentially close to 1.

Lemma 2.2. For any n ∈ R + and N ∈ N * , let us write r = r n,N = N bn . Then we have

P B n 1 + r -r 3/2 1/d ⊂ DA N B[n] ≥ 1 -exp -C 3 nr 3/2 .
We will use this lemma in the window where nr 3/2 is large but r is small, so that r -r 3/2 ≥ 0 (otherwise the lemma is true but useless).

Proof. Our lemma is almost exclusively a consequence of the many ideas provided in [START_REF] Asselah | From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models[END_REF]. Therefore, we refer the reader to the appendix, in which we give a guide to the modifications one needs to do in [START_REF] Asselah | From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models[END_REF] to get this result.

The next two lemmas propose stochastic dominations between standard IDLA and the uIDLA process. We start with a lemma that compares one step.

Lemma 2.3. There exists a constant C 4 > 0 such that, if B[n] ⊂ S ⊂ T then A 1 (T ) stochastically dominates DA δ (S), where δ is a Bernoulli variable with parameter |B[n]| |T | (1 - C 4 n ).
Proof. First, remark that our new point falls inside

B[n] with probability |B[n]| |T | .
Once more, we consider the stopping time τ A = inf {t : ξ x (t) / ∈ A} and the function

h y (x) = P x (ξ x (τ A ) = y) .
This function is harmonic in x on B[n]. We are now interested in an averaging property for this harmonic function; namely, is h y (0) close to 1

|B[n]| x∈B[n] h y (x) ?
The study on this averaging property is linked to that of quadrature domains and the divisible sandpile model, and, in particular, one shape on which a relation is known between the two terms is the shape taken by the divisible sandpile after toplings, with all the initial mass started at the origin, as defined in [START_REF] Levine | Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile[END_REF]. Let m(x) be the final mass distribution corresponding to an initial mass M at the origin, then we have, for all harmonic functions h, M h(0) =

x∈Z d m(x)h(x).
Recall that the final mass distribution m is equal to 1 on a given shape, has value between 0 and 1 at distance one from this shape, and is zero at distance more than one of this shape. It is hence a consequence of Levine and Peres's shape theorem (see [START_REF] Levine | Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile[END_REF]) that there is a constant c > 0 depending only on the dimension such that

h y (0) = 1 |B[n]| x∈Z d m(x)h y (x), with m(x) = 1 on B[n -c] and m(x) = 0 outside B[n + c].
Combining the facts that m has values between 0 and 1 everywhere; and that y h y (x) = 1, allows to bound the error given by replacing m with 1 B . We get that there is a constant C 4 depending only on the dimension, such that:

y∈∂A h y (0) - 1 |B[n]| x∈B[n] h y (x) ≤ C 4 n .
Hence, our two particles can be coupled with probability

|B[n]| |A| (1 -C 2 n ), which yields the result.
Assume E is some subset of our aggregate F . As F evolves, there is a natural increasing subset E n ⊂ A n (F ) which corresponds to E and is in fact a time change of an uIDLA started from E. Basically, one traces only particles which started on E n and follows them only until they exit E n . Further, it is not necessary to know anything about the structure of F , it is enough to know its size. Formally, the definition is as follows: Let E 0 = E. Next, for every n define

E n+1 := Add[ξ Xn n+1 , E n ] with (independent) probability |En| |F |+n E n otherwise
where X n is a point chosen uniformly on E n , and ξ Xn n+1 is a random walk starting at X n and independent of ξ X 1 1 , . . . , ξ

X k-1 k
. Finally, the Bernoulli events which determine whether the point will be added or not are independent of the walks (and of one another). We see that the process depends only on the size of F and not on its structure. This leads to the following definition Definition. For E ⊂ Z d and m ≥ |E| we let A n (E; m) be the E n defined in the previous paragraph, for some F with |F | = m. We call A n (E; m) the subset uIDLA.

Clearly A n (E; |E|) is the same as A n (E) and, in general, if E ⊂ F , then A n (F ) stochastically dominates A n (E; |F |). A little more than that is, in fact, true:

Lemma 2.4. For any sets E ⊆ F , we have that A n (F ) stochastically dominates DA F \E A n (E; |F |) .
Proof. We will colour A n (F ) in 3 colours, blue, red and black, such that the blue part has the same distribution as A n (E; |F |), the union of the red and the blue has the same disribution as DA F \E (A n (E; |F |)) and black is the rest. Here is the colouring scheme:

We start the process with A 0 = F coloured as follows: E is coloured blue and F \ E is coloured red. Suppose we already constructed (and coloured) A n . We choose a vertex x of A n randomly to start the random walk from.

• If x is blue, perform the random walk until the particle exits the blue set. When it does, the site where it lands is coloured blue. If there was already a particle at that site, "wake it up" -it continues walking according to the rules in the following clauses.

• Now assume we have a red particle walking (which can only happen if a red particle was woken by a blue one, as in the previous clause). Perform the random walk until the particle exits the union of the red and the blue. When it does, that site will be coloured red. If there is a black particle there, wake it up and let it continue walking according to the rule in the next (and last) clause.

• If x is red or black, let the new particle be black. Let it perform simple random walk until the it exits the entire aggregate, and colour that site black.

Thus, for example, a particle might start from a blue site, walk until reaching a red site, change that site to blue, continue walking until reaching a black site, change that site to red, and then walk until exiting. This ends the description of the colouring. Now, the fact that the blue part of the aggregate has the same distribution as A n (E; |F |) is evident. The fact that the union of the red and the blue has the same distribution as DA F \E (A n (E; |F |)) is also simple, because the red part starts with F \ E and then each red particle does a random walk and ends outside the eventual blue part.

One might claim that, even though each red particle does simple random walk, they are stopped and woken up mixing up their order. It is well-known that this does not affect the distribution of the final aggregate. For the convenience of the reader, let us recall the argument. One attaches labels to each red particle, and when a particle with a lower label steps over a particle with a higher label, they exchange labels so that the higher label continue to walk. This, of course, does not change the red part, but now each label does simple random walk until its final resting point, and only then does the next label start to walk. So the union of the blue and the red part has indeed the same distribution as DA F \E (A n (E; |F |)) and the lemma is proved.

The following lemma is extracted from [START_REF] Duminil-Copin | Containing Internal Diffusion Limited Aggregation[END_REF]. It states that a random walk has a small probability of passing through an area of small density, and will be used to couple our process with a First Passage Percolation process. Rather than refer to the proof of [START_REF] Duminil-Copin | Containing Internal Diffusion Limited Aggregation[END_REF] which holds in a more general setting, we give a shorter proof specific to Z . By [19, Lemma 6.3.7], every x ∈ ∂B[r] has probability ≤ Cr 1-d that random walk started from 0 will exit B[r] at x. Summing over x ∈ S ∩ ∂B[r] gets that the probability that random walk started from 0 will exit B[r] at S is less than Cε. Thus for small enough, ξ exits S before reaching ∂B[n] with probability greater than 1 -p.

Stability of the Euclidean ball 3.1 Inner stability of the ball

In this section ( §3) we show that, if you start a uIDLA from a large ball, it remains an approximate ball, with high probability. We first ( §3.1) show inner stability, i.e. that the aggregate contains a ball of the approximately correct size. In a formula,

A bm-bn (B[n]) ⊇ B[m(1 -Cn -1/4 )]
with high probability. In other words, the only error is the missing Cn -1/4 in the diameter.

It will be convenient, though, to formulate the claim slightly more generally: if B[n] ⊆ S then A M (S) contains a ball of the correct size. We will use the notation A M (E; N ) introduced on page 7 -recall that A M (E; N ) is the way E evolves when you embed it in some set of size N , add M particles in a uIDLA fashion to that set, and examine only particles that landed on E. We first formulate a lemma for adding a relatively small number of particles, an n -1/2 proportion: Lemma 3.1. There exist δ 2 , C 5 > 0 such that for any M ≥ b n ,

P B n 1 + n -1/2 -C 5 n -3/4 1/d ⊆ A M n -1/2 (B[n]; M ) ≥ 1 -C exp(-n δ 2 ).
Remark in particular that the probability does not depend on M .

Proof. The definition of A M n -1/2 (B[n]; M ) gives that it is the same as A K (B[n]) where K is a random variable which stochastically dominates a binomial distribution with M n -1/2 trials and probability b n /(M + M n -1/2 ) for success.

