Nicolas Chenavier

Christian Hirsch

EXTREMAL LIFE TIMES OF PERSISTENT LOOPS AND HOLES

Keywords: Mathematics Subject Classification. 60K35; 82C22 topological data analysis, persistent Betti numbers, Poisson approximation

Persistent homology captures the appearances and disappearances of topological features such as loops and holes when growing disks centered at a Poisson point process. We study extreme values for the life times of features dying in bounded components and with birth resp. death time bounded away from the threshold for continuum percolation. First, we describe the scaling of the minimal life times for general feature dimensions, and of the maximal life times for holes in the Čech complex. Then, we proceed to a more refined analysis and establish Poisson approximation for large life times of holes and for small life times of loops. Finally, we also study the scaling of minimal life times in the Vietoris-Rips setting and point to a surprising difference to the Čech complex.

Introduction

One of the core challenges in statistics is to provide scientifically rigorous methods to detect structure among noise. While for low-dimensional datasets, we do indeed have a versatile toolkit at our disposal, the situation is radically different in high dimensions. In this context, topological data analysis (TDA) is emerging as an exciting novel approach. The simple and effective idea is to leverage invariants from algebraic topology to extract insights from data. While initially TDA started off as an eccentric mathematical idea, it is now applied in a wide variety of disciplines, ranging from astronomy and biology to finance and materials science [START_REF] Gidea | Topological data analysis of financial time series: Landscapes of crashes[END_REF][START_REF] Pranav | The topology of the cosmic web in terms of persistent Betti numbers[END_REF][START_REF] Rocks | Revealing structure-function relationships in functional flow networks via persistent homology[END_REF][START_REF] Saadatfar | Pore configuration landscape of granular crystallization[END_REF].

A core component of TDA is persistent homology. Here, we start growing balls centered at the points of a dataset. At the beginning, all points are in isolated components, but as the balls grow, new topological features such as loops or holes may appear. As the radii increase, these features disappear again, until finally everything is contained in a single connected component without loops or holes. Through this mechanism, we can associate a birth time and a death time with each such feature.

Despite the spectacular successes of TDA in the application domains, major parts of it are still lacking a rigorous statistical foundation. For instance, when analyzing persistent homology, practitioners often look for features living for exceptionally long periods of time, and then draw conclusions if they do occur. However, how can we decide whether the observed long life times come from genuinely interesting phenomena and are not a mere incarnation of chance?

To answer this question, we need to understand how long life times behave under the null model of complete spatial randomness, i.e., a Poisson point process. Figure 1 illustrates cycles of a maximal, typical and minimal life times in a bounded sampling window. In this regard, a breakthrough was achieved in [START_REF] Bobrowski | Maximally persistent cycles in random geometric complexes[END_REF], where the order of the longest life time occurring in a large window was determined on a multiplicative scale. Despite the significance of this discovery, it is still difficult to build a rigorous statistical test, knowing only the order of the life time. A further interesting cross-connection is that for 0-dimensional features, the longest life time corresponds to the longest edge in the minimum spanning tree whose asymptotic was established in [START_REF] Penrose | The longest edge of the random minimal spanning tree[END_REF].

In the present paper, we move one step closer to statistical applications and establish Poisson approximation results for extremal life times of loops and holes in large sampling windows. More precisely, in Theorem 2.1 we determine the scaling of the maximal life time of (d -1)-dimensional features, i.e., holes, in the Čech filtration. Moreover, we show through the Chen-Stein method that after suitable normalization, the locations of exceedances over suitable thresholds converge to a homogeneous Poisson process. Thus, the investigation is in a similar spirit as the extreme value analysis of geometric characteristics of inradii in tessellations [START_REF] Calka | Extreme values for characteristic radii of a Poisson-Voronoi tessellation[END_REF][START_REF] Chenavier | A general study of extremes of stationary tessellations with examples[END_REF]. However, after proceeding from the general set-up of the Chen-Stein method to more concrete tasks, the geometric analyses of the two problems are entirely different.

Another vigorous research stream in spatial extremes concerns limit results for exceptionally small structures [START_REF] Schulte | The scaling limit of Poisson-driven order statistics with applications in geometric probability[END_REF]. Our second and third main results, concern this setting. More precisely, we determine the minimal life time of features in any dimension both in the Vietoris-Rips (VR) and in the Čech ( Č) filtration. Our analysis reveals a striking difference in the scaling of minimal features between the VR-and the Č-filtration. Often the VR-filtration is chosen as an approximation to the Č-filtration, when the latter becomes prohibitively time-consuming to compute. In this respect, our result provides a clear example where this approximation fails on a fundamental level.

The rest of the manuscript is organized as follows. First, in Section 2, we fix the notation and state the main results. Section 3 illustrates in broad strokes how the Chen-Stein method can be used to tackle the assertion. This involves two critical steps, namely identifying the correct scaling and excluding multiple exceedances, which will be established in Sections 4 and 5, respectively. To ease the flow of reading, we have postponed overly technical volume computations into an appendix.

Model and main results

In this section, we describe extremal life times in TDA. First, we review briefly the fundamental concepts of the VR-and Č-filtrations, referring the reader to [START_REF] Boissonnat | Geometric and Topological Inference[END_REF] for an excellent textbook treatment on this topic.

Both the VR-as well as the Č-filtration describe collections of families of simplices in R d . A k-simplex {x 0 , . . . , x k } ⊂ R d is contained in the VR-filtration at a level r > 0 if the distance between any pair of points is smaller than 2r. It is included in the Č-filtration if ∩ i k B r (x i ) = ∅, where B r (x i ) denotes the Euclidean ball with radius r centered at x i .

Having explained these basic building blocks, we now introduce more precisely the concept behind birth and death times when dealing with Poisson point processes in the entire Euclidean space. For finite complexes there is a precise algorithmic description of when a birth occurs and when a death occurs. Each time we add a simplex to the filtration it either increases the homology in the dimension of the feature or it decreases the homology in one dimension lower [START_REF] Boissonnat | Geometric and Topological Inference[END_REF]Algorithm 9]. In the first case, we add a positive simplex, which corresponds to the birth of a feature. In the second case, we add a negative simplex corresponding to the death of a feature. Defining the life time L i,k of the ith feature as the difference between the death and the birth time, this algorithm gives rise to a collection of birth-and-death times {L i,k } i 1 . Assigning to the ith feature the center of gravity Z i,k of the simplex at its death time, we may therefore consider features as a marked point process.

Let P := {X i } i 1 be a Poisson point process in R d with intensity 1 and W n := [-n 1/d /2, n 1/d /2] d be a sampling window of volume n. In the Č-filtration, persistent homology tracks the topological changes of the Boolean model

O r (P) := i 1 B r (X i )
for increasing values of the radius r. For instance, 1-dimensional features are loops, and (d -1)dimensional features correspond to holes, i.e., to bounded connected components in the vacant set R d \ O r (P).

Before moving to the general setting, we first describe life times for the specific case of (d-1)dimensional features. When the growing radii cause an existing hole to split into two at a radius r, we say that r is the birth time of the new hole. Moreover, the death time of a hole is the smallest radius r > 0 when it is covered completely by the Boolean model O r (P). So far, this definition is still ambiguous: when a new hole is born, it is not clear, which of the newly created vacant components should determine its death time. Therefore, it is a common convention in TDA to impose that features born first die last. Now, we enumerate the holes as {H * i } i 1 and associate with each such hole its life time L * i > 0 and the point Z * i ∈ R d as the last point that is covered at the death time. In this way, {(Z * i , L * i )} i 1 defines a stationary marked point process of holes together with their life times. Long-range interactions coming from percolation effects make it difficult to study this marked point process in full generality. For holes born in large vacant components, the seemingly innocent convention that features born first die last may induce long-range interactions in order to decide which birth times correspond to which death times. This makes analyzing extremes prohibitively challenging: we are not aware of any form of Poisson approximation reflecting critical phenomena in percolation contexts.

To state a clean Poisson-approximation theorem that applies to the bulk of the considered holes, we restrict {H * i } i 1 to a smaller family of holes {H i } i 1 as follows. Let r V c > 0 denote the critical threshold of vacant continuum percolation and fix a time window

[r V c -ε c , r V c +ε c ] around r V c .
Then, we consider only holes with birth time outside

[r V c -ε c , r V c + ε c ].
As made precise in Section 3 below, the striking advantage of this restriction is that the bounded connected components are of logarithmic size [START_REF] Meester | Continuum Percolation[END_REF].

In the first main result, we describe the limiting behavior of exceedances of life times in the Č-filtration centered in the sampling window W n . To do it, let FČ be the survival function of the typical life time of a hole in the Č-filtration, i.e., FČ (

) = E #{Z i ∈ W 1 : L i } . (1)
We show in Appendix A that the function FČ : (0, ∞) → (0, 1) is invertible. In what follows, we deal with the asymptotic behavior of the rescaled point process of exceedances

Φ max n, Č := (n -1/d Z i , n FČ (L i )) Z i ∈Wn ⊂ W 1 × (0, ∞).
We consider this point process because it can capture both the location as well as the magnitude of large life times of features.

Our first main result shows that the maximal life time of holes in the Č-filtration is asymptotically equivalent to γ n := (κ -1 d log n) 1/d , and in particular is asymptotically independent of the feature dimension 1 k d -1, where κ d is the volume of the unit ball in R d . Moreover, Φ max n, Č is approximately a homogeneous Poisson point process Φ with intensity 1 in W 1 × (0, ∞). We write ⇒ for convergence in distribution. In the next theorems, we estimate quantities of the form F -1 (τ /n). Considering this type of quantities is classical in Extreme Value Theory. Indeed, if {Y i } i denotes a family of iid random variables with survival function F , then P(max{Y 1 , . . . , Y n } u n ) → exp(-τ ) if and only if n F (u n ) → τ . Theorem 2.1 (Maximal life times, Č-filtration). For every τ > 0,

lim n→∞ F -1 Č (τ /n) γ n = 1 and Φ max n, Č ⇒ Φ.
As a corollary, we obtain the scaling of the maximal life time.

Corollary 2.2 (Scaling of maximal life times -Č-filtration). It holds that

γ -1 n max Z i ∈Wn L i n→∞ ---→ 1 in probability. Proof. First, max Z i ∈Wn L i (1 + ε)γ n means that {n FČ (L i )} Z i ∈Wn ∩ [0, n FČ ((1 + ε)γ n )] = ∅.
Now, we claim that n FČ ((1 + ε)γ n ) tends to 0. Indeed, otherwise there exists τ > 0 such that F -1 Č (τ /n) (1 + ε)γ n for infinitely many n, contradicting the scaling in Theorem 2.1. Thus, we conclude by the point-process convergence in Theorem 2.1.

Similarly, max

Z i ∈Wn L i (1-ε)γ n means that {n FČ (L i )} Z i ∈Wn ∩[0, n FČ ((1-ε)γ n )] = ∅. We claim that n FČ ((1 -ε)γ n ) tends to ∞. Otherwise, there exists τ < ∞ such that F -1 Č (τ /n) (1 -ε)γ n for
infinitely many n, again contradicting Theorem 2.1, and we conclude by the convergence in Theorem 2.1.