Recall that Lemma 2.3 says that adding a single particle to uIDLA stochastically dominates adding a single particle to standard IDLA, with an appropriate probability. Applying Lemma 2.3 K times gives that A K (B[n]) stochastically dominates a standard IDLA with initial set B[n] and with a random number L of particles (started at the origin), where L stochastically dominates a binomial distribution with K trials and probability

(b n /(b n + K))(1 -C 4 /n) for success.
As a first step we need to make sure that K is not too large, so that the factor (b n /(b n + K)) does not impact the probability for success too much. Since the expected value of K is n -1/2 b n /(1 + n -1/2 ), and b n = ω d n d + O(n d-1 ), we know that K should only be of order n d-1/2 . We use a Chernoff bound to ensure that:

K ≤ 2n -1/2 b n (1 + n -1/2 )
with probability larger than 1 -C exp(-n d-1/2 /3) (note that d ≥ 2). We now assume that this bound for K is verified, and therefore b n /(b n + K) ≥ 1/(1 + 2n -1/2 ).

Combining these facts shows that A M n -1/2 (B[n]; M ) stochastically dominates standard IDLA started from a ball with the number of particles L following a binomial distribution Bin(s, p) with

s = M n -1/2 and p = b n M (1 + n -1/2 )(1 + 2n -1/2 ) 1 - C 4 n . Since b n = ω d n d + O(n d-1
), another Chernoff bound directly yields that L satisfies:

L - n -1/2 b n (1 + n -1/2 )(1 + 2n -1/2 ) 1 - C 4 n ≤ n d-3/4
with probability larger than 1 -exp(-Cn d-1 ), for a constant C > 0 (note that d ≥ 2). We now assume that this bound for L is verified. We then apply Lemma 2.2 with B[n] already occupied and L new particles started at the origin. With the notations of the lemma, r n,N ∼ n -1/2 and we get that the ball of radius n(1 + r -r 3/2 ) 1/d is included in the cluster DA L (B[n]) with probability at least 1 -exp(-C 3 n 1/4 ). We estimate

1 + r -r 3/2 ≥ 1 + n -1/2 (1 + n -1/2 )(1 + 2n -1/2 ) 1 - C 4 n - n d-3/4 b n -Cn -3/4 ≥ 1 + n -1/2 -C 5 n -3/4
where the first inequality is the lower bound on L, and these inequalities hold with probability 1 -C exp(-n δ 2 ).

The case where the number of particles we add is proportional to the volume (or more) is a corollary: Corollary 3.1. There exist δ 3 , C 6 > 0 such that for any M ≥ b n ,

P ∀a ≥ 1 B na 1/d 1 -C 6 n -1/4 ⊆ A (a-1)M (B[n]; M ) ≥ 1 -C exp(-n δ 3 ).
Proof. Examine first the case that a ≤ 2. We apply the previous lemma repeatedly K times, i.e. define

S 0 = B[n], M 0 = M, S i+1 = A M i n -1/2 (S i ; M i ), M i+1 = M i (1 + n -1/2 ),
with K chosen in such a way that we obtain additional |S| particles. Since each time we add M i n -1/2 particles and M i ≥ M , we deduce that K ≤ n 1/2 Therefore, with probability larger than 1 -n 1/2 exp(-n δ 2 ) the aggregate A (a-1)M (B[n]; M ) contains the Euclidean ball of radius

na 1/d 1 -Cn -3/4 K ≥ na 1/d 1 -Cn -3/4 n 1/2 ≥ na 1/d 1 -Cn -1/4 . (3.1)
This takes care of a along a sequence. For a general a ∈ [1, 2], we find some i such that M i < (a -1)M < M i+1 and the inequality still holds from monotonicity of the aggregate (we lose Cn d-1/2 particles from the approximation, but this only changes the value of the constant in (3.1)).

For general a (i.e. a > 2) we repeat the last calculation for 2n, 4n etc. We get that the claim holds for all a except for an event whose probability is smaller than

C n i=0 2 i n 1/2 exp(-2 i n δ 2 ).
Since this sum converges exponentially, we may bound it by C exp(n -δ 3 ) for an appropriate δ 3 . Similarly, the errors in (3.1) converge exponentially, so they only change the constant. So we get that the radius is bounded by

na 1/d (1 -C 6 n -1/4 )
for a suitable constant C 6 .

Genealogical construction and outer stability

Our aim in this section is to prove a converse to Corollary 3.1 for the outer stability of the ball. We begin by comparing the process started from any set S ⊂ B[n] with the process started from B[n], on an event of high probability. Here we are comparing uIDLA to another uIDLA (and not to standard IDLA, as in the previous section), so the argument is much simpler.

Lemma 3.2. There exist δ 5 > 0 and C 7 > 0 such that for any set S with S ⊆ B[n], and for any 1 ≤ a ≤ 2, there is a coupling of A (a-1)|S| (S) and A (a-1)bn(

1+C 7 |S| -1/4 ) (B[n]) such that P A (a-1)|S| (S) ⊆ A (a-1)bn(1+C 7 |S| -1/8 ) (B[n]) ≥ 1 -C exp(-|S| δ 5 ).
Proof. We may assume |S| is sufficiently large. Recall the definition of subset uIDLA on page 7 and the natural coupling of A i (S; b n ) and A i (B[n]), with the property that

A i (S; b n ) ⊆ A i (B[n]). Examine first the first b n |S| -1/2 particles added to A i (B[n]
). Each of these is added to A i (S; b n ) with probability at least |S|/(b n + b n |S| -1/2 ). A Chernoff bound therefore shows that

P |A bn|S| -1/2 (S; b n )| -|S| > |S| 1/2 1 + |S| -1/2 -|S| 3/8 > 1 -C exp(-c|S| 1/8 ).
By repeating this procedure at most |S| 1/2 times (here we use the assumption that a ≤ 2), we get that on an event of probability at least

1 -C|S| 1/2 exp(-C|S| 1/8 ) ≥ 1 -C exp(-|S| 1/9 ), we have |A (a-1)bn(1+C 7 |S| 1/8 ) (S; b n )| > (a -1)
|S|. This finishes the lemma: we construct the coupling by letting

A i (S) = A j(i) (S; b n ) where j(i) is the first time that |A j(i) (S; b n )| = |S| + i and then with probability at least 1 -C exp(-|S| 1/9 ) we have j((a -1)|S|) ≤ (a -1)b n (1 + C 7 |S| 1/8 ) so A (a-1)|S| (S) = A j((a-1)|S|) (S; b n ) ⊆ A (a-1)bn(1+C 7 |S| 1/8 ) (S; b n ) ⊆ A (a-1)bn(1+C 7 |S| 1/8 ) (B[n]).
As needed.

We will now prove that the uIDLA started from a ball is contained in a suitable ball with high probability.

Proposition 3.1. There exist C 8 > 0 such that for any 1 ≤ a ≤ 2 the event

A (a-1)bn (B[n]) ⊆ B na 1/d (1 + C 8 n -1/5 )
occurs with superpolynomially large probability.

Here and below, when we say about a sequence of events E n that they "occur with superpolynomially large probability" we mean that there exists a function φ decreasing to 0 faster than any power of n such that P(E n ) > 1 -φ(n). We might also use the phrase "P(E n ) grows superpolynomially" (and we do not insinuate by that the the probabilities increase as a function of n, just the bound above).

In order to prove this proposition, we first remark that as a consequence of Corollary 3.1, the ball of radius na 1/d (1 -C 6 n -1/4 ) is included in the uIDLA cluster A (a-1)bn (B[n]) with stretched exponentially small probability. Hence we only need to control a number of particles of order an d-1/4 . However, these particles could in principle cover a thin spike that would reach very far. We know this cannot happen in regular IDLA, but in our case, a new particle may start on the furthermost point of the cluster, which complicates the situation. We therefore need to consider the genealogy of the particles in the process.

Recall that a rooted tree is a graph with no cycle and one marked point called the root. A rooted forest is a family of disjoint rooted trees.

We construct the uIDLA starting from a set S in a new fashion. Consider a rooted forest whose vertices are indexed by integers and constructed as follows. At time 0, T 0 (S) is given by |S| isolated sites indexed by 1, 2, . . . , |S|, which are the roots of the trees. At each step the vertex set of T k is A k (S) and the edges of T k are constructed inductively as follows: T k+1 has all the edges of T k and one more, from the starting point of the random walk which constructed A k+1 to its end i.e. to A k+1 \ A k . We will call this construction the genealogical construction of the uIDLA cluster, and T k the genealogical tree encoding it. We will look closely at the forest structure of T k , not at its embedding in Z d : rather, we think of a particle in the cluster as having both a position in Z d and a position in the genealogical tree (or forest) associated with the cluster.

We start by an elementary lemma which is a generalisation of [START_REF] Fayolle | Growth rate and ergodicity conditions for a class of random trees[END_REF]Lemma 2.1]. As in First Passage Percolation, we attribute to every edge of the forest a geometric random variable with parameter 1/2, independent of the random variables of other edges. We define the passage time between two vertices as the sum of the random variables over edges on the geodesic between those two vertices (note that in this case, there is only one choice for the minimal path). The reaching time of a vertex is the passage time between the root and this vertex. Proof. Let us first consider a slightly different model. Let Tt be the tree obtained from the same rule as for T n , but in continuous time (meaning that a new edge appears on each vertex according to an exponential clock of mean 1). Rather than explicitly writing the coupling between Tt and T n , we embed Tt in our probability space so that it is independent from T n . This model is exactly the model studied in [START_REF] Fayolle | Growth rate and ergodicity conditions for a class of random trees[END_REF]. In particular, if X t (k) is the number of vertices at graph distance k from the root, Lemma 2.1 of [START_REF] Fayolle | Growth rate and ergodicity conditions for a class of random trees[END_REF] shows that

E[X t (k)] = t k k! .
Choose C 9 > 1 large enough and c 7 > 0 small enough so that E[X t (k)] ≤ e -c 7 k for k ≥ C 9 t. Let Dt (h) be the number of sites with passage time larger than h in Tt . We find

E[ Dt (h)] = ∞ k=0 p k,h E[X t (k)] = ∞ k=0 p k,h t k k! ,
where p k,h is the probability that the sum of k independent geometric random variables of mean 1/2 is larger than h -k. There exists c 8 > 0 such that p k,h ≤ e -c 8 h for any k ≤ h/3. For simplicity, let us assume that c 8 < 2/3. By dividing the sum between k ≤ c 8 h/2 and k ≥ c 8 h/2, we find that for h ≥ 2C 9 t/c 8 ,

E[ Dt (h)] = c 8 h/2 k=0 p k,h t k k! + ∞ k=c 8 h/2 p k,h t k k! ≤ c 8 h/2 k=0 p k,h t k k! + ∞ k=c 8 h/2 t k k! ≤ c 8 h 2 e c 8 h/2C 9 e -c 8 h + e -c 7 c 8 h/2
1 -e -c 7 .

In the second line, we used that p k,h ≤ 1, and in the third line both the bound on

t k
k! ≤ e -c 7 k obtained by assuming that k ≥ c 8 h/2 ≥ C 9 t, and the bounds t k k! ≤ e t ≤ e c 8 h/2C 9 and p k,h ≤ e -c 8 h when k ≤ c 8 h/2.

It only remains to go back from continuous time to discrete time. Let D n (h) be the number of sites with reaching time larger than h in T n . From our construction, conditionally on the event { Tt has k sites}, Dt has the same law as D k .

Since with probability at least 1/2, the aggregate T2 log n has more than n particles, we deduce that

P[D n (h) > 0] ≤ E[D n (h)] ≤ 2E[D n (h)]P( T2 log n has more than n sites) ≤ 2 k≥n E[D k (h)]P( T2 log n has k sites) ≤ 2 k≥n E[ D2 log n (h)| T2 log n has k sites]P( T2 log n has k sites) ≤ 2E[ D2 log n (h)] ≤ exp(-c 9 h)
for any h ≥ (4C 9 /c 8 ) log n, and c 9 sufficiently small.

We are now ready to prove Proposition 3.1. Recall that it stated that with superpolynomially large probability,

A (a-1)bn (B[n]) ⊂ B[a 1/d n(1 + C 8 n -1/9d )].
Proof. By Corollary 3.1, we know that with stretched exponential probability,

A (a-1)bn (B[n]) contains B[a 1/d n(1 -C 6 n -1/4 )].
But that leaves only C 10 n d-1/4 particles unaccounted for. Consequently, there are at most C 10 n 3d/4 particles outside B[a 1/d n] at the end of the construction and therefore also at every previous step.

Recall that Lemma 2.5 states that it is difficult to traverse any annulus B[m] \ B[n] containing less than εb m-n vertices. In our setting, this means that there exists some constant β d such that for each of the annuli

R k = B[n + (k + 1)β d n 3/4 ] \ B[n + kβ d n 3/4 ],
the conclusion of Lemma 2.5 holds in this annulus, with p Lemma 2.5 = 1/2, if it is filled with less than C 10 n d-1/4 particles (note that 3d/4 ≤ d -1/4 since d ≥ 2). Remark that β d is a constant that depends only on the dimension.

Since all the R k 's are outside B[n], each of them contains at most C 10 n d-1/4 particles at any point in the construction of the cluster. Hence, the number of annuli that a particle can cross between its starting point and its exit point is stochastically dominated by a geometric variable of parameter 1/2, and all these geometric variables can be taken to be independent.

The above discussion shows that a single particle may not go further than C log n annuli from its starting point. To get from this a bound on the size of the aggregate is a question about the forest T . Precisely, the maximum k that we consider is stochastically dominated by the maximum reaching time in the forest T (a-1)bn (B[n]).

We now apply Lemma 3.3. Recall that it stated that for T bn ({0}), the probability that it has a vertex with reaching time bigger than log(n) 2 is smaller than exp -c log(n) 2 . For every x ∈ B[n], the corresponding tree in T (a-1)bn (B[n]) is stochastically dominated by T bn ({0}) (recall that a ≤ 2) so we get that, with superpolynomially large probability, the reaching time of every x in every tree of T (a-1)bn (B[n]) is smaller than log(n) 2 . Now, the reaching time was defined using geometric random variables independent of the forest T (a-1)bn (B[n]), so we can use the number of R k crossed by the corresponding particles, because the events that "there are at most C 10 n d-1/4 particles outside B[a 1/d n] and yet our particle crossed annulus i" have probability bounded above by 1 2 , independently of the tree structure.

We conclude that with superpolynomially large probability, the annulus R log(n) 2 is not reached by any particle, so that A (a-1)bn (B This follows from Lemma 3.2, which states that

[n]) ⊆ B[na 1/d (1 + β d log(n) 2 n -1/4 ],
A (a-1)|S| (S) ⊂ A (a-1)bn(1+Cn -1/2 ) (B[n])
and Proposition 3.1 for a ≤ 2. Iterating (as in the end of the proof of Corollary 3.1) gives the result for general a. We omit the details.

4 Proof of Theorem 1.1

Inner bound

Our proof requires rough initial bounds before better bounds can be proved. Our rough outer bound is the obvious remark that A n ⊂ B[n] because it is connected. For a rough inner bound, we have the following lemma: Lemma 4.1. There exists δ 6 > 0 such that or all n big enough,

P B[n/2] ⊆ A e bn ≥ 1 -exp(-n δ 6 ).
Proof. We consider the particles that start from the origin. At step k, the event that the new particle starts from the origin has probability 1/k, hence the number of particles started from the origin by time e bn has expected value b n and is bigger than 2b n /3 with probability e -bn/18 from a Chernoff bound. Classical IDLA results (in particular, the explicit bound in [3, paragraph 3.1.3] is much stronger than what is needed here) guarantee that the standard IDLA with 2b n /3 particles started from the origin covers at least B([n/2]) except on an event of stretched exponentially small probability. Now that we have a rough bound, we are in a position to improve it. The following proposition states that a rough inner bound can always be improved, provided we have an outer bound as well. with superpolynomially large (in εn) probability.

There are two steps in the proof. We first look at the growth of B[n] while ignoring completely the sites in S \ B[n]. Then, we use sites of S \ B[n] (which are not too far from the origin). These sites represent a tiny proportion of A ((3λ) d -1)|S| (S), but it is more than sufficient to counter the loss of the first step. Lemma 2.4 is crucial in this argument.

Proof. We know from Corollary 3.1 that the ball of radius 3λn(1 -C 6 n -1/4 ) is included in the subset uIDLA aggregate A ((3λ) d -1)|S| (B[n]; |S|) with probability greater than 1exp(-n δ 3 ).

Lemma 2.4 now yields that the aggregate we are interested in stochastically dominates the one built by adding the particles of S \ B[n] to the subset uIDLA aggregate A ((3λ) d -1)|S| (B[n]; |S|). In a formula,

P E ⊆ A ((3λ) d -1)|S| (S) ≥ P E ⊆ DA S\B[n] A ((3λ) d -1)|S| (B[b]; |S|) ≥ P E ⊆ DA S\B[n] B[3λn(1 -C 6 n -1/4 )] -exp(-n δ 3 ).
(this holds for any set E but, for the curious, we will eventually use it for E = B 3λn(1 + η 2 ε/λ d ) 1/d , i.e. the set from the statement of the lemma).