Remark 2.3. We stress that the point-process convergence in Theorem 2.1 gives much more than just the scaling of the maximal life time: it allows us to describe the joint distribution of life times exceeding different thresholds. A downside in our definition of Φ max n, Č is that the normalization of life times of the form n FČ (L i ) is rather implicit. This is because even approximating the survival function of the life time of a typical hole sufficiently accurately is challenging. If this could be achieved, then one could proceed as in [START_REF] Calka | Extreme values for characteristic radii of a Poisson-Voronoi tessellation[END_REF] to obtain a more explicit transformation. In that sense, the leading-order scaling derived in Theorem 2.1 is a promising first step, but a finer analysis would be highly desirable.

Remark 2.4. Theorem 2.1 reveals a striking difference to the multiplicative persistence approach of measuring life times in [START_REF] Bobrowski | Maximally persistent cycles in random geometric complexes[END_REF], where the scaling is (log n/ log log n) 1/(d-1) γ n . That work considers the ratio between the death time and the birth time, whereas we work with the difference. The two scalings are different as the multiplicative approach is sensitive with respect to cycles with a very small birth time. Depending on the application context this may or may not be desirable. Hence, it is valuable to know the scaling of both variants and especially how they differ.

Next, we proceed to minimal life times, where we also derive Poisson approximation results. Similarly to the setting of holes, also for general feature dimensions, the long-range correlations close to the critical radius r O c of continuum percolation of the Boolean model would pose a formidable obstacle for proving Poisson approximation. Therefore, we henceforth fix again a time window [r O c -ε c , r O c +ε c ] and consider only features with death time outside [r O c -ε c , r O c +ε c ] and that die in a bounded connected component. Furthermore, for instance by taking the center point of the (k + 1)-simplex whose insertion kills the k-dimensional feature, we attach to any such feature with life time L i,k a center point Z i,k ∈ R d . Hence, we again obtain a stationary marked point process

{(Z i,k , L i,k )} i 1 .
Fixing a feature dimension 1 k d -1, we define the rescaled point process of undershoots

Φ min n,k, Č := (n -1/d Z i,k , nF k, Č(L i,k )) Z i,k ∈Wn ⊂ W 1 × (0, ∞),
where

F k, Č( ) = E #{Z i,k ∈ W 1 : L i,k } . (2)
denotes the distribution function of the typical life time in the Č-filtration. In the next result, we show that the minimal life time of k-dimensional features is of order n -2 , independent of k. Moreover, the rescaled point process of undershoots is approximately a homogeneous Poisson point process with intensity 1 in W 1 × (0, ∞). However, again due to involved geometrical configurations occurring in the Č-filtration, we prove the point-process convergence only for a specific feature dimension, namely k = 1 (loops).

Theorem 2.5 (Minimal life times, Č-filtration). Let 1 k d -1. Then, for every τ > 0, (i)

lim n→∞ log F -1 k, Č(τ /n) log n = -2 and (log n) -1 min Z i,k ∈Wn log L i,k ⇒ -2. (ii) Moreover, Φ min n,k, Č ⇒ Φ for k = 1.
The construction of Φ min n,k, Č outlined above does not only work for the Čbut also for the VRfiltration Φ min n,k,VR . The difference in filtration manifests itself prominently in a different scaling for the threshold.

Theorem 2.6 (Minimal life times -VR-filtration). Let 1 k d -1. Then, for every τ > 0, lim n→∞ log F -1 k,VR (τ /n) log n = -1 and Φ min n,k,VR ⇒ Φ.
As a corollary, we obtain again the scaling of the minimal life time.

Remark 2.7. A priori, it is not clear whether typical life times in the VR-filtration should be shorter or longer than those in the Č-filtration. However, the specific construction in the proof of Theorem 2.1 suggests it seems plausible that asymptotically the corresponding life-time scale F -1 d-1,VR (τ /n) should be smaller than F -1 Č (τ /n) by a constant factor. Loosely speaking, the characteristic of persistent features is a very late death time rather than a very early birth time, and features are dying later in the Čthan in the VR-filtration.

The Chen-Stein method

The proofs of the Poisson approximation in Theorems 2.1-2.6 rely fundamentally on the Chen-Stein method in the form of [START_REF] Arratia | Poisson approximation and the Chen-Stein method[END_REF]Theorem 1]. To make the presentation self-contained, we sketch now how to tune the general machinery to the setting of extremal life times. The delicate problem-specific conditions are then verified in Sections 4 and 5.

More precisely, proceeding as in [START_REF] Chenavier | A general study of extremes of stationary tessellations with examples[END_REF], we discretize the problem to the Chen-Stein framework in the form of [1, Theorem 1]. For the convenience of the reader, we outline the most important steps for the maximal life times, noting that the arguments carry over seamlessly to the setting of minimal life times.

The first central ingredient is to extract a dependency graph with bounded degrees. It is at this point that the percolation conditions described in Section 2 enter the stage. Loosely speaking, we introduce a discretization that is coarse enough so that the life times of holes in nonadjacent blocks are independent. Hence, the size of the discretization is essentially determined by the diameter of the largest hole that we expect to see in the window W n . However, due to the convention that features born first should die last, the actual block size that needs to be considered is a bit larger. More precisely, writing S(x, r) for the diameter of the connected component of the vacant set R d \ O r (P) containing x ∈ R d , we set

S i := sup r ∈[r V c -εc,r V c +εc]: S(Z i ,r)<∞ S(Z i , r)
as the diameter of the largest bounded hole containing Z i . Now, in Lemma B.1 in the appendix, we show that when defining the event

E V n := {for every radius r ∈ [r V c -ε c , r V c + ε c ] all vacant bounded connected components centered in W n have diameter at most (log n) 2 },
then, for all large n,

P(E V n ) 1 -n -4d . (3)
Notice that, on the event E V n for each hole with Z i ∈ W n , we have L i < (log n) 2 . Therefore, we partition the sampling window

W n into N n ∈ O(n/(log n) 2d ) sub-boxes {C j } j≤Nn of side length s n := 2(log n) 2 .
To prove that the point process of exceedances converges to a Poisson point process, we have to estimate the number of exceedances. To do it, we first give some notation. For each j ≤ N n and B ⊂ W 1 , we write

M j,n (B) := max Z i ∈C j ∩(n 1/d B) L i : S i (log n) 2
for the maximum life time of a hole in C j with diameter-bound at most (log n) 2 . Finally, we denote by

N n (τ, B) := Φ max n, Č B × (0, τ ) and N n (τ, B) := #{j N n : n FČ (M j,n (B)) τ }.
the number of exceedances w.r.t. B (resp. the number of sub-boxes intersected by n 1/d B for which there exists at least one exceedance with diameter-bound lower than (log n) 2 ). Setting s n,+ = (3s n ) d , the main difficulty is to exclude multiple exceedances in small boxes of size s n,+ .

Proposition 3.1 (No multiple exceedances). Let τ > 0. (i) Let q n := P N n (τ, W s n,+ ) 2 . Then q n ∈ O(n -1
).

(ii) The same property holds for the minimal life time in the VR-filtration with 1 k d-1, and for the minimal lifetime in the Č-filtration with k = 1.

We are now prepared to prove the Poisson approximation in Theorem 2.1.

Theorem 2.1, Poisson approximation. Let τ > 0 and B ⊂ W 1 Borel. According to Kallenberg's theorem (see e.g. Theorem A.1, p.309 in [START_REF] Leadbetter | Extremes and related properties of random sequences and processes[END_REF]), it is sufficient to prove that,

E[N n (τ, B)] = τ |B| (4)
and

P(N n (τ, B) = 0) -→ n→∞ e -τ |B| . (5)
Equation ( 4) holds since

E[N n (τ, B)] = E #{Z i ∈ n 1/d B : n FČ (L i ) < τ } = n|B|E #{Z i ∈ W 1 : L i > ( FČ ) -1 (τ /n)} = n|B| FČ ( FČ ) -1 (τ /n) = τ |B|.
To deal with (5), we first notice that

P N n (τ, B) = N n (τ, B) P((E V n ) c ) + j Nn P #{Z i ∈ C j : n FČ (L i )<τ } 2 P((E V n ) c ) + N n q n
, where the last uses that P is stationary. Therefore, thanks to Equation (3) and Proposition 3.1, we have P N n (τ, B) = N n (τ, B) → 0. Thus, it is enough to prove that P(N n (τ, B) = 0) converges to e -τ |B| . We will actually prove that N n (τ, B) converges to a Poisson random variable with parameter τ |B|. To do it, we denote by F n a Poisson random variable with parameter

E[N n (τ, B)].
With the same computations as [START_REF] Chenavier | A general study of extremes of stationary tessellations with examples[END_REF], we can easily prove (thanks to Proposition 3.1) that

E[F n ] = E[N n (τ, B)] ∼ n→∞ E[N n (τ, B)] = τ |B|.
Thus, it is sufficient to approximate N n (τ, B) by F n . The main idea to do it is to apply the Chen-Stein method. Indeed, thanks to a result due to Arratia, Goldstein and Gordon (see [1, Theorem 1]), and because each box C j is adjacent to at most 3 d -1 other boxes, we have sup

A⊂Z 0 |P(N n (τ, B) ∈ A) -P(F n ∈ A)| 3 d 2τ p (1) n + 2n (2(log n) 2 ) d • p (2) n , (6)
where p (1) n := P(n FČ (M j,n (W 1 )) τ ) and p (2) n := P(n FČ (M j 1 ,n (W 1 ) ∧ M j 2 ,n (W 1 )) τ ) denote the exceedance probabilities in a single box C j or in two adjacent boxes C j 1 and C j 2 . By stationarity and symmetry, notice that p n do not depend on j, j 1 or j 2 . The first term of the right-hand side converges to 0 since, by stationarity,

p (1) n E #{Z i ∈ C j : L i > F -1 Č (τ /n)} = |C j |τ n .
The second one converges to 0 according to Proposition 3.1. This concludes the proof of Theorem 2.1.

In a similar way, we obtain the Poisson approximation asserted in Theorem 2.5 and Theorem 2.6, relying on a correspondingly defined

E O n instead of E V n . 4. Scalings of FČ , F k,VR and F k, Č
In the present section, we derive the scalings of FČ , F k,VR and F k, Č asserted in Theorems 2.1-2.6.

To begin with, we deal with FČ . As observed in [4, Remark 3.2], the upper bound follows from the results in [START_REF] Kahle | Random geometric complexes[END_REF]: according to the scaling of the coverage radius in [START_REF] Hall | On the coverage of k-dimensional space by k-dimensional spheres[END_REF], above γ n the union of balls covers W n , which is contractible. In particular, by the nerve theorem [3, Theorem 10.7], the Č-complex has the same Betti numbers as the union of balls, so that any cycle is trivial. The lower bound resorts to a construction inspired by [START_REF] Bobrowski | Maximally persistent cycles in random geometric complexes[END_REF]Lemma 5.1].

Scaling of FČ (proof of Theorem 2.1, scaling).