Next, since S ⊂ B[λn], all the points in S \ B[n] are inside the half-radius of B[3λn(1 -C 6 n -1/4 )], and we are in a position to apply Lemma 2.1. It yields the following:

P E ⊆ DA S\B[n] B[3λn(1 -C 6 n -1/4 )] ≥ P E ⊆ DA κ B[3λn(1 -C 6 n -1/4 )] ,
where κ is a random variable following a binomial law with |S| -b n > εb n trials and probability of success η, and η > 0 is the constant defined in Lemma 2.1. Now, applying Chernoff's bound yields that κ > 3 4 ηεb n with probability higher than 1 -exp(-cεb n ). This means that the number of particles added is not too small. This fact, together with the inner bound for standard IDLA (from [START_REF] Asselah | Sublogarithmic fluctuations for internal DLA[END_REF][START_REF] Jerison | Logarithmic fluctuations for internal DLA[END_REF] once again), guarantees that with exponentially high probability, DA κ B[3λn(1 -C 6 n -1/4 )] contains a ball of radius 3λn(1 + ηε/(8 • 3 d-1 λ d )) 1/d . The lemma thus holds with the value η 2 = η/(8 • 3 d-1 ). This method for improving inner bounds enables us to prove the inner part of Theorem 1.1. for any a. In other words, for some τ = τ (m 0 ), A n ⊆ B[τ n 1/d ] for all n. Let us also assume that τ is sufficiently large so that b τ n 1/d ≥ 2n (though it would have probably held even if we had not assumed it explicitly).

We now repeatedly apply Proposition 4.1, starting from m 0 . Recall that it states that if B[r] ⊆ A M ⊆ B[λr] and if M > (1 + ε)b r , then with high probability, for some other r and M we have B[r ] ⊂ A M . So applying the proposition repeatedly gives sequences r i and M i such that, with high probability, B[r i ] ⊆ A M i . Let us list all relevant parameters:

r 0 = m 0 r i+1 = 3λ i r i 1 + η 2 ε i λ d i 1/d M 0 = e 2bm 0 M i+1 = (3λ i ) d M i ε i = M i /b r i -1 λ i = τ M 1/d i /r i
The constant η 2 comes from Proposition 4.1, but we may assume that it is small enough, so let us assume η 2 ≤ 1 2 . Let us now analyse these parameters. We first note that λ i is decreasing -indeed, M 1/d i is increased at each step by 3λ i while r i is increased by more. On the other hand, we always have B[r i ] ⊆ A M i and b τ M 1/d i ≥ 2M i so

λ d i = τ M 1/d i r i d = b τ M 1/d i b r i ≥ 2M i M i = 2
so λ i ≥ 2 1/d for all i. More important is the behaviour of ε i . Putting together the formulas for r i+1 and M i+1 gives

ε i+1 + 1 = M i+1 b r i+1 = (3λ i ) d M i (3λ i ) d b r i (1 + η 2 ε i /λ d i ) = ε i + 1 1 + η 2 ε i /λ d i rearranging gives ε i+1 ≤ ε i - cε i λ d i
and since λ i is bounded above, we get that ε i decreases exponentially in i.

On the other hand, ε i does not decrease too fast: since we assumed η 2 ≤ 1 2 , and since λ d i ≥ 2 we get that ε i+1 ≥ min{1, 1 2 ε i }. Since M i+1 ≥ 3M i , we get that M i increases faster than ε i decreases. This is important because the bad event of Proposition 4.1 happens with probability superpolynomially small in M i ε i . Thus we have just shown that these bad events have summable probabilities, and further, the sum is superpolynomially small in m 0 .

This establishes the proposition on a sequence. Indeed, since M i increases no more than exponentially, then we get ε i ≤ CM -c i . So we get B[r i ] ⊆ A br i (1-r -c i ) , which is equivalent to the claim. To extend from a subsequence to all integers, we use Corollary 3.1. We get that between M i and M i+1 the contained ball still follows the volume of the

Figure 1 :

 1 Figure 1: Symmetric difference between the two-dimensional uIDLA aggregate and the Euclidean ball, with 10 6 particles. Blue points are present in the aggregate but not the ball, whereas it is the other way around for red points.

  d . Recall that we defined b n = |B[n]| the volume of the Euclidean ball of radius n intersected with Z d .Lemma 2.5. Let p > 0. There exists ε > 0 such that for any n, m ≥ 1 large enough,P x ξ exits S ∪ B[m] through ∂B[m + n] ≤ p uniformly in x ∈ B[m] and S ⊆ B[m + n] satisfying |S| ≤ εb n .Proof. By Markov's property, it is sufficient to boundP y (ξ exits S ∩ B y [n/3] through ∂B y [n/3]) for starting points y ∈ S ∩ B[m + 2n 3 ] \ B[m + n 3 ]. Similarly, by shifting y to zero and replacing n/3 by n, it is enough to prove thatP 0 (ξ exits S ∩ B[n] through ∂B[n]) ≤ puniformly in any set S ⊂ B[n] such that |S| ≤ εb n , for ε small enough (thus our new ε is multiplied by 3 d , which does not affect the rest of the proof). Now, if |S| ≤ εb n then for some r ≤ n we must have that S ∩ ∂B[r] ≤ Cεr d-1

Lemma 3 . 3 .

 33 Let n, h > 0. Consider T n ({0}) constructed as above when starting from A = {0}. There exist c, C > 0 such that for any h ≥ C log n, then P[∃ a vertex with reaching time larger than h] ≤ e -ch .

1 : 3 . 2 .

 132 which concludes the proof. Proposition 3.1 and Lemma 3.2 together imply the following corollary, which is a converse to Corollary 3.Corollary There exist C 11 > 0 such that for any set S ⊆ B[n],P ∀a ≥ 1 A (a-1)|S| (S) ⊆ B([na 1/d (1 + C 11 n -1/5 ])grows superpolynomially in |S|.

Proposition 4 . 1 .⊆

 41 Let λ > 1 and ε > 0 be two parameters. Suppose that B[n] ⊆ S ⊆ B[λn] and that |S| > (1+ε)b n . For a constant η 2 depending only on the dimension, and uniformly in SA ((3λ) d -1)|S| (S)

Proposition 4 . 2 .

 42 Let d ≥ 2. There exists constants c 1 , C 1 depending only on the dimension such that almost surely, B n(1 -C 1 n -c 1 ) ⊆ A bn , for n large enough. Proof. Lemma 4.1 and the remark before it provide us with the following bounds: for arbitrarily large m 0 , with probability at least 1 -exp(-m δ 6 0 ), B[m 0 ] ⊆ A e 2bm 0 ⊆ B[e 2bm 0 ]. Corollary 3.2 then guaranties that conditionally on the previous event, with superpolynomially (in m 0 ) large probability, A (a d -1)e 2bm 0 ⊆ B[ae 2bm 0 (1 + C 11 e -2bm 0 /9d )]

Acknowledgements The authors would like to thank Vincent Beffara and Vladas Sidoravicius for many fruitful discussions, and Nicolas Curien for referring us to [11]. This paper was partly written during the visit of the second and fourth authors to the Weizmann Institute in Israel. The first author is the incumbent of the Renee and Jay Weiss Professorial Chair. The second author was supported by the EU Marie-Curie RTN CODY, the ERC AG CONFRA, as well as by the Swiss FNS. The third author was supported by the Israel Science Foundation.

aggregate up to a polynomial probability, with superpolynomially large probability (in M i ). These probabilities may be summed.

All in all we get that for some c 10 independent of m 0 and some C(m 0 ), the event B[n(1 -C(m 0 )n -c 10 )] ⊂ A bn ∀n holds with superpolynomially large probability. This means that almost surely, it does indeed hold for some m 0 , and this means that the proposition holds with c 1 = 1 2 c 10 , and an arbitrary C 1 .

Outer bound

Proof. The proof is identical to the proof of Proposition 3.1, but using the inner bound (Proposition 4.2) as a basis. Let us recall quickly the argument. Proposition 4.2 ensures that B[n(1 -Cn 1-c )] ⊂ A bn but that leaves only Cn d-c particles unaccounted for (and possibly outside B[n]). Hence the same holds during the entire process up to time b n . Lemma 2.5 then ensures that annuli of width Cn 1-c around B[n] are difficult to cross for a random walker, hence none of the particles cross more than log n of them. Finally, Lemma 3.3 ensures that none of the trees of T bn ({0}) has depth larger than log 2 n, so no particle may end up further than Cn 1-c log 3 n. This ends the proof.

Appendix : Proof of Lemma 2.2

The following is a guide on how to read [START_REF] Asselah | From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models[END_REF] and modify the authors' proof to obtain the desired result. Note than any reference used in the following is to be found inside [START_REF] Asselah | From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models[END_REF]. First, we start with an already occupied region B[n]. The flashing process introduced by the authors is the same, except it only starts flashing after exiting B[n], and the coupling they describe holds true in our setting. Therefore an interior bound like the one we wish to prove can be proven in the flashing process setting. Then, in paragraph 4.2, we will use ξ = nr 3/2 (instead of log(n)) in equation (4.10). Moreover we follow the authors' recommandation to use a constant h k = h when dealing with IDLA.