Henceforth, we write

N max n, Č (τ ) := Φ max n, Č (W 1 × [0, τ ]) for the number of features in W n with life time at least F -1 Č (τ /n). F -1 Č (τ /n)/γ n 1 + ε. For ε > 0 write max,+ n := (1 + ε)γ n and τ - n := n FČ ( max,+ n ).
To prove that lim n→∞ τ - n = 0, we have to prove that

E N max 1, Č (τ - n /n) = FČ ( max,+ n ) ∈ o(1).
In order to bound the expected number of holes with death time at least max,+ n that are centered in W 1 , let R := min{|x| : x ∈ P} be the distance of the closest point of P to the origin. In particular, the death time of any hole centered in W 1 is at most R + d, as W 1 is covered completely at the latest at this time. Next, there can be at most one feature with life time exceeding max,+ n and centered in W 1 . Indeed, if there were at least two such features H 1 , H 2 , then at least one of them is born after time R -d, as otherwise, H 1 and H 2 would still be the same hole at the corresponding birth time. Combining this with the previous observation on the death time leads to the contradiction that such a hole would have life time at most 2d. Hence,

E N max 1, Č (τ - n /n) = P Φ max 1, Č (W 1 × [0, τ - n /n]) = 0 P P ∩ B max,+ n -d (o) = ∅ .
Invoking the void probability of the Poisson process P, the right-hand side is at most

exp -κ d ( max,+ n -d) d = exp -(1 + ε) d log(n)(1 -d/ max,+ n ) d ∈ o(n -1 ). F -1 Č (τ /n)/γ n 1 -ε. With max,- n := (1 -ε)γ n and τ + n := n FČ ( max,- n ), we claim that lim n→∞ τ + n = ∞, i.e., lim n→∞ nE N max 1, Č (τ + n /n) = lim n→∞ E N max n, Č (τ + n ) = ∞.
As indicated at the beginning of this section, we modify a construction of [START_REF] Bobrowski | Maximally persistent cycles in random geometric complexes[END_REF]Lemma 5.1].

Loosely speaking, we cover a discretized annulus of radius max,- n by Poisson points, see Figure 2.

P(E b ∩ E d ) = P(E b )P(E d ) e -cmn n -1+ε
for some c > 0. Finally, we can take the template described by E b ∩ E d and shift it to a different location in the window W n . Since those configurations are of logarithmic extent, we can arrange at least a n := n 1-ε /2 of them disjointly in W n . In particular, since

m n ∈ O((log n) (d-1)/d ), lim n→∞ E N max n, Č (τ + n ) lim n→∞ a n e -cmn n -1+ε = ∞.
Next, we analyze the scaling of F k, Č. Since we now deal with short rather than long life times, the proof structure is converse to that for FČ . That is, in the lower bound we analyze configurations leading to small life times, whereas the upper bound relies on a specific construction. We also write

N min n,k, Č(τ ) := Φ min n,k, Č(W 1 × [0, τ ]
) for the number of features in W n with life time at most

F -1 k, Č(τ /n). Henceforth, let d 0•••k := d(x 0 , . . . , x k ) := inf{r > 0 : x∈{x 0 ,...,x k } B r (x) = ∅} (7)
denote the filtration time of the simplex {x 0 , . . . , x k }, i.e., the first time it appears in the Čfiltration. In particular,

d ij = |x i -x j |/2.
Moreover, if x 0 , . . . , x k-1 ∈ R d are affinely independent and r > d(x 0 , . . . , x k-1 ), then in every k-dimensional hyperplane containing x 0 , . . . , x k-1 , there are precisely two k-dimensional balls with radius r containing x 0 , . . . , x k-1 on their boundary, and we let D r (x 0 , . . . , x k-1 ) denote the union of all such balls. Then, for r, > 0, we introduce the crescent D r, (x 0 , . . . , x k-1 ) := D r+ (x 0 , . . . , x k-1 ) \ D r (x 0 , . . . , x k-1 ), illustrated in Figure 3 for k = 2. Having a k-simplex x 0 , . . . , x k with filtration time s means that there exist i 0 , . . . , i m k and an m-dimensional ball B s (P ) centered at some P ∈ R d such that x 0 , . . . , x k ⊂ B s (P ) and x i 0 , . . . , x im ∈ ∂B s . In particular, s ∈ [r, r + ] means that x im ∈ D r, (x i 0 , . . . , x i m-1 ). Scaling of F k, Č (proof of Theorem 2.5, scaling).

F -1 k, Č (τ /n) n -2-ε and P min Z i,k ∈Wn L i,k n -2-ε → 0. It
) = 0.
When a Č-feature with life time > 0 is born at time r = d 0•••k , then there exist Poisson points X 0 , . . . , X k such that ∪ i k B r (X i ) covers the simplex spanned by these points. If a feature dies at time t d = d 0•••k + , then there is m k and Poisson points X 0 , . . . , X m+1 such that the associated simplex is covered at time d 0•••k + . After reordering and choosing m to be the smallest such value, there exists j m such that X i = X i for i j and X i = X i for i j + 1. Then, the previous observations can be succinctly summarized as

X m+1 ∈ D d 0•••k , (X 0 , . . . , X j , X j+1 , . . . , X m ).
Now, we invoke the Slivnyak-Mecke formula from [START_REF] Last | Lectures on the Poisson Process[END_REF]Theorem 4.4]. It shows that the expected number of Poisson points X 0 , . . . , X j , X j+1 , . . . , X m+1 satisfying

X m+1 ∈ D d 0•••k , (X 0 , . . . , X j , X j+1 , . . . , X m ) is at most W 2n ×Bs n (x 0 ) j ×Bs n (x 0 ) m-j D d 0•••k , (x 0 ,...,xm)
1dx m+1 d(x 0 , . . . , x j , x j+1 , . . . , x m ) = W 2n ×Bs n (x 0 ) j ×Bs n (x 0 ) m-j D d 0•••k , (x 0 , . . . , x m ) d(x 0 , . . . , x j , x j+1 , . . . , x m ). Now, Lemma C.1 in the appendix shows that the volume of the integrand is bounded above by cs d n √

, so that

W 2n ×Bs n (x 0 ) j ×Bs n (x 0 ) m-j D d 0•••k , (x 0 , . . . , x m ) d(x 0 , . . . , x j , x j+1 , . . . , x m ) cκ 2m-j d s (2m-j+1)d n W 2n √ dx 0 ,
and for min,n the latter expression tends to 0 as n → ∞.

F -1 k, Č (τ /n) n -
) = ∞.
The proof idea is to construct features with a short life time by relying on perturbations from a deterministic template. We first define suitable templates of constant size, which in the final argument will then be copy-pasted throughout the entire window W n . To lower the technical barriers, we give the proof for k = d -1. The arguments for the lower-dimensional settings are very similar. First, for affinely independent points x 0 , . . . , x d-1 ∈ R d let Q(x 0 , . . . , x d-1 ) ∈ R d denote one of the two possible points projecting onto the center O of the (d -1)-dimensional circumball of x 0 , . . . , x d-1 , and whose distance from that center is given by the radius of that sphere. Figure 4 illustrates this configuration in dimension d = 2. To fix a distinguished template, we consider the unit vectors P i := e i+1 ∈ R d , 0 i d -1, forming a regular (d -1)-simplex of side length √ 2 with circumradius ρ d-1 = (d -1)/d. We claim that at time ρ d-1 the faces of the simplex {P 0 , . . . , P d-1 , Q} are all covered. Since ρ d-1 is the coverage radius of a regular simplex, it suffices to show by symmetry that |P 0 -Q| √ 2. Indeed, remark that OP 0 Q is an isosceles rectangular triangle and write

x 0 x 1 O Q(x 0 , x 1 )
|P 0 -Q| 2 = 2ρ 2 d-1 = 2(d -1)/d, which gives |P 0 -Q| = √ 2 (d -1)/d < √ 2.
Next, we consider random perturbations of this template of a small magnitude δ ∈ (0, 1/2). More precisely, we define the event

E := P(B δ (P i )) = 1 for all i d -1 ∩ P(B 4d (o)) = d + 1
that P has precisely one point in the δ-neighborhood of each of the points in the template in dimension k and no other points in the ball B 4d (o). Conditioned on E, we write P i for the Poisson point contained in B δ (P i ), which is then uniformly distributed at random in B δ (P i ). Moreover, under the event E the Poisson process P contains precisely one further point inside B 4d (o), which we denote by Q . Hence, writing d 0 •••(d-1) = inf{r > 0 : x∈{P 0 ,...,P d-1 } B r (x) = ∅}, conditioned on E, the probability that this feature has life time at most is at least Scaling of F k,VR (proof of Theorem 2.6, scaling).

P Q ∈ D d 0 •••(d-1) , (P 0 , . . . , P d-1 ) ∩ B δ Q(P 0 , . . . , P d-1 ) = 1 κ d ((4d) d -dδ d ) E D d 0 •••(d-
F -1 k,VR (τ /n) n -1-ε . Writing min,- n := n -1-ε and τ min,- n := nF k,VR ( min,- n ), we now assert that lim n→∞ τ min,- n = 0, i.e., lim n→∞ E N min n,k,VR (τ min,- n ) = 0.
First, we may assume that E O n occurs. Indeed, by Cauchy-Schwarz,

E N min n,k,VR (τ min,- n )1{(E O n ) c } E P(W 2n ) d 1{(E O n ) c } E P(W 2n ) 2d P((E O n ) c ) 1/2 .
Since P is a Poisson point process, the expectation on the right-hand side is of order O(n 2d ), so that it remains to invoke a property which is similar to (3). For the remaining argument, recall that a simplex belongs to the VR-filtration at time t if the distance between any pair of its vertices is at most 2t. In particular, when a feature with life time is born at time t b , there exist X 1 , X 2 ∈ P with d 12 = t b , and when it dies at time t d = t b + , there are Poisson points X 3 , X 4 with d 34 = t b + . In other words, X 4 ∈ B 2d 12 ,2 (X 3 ). Moreover, since the event E O n occurs whp, we may assume that X 1 ∈ W 2n and that max 2 i 4 d 1i s n . Here, {X 1 , . . . , X 4 } consists of at least three different points. First, if they are all pairwise distinct, then the Slivnyak-Mecke formula bounds the expected number of undershoots as

W 2n Bs n (x 1 ) 2 B 2d 12 ,2 min,- n (x 3 ) d(x 2 , x 3 )dx 1 = κ 2 d s d n W 2n Bs n (x 1 ) (2d 12 + 2 min,- n ) d -(2d 12 ) d dx 2 dx 1 4 d dκ 3 d s 3d n W 2n min,- n dx 1 = 8 d dκ 3 d s 3d n n -ε
, which tends to 0 as n → ∞. Second, if say X 2 = X 3 , then we proceed similarly to obtain the bound

W 2n Bs n (x 1 ) B 2d 12 ,2 min,- n (x 2 ) dx 2 dx 1 = κ d W 2n Bs n (x 1 ) (2d 12 + 2 min,- n ) d -(2d 12 ) d dx 2 dx 1 4 d dκ 2 d s 2d n W 2n min,- n dx 1 = 8 d dκ 2 d s 2d n n -ε ,
which again tends to 0 as n → ∞.

F -1 k,VR (τ /n) n -1+ε
) = ∞.
To that end, we construct a feature with a short life time for each feature dimension 1 k d -1. The features depend on a deterministic template illustrated in Figure 5.

A -A + B (-1 2 , 1 2 ) C 1 + δ Figure 5
. Template for VR-feature of minimal life time.

We fix a small value δ = δ(k, d) > 0. Then, we set

A ± := ± (1 + δ) 2 -1e 1 , B := e 2 ,
so that |A --B| = |A + -B| = 1 + δ. We also fix a point C := (η, 1/2), where η < 0 is chosen, so that |C -

A + | > 1 + δ > |C -B|.
To convey the idea, we sketch how this template gives rise to the desired feature for k = 1 before moving to higher k. If we consider the VR-complex on {A -, A + , B, C} at level 1 + δ, then A -A + BC is a loop. Moreover, removing the edge A -B and the attached higher simplices, the complex does not contain any triangles as by construction |C -A + | > 1 + δ. In particular, the loop has positive life time. However, after adding the edge A -B, the loop becomes the boundary of the triangles A -A + B and BCA -.

To generalize the construction to higher k, we introduce additional points {P 1,± , . . . , P k-1,± } to the complex as follows. First, set

P i,± := 1 2 (e 2 -e 1 ) ± (0.5 + 4δ)e i+2 .
In particular, the P i,± are at distance at most 1 + δ from A -, A + , B and C. Now, we generalize the above consideration for the VR-complex at level |A + -B| but with the edge {A -, B} removed. In particular, the complex contains all k-simplices of the form

{P 1,ε 1 , P 2,ε 2 , . . . , P k-1,ε k-1 , P, P },
where ε i ∈ {-, +} and {P, P } is one of {A -, A + }, {A + , B}, {B, C} or {C, A -}. Then, these simplices form a cycle. Indeed, when removing the vertex P i,ε i , then the resulting face is also contained in the k-simplex with P i,ε i replaced by P i,-ε i . On the other hand, if for instance {P, P } = {A -, A + } and if we remove the point A -, then we find the corresponding face also in the simplex with {P, P } = {A + , B}. However, this complex does not contain any (k + 1)simplices. Indeed, |P i,--P i,+ | > 1 + δ are at distance larger than 1 + δ, and for any triple from {A -, A + , B, C} at least one edge is also not in the complex. This situation changes drastically if A -B belongs to the complex. Then, the boundaries of the (k + 1)-simplices

{P 1,ε 1 , P 2,ε 2 , . . . , P k-1,ε k-1 , A -, A + , B} and {P 1,ε 1 , P 2,ε 2 , . . . , P k-1,ε k-1 , B, C, A -}
yield the cycle constructed before. It remains to connect this template to random features induced by P. Similarly as in the Čech case, set δ 1 = δ/8 and consider the event

E := P(B δ 1 (P )) = 1 for all P ∈ {A -, A + , B, C} ∪ {P i,± } i k-1 ∩ P(B 3 (o)) = 2k + 2
that P has precisely one point in the δ 1 -neighborhood of each of the template points and no other points in the ball B 3 (o). Conditioned on E , we write A -, A + , B , C and (P i,± ) for the Poisson points lying in the corresponding neighborhoods of the template points. In particular, under E , the primed points are distributed uniformly at random in the corresponding δ 1neighborhoods. Moreover, the construction of the template is sufficiently robust with respect to δ 1 -perturbations to show that the adjacency findings for the VR-complex continue to hold upon replacing the template points by the perturbed points.

Conditioned on E , the probability that this feature has a short life time is at least

P(0 < |A --B | -|A + -B | < min,+ n | E ) = (κ d δ d 1 ) -3 (A ± , B ) : max P ∈{A ± ,B} |P -P | δ 1 and 0 < |A --B | -|A + -B | < min,+ n .
Now, we set u := A + -B and v := A --B , and for α ∈ [0, 1] and

η ∈ R d A +,α,η := B + (|v| -α min,+ n )(u + η)/|u + η|, so that |A +,α,η -B | = |v| -α min,+ n . In particular, |A --B | -|A +,α,η -B | = α min,+ n < min,+ n . Hence, writing δ 2 := δ 1 /8, it suffices to show that |A +,α,η -A + | δ 1 for every A -∈ B δ 2 (A -), B ∈ B δ 2 (o), α ∈ [0, 1]
and η ∈ B δ 2 (o), because then every such point A +,α,η is an admissible choice for A + . To that end, we leverage the bound

|A +,α,η -A + | -u + |v| u |u| + |v| u + η |u + η| - u |u| + min,+ n |u| -|v| + |v| u + η |u + η| - u |u| + min,+ n .
The first expression is at most 3δ 2 and the second one at most 4δ 2 , so that taken together, the right-hand side is indeed at most δ 1 . Now, note that

{A +,α,η : α ∈ [0, 1], η ∈ B δ 2 (o)} min,+ n κ d-1 δ d-1
2 , so that the probability is much larger than n -1 , and consequently that the expectation tends to infinity.

Proof of Proposition 3.1

In this section, we prove that there are no multiple exceedances in W s n,+ .

Proof of Proposition 3.1 (i). By the scaling derived in Section 4, it suffices to prove that

P N max s n,+ (s n,+ FČ ( max,- n )) 2 ∈ O(n -1 ).
By the scaling in Section 4, both features H i and H j die at time at most max,+ n , so that their birth times are at most 2εγ n . In particular, for the holes H i and H j to be contained in different connected components at the birth time of the later hole, say H i , we have

d(Z i , ∂H i ) (1-ε)γ n and d(Z j , ∂H j ) (1 -3ε)γ n . Since d(Z i , Z j ) d(Z i , ∂H i ) + d(Z j , ∂H j ) at the birth time of H i , this implies that |v -v | (2 -5ε)γ n γ n , for ε small enough, where W (v) := v + W 1 , W (v ) := v + W 1
are the lattice boxes containing Z i and Z j , respectively. Therefore,

B max,- n (v) ∪ B max,- n (v ) κ -1 d log(n)|B 1-ε (o) ∪ B 1-ε (e 1 )|.
Then,

P N max s n,+ (s n,+ FČ ( max,- n )) 2 v,v ∈Ws n,+ ∩Z d |v-v | γn P L i ∧ L j max,- n for some Z i ∈ W (v), Z j ∈ W (v ) v,v ∈Ws n,+ ∩Z d |v-v | γn P B max,- n (v) ∪ B max,- n (v ) ∩ P = ∅ |W 2 s n,+ | exp -κ -1 d log(n)|B 1-ε (o) ∪ B 1-ε (e 1 )| .
The last term is in O(n -1 ) provided that ε > 0 is sufficiently small.

Next, we move to the minimum in the Č-filtration.

Proof of Proposition 3.1 (ii); Č. Again, it suffices to show that

P(N min, Č s + n ,1 ( min,+ n ) 2) ∈ O(n -1 ), (8)
where we now set min,+ n := n -2+ε . We need to understand the geometric implications of finding features Z i , Z j ∈ W s n,+ with life times shorter than min,+ n . We recall from Section 4 that one such undershoot gives Poisson points {X 0 , . . . , X 4 } such that the triangle X 2 X 3 X 4 is covered by the union of balls centered at X i , 2 i 4. First, we observe that if at least one of X 0 , X 1 is not contained in {X 2 , X 3 , X 4 }, then (8) holds even without taking the second feature Z j into account. Indeed, if for instance X 1 = X 2 , but X 0 ∈ {X 2 , X 3 , X 4 }, then the Slivnyak-Mecke formula yields the bound

W 3 s n,+ |B 2d 234 -2 min,+ n ,2 min,+ n (x 2 )|d(x 2 , x 3 , x 4 ) ∈ O(s 4d n min,+ n ).
We may hence assume to have points {X 0 , X 1 , X 2 } with d 01 < d 012 d 01 + min,+ n for the first feature and similarly points {X 0 , X 1 , X 2 } = {X 0 , X 1 , X 2 } with d 0 1 < d 0 1 2 d 0 1 + min,+ n . Again, several configurations are possible, where the most challenging ones correspond to the cases where {X 0 , X 1 , X 2 } and {X 0 , X 1 , X 2 } differ in one variable.

First, assume that .

X 0 = X 0 , X 1 = X 1 , but X 2 = X 2 .
Noting that the last line is of order O(n -1 ) concludes the proof of the first case. Second, assuming that

X 1 = X 1 , X 2 = X 2 , but X 0 = X 0 ,
{x 0 ∈ W s n,+ : d 0 2 < d 0 21 d 0 2 + min,+ n } dx 2 d(x 0 , x 1 )
, so that we now conclude as in the previous case.

Finally, we deal with the minimum in the VR-filtration.

Proof of Proposition 3.1 (ii); VR. Again, it suffices to prove that

P(N min,VR s + n ,k ( min,+ n ) 2) ∈ O(n -1 ), (9)
where we now set min,+ n = n -1+ε . Now, suppose two features centered at Z i , Z j ∈ W s n,+ live shorter than min,+ n . By the VR-filtration, this means that for the feature centered in Z i , there exist Poisson points X 0 , . . . , X 3 ∈ W s n,+ such that

d 01 < d 23 < d 01 + min,- n .
Similarly also the feature centered in Z j gives rise to Poisson points X 0 , . . . , X 3 ∈ W s n,+ such that d 0 1 < d 2 3 < d 0 1 + min,- n . Again, not all points need to be distinct, but both {X 0 , . . . , X 3 } and {X 0 , . . . , X 3 } consist of at least 3 elements. Moreover, {X 0 , X 1 } = {X 0 , X 1 } and {X 2 , X 3 } = {X 2 , X 3 } since the features are distinct.

We now distinguish two cases. The first one being that {X 0 , . . . , X 3 } = {X 0 , . . . , X 3 }. Say, for instance X 3 ∈ {X 0 , . . . , X 3 }. Then, we may apply the Slivnyak-Mecke formula to see that the expected number of configurations is at most

W sn+ E # X 0 , . . . , X 3 , X 0 , . . . , X 2 ∈ P ∩ W s n,+ : d 01 < d 23 < d 01 + min,- n and x 3 ∈ B 2d 01 ,2 min,- n (X 2 ) dx 3 2 d+1 ds 4d n,+ min,- n E # X 0 , . . . , X 3 ∈ P ∩ W s n,+ : d 01 < d 23 < d 01 + min,- n 2c(2 d ds 4d n,+ ) 2 ( min,- n ) 2 ∈ O(n -1 ),
for a suitable c > 0. It remains to deal with the case {X 0 , . . . , X 3 } = {X 0 , . . . , X 3 }, which necessarily means that they consist of 4 elements. For instance, it may occur that X 0 = X 0 ,

X 1 = X 3 , X 2 = X 2 and X 3 = X 1 .
d ij = |X i -X j |/2, there would exist points X 0 , X 1 , X 2 , X 3 ∈ P such that d 23 = d 01 + with X 3 ∈ {X 0 , X 1 , X 2 }. Then, E #{X 3 : X 3 ∈ ∂B 2d 01 +2 (X 2 ) for some X 0 , X 1 , X 2 ∈ P} = R d P x ∈ ∂B 2d 01 +2 (X 2 ) for some X 0 , X 1 , X 2 ∈ = E X 0 ,X 1 ,X 2 ∈P ∂B 2d 01 +2 (X 2 ) = 0.
For the Č-filtration, the argumentation is similar. Indeed, we recall from Section 4 that a Čfeature gives Poisson points X 0 , . . . , X k ∈ P and X 0 , . . . , X k+1 ∈ P with X k+1 ∈ {X 0 , . . . , X k } such that relying on the filtration times from (7), we have

d 0 •••(k+1) = d 0•••k + .
Hence, similarly as in the VR-filtration,

E #{X k+1 : d 0 •••(k+1) = d 0•••k + for some X 0 , . . . , X k , X 0 , . . . , X k ∈ P} = E X 0 ,...,X k ∈P X 0 ,...,X k ∈P ∂D d 0•••k (X 0 , . . . , X k ) = 0.
Strict monotonicity. Strict monotonicity means that for < there is a positive probability to observe a feature with life time in ( , ). Note that scaling all points by a factor also scales the life time by that factor. Hence, it suffices to show that for some fixed b and every ε > 0, there is a positive probability to have a feature with life time in (b -ε, b + ε).