Recall that W k (T ) is the number of unsettled explorers (out of our N initial explorers) that stand in a cell T when the cluster is built up to radius r k , and that M k (T ) is the number of explorers that exit B[r k ] through T . Bounding the number of settled explorers that exit B[r k ] through T by saying at most one can have settled on each site of B[r k -h] \ B[n], we get the usual equation:

Note that here we stray from the authors' proof since our L k is not the same as theirs. Hence we need to check that we still have the desired value for

The computation is done in a subparagraph called Step 1 which we emulate here. Recall that for an integer-valued function f , M (f, r k , T ) is the number of particles that exit B[r k ] through T when f (x) particles are started at each point x independently. We follow the authors in also denoting

The absolute values of the differences on the second and third lines are then each bounded using Corollary 5.4, so that we get for some constants C, κ depending only on the dimension,

Note that as long as r k -h ≤ n(1 + r -r 3/2 ) 1/d , the first factor on the right hand side (that is, the difference in volumes) is bigger than κ d n d r 3/2 for a constant κ d depending only on the dimension.

We then resume the course of the authors' proof, noting that the bound (4.14) does not concern us since the order of the left-hand-side is at most logarithmic (see (4.18)). Finally, we conclude in (4.15) by replacing the authors' log(n) with nr 3/2 .


Mérillet Laurène
email: laurene.merillet@gmail.com

Mouchet Maud

Robert Marianne

Salaun Michèle

Schuck Lucie

Vaz Sandrine

Kopp Dorothée

Using underwater video to assess megabenthic community vulnerability to trawling in the Grande Vasière (Bay of Biscay)

Keywords: Bay of Biscay, biodiversity, bottom trawl, fishing, underwater video, vulnerability

Trawling activities are considered to be one of the main sources of disturbance to the seabed worldwide. We aimed to disentangle the dominance of environmental variations and trawling intensity in order to explain the distribution of diversity patterns over 152 sampling sites in the French trawl fishingground, the Grande Vasière. Using a towed underwater video device, we identified 39 taxa to the finest taxonomic level possible, which were clustered according to their vulnerability to trawling disturbance based on functional traits. Using generalized linear models, we investigated whether the density distribution of each vulnerability group was sensitive to trawling intensity and habitat characteristics. Our analyses revealed a structuring effect of depth and substratum on community structure. The distribution of the more vulnerable group was a negative function of trawling intensity, while the distributions of the less vulnerable groups were independent of trawling intensity. Video monitoring coupled with trait-based vulnerability assessment of macro-epibenthic communities might be more relevant than the traditional taxonomic approach to identifying the areas that are most vulnerable to fishing activities in conservation planning.

INTRODUCTION

To develop an ecosystem approach to fisheries, precise identification and quantification of fishing impacts is essential as contact fishing has a direct physical impact on benthic communities [START_REF] Gray | On effects of trawling, benthos and sampling design[END_REF]. In particular, bottom trawling is one of the main sources of seabed disturbance worldwide, impacting the structure and the functioning of benthic communities [START_REF] Kaiser | Global analysis of response and recovery of benthic biota to fishing[END_REF]. Through frequent ploughing of the seabed, resuspension and homogenization of sediments, and removal of parts of the habitat and benthic fauna, bottom trawling leads to a decrease in biodiversity, abundance and distribution of species [START_REF] Thrush | What Can Ecology Contribute to Ecosystem-Based Management?[END_REF][START_REF] Palanques | Effects of bottom trawling on the Ebro continental shelf sedimentary system (NW Mediterranean)[END_REF][START_REF] Clark | The impacts of deep-sea fisheries on benthic communities: a review[END_REF]. Because of their specific biological traits, species are not equally vulnerable to trawling (Bremner et al. 2006a(Bremner et al. , 2006b) ) and can respond differently to disturbances (de Juan & Demestre 2012). Species' biological traits are linked to their functions within the ecosystem [START_REF] Snelgrove | The biodiversity of macrofaunal organisms in marine sediments[END_REF] and changes in observed trait patterns can make it possible to identify functional shifts [START_REF] Tillin | Chronic bottom trawling alters the functional composition of benthic invertebrate communities on a seabasin scale[END_REF]. Assessing species and community vulnerability is thus critical to identify and prioritize areas requiring conservation measures in seascape and fisheries planning [START_REF] Clark | The impacts of deep-sea fisheries on benthic communities: a review[END_REF]. The list of Vulnerable Marine Ecosystems (VMEs; UNGA Resolution 61/105 2006), encompasses various ecosystems characterised by high structural complexity and a low recoverability, and recognized as strongly vulnerable to fishing disturbance, e.g. seamounts, hydrothermal vents and cold corals. VMEs house rare and/or fragile species of functional significance.

However, delineating VMEs remains problematic due to the lack of reliable and consistent methodology to identify indicator species and to quantify adverse fishing impacts [START_REF] Auster | Definition and detection of vulnerable marine ecosystems on the high seas: problems with the 'move-on' rule[END_REF]. Vulnerable mega-epibenthic species, such as large hydrozoans, seapens, crinoids and anemones, are usually used as indicator species (ICES 2016a). However, they are hardly sampled by traditional and scientific fishing gears [START_REF] Auster | Definition and detection of vulnerable marine ecosystems on the high seas: problems with the 'move-on' rule[END_REF]) such as trawls, grabs or dredges, due to their overdispersed distribution on the seabed and low catchability. The recent development of underwater video devices [START_REF] Mallet | Underwater video techniques for observing coastal marine biodiversity: A review of sixty years of publications (1952-2012)[END_REF] could provide an alternative sampling method.

The vulnerability of mega epibenthic species based on video data can be difficult to assess in the absence of empirical knowledge of species' responses to trawling [START_REF] Certain | Mapping the vulnerability of animal community to pressure in marine systems: disentangling pressure types and integrating their impact from the individual to the community level[END_REF], compounded by difficulties in identifying fauna to species level. Using traits that can be visually assessed (e.g. size or position on the substratum), at both fine (species) and high (often genus) taxonomic levels, can circumvent these limitations.

Several biological and life history traits allow assessment of vulnerability to trawling activities, such as feeding type, mobility and fragility for instance [START_REF] Tillin | Chronic bottom trawling alters the functional composition of benthic invertebrate communities on a seabasin scale[END_REF][START_REF] Hily | Soft bottom macrobenthic communities of North Biscay revisited: Long-term evolution under fisheriesclimate forcing[END_REF][START_REF] Thrush | What Can Ecology Contribute to Ecosystem-Based Management?[END_REF], and have been used in different vulnerability assessment frameworks (Bremner et al. 2006a;[START_REF] Tyler-Walters | A method to assess the sensitivity of sedimentary communities to fishing activities[END_REF][START_REF] De Juan | Defining ecological indicators of trawling disturbance when everywhere that can be fished is fished: A Mediterranean case study[END_REF][START_REF] Bolam | Sensitivity of macrobenthic secondary production to trawling in the English sector of the Greater North Sea: A biological trait approach[END_REF][START_REF] Costello | Biological and ecological traits of marine species[END_REF]. Traits can be partitioned into response traits, characterising species responses to environmental disturbances or changes in resources availability, and effect traits, defining taxa that affect ecosystem properties [START_REF] Lavorel | Predicting changes in community composition and ecosystem functioning from plant traits: revisiting the Holy Grail[END_REF][START_REF] Beauchard | The use of multiple biological traits in marine community ecology and its potential in ecological indicator development[END_REF]). Among these frameworks, de Juan & Demestre's (2012) focus on traits that can be visually assessed rather than life-history traits, appears to be particularly suited to video surveys in areas with strong knowledge gaps on species biology and demography.

The framework assesses a taxon's response to trawling disturbance and clusters them into vulnerability groups, according to: (1) position on the substratum, determining exposure to the gear; (2) feeding type (e.g. scavengers can feed on dead or damaged individuals, or filter feeding may be clogged by resuspended sediments); (3) mobility, characterising the ability of individuals to escape; (4) size, dictating selectivity of a trawl net; and (5) presence of a hard external body structure, preventing the individuals from being damaged [START_REF] Tillin | Chronic bottom trawling alters the functional composition of benthic invertebrate communities on a seabasin scale[END_REF][START_REF] Hily | Soft bottom macrobenthic communities of North Biscay revisited: Long-term evolution under fisheriesclimate forcing[END_REF][START_REF] De Juan | Defining ecological indicators of trawling disturbance when everywhere that can be fished is fished: A Mediterranean case study[END_REF][START_REF] Thrush | What Can Ecology Contribute to Ecosystem-Based Management?[END_REF] (Table S1).