We start with the VR-filtration. Consider a feature described by a cross-polytope {±e i } 1 i k+1 . Then, this feature has birth time 1/ √ 2 and death time 1. Therefore its life time is b := 1-1/ √ 2. If we allow the vertices of the feature to be perturbed by at most ε/2, the life time is in the interval (b -ε, b + ε).

Finally, we deal with the Č-filtration. Consider a feature described by a regular simplex {e i } 1 i k+1 . Then, this feature has birth time ρ := (k -1)/k and death time ρ := k/(k + 1). Therefore, its life time is b := ρ -ρ, and we conclude as in the VR-filtration.

Appendix B. Percolation properties

In this section, we show that occupied or vacant components in continuum percolation whose birth time is bounded away from the critical threshold are of poly-logarithmic size with high probability. More precisely, recalling that we center components at their center of gravity, we fix ε c > 0 and put denote the interval consisting of all points inside D a, projecting onto x. Then, it suffices to show that there exists a constant c > 0 such that for each such x we have |I(x)| c √ . Again, after rotation, we may reduce to the two-dimensional setting, i.e., assume that x = (0, x, 0, . . . , 0) for some x δ.

E O, * n := {for every radius r ∈ [r O c -ε c , r O c + ε c ] all occupied bounded connected components centered in W n have diameter at most (log n) 2 }. and E V, * n := {for every radius r ∈ [r V c -ε c , r V c + ε c ] all vacant bounded connected components centered in W n have diameter at most (log n) 2 },
To derive the lower bound on |I(x)|, note that the first coordinate of one of the midpoint of the circle through ±ae 2 with radius a + equals b 0 := (a + ) 2 -a 2 = √ √ 2a + .

Thus, as illustrated in Figure 6,

|I(x)| = (a + ) 2 -x 2 + b 0 -a 2 -x 2 = b 0 + 2a + (a + ) 2 -x 2 + √ a 2 -x 2 ,
which is bounded below by a scalar multiple of √ since b 0 √ 2a √ .

-ae 2 Proof. First, by rotating x 3 into the plane spanned by x 0 , x 1 and x 2 , we may reduce to the two-dimensional setting. Second, we may assume that min i =j d ij n -1/16 . Indeed, for instance if i = 0 and j = 1, then (x 0 , . . . , x 3 ) ∈ W where we set n := n -1+ε . Similarly, we may also assume that the angles ∠x i x j x k are at least n -1/16 for all pairwise distinct i, j, k. Note that we did not attempt to optimize the exponent -1/16.

After these simplifications, it remains to bound the annuli-intersection area |B Hence, if we write P, Q for the endpoints of one of the two arcs ∂B r (x 0 ) ∩ B d 01 , n (x 2 ) and α for the enclosed angle, it suffices to show that α is of order O(n -1/2 ). Since α = ∠x 2 x 0 Q -∠x 2 x 0 P , the mean-value theorem yields some α ∈ [∠x 2 x 0 P, ∠x 2 x 0 Q] with α = cos(∠x 2 x 0 Q) -cos(∠x 2 x 0 P ) sin(α ) = n 2d 01 + n 2d 02 r sin(α ) .

To finish the proof, note that denominator is of order at least n -1/4 by the assumptions at the beginning of proof.

We conclude the appendix with a final elementary geometric volume bound. Proof. To ease notation, we put n := n -2+ε . Arguing similarly as in the proof of Lemma C.2, we may leverage rotational symmetry around the axis formed by x 1 x 2 to reduce the proof to the setting d = 2. Similarly, we may assume that d 12 2 n . Then, for fixed a > d 12 , as observed in the proof of Lemma C.1, the location of all x 0 with d 012 = a is the union of the two circles with radius a passing through x 1 , x 2 . In particular, the location of x 0 with d 01 an is given by an arc in each of these circles, see Since the gradient of u is bounded away from 0, it suffices to show that the length of these arcs is of order O( √ a n ). By construction, the angle α associated with one of these arcs satisfies cos(α/2) = (2a -2 n )/2a, so that sin 2 (α/2) = 1 -cos 2 (α/2) n a ,

Since n /a 1/2, we deduce the asserted aα a Email address: nicolas.chenavier@univ-littoral.fr (Christian Hirsch) University of Groningen, Bernoulli Institute, Nijenborgh 9, 9747 AG Groningen, The Netherlands.

Email address: c.p.hirsch@rug.nl

Figure 1 .

 1 Figure 1. Čech-features of a maximal (left), typical (right top) and minimal (right bottom) life times.

1 Figure 2 .

 12 Figure 2. Template for maximal life time feature.

Figure 3 .

 3 Figure 3. The crescent set D r, (x 0 , x 1 ).

  suffices to prove the first assertion. We write min,- n := n -2-ε and τ min,- n := nF k, Č( min,- n ) and claim that lim n→∞ τ min,-

Figure 4 .

 4 Figure 4. Template for Č-feature of minimal life time in dimension d = 2.

1 )

 1 , (P 0 , . . . , P d-1 ) ∩ B δ Q(P 0 , . . . , P d-1 ) .Then, Lemma C.1 in the appendix shows that the volume on the right-hand side is at least c √ for a suitable c > 0. Now, for a suitable c > 0, we can fit at least c n such templates into the window W n . Hence, for = min,+ as asserted. To prove that P min Z i,k ∈Wn L i,k n -2+ε → 0, we note that by the independence of the Poisson point process, the probability that none of the shifted templates yields a feature with life time at most min,+ n is at most (1-c min,+ n ) c n , which tends to 0, since n min,+ n → ∞. Finally, we move to the scaling of F k,VR . Henceforth, B r, (x) := B r+ (x) \ B r (x) denotes the annulus with center x ∈ R d , inner radius r and thickness .

  where r O c , r V c are the critical radii for occupied and vacant continuum percolation, respectively. We show that the events E O, * n and E V, * n occur with high probability. For a fixed sub-or supercritical radius r, this follows from classical continuum percolation theory[START_REF] Meester | Continuum Percolation[END_REF] Chapter 4]. However, since in the definitions of E O, * n and E V, * n the radius may vary, we need a small discretization argument.Now, r + (b) d-1 -r -(b) d-1 is at most d(r + ) d-2 (r + (b) -r -(b)), so that D r, (x 0 , . . . , x d-1 ) dκ d-1 (r+ ) d-2 2r+2 0 (r + (b)-r -(b))db = d 2 κ d-1 (r+ ) d-2 |D r, (-ae 2 , ae 2 )|.Hence, we have now reduced the upper bound to the special setting where d = 2. Here, an elementary geometric argument that is elucidated in [2, Lemma 9.8] concludes the proof. Lower bound. For a point x ∈ R d-1 with |x| δ, we let I(x) := {b 0 : (b, x) ∈ D a, (x 0 , . . . , x d-1 )}

Figure 6 .

 6 Figure 6. Lower bound on crescent volume

  4 s n,+ : d 01 n -1/16 and |d 23 -d 01 | n ∈ o(n -1 ),

  d 12 , n (x 0 ) ∩ B d 01 , n (x 2 )|, as illustrated in Figure7. To bound this quantity, we rely on the co-area formula from[START_REF] Morgan | Geometric Measure Theory: A Beginner's Guide[END_REF] Chapter 3]. More precisely, we have|B d 12 , n (x 0 ) ∩ B d 01 , n (x 2 )| = d 01 + n d 01 |∂B r (x 0 ) ∩ B d 01 , n (x 2 )|dr.

1 2a -2 n α 2 2aFigure 7 .

 127 Figure 7. Intersection area of annuli (left); Arc describing those x 0 with d 01 d 012n (right)

Lemma C. 3 .

 3 If ε > 0 is sufficiently small, then{(x 0 , x 1 , x 2 ) ∈ W 3 s n,+ : d 01 < d 012 d 01 + n -2+ε } ∈ O(n -1/4).

Figure 7 (

 7 right). Hence, applying the co-area formula[START_REF] Morgan | Geometric Measure Theory: A Beginner's Guide[END_REF] Chapter 3] to the level sets of the function u(x 0 ) := d 012 , we obtain that{(x 0 , x 1 , x 2 ) ∈ W 3 s n,+ : d 01 d 012n } = ∞ n {(x 0 , x 1 , x 2 ) ∈ W 3 s n,+ : d 01 an } ∩ {d 012 = a} |∇u(x 0 )| -1 d

  2+ε and P min Z i,k ∈Wn L i,k n -2+ε → 0. Next, writing min,+

					n	:=
	n -2+ε and τ min,+ n	:= nF k, Č( min,+ n	), we claim that lim n→∞ τ min,+ n	= ∞, i.e.,
		lim n→∞	E N min n,k,VR (τ min,+ n

  . Next, writing min,+

			n	), we claim
	that lim n→∞ τ min,+ n	= ∞, i.e.,
		lim n→∞	E N min n,k,VR (τ min,+ n

n := n -1+ε and τ min,+ n := nF k,VR ( min,+

  Then, invoking Lemma C.1 in Appendix C gives the volume bound (x 0 , x 1 , x 2 , x 2 ) ∈ W 4 s n,+ : d 012 , d 012 ∈ (d 01 , d 01 + min,+

	n	)	s n,+ W 2	D d 01 , min,+ n	(x 0 , x 1 )	2 d(x 0 , x 1 )
			c 2 cresc s 4 n,+	min,+ n

  Lemmas C.1 and C.3 in Appendix C give that (x 0 , x 1 , x 2 , x 0 ) ∈ W 4 s n,+ : d 01 < d 012 d 01 + min,+ n and d 0 1 < d 0 12 d 0 1 + min,+{x 0 ∈ W s n,+ : d 0 1 < d 0 12 d 0 1 + min,+ -1 ). Finally consider the case whereX 1 = X 2 , X 2 = X 1 , but X 0 = X 0 . Then,similarly, Lemmas C.1 and C.3 in Appendix C give that (x 0 , x 1 , x 2 , x 0 ) ∈ W 4 s n,+ : d 01 < d 012 d 01 + min,+

							n
	s n,+ W 2	D d 01 ,	min,+ n	(x 0 ,x 1 )		n	} dx 2 d(x 0 , x 1 )
	cn -1/4	s n,+ W 2	D d 01 , min,+ n	(x 0 , x 1 ) d(x 0 , x 1 )
	cc s 3 n,+ n -1/4 min,+ n	,
	which is again in O(n n	and d 0 2 < d 0 21 d 0 2 + min,+ n
	W 2 s n,+	D d 01 ,	min,+ n	(x 0 ,x 1 )

  Then, d 01 < d 23 < d 01 + min,+ n and d 03 < d 12 < d 03 + min,+

n . Applying the Slivnyak-Mecke formula, the expected number of such configurations in W s n,+ is at most (x 0 , . . . , x 3 ) ∈ W 4 s n,+ : |d 23 -d 01 | ∨ |d 03 -d 12 | min,+ n . Lemma C.2 in Appendix C shows that the latter expression is in O(n -1 ), as asserted.

  Université du Littoral Côte d'Opale, EA 2597, LMPA, 50 rue Ferdinand Buisson, 62228 Calais, France

	8 n a =	√	8a n .

(Nicolas Chenavier)

Appendix A. Invertibility of F k,VR and F k,

Č

In this section, we show that F k,VR , F k, Č : (0, ∞) → (0, 1) are invertible for every 1 k d-1.

Lemma A.1 (Invertibility of F k,VR and F k, Č). Let 1 k d -1. Then, F k,VR , F k, Č : (0, ∞) → (0, 1) are invertible.