The Grande Vasière in the northern part of the Bay of Biscay is a historical fishing ground and has been exploited for many years. Although negative impacts of trawling have been documented worldwide [START_REF] Watling | Disturbance of the Seabed by Mobile Fishing Gear: A Comparison to Forest Clearcutting[END_REF][START_REF] Kaiser | Global analysis of response and recovery of benthic biota to fishing[END_REF][START_REF] Gray | On effects of trawling, benthos and sampling design[END_REF][START_REF] Clark | The impacts of deep-sea fisheries on benthic communities: a review[END_REF], the few studies that investigated biodiversity in this area found no direct and tangible link between the observed biodiversity patterns and the fishing pressure [START_REF] Blanchard | Fishing effects on diversity, size and community structure of the benthic invertebrate and fish megafauna on the Bay of Biscay coast of France[END_REF][START_REF] Vergnon | Evaluation of trawling disturbance on macrobenthic invertebrate communities in the Bay of Biscay, France: Abundance biomass comparison (ABC method)[END_REF]). This raises the question of whether the magnitude of environmental variation masks correlations between biodiversity patterns and trawling intensity. The main goal of this study was therefore to (1) map patterns of vulnerability to trawling, based on response traits of organisms encountered on underwater video footage routinely acquired for the Nephrops stock assessment, (2) analyse the relative density of vulnerability groups to evaluate whether there are associations among groups or whether some tend to be dominant and (3) disentangle the relative impacts of environment and trawling in explaining vulnerability patterns.

METHODS

Study area

The Grande Vasière is located on the continental shelf, in the Northeastern Bay of Biscay. The area is 225 km long by 40 km wide and stretches from Penmarc'h cape (47°48N) to Rochebonne cape (46°10N), covering around 12000 km 2 [START_REF] Hily | Soft bottom macrobenthic communities of North Biscay revisited: Long-term evolution under fisheriesclimate forcing[END_REF]. The sampling sites had a depth range from 66 m to 133 m. The substratum is composed of a mosaic of sediments [START_REF] Bouysse | Carte des sédiments superficiels du plateau continental du Golfe de Gascogne -Partie septentrionale[END_REF]): fine sediments containing > 75% fine mud ("mud>75%"), > 25% fine mud ("mud>25%"), carbonate mud ("carbo_mud"), calcareous mud ("calc_mud") and lithoclastic mud (i.e. sandy mud, "litho_mud") (Fig. 1). Average temperature, salinity and speed current on the seabed were extracted from the model MARS3D [START_REF] Lazure | Development of a hydrodynamic model of the Bay of Biscay. Validation of hydrology[END_REF] for the year 2014.

Data collection

Videos of the seabed were recorded using a camera (Kongberg oe 14-366, 0.48 megapixel) fixed on a sledge dropped onto the seabed and towed behind the Celtic Voyager RV for 10 min in September 2014. The sledge was equipped with CTD and GPS devices, recording depth and geographic position as well as two lasers, spaced 0.75 m apart, that delineated the area covered by the camera (calibrated for a consistent spacing of the area filmed). The locations of 152 sampling sites were chosen along a regular square grid of 4.7x4.7 miles resolution, built from a first point picked randomly inside the limits of the study area. At each site, a video transect was recorded at an average speed of 0.85 knots. Transects covered a mean length of 183.7 m corresponding to a mean area of 137.7 m 2 .

Each individual recorded on the videos was identified to the lowest possible taxonomic level. The first seven complete minutes with sufficient visibility for taxonomic identification were used to count individuals [START_REF] Lordan | Aran, Galway Bay and Slyne Head Nephrops Grounds (FU17) 2011[END_REF]. A total of 39 taxa were identified (Table S2). When identification at the lowest taxonomic level was not possible, we clustered individuals at a higher taxonomic level ensuring a sufficient consistency in biological traits. For instance, individuals clustered as Actinopterygii are all mobile, swimming in the water column and without a hard external structure. Abundances per min were summed to obtain the total abundance of each taxon per transect. These values were then divided by the distance covered by the sledge over the selected minutes multiplied by 0.75 m, to obtain a standardized density value per m 2 . Taxonomic richness was also estimated for each transect.

Vulnerability assessment

The methodology of de Juan & Demestre (2012) was used to assess the vulnerability of benthic communities to trawling disturbance. This framework is based on the use of functional traits from morpho-anatomical and behavioural characteristics (i.e. position on the substratum, mobility, size, feeding habits and fragility). Scores from 0 to 3 were attributed to each biological trait and then summed for each taxon to obtain a total vulnerability score per taxon (Table S1 andS2). The taxa that were the least vulnerable to trawling disturbance had the lowest scores, and were classed in group A while the most vulnerable taxa, with the highest scores, were classed in group D. Assessment of scores may vary across the sources of information (e.g. online database, local expert knowledge), and following the plasticity of traits, but is robust to minor changes in scores [START_REF] Hewitt | Chatham -Challenger Ocean Survey 20 / 20 Post-Voyage Analyses: Objective 10 -Biotic habitats and their sensitivity to physical disturbance[END_REF]).

Consequently, one scoring is likely specific to a dataset and an environmental context.

Densities of the taxa were summed per vulnerability group at each sampling station. In absence of biomass data, all species are thus considered as equivalent regardless of their size. To map the vulnerability patterns, densities of each vulnerability group were interpolated among stations using the Inverse Distance Weighting interpolation method under QGIS versions 2.12.1-Lyon (Fig. 2).

A visual investigation of how vulnerability groups occuring together was then conducted, following [START_REF] Jørgensen | Vulnerability of megabenthic species to trawling in the Barents Sea[END_REF]. Percentages of each group were plotted on pie charts to display how the groups were distributed at each site.

Explaining spatial patterns of vulnerability

We investigated the patterns of density distribution of the vulnerability groups as a function of environmental variables (depth, sediment type, longitude, latitude, and current, salinity and temperature at the seabed level) and trawling intensity. We used fishing effort (hours) as a proxy of trawling intensity. Access to official data was provided by the French Direction of Maritime Fisheries and Aquaculture (DPMA), for fishing vessels > 12m. These data were standardized outputs from the IFREMER SACROIS algorithm [START_REF] Demaneche | Projet SACROIS "IFREMER/DPMA" -Rapport final -Convention SACROIS 2008-2010[END_REF] that filter VMS data according to vessel speeds (< 4.5 knots) and the distance to the nearest port to select records assumed to correspond to fishing operations. Data can be delivered at a 10'x10', 3'x3' or 1'x1' resolution. We selected the 3'x3' resolution, which was the closest to the resolution of our biological sampling grid (4.7x4.7 miles). Spatial mismatch between the very local scale of biological data collected at one transect and the larger scale of fishing effort data is not considered as critical here since the mismatch can be reduced by the homogenization of effort occurring over the years inside the same grid cell [START_REF] Ellis | Scaling up experimental trawl impact results to fishery management scales -a modelling approach for a 'hot time[END_REF]) and trawling effort used is representative of the recent period (average landings and exploitation patterns were stable -ICES, 2016b).

We considered the period from the beginning of April to the end of September, which corresponds to the Nephrops fishing season (the dominant fishing activity in terms of trawling in the area) and sum the monthly fishing hours of trawling in each grid cell.

Collinearity among variables was investigated using variance inflation factor (VIF) coupled with a stepwise selection and a release threshold of VIF>2.5 when both variables were quantitative, and a Kruskal-Wallis test for relationships among quantitative and qualitative variables. Then, a generalized linear model (GLM), fitted with a negative binomial error distribution to handle over dispersion, was used to investigate which environmental and trawling variables influenced the density distribution of each vulnerability group. Commercial and non-commercial taxa have different relationships with trawling that targets commercial species. For this reason, we performed GLM models on commercial and non-commercial taxa separately, in the case of groups A and B, which are the only two groups gathering both fractions (see Table S3 and Fig. S1 in Supplementary material). We expressed the density distribution of each vulnerability group as a function of the explanatory variables and their quadratic terms. We also included interaction terms and the spatial coordinates of the sites (i.e. linear and quadratic terms of longitude and latitude) to account for the significant spatial autocorrelation of the density (all groups together) distribution (Moran's I = 0.58; p = 1.00.e -3 ). We also included an offset of the log-transformed area sampled to account for bias in the estimation of the density per station.

For each GLM model, the most significant variables were selected using stepwise procedures (stepAIC function in R). We compared the results of two stepwise procedures, based on the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), respectively [START_REF] Lelièvre | Delineating recurrent fish spawning habitats in the North Sea[END_REF]. The two models selected were then compared with a Taylor diagram [START_REF] Taylor | Summarizing multiple aspects of model performance in a single diagram[END_REF] to select the best model. Finally, the goodness of fit of the selected models was assessed by performing a χ² test between the null and the selected model. Analyses were performed in R version 3.2.3 (R Development Core Team, 2015) (see supplementary for details on the packages used).