Proof. We prove invertibility by showing that F k, Č/VR is a strictly decreasing function, which is continuous and satisfies lim →0 F k, Č/VR ( ) = 0 and lim →∞ F k, Č/VR ( ) = 1.

, where F k, Č/VR (0) describes the expected number of features with life time 0 centered in W 1 . However, by definition, the life time of a feature is always strictly positive, so that F k, Č/VR (0) = 0.

describes the expected number of features in W 1 with an infinite life time. However, this number vanishes. Indeed, in the Č-filtration the feature must have died by the time that a representing cycle is contained in a ball, since the latter is contractible. If we take a Č-boundary chain as a witness for the death of the feature in the Č-filtration, then this chain is also a witness for the death in the VR-filtration. Thus, with probability 1, all life times are finite.

Continuity. Let > 0 be arbitrary. To show continuity, we establish that there are no features with life time exactly . We start by proving the claim for VR-filtration. In that case, writing Lemma B.1 (E O, * n and E V, * n occur with high probability). For all sufficiently large n 1 it holds that

We explain how to proceed in the occupied setting, noting that the arguments in the vacant case are almost identical. We say that an event occurs whp if its complement occurs with probability at most n -5d for large n. First, below r O c all connected components are almost surely bounded, and the size of these components increases with the radius. Hence, we conclude from [START_REF] Meester | Continuum Percolation[END_REF]Lemma 3.3] that the corresponding components are of logarithmic size whp. Second, above r V c all vacant components are bounded and their size decreases with increasing radius. Since every bounded occupied component lies within a vacant component, we conclude from [13, Lemma 4.1] that again the components are of logarithmic size whp.

Hence, it remains to consider r ∈ (r O c + ε c , r V c ), noting that this interval is non-empty only if d > 2. Here, we discretize the possible radii that can occur. More precisely, we subdivide the interval

. Then, by [?, Theorem 10.20], whp for each r i , all bounded connected components at level r i centered in W n are of diameter at most (log n) 2 . Hence, it suffices to show that for any r ∈ (r O c + ε c , r V c ) the components at level r correspond to the components at level r i for some i K n . In other words, we claim that whp, for each i K n there exists at most one pair

To that end, note that for fixed i, the Slivnyak-Mecke formula shows that the probability that there exist distinct pairs (1-8d) ). Hence, by the union bound, the probability that some i K n has this property is at most O(n 2-8d ), as asserted.

Appendix C. Volume computations

In this section, we compute volume bounds for the specific configurations occurring in the proofs in Sections 4 and 5. All computations rely only on findings from elementary geometry, but are still bit tedious when written out in detail.

First, we bound the volumes of crescents. As in the proof of Theorem 2.5, for any affinely independent points x 0 , . . . , x d-1 ∈ R d let Q(x 0 , . . . , x d-1 ) ∈ R d denote one of the two possible points projecting onto the center of the (d -1)-dimensional circumsphere of x 0 , . . . , x d-1 , and whose distance from that center is given by the radius of that sphere. As in Sections 4 and 5, we write d i 0 •••i k for the filtration time when the simplex {x i 0 , . . . x i k } appears in the Č-filtration.

Lemma C.1 (Volume of crescents).

There exists c cresc = c cresc (d) > 0 with the following properties. Let r > 0, < 1, x 0 , . . . , x d-1 ∈ R d be affinely independent. Then, D r, (x 0 , . . . , x d-1 )

Proof. To ease notation, set a :

. By rotating and shifting we may assume that x 0 , . . . , x d-1 are contained in {0} × R d-1 and that their circumcenter is the origin. Furthermore, the set D r, (x 0 , . . . , x d-1 ) is rotationally symmetric around the axis Re


Muhammad Haseeb Iqbal

Fabienne Quilès

Emeline Pradel

Sarah Benmalek-Kehili

Christine Lancelon-Pin

Laurent Heux

Florent Meyer

Lydie Ploux

Gregory Francius

Fouzia Boulmedais
email: fouzia.boulmedais@ics-cnrs.unistra.fr

Grégory Francius

Bactericidal and Bioinspired Chitin-Based Anisotropic Layer-by-Layer Brushed-Nanocoating

Keywords: Orientation, multilayers, bio-sourced, nanotopography, nanostructured surfaces, antibacterial

may come

Bactericidal and bioinspired chitin-based anisotropic layer-by-layer brushed-nanocoating

Introduction

Nanostructured surface coatings have attracted great attention in the last decade to develop advanced next-generation materials. Antibacterial surface modification is of paramount importance to address bacterial infections, a major threat to human health worldwide. Antibacterial coatings are developed either by top-down i.e., physicomechanical structuring of surfaces, [START_REF] Fadeeva | Bacterial Retention on Superhydrophobic Titanium Surfaces Fabricated by Femtosecond Laser Ablation[END_REF] or bottom-up approaches that typically involve chemical modification of surfaces. [START_REF] Chen | A Review on Antimicrobial Coatings for Biomaterial Implants and Medical Devices[END_REF] Top-down biomimetic surfaces with micro/nano-topographical features have emerged as a strategy to prevent bacterial attachment and biofilm formation. [START_REF] Ivanova | Natural Bactericidal Surfaces: Mechanical Rupture of Pseudomonas aeruginosa Cells by Cicada Wings[END_REF] These nanostructured surfaces are inspired by the nano-micro-topographic features found naturally on cicada wings, [START_REF] Ivanova | Natural Bactericidal Surfaces: Mechanical Rupture of Pseudomonas aeruginosa Cells by Cicada Wings[END_REF] lotus leaves, gecko feet, and shark skin. [START_REF] Linklater | Mechanobactericidal actions of nanostructured surfaces[END_REF] However, their development requires expensive equipment and complex processes such as laser ablation, [START_REF] Fadeeva | Bacterial Retention on Superhydrophobic Titanium Surfaces Fabricated by Femtosecond Laser Ablation[END_REF] electropolishing, and photolithographic techniques. [START_REF] Linklater | Mechanobactericidal actions of nanostructured surfaces[END_REF][START_REF] Tripathy | Natural and bioinspired nanostructured bactericidal surfaces[END_REF] The chemical modification includes surface functionalization with antimicrobial peptides, [START_REF] Etienne | Multilayer polyelectrolyte films functionalized by insertion of defensin: a new approach to protection of implants from bacterial colonization[END_REF][START_REF] Cado | Self-defensive biomaterial coating against bacteria and yeasts: polysaccharide multilayer film with embedded antimicrobial peptide[END_REF] silver nanoparticles, antibiotics, and synthetic or bio-based polymers. [START_REF] Chen | A Review on Antimicrobial Coatings for Biomaterial Implants and Medical Devices[END_REF] Among the numerous bottom-up surface functionalization strategies, the layer-by-layer (LbL) method is one of the simplest and most versatile techniques, based on the alternate deposition of positively and negatively charged polyelectrolytes on a substrate via dipping, [START_REF] Decher | Fuzzy Nanoassemblies: Toward Layered Polymeric Multicomposites[END_REF] spin coating, [START_REF] Jiang | Collective and individual plasmon resonances in nanoparticle films obtained by spin-assisted layer-by-layer assembly[END_REF] or spray assembly. [START_REF] Schlenoff | Sprayed Polyelectrolyte Multilayers[END_REF] Because of its ease of application to almost any substrate shape and size, and its mild preparation conditions (aqueous solutions, room temperature), it has been widely used to develop anti-adhesive or bacteriostatic, contact-killing, and release-killing antimicrobial surfaces. [START_REF] Séon | Polyelectrolyte Multilayers: A Versatile Tool for Preparing Antimicrobial Coatings[END_REF] Recently, a brush-based LbL method has been introduced using paint brushes, synthetic polyelectrolytes, and soluble polysaccharides. [START_REF] Park | Nanostructured Polymer Thin Films Fabricated with Brush-based Layer-by-Layer Self-assembly for Site-selective Construction and Drug release[END_REF] The shear effect of the brushing technique has been reported to align the monolayer of clay nanotubes, [START_REF] Zhao | Clay Nanotubes Aligned with Shear Forces for Mesenchymal Stem Cell Patterning[END_REF] hydroxyapatite micro-to nano-fibers, [START_REF] Chen | Biomimetic twisted plywood structural materials[END_REF] and collagen to develop human muscle fibers in-vitro. [START_REF] Iqbal | Brush-Induced Orientation of Collagen Fibers in Layer-by-Layer Nanofilms: A Simple Method for the Development of Human Muscle Fibers[END_REF] Extracted by acid hydrolysis from crustacean exoskeletons, [START_REF] Revol | In vitro chiral nematic ordering of chitin crystallites[END_REF] highly anisotropic crystalline chitin nanocrystals and nanofibers are commonly used for their excellent mechanical properties to design biodegradable three-dimensional (3D) materials [START_REF] Tian | Fabrication of Nanocomposite Bioelastomer Porous Scaffold Based on Chitin Nanocrystal Supported Emulsion-Freeze-Casting[END_REF] or films. [START_REF] Duan | Highly Biocompatible Nanofibrous Microspheres Self-Assembled from Chitin in NaOH/Urea Aqueous Solution as Cell Carriers[END_REF] Using expensive types of equipment, chitin-based oriented nanostructures have been designed using electric [START_REF] Boltoeva | Electric-Field Alignment of Chitin Nanorod-Siloxane Oligomer Reactive Suspensions[END_REF] and magnetic fields, [START_REF] Nge | Synthesis and orientation study of a magnetically aligned liquid-crystalline chitin/poly(acrylic acid) composite[END_REF] shear, [START_REF] Mendoza-Galvan | Birefringence of nanocrystalline chitin films studied by Mueller-matrix spectroscopic ellipsometry[END_REF] and wet stretching [START_REF] Torres-Rendon | Mechanical Performance of Macrofibers of Cellulose and Chitin Nanofibrils Aligned by Wet-Stretching: A Critical Comparison[END_REF] to develop optically anisotropic or mechanically robust films. However, despite the well-known antibacterial properties of chitosan, [START_REF] Sudarshan | Antibacterial action of chitosan[END_REF] a deacetylated compound of chitin, few studies have reported chitin-based membranes or 3D materials with antifungal [START_REF] Salaberria | Processing of α-chitin nanofibers by dynamic high pressure homogenization: Characterization and antifungal activity against A. niger[END_REF] or antibacterial properties. [START_REF] Ma | Structure and properties of chitin whisker reinforced chitosan membranes[END_REF][START_REF] Zhang | Preparation, assessment, and comparison of α-chitin nano-fiber films with different surface charges[END_REF] Fungal and bacterial inhibitions have rarely been achieved with pristine chitin, [START_REF] Salaberria | Processing of α-chitin nanofibers by dynamic high pressure homogenization: Characterization and antifungal activity against A. niger[END_REF] but mostly with chemically modified chitin, [START_REF] Zhang | Preparation, assessment, and comparison of α-chitin nano-fiber films with different surface charges[END_REF] or by incorporating chitosan [START_REF] Ma | Structure and properties of chitin whisker reinforced chitosan membranes[END_REF] or other antipathogenic agents such as copper. [START_REF] Solairaj | Studies on electrochemical glucose sensing, antimicrobial activity and cytotoxicity of fabricated copper nanoparticle immobilized chitin nanostructure[END_REF] Furthermore, none of the few studies on chitin-based LbL obtained by the dipping method, [START_REF] Villares | Chitin Nanocrystal-Xyloglucan Multilayer Thin Films[END_REF][START_REF] Qi | Multifunctional Coating Films by Layer-by-Layer Deposition of Cellulose and Chitin Nanofibrils[END_REF][START_REF] Manabe | Chitin Nanofibers Extracted from Crab Shells in Broadband Visible Antireflection Coatings with Controlling Layer-by-Layer Deposition and the Application for Durable Antifog Surfaces[END_REF][START_REF] Taniguchi | Layer-by-layer self-assembled thin films of chitin fibers and heparin with anti-thrombus characteristics[END_REF] reported antibacterial properties of the bio-sourced film.