RESULTS

Trawling intensity

From the beginning of April to the end of September 2014, fishing effort ranged from 0.16 to 979.20 fishing hours over the Grande Vasière, with an average per grid cell of 236.30 fishing hours. This corresponds to a swept area ranging from 0.05 km² to 342.70 km² and a trawling frequency ranging from 0.003 to 16.20 trawl.y - 1 [START_REF] Eigaard | Estimating seabed pressure from demersal trawls, seines, and dredges based on gear design and dimensions[END_REF], see supplementary). Trawling intensity was not homogeneous over the area with the highest trawling intensity exerted along the coastline (Fig. 2).

In contrast, intensity was lower offshore (< 50 fishing hr over the studied period), leading to a relationship between trawling intensity, depth and current.

Distribution and co-occurrence of vulnerability groups

Group A, composed of the taxa least vulnerable to trawling disturbance, was mainly present in the North and at southern end of the area, where the trawling pressure is the strongest (Fig. 2,A); it included Nephrops, angular crab and other non-identified crabs and shrimps. Group B was observed in the central and southern areas (Fig. 2,B); it contained slightly more vulnerable taxa, mostly fish (Table S2).

Group C was composed of non-mobile taxa, crinoids and Pennatulacea for instance, and was clustered on the external border (Fig. 2,C). Finally, group D consisted of the most vulnerable taxa, i.e. hydrozoans and Alcyonacea, which mainly occurred in the central and southern external border (Fig. 2,D).

Group C tended to be dominant at the sites in which it occurred (Fig. 3). Individuals of this group were mostly crinoids (85% of the group total density). In a similar way, group D was dominated by hydrozoans (98% of total density). Density of group A and B was more balanced between the dominant species: crabs, shrimps and Nephrops accounted for 90% of group A total density, while Actinopterygii and squat lobster represented 89% of group B total density. Individuals from group A and B often occurred together at the same site, as did individuals from group B and D.

Conversely, individuals from group C and D were rarely found together in high proportions.

Factors influencing vulnerability patterns

All final models appeared to be significantly better than null expectations and no interactions were significant regardless of the vulnerability group (Table 1). The stepwise selection procedure retained at least depth or longitude in each GLM model (Table 1). Overall, selected models included quadratic terms, highlighting the complexity and non-linearity of relationships between the distribution of each group and environmental and fishing variables.

The distribution of the least vulnerable group, group A, was influenced, negatively but non-linearly, by depth (Table 1, Fig. 4). Likewise, distribution of group B density was positively and non-linearly influenced by longitude (Fig. 4). Only groups A and B clustered commercial and non-commercial species. When testing environmental and fishing variables against density patterns of each fraction separately, only patterns of commercial taxa differed from previous results (Table S3 and Fig. S1). In both groups A and B, density patterns of commercial taxa as expected depended on trawling intensity only (Table S3 and Fig. S1). Both depth and its quadratic term and current significantly predicted the density patterns of group C.

Finally, the distribution of group D density varied according to all variables, except current, and was the only group significantly influenced by trawling intensity and sediment type (Table 1, Fig. 4).

DISCUSSION

The occurrence of taxa at a site results from environmental constraints and competitive interactions (Weiher & Keddy 2001). The vulnerability assessment made here relates to the former, with depth, longitude, current, sediment type and trawling intensity shown to be important constraining environmental conditions in the Grande Vasière.

The distribution of the least vulnerable taxa (group A) seems weakly related to trawling intensity. These small mobile taxa with a hard shell are less vulnerable to trawling disturbance than other taxa and dominated where trawling intensity was the strongest. Many taxa from this group are scavengers such as Nephrops, sea stars and some crustaceans [START_REF] Thiel | The Natural History of the Crustacea[END_REF]. Trawling damages animals living on the seabed, thus providing a food source for scavengers [START_REF] Kaiser | The effects of beam-trawl disturbance on infaunal communities in different habitats[END_REF].

The slightly more vulnerable group B is mainly composed of fish taxa, and occured in higher density in the South. This observed longitudinal gradient in the group B pattern most likely originated from complex relationships among unmeasured factors (see discussion below).

Overall density patterns of groups A and B were likely driven by noncommercial taxa, which represented 73% of the individuals encountered in group A and 92% in group B, the latter being dominated by squat lobster and small noncommercial Actinopterygii. Density patterns of the commercial taxa were significantly driven by trawling intensity. We assume that the positive relationship in the case of group A comes from an expected spatial overlap between fishery activities and density patterns of Nephrops that happens to be the only commercial species of this group, and one of the most targeted species in this area. Conversely, the negative relationship of commercial taxa of group B with trawling intensity might either originate from the ability of these mobile taxa to avoid heavily trawled areas or may result in a local depletion of their numbers due to their catch by trawlers.

The two most vulnerable groups, group C and D, included non-mobile taxa and large filter feeders. Density of group C tended to increase in areas where the current was the strongest which is typical for the feeding type of this group. This result has to be considered in the light of the correlations between depth, current and trawling intensity that remain strong after the VIF selection. The stepwise selection can be sensitive to collinearity among explanatory variables. Thus, we suspect that the significant negative effect of trawling intensity on group C in the complete model (results not shown) could have been masked by depth and/or current during the stepwise procedure. Unravelling the pure effect of trawling intensity from the effect of depth and current is difficult in the absence of a reference state, as variables are interrelated [START_REF] Handley | The importance of benchmarking habitat structure and composition for understanding the extent of fishing impacts in soft sediment ecosystems[END_REF]. Indeed, the distribution of group C is limited to the external border of the Grande Vasière, where depth reaches 120 m and trawling intensity is very low due to hard substrates hindering trawling activities. Distribution of group D showed marked environmental preference for deeper (100-120m depth) southern and less trawled stations of lithoclastic and carbonated mud. The spatial mismatch between trawling intensity and the level of vulnerability of communities agrees with other studies (Bremner et al. 2006a(Bremner et al. , 2006b;;[START_REF] Tillin | Chronic bottom trawling alters the functional composition of benthic invertebrate communities on a seabasin scale[END_REF][START_REF] De Juan | A Trawl Disturbance Indicator to quantify large scale fishing impact on benthic ecosystems[END_REF][START_REF] Hinz | Trawl disturbance on benthic communities: Chronic effects and experimental predictions[END_REF][START_REF] Jørgensen | Vulnerability of megabenthic species to trawling in the Barents Sea[END_REF]. The negative co-occurrence between group C and D likely results from diverging environmental preferences since taxa forming group C were found on carbonated mud while the highest densities of taxa forming group D were on lithoclastic mud.

Although our results show that environmental factors mainly drive density patterns of all groups, we cannot exclude that the North-South gradient observed for groups B and D partly encapsulates unmeasured environmental parameters like dissolved oxygen, the mobility of sediments or the amount of organic carbon (Bremner et al. 2006b). Extreme events (e.g. storms) can impact seabed [START_REF] Watling | Disturbance of the Seabed by Mobile Fishing Gear: A Comparison to Forest Clearcutting[END_REF]), but during summer, the more intensive fishing season, resuspension generated by trawling locally exceeds storm impact [START_REF] Mengual | Influence of bottom trawling on sediment resuspension in the 'Grande-Vasière' area (Bay of Biscay, France)[END_REF].

Trawling activities could alter the physical structure of the habitats. In the Grande Vasière, [START_REF] Hily | Soft bottom macrobenthic communities of North Biscay revisited: Long-term evolution under fisheriesclimate forcing[END_REF] noted a significant decrease in the mud and coarse fraction, from 19.8% to 9.1% of the total sampled area since the first extensive study of the area by [START_REF] Glémarec | Le Plateau Continental Nord-Gascogne Et La Grande Vasière Etude Bionomique[END_REF]. Simultaneously, the sand fraction increased from 50.5% to 68.3%. A homogenisation of the type of substrata occurred and an 'intermediate' sediment, which was neither very muddy nor very coarse, became dominant. Penetrable substrata are the preferred habitat of burrow builders like Nephrops or crabs (Bremner et al. 2006b) that could be disadvantaged by the shift toward an increase in the sandy fraction [START_REF] Campbell | Investigating the link between Nephrops norvegicus burrow density and sediment composition in Scottish waters[END_REF]). In the same way, the increase in the sandy fraction would be a threat to filter feeders, such as Pennatulacea or sea anemones that are present on muddy/sandy sediments.

Underwater video can offer a valuable complement to dredge and trawl sampling techniques in the evaluation of trawling impacts [START_REF] Smith | Improving the quantitative estimation of trawling impacts from sidescan-sonar and underwater-video imagery[END_REF]) and allows assessing changes over larger areas [START_REF] Hewitt | Chatham -Challenger Ocean Survey 20 / 20 Post-Voyage Analyses: Objective 10 -Biotic habitats and their sensitivity to physical disturbance[END_REF]. In particular, the direct visualisation of the seabed habitats provides access to unprecedented information, such as unexpected occurrence of species that are either damaged by conventional trawling surveys or able to escape. Using non-extractive methods during scientific surveys should be encouraged especially when fragile species are present. This idea is illustrated by the recording of the sea pen Pennatula phosphorea, considered rare, at 76 stations among the 152 sampled in this study.