Here, inspired by chitin-based butterfly wings [START_REF] Elbaz | Chitin-Based Anisotropic Nanostructures of Butterfly Wings for Regulating Cells Orientation[END_REF] and iron-rich crustacean exoskeletons, [START_REF] Gordon | Nanoscale chemical tomography of buried organic-inorganic interfaces in the chiton tooth[END_REF] we report the simple design of fully bio-sourced, anisotropic, and antibacterial LbL coatings based on chitin nanocrystals (ChNCs) using the brushing method (Figure 1 and Supporting Information Figure S1). Composed exclusively of FDA-approved materials (FDA: United States Food and Drug Administration), [START_REF] Wang | Chitosan Derivatives and Their Application in Biomedicine[END_REF][START_REF] Shukla | Hemostatic Multilayer Coatings[END_REF] the LbL nanocoatings were obtained by manual brushing of ChNC and tannic acid (TA), an abundant natural polyphenol with intermediate dipping in an iron III (Fe 3+ ) solution to ensure coordination bonding with both TA and ChNC. [START_REF] Ejima | One-Step Assembly of Coordination Complexes for Versatile Film and Particle Engineering[END_REF] Figure 1: The LbL brushing method. Schematic representation of anisotropic ChNC film's buildup by the brushing LbL method based on the cycling deposition of (TA/Fe 3+ /ChNC/Fe 3+ ). After the deposition of TA or ChNC by the paintbrush, an adlayer of Fe 3+ is deposited by dipping alternated by the rinsing step with ultrapure water and the drying step by compressed air along the brushing direction. Paintbrush image from stockio.com for commercial use.

Tannic acid Fe 3+ Chitin nanocrystals

Rinsing Drying

Fe 3+

Rinsing Drying

Rinsing Drying

Rinsing Drying

In contrast to those obtained by the dipping method, the brushed TA/Fe 3+ /ChNC coatings exhibit highly oriented ChNC nanostructures, which enable contact-killing antibacterial properties against

Staphylococcus aureus (S. aureus, a Gram-positive bacterium) and Escherichia coli (E. coli, a Gramnegative bacterium). The main focus of this study is on a comprehensive evaluation of the physicochemical and antibacterial properties of brushed and dipped TA/Fe 3+ /ChNC/Fe 3+ coatings to decipher their mechanism of action.

Results and discussion

The brushing method aligns chitin nanocrystals in LbL films

Anisotropic elongated nanostructured ChNCs suspension was obtained from chitin powder using a previously described protocol. [START_REF] Revol | In vitro chiral nematic ordering of chitin crystallites[END_REF] The ChNC nanostructure size was measured at 116 nm by dynamic light scattering measurements and confirmed by atomic force microscopy (AFM) with 116 ± 8 nm in length and 10 ± 2 nm in width (Supporting Information Figure S2). They showed strong cationic character with a zeta potential of +48.3 mV due to the presence of protonation of some amino groups that have been formed by deacetylation of surface N-acetyl amido groups. [START_REF] Tian | Fabrication of Nanocomposite Bioelastomer Porous Scaffold Based on Chitin Nanocrystal Supported Emulsion-Freeze-Casting[END_REF] Since the ChNC suspension was at pH 3 to ensure its long-term stability during storage, all the LbL buildups were performed with all solutions prepared at pH ≤ 3. Based on chitosan/TA [START_REF] Huang | Chitosan/tannic acid bilayers layer-by-layer deposited cellulose nanofibrous mats for antibacterial application[END_REF] and chitosan/Fe 3+ [START_REF] Nieto | Preparation and Characterization of a Chitosan-Fe(Iii) Complex[END_REF] interactions, ChNC/Fe 3+ and ChNC/TA LbL films were built up on a poly(ethylene imine) (PEI)

precursor layer by manual brushing, which showed limited thickness growth (Supporting Information Figure S3-S4). Thus, the deposition of (TA/Fe 3+ /ChNC/Fe 3+ ) quadrilayers was enabled by the coordination bonds known to allow TA to interact with Fe 3+ , [START_REF] Ejima | One-Step Assembly of Coordination Complexes for Versatile Film and Particle Engineering[END_REF] resulting in the so-called brushed ChNC films. The dipping method was used for a control surface, which was named dipped ChNC films. The thickness of the brushed ChNC film increases linearly as a function of the last deposition step reaching up to 24.8 ± 0.8 nm after 8 quadrilayers, i.e. PEI-(TA/Fe 3+ /ChNC/Fe 3+ )8. In contrast, irregular growth of the dipped films was obtained reaching a plateau of 121.3 ± 2.8 nm after 7 quadrilayers (Figure 2a). The main thickness increase comes from ChNC (about 15-20 nm corresponding to 1-2 ChNC thick). The reduced amount of suspension and the high shear effect of the brushing method lead to a smaller thickness increment of ChNC compared to the dipping method as reported for other systems. [START_REF] Park | Nanostructured Polymer Thin Films Fabricated with Brush-based Layer-by-Layer Self-assembly for Site-selective Construction and Drug release[END_REF] The small increment may also result from the different refractive index values in a direction parallel or perpendicular to their long axis [START_REF] Cranston | Morphological and Optical Characterization of Polyelectrolyte Multilayers Incorporating Nanocrystalline Cellulose[END_REF] which can explain the small thickness increment of ChNC. The dipped and brushed ChNC/TA films were stable in PBS with no significant loss of dry thickness (Figure 2a). The low magnification light microscopy and scanning electron microscopy (SEM) images of the brushed ChNC films show the presence of parallel patterns in the brushing direction (Supporting Information, Figure S5). In contrast, the dipped films appeared homogeneous with a dense coverage of the surface by ChNCs. The ChNC films were then imaged by atomic force microscopy (AFM) in the dry state to characterize their topography (Figure 2b-c). In contrast to the dipped films, the ChNCs of the brushed ChNC/TA films were preferentially oriented along the brushing direction. To characterize their degree of orientation, the AFM topography images were analyzed using OrientationJ (Supporting Information Figure S6). The orientation distribution curve of the brushed ChNC/TA films shows a maximum at 0° (i.e. along the brushing direction) and nanocrystals oriented between -40° and +40° (Figure 2b). In contrast, the dipped films result in randomly distributed nanocrystals with an almost flat distribution curve (Figure 2c). The shearing effect of the brushing allowed an efficient orientation of the ChNC with the order of orientation "S2D" (values ranging from 0 to 1, random orientation when S2D = 0, perfect alignment when S2D = 1) at 0.40 and 0.14 for the brushed and dipped films, respectively. [START_REF] Zhao | Clay Nanotubes Aligned with Shear Forces for Mesenchymal Stem Cell Patterning[END_REF] The cross-section of the AFM topography images shows peaks of 20-30 and 15-25 nm in height for the brushed and dipped films, respectively (Figure 2d).

The thickness of the brushed ChNC films was verified by AFM imaging of the cross-section of a scratched film. The thickness of the 8 brushed quadrilayers was 23.1 ± 2.7 nm (Supporting Information Figure S7) in agreement with the value obtained by ellipsometry. TA, TA/Fe 3+ , and ChNC/Fe 3+ -terminated brushed films showed similar topography features with S2D values of 0.57, 0.34, and 0.44, respectively (Supporting Information Figure S8). In the following, the brushed and the dipped ChNC films of the eight ChNC terminated quadrilayers, i.e. PEI-(TA/Fe 3+ /ChNC/Fe 3+ )7-TA/Fe 3+ /ChNC, will be referred to as oriented and nonoriented ChNC films, respectively.

Antibacterial activity of ChNC LbL films

Bacterial adhesion is the first step before surface colonization leads to biofilm (with a polysacchariderich matrix), which protects bacteria from the action of antibiotics. S. aureus is known to cause nosocomial and device-associated infections [START_REF] Jensen | Increasing frequency of vertebral osteomyelitis following Staphylococcus aureus bacteraemia in Denmark 1980-1990[END_REF] and E. coli is the most common cause of catheterassociated infections. [START_REF] Jacobsen | Complicated catheter-associated urinary tract infections due to Escherichia coli and Proteus mirabilis[END_REF] Therefore, the antibacterial properties of the oriented and nonoriented ChNC films were evaluated against S. aureus and E. coli by assessing bacterial adhesion at a short time (1.5 h) and proliferation during 24 h as well as biofilm formation after 48 h of S. aureus incubation. A Live/Dead assay was performed to observe bacterial growth on the surface of the films, with live bacteria stained green (SYTO®-9 stain) and damaged bacteria strained red (propidium iodide stain, PI). Note that no reduction in bacterial growth of both species was observed in the supernatant of either ChNC film compared to uncoated glass (Supporting Information Figure S9).

Oriented ChNC film prevents bacterial adhesion

For both bacterial species, the confocal optical microscopy images clearly show fewer bacteria adhering to the oriented ChNC films compared to the nonoriented films and uncoated glass (Figures 3a andc). The oriented ChNC films show approximately 86% reduction in the number of live S.

aureus (green staining) compared to uncoated glass (with p-value ≤ 0.0001) (Figure 3a). The nonoriented films present only about 43% reduction vs the uncoated glass (with p-value ≤ 0.01). The viability of bacteria was evaluated by calculating the ratio of the number of permeabilized and damaged bacteria (red staining) and all bacteria (SYTO®-9 green staining + PI red staining).

Approximately 46% of adherent S. aureus were damaged on the oriented ChNC films compared to only 26% and 16% on the nonoriented films and uncoated glass, respectively. For E. coli, bacterial adhesion on the oriented ChNC films was reduced by 46% compared to the uncoated glass (with pvalue ≤ 0.01), with 50% of the bacteria damaged (vs. 20% for uncoated glass) (Figure 3c). The nonoriented films showed similar results with a 41% reduction in live bacteria and the presence of 44% damaged bacteria. The oriented ChNC films show better prevention of bacterial adhesion than the nonoriented films for both bacterial strains. In this effect, contact-killing plays an important role since a significant fraction of bacteria observed on the surface are damaged, while no inhibition of bacterial growth is observed in the supernatant (Supplementary Information Figure S9ac).

Oriented ChNC film inhibits bacterial proliferation on the film surface

After 1.5 h of initial incubation followed by 24 h of incubation in fresh culture media, confocal microscopy images showed a dramatic effect of the oriented ChNC films with only a few S. aureus and E. coli bacteria on the surface, in contrast to that observed on the nonoriented films and uncoated glass (Figure 3b,d). all bacteria green and respiratory active bacteria red (CTC) (Supplementary Information Figure S10) after 24 h of proliferation. As observed previously, few bacteria adhered to the oriented ChNC films with 48% of S. aureus having a respiratory activity (90% on the uncoated glass), thus confirming the detrimental effect on live bacteria (Figure 3b). Taken together, these results confirm that the reduction in colonization on the oriented ChNC is due, at least in a large part, to a surface-localized contactkilling effect rather than an antiadhesive effect, and demonstrate that it also affects the bacterial cells resulting from the proliferation of the pioneer sessile bacteria. In addition, the results show that contact-killing not only damages the cell wall but also a viable function such as respiration.