Thirty-nine taxa observed, including 33 from groups A and B, found in trawled areas, indicate that the number of taxa is not affected by trawling disturbance. However, underwater video samples only large (> 5cm) epifauna and often cannot be used to identify taxa to species. The use of morphological traits from individuals identified at different taxonomic levels conserves the community structure and richness patterns and leads to results in community analysis similar to those at higher taxonomic resolution [START_REF] Brind'amour | Morphospecies and taxonomic sufficiency of benthic megafauna in scientific bottom trawl surveys[END_REF]. The vulnerability of individuals thus appears to depend more on their morpho-anatomical characteristics than on their taxonomic classification. The assessment of vulnerability conducted in this study, based on biological traits is therefore not penalized by the use of different taxonomic levels and can be compared to the approach by morphospecies of [START_REF] Howell | Mounting evidence: near-slope seamounts are faunally indistinct from an adjacent bank[END_REF].

Our assessment contributes to an integrated assessment of resilience and resistance of this system. The European Marine Strategy Framework Directive for the evaluation of the overall state of marine ecosystems requires describing the dynamics, the resilience and the resistance of communities to trawling disturbance using more biological traits (e.g. life-history traits such as longevity or reproduction)

and biomass [START_REF] Kaiser | Global analysis of response and recovery of benthic biota to fishing[END_REF][START_REF] Tillin | Chronic bottom trawling alters the functional composition of benthic invertebrate communities on a seabasin scale[END_REF], unavailable for many taxa considered here. The year 2014 can be considered as representative of the recent period (ICES 2016b), however fishing intensity based on a single year is likely not representative of mid-or long-term trends in fishing activities. Thus the time scale considered here does not permit inferring processes linking biodiversity patterns, environmental conditions and fishing activities, which was beyond the scope of our study.

An accurate evaluation of the health status of ecosystems would require the definition of a reference state before the intensification of anthropogenic pressures.

However, no usable information about historical states is available for this area and any decline in the abundance of vulnerable taxa might have occurred before any modern monitoring [START_REF] Roberts | The unnatural history of the sea[END_REF]. Communities of the Grande Vasière might have already been shaped by fishing activities that can be considered as moderate to high compared to adjacent areas [START_REF] Eigaard | Estimating seabed pressure from demersal trawls, seines, and dredges based on gear design and dimensions[END_REF]. Without a documented reference state, it is challenging to state that a species is absent from an area because it disappeared in response to excessive fishing pressure, or if the species was never present due to habitat preferences and ecosystem interactions.

This study is a first step in the direction of an integrated management of fisheries in the Grande Vasière based on knowledge of the distribution of vulnerable species. To our knowledge, this is the first study evaluating biodiversity vulnerability based on recorded videos routinely acquired for stock assessment. Such trait-based approaches to map the vulnerability level of species and communities could easily be generalised to other fishing grounds surveyed for stock assessment using video and should be very relevant to identify priority areas for management intervention.

  Weiher E. & Keddy P. (2001) Ecological assembly rules: perspectives, advances, retreats. eds. E. Weiher & P. Keddy, pp: 451. Cambridge, UK: Cambridge University Press.

FIGURE CAPTIONS Figure 1

 CAPTIONS1 FIGURE CAPTIONS

Figure 2

 2 Figure 2 Spatial patterns of vulnerability and trawling intensity in the Grande Vasière.

Figure 3

 3 Figure 3Relative density of vulnerability groups, expressed as the number of

Figure 4

 4 Figure 4 Relationships between the fitted values of the final GLM and the

Fig

  Fig S1 Relationships between the fitted values of the final GLM and the explanatory

Table 1

 1 Outcomes of the stepwise selection procedure on the GLM models. The departure of the deviance of the model from deviance of the null model was tested with a χ 2 test. The significance of each variable was then tested with a χ 2 test whose p-value is given in the column "Significance".

		Explanatory variable	Estimate	Deviance	Significance
	GROUP A	Depth	2	-1.07.e -4	12.61	3.84.e -4
		χ 2 between null and selected model = 12.08, p=5.10.e	-4
	GROUP B	Lon 2		-5.35.e -2	12.30	4.52.e -4
		χ 2 between null and selected model = 11.90, p=5.62.e	-4
	GROUP C	Depth	2	2.60.e -3	154.89	<2.20.e	-16
		Depth		-4.66.e -1	21.55	3.45.e -6
		Current	25.81			12.40	1.03.e	-2
		χ 2 between null and selected model = 129.48, p<2.00.e	-16
	GROUP D	Sediments				42.22	1.51.e -8
			carbo_mud	2.22.e -1
			litho_mud	9.12.e -1
			mud>25%	3.20.e -1
			mud>75%	-2.48.e	1
		Trawl 2	-4.39.e	-6	11.23	8.04.e -4
		Lon		5.18.e 0	7.38	6.60.e -3
		Depth		5.63.e -1	6.18	1.30.e -2
		Depth	2	-2.52.e -3	9.43	2.14.e -3
		Lon 2		6.11.e -1	3.36	6.78.e -2
		χ 2 between null and selected model = 65.38, p=1.21.e -10

ACKNOWLEDGEMENTS

This work was part of the TETRIS project supported by France Filière Pêche. The authors are deeply indebted to the National Committee for Maritime Fisheries and Fish Farming (CNPMEM) and to the French Directorate of Marine Fisheries and Aquaculture (DPMA) who accepted to provide the VMS data to the team from the French Fisheries Information System (SIH) who processed the data. All data used were aggregated so that the confidentiality of vessels was not compromised. Thanks are expressed to the LANGOLF-TV team for video acquisition as well as the staff of the R/V Celtic Voyager and the Marine Institute of Ireland for their contribution in data collection. We also thank Mickaël Vasquez (IFREMER-DYNECO-LEBCO) for the data extraction from the MARS3D model.

SUPPLEMENTARY MATERIAL

For supplementary material accompanying this paper, visit http://www.journals.cambridge.org/ENC Supplementary material for: Using underwater video to assess megabenthic community vulnerability to trawling in the Grande Vasière (Bay of Biscay) Laurène Mérillet* 1,2 , Maud Mouchet 1 , Marianne Robert 2 , Michèle Salaün 2 , Lucie Shuck 2 , Sandrine Vaz 2 , Dorothée Kopp 2 . The following documents give precisions about: (1) the scoring of the vulnerability for the 5 biological traits considered (Tables S1 andS2); (2) the calculation of the swept area and trawling frequency in each grid cell; (3) the packages used to perform analysis and the outcomes of the GLM models for the commercial and noncommercial species separately (Table S3) and the plotted relationship between the GLM models fitted values and the explanatory variables selected in these models (Fig. S1).

(1) Scoring of the vulnerability for the 5 biological traits considered The lowest scores were assigned to the less vulnerable taxa. For a total score value under 8, taxa were assigned to the least vulnerable group, group A. Likewise, taxa were allocated to group B if their total score value fell between 8 and 9, to group C if it was between 10 and 13, or to group D if it was above 13.

Table S2

Vulnerability scoring for each biological traits and their sum per taxa. The higher scores indicate a higher vulnerability to trawling. (2) the calculation of the swept area and trawling frequency in each grid cell Fishing hours in a 3'x3' grid cell was taken as a proxy of fishing intensity. However, this metric does not give information on the area of the seabed impacted. Following [START_REF] Eigaard | Estimating seabed pressure from demersal trawls, seines, and dredges based on gear design and dimensions[END_REF], the swept area and the trawling frequency were calculated in each grid cell. Fig. 11 gives a value of the hourly swept area (surface impact) for otter trawl targeting Nephrops (OT_CRU) of 0.35km 2 for an average vessel. We checked Where W the average distance between the doors (considering the gear path width impacts 100% of the surface), and S the average fishing speed.

For an average fishing speed of 3 knots, the average width is 63m. These values are in line with the ones of the standard fishing practices of OT_CRU vessels in the Grande Vasière.

Swept area was obtained multiplying fishing hours data by 0.35km 2 .h -1 , following:

A swept = A swept per hour * E, where E is the trawling effort in fishing hours.

Area of each grid cell was computed under Qgis versions 2.12.1-Lyon according to the variation with latitude. Trawling frequency was calculated as swept area divided by grid cells area.

Supplementary Reference: Table S3 Outcomes of the stepwise selection procedure on the GLM models for commercial and non-commercial taxa in sensitivity groups A and B. The departure of the deviance of the model from deviance of the null model was tested with a χ 2 test.

The significance of each variable was then tested with a χ 2 test whose p-value is given in the column "Significance".