Oriented ChNC film prevents biofilm formation

To investigate the long-term surface activity, biofilm assays were performed with 48 h incubation of high S. aureus density. The SYTO®-24/CTC staining showed the presence of only a few bacteria on the oriented ChNC films in contrast to the nonoriented films and the uncoated glass (Figure 4a). The biomass (i.e. the number of cells) on the oriented ChNC films was approximately 96% lower than on the uncoated glass (with a p-value ≤ 0.001) (Figure 4b).

The crucial role of the terminating layer of ChNC-oriented films on the antibacterial activity

Knowing that they have a similar nanotopography (Supplementary Information Figure S8), the antibacterial properties of ChNC-based oriented films with different terminating layers were evaluated against S. aureus. Bacterial adhesion and proliferation were assessed using the SYTO®-9/PI Live/Dead assay on the oriented ChNC/Fe 3+ , TA, and TA/Fe 3+ -terminated films corresponding to (TA/Fe 3+ /ChNC/Fe 3+ )8, (TA/Fe 3+/ ChNC/Fe 3+ )7-TA, and TA/Fe 3+/ ChNC/Fe 3+ )7-TA/Fe 3+ LbL films, respectively. The oriented ChNC/Fe 3+ , TA, and TA/Fe 3+ films showed higher colonization with more adherent bacteria compared to the oriented ChNC films terminated by ChNC (Figure 5a-b). The percentage of damaged bacteria after 1.5 h of culture was between 11-14 % (similar to glass at 16%).

In terms of proliferation, a weaker reduction effect was obtained on the three oriented films compared to the ChNC terminated films, but with a statistically lower number of bacteria than on the Thus, the results showed a clear effect of the ChNC terminated layer on the antibacterial properties of the oriented ChNC films. In the case of Fe 3+ -terminated films, iron could affect the antibacterial activity as it is an essential nutrient for bacteria. On TA-terminated films, even though TA is known to be antibacterial in solution, [START_REF] Payne | Tannic acid inhibits Staphylococcus aureus surface colonization in an IsaA-dependent manner[END_REF] but its immobilized form is usually not.

[44]

Insight into the mode of action of the oriented ChNC films

ChNC-terminated oriented films showed strong contact-killing antibacterial activity with a possible antiadhesive property against S. aureus and E. coli. Few studies have reported the antibacterial property of chitin-based films or materials. ChNC/starch blend membranes have been reported to be bacteriostatic against L. monocytogenes (Gram-positive) but not against E. coli (Gram-negative). [START_REF] Qin | Effects of chitin nano-whiskers on the antibacterial and physicochemical properties of maize starch films[END_REF] The antibacterial activity of ChNC is described as being due to its interaction with the negatively charged bacterial cell wall, altering permeability and causing membrane rupture and leakage of intracellular material, followed by cell death. [START_REF] Nagy | Silver nanoparticles embedded in zeolite membranes: release of silver ions and mechanism of antibacterial action[END_REF] To gain more insight into the mechanism of action, we characterized the properties of adherent S. aureus on ChNC films and the surface properties of the films in comparison to the glass substrate.

Membrane integrity and mechanical properties of adherent S. aureus

The membrane integrity of adherent S. aureus was assessed by SEM after their proliferation on the different ChNC-based films compared to uncoated glass. The few adherent bacteria found on oriented ChNC films had severely damaged cell wall membranes (Figure 6a). More bacteria were present on the nonoriented ChNC films, oriented ChNC/Fe 3+ , TA/Fe 3+ , and TA-terminated films as well as uncoated glass with less damaged bacteria (Supplementary Information Figure S11). Interestingly, the cell-wall thickness of the adherent S. aureus was significantly higher on the oriented ChNC films (86 ± 17 nm, p values < 0.0001) than on glass (38 ± 5 nm) and about 60 nm on the nonoriented ChNC films and the other oriented films (Figure 6b). This increase in cell-wall thickness can probably explain the higher stiffness values of bacteria in contact with the oriented ChNC film (2.00 ± 0.05 MPa), and the other films tested, compared to those on the uncoated glass (0.80 ± 0.05 MPa) as shown in Figure 6c. Thickening of the S. aureus cell wall is commonly associated with antibiotic resistance. [START_REF] Cui | Cell wall thickening is a common feature of vancomycin resistance in Staphylococcus aureus[END_REF] This bacterial strain can reversibly switch from the wild-type to a different phenotype (small colony variants) resulting in an almost twofold increase in cell wall thickness in an attempt to survive in harsh environmental conditions, usually after a few days. [START_REF] Loss | Staphylococcus aureus Small Colony Variants (SCVs): News From a Chronic Prosthetic Joint Infection[END_REF] Nanopillar-structured surfaces have been reported to cause stiffening of bacterial cell walls due to induced mechanical stresses superseding the cell wall elasticity, leading to membrane perforation, loss of cell contents, and ultimately cell death. [START_REF] Deng | Direct measurement of cell wall stress stiffening and turgor pressure in live bacterial cells[END_REF] The increase in cell wall thickness on the oriented ChNC films may therefore be due to an adaptation and survival strategy of S. aureus to the harsh environment provided by the films. The increase in cell wall stiffness may be the result of this change in thickness or the direct effect of the mechanical stresses exerted on the cell membrane by the ChNC on the film surface.

Nanotopography characterization of ChNC-based films

Spike-like topographic features of the oriented ChNC films were observed from the threedimensional (3D) AFM images as well as the cross-sectional profiles, where the oriented ChNC films We evaluated the difference in nanotopography between the ChNC LbL films using the bearing analysis on the AFM topography images. This method allows analyzing the distribution of the heights over a surface by plotting the percentage of the surface (y-axis), i.e. the bearing area ratio, which is above the selected height (x-axis) (Figure 7b). [START_REF] Santos | Atomic Force Microscopy[END_REF] A sharp increase in the bearing area ratio was observed at 60 nm for all the oriented films regardless of the final layer, 20 nm for the nonoriented CNC films, and 2 nm for uncoated glass. The increase in bearing area ratio at higher values of heights is related to a rougher topography. Infrared reflection absorption spectra (IRRAS) were recorded on the oriented and nonoriented ChNC films, deposited on a gold substrate, to have information on the molecular orientation of ChNC (Figure 7c-d and Supplementary Information Figure S12). The bands' wavenumbers and assignments are gathered in Supplementary Information Table S1. The band assignments were found in the literature [START_REF] Falcao | Application of ATR-FTIR spectroscopy to the analysis of tannins in historic leathers: The case study of the upholstery from the 19th century Portuguese Royal Train[END_REF][START_REF] Pearson | Infrared spectra of crystalline polysaccharides. V. Chitin[END_REF] and the spectra of pure ChNC and TA compounds (Supplementary Information Figure S12). The spectral features show the presence of ChNC and TA in both films with different overall intensities. In the 1800-1480 cm -1 region, the spectra are dominated by band absorptions at 1717 cm -1 from TA (C=O, esters of gallic acid units), 1662 cm -1 from ChNC (C=O, amide I band), 1625 cm -1 from ChNC (C=O, acetyl group) and TA (C=C aromatic skeletal rings), and 1559 cm -1 from ChNC (amide II band) (Figure 7c-d). The high reflectivity of p-polarized light on metals allows the measurement of only vibrations with a component of the transition dipole moment oriented perpendicular to the surface. [START_REF] Guo | Analyzing the Molecular Orientation of Ultrathin Organic Films by Polarized Transmission and Grazing Incidence Reflection Absorption IR Spectroscopy[END_REF] Considering the fingerprint region (1900-700 cm -1 ) without polarized or with a p-polarized IR beam, the calculated integrated intensities were about 2 times higher for the nonoriented ChNC films compared to the oriented ones. With the film thickness measured to be 121.3 ± 2.8 nm for the nonoriented films and 24.8 ± 0.8 nm for the oriented ones (Figure 1a), the integrated intensity ratio should be about 5 times higher instead of 2. Thus, despite a lower amount of ChNC deposited, the oriented films have more vibrational modes oriented perpendicular to the surface plane than the nonoriented films, showing a higher spatial organization in this direction. This confirms the spike-like topography of the oriented ChNC films.

Surface charge plays a critical role in bacterial death

The nature of the terminal layer plays a crucial role as the most effective antibacterial activity is obtained by terminating with the ChNC layer. Chemical force microscopy (CFM) was used to characterize the surface charge density of the different studied films. [START_REF] Mi | Virus Isoelectric Point Determination Using Single-Particle Chemical Force Microscopy[END_REF] The chemical interaction between the tip and the surface was measured by converting the deflection of the AFM cantilever during the AFM tip withdrawal into adhesion force. [START_REF] Santos | Atomic Force Microscopy[END_REF][START_REF] Kámán | Young's Modulus and Energy Dissipation Determination Methods by AFM, with Particular Reference to a Chalcogenide Thin Film[END_REF] The positive surface charge of the films was characterized using a phosphate-functionalized AFM probe, to mimic the bacterial cell membranes, in PBS at pH 7.4. Since the radius of the contact area of S. aureus on flat surfaces is in the range of several hundred nanometers, [START_REF] Spengler | Determination of the nano-scaled contact area of staphylococcal cells[END_REF] the mapping of the adhesion force was determined on the 5 × 5 µm 2 surface area of the samples. A statistically higher adhesion force, i.e. interactions between the phosphate tip and the surface, was observed for the non-oriented and oriented ChNCbased films independently of the final layer compared to the uncoated glass (Figure 8a, Supplementary Information Table S2). With a median value of 0.35 nN (p-value < 0.001), a higher density of positive charges and adhesion force were observed on the oriented ChNC films compared to the nonoriented films (Figure 8 and Supplementary Information Table S2). This result is probably related to the rougher topography of the oriented ChNC films, resulting in the availability of a higher amount of surface charges of the deposited ChNC compared to the nonoriented ChNC films. The density of charges on ChNC/Fe 3+ , TA/Fe 3+ , and TA-terminated films was either similar to or higher than the ChNC-terminated film with statistically similar adhesion force (Figure 8b). This could be explained by electrostatic interactions of Fe 3+ and hydrogen bonding interactions of TA with the phosphate-functionalized tip, respectively.[59] The bactericidal effect of nanostructured surfaces, consisting of metals, organic/inorganic materials, or synthetic polymers, is usually associated with the mechanical rupture of cells, independent of the surface chemistry and possible antiadhesive properties resulting from the nanoscale topography. Two mechanisms of bacterial cell rupture on nanostructured surfaces have been described in the literature:

(i) compression of the bacteria on the nanopillars under sedimentation forces, leading to puncture or rupture of the bacterial cell wall [START_REF] Ivanova | Natural Bactericidal Surfaces: Mechanical Rupture of Pseudomonas aeruginosa Cells by Cicada Wings[END_REF] and (ii) a combination of strong adhesion between the cells and the nanopillars, and a shear force exerted on the adherent bacteria as they move away from their initial adhesion site. The shear force can be due to their motility or to the hydrodynamic or thermodynamic motion of the fluid environment. Under this stress, the cell wall of the highly adherent bacteria is thought to rupture. Recently, Elena P. Ivanova and coworkers demonstrated that the mechanical injury caused is not sufficient to kill the bacteria immediately, due to the survival of the inner plasma membrane, but leads to apoptosis-like death. This work also showed that the mechano-bactericidal actions have sustained physiological effects on the bacterium, due to the self-accumulation of reactive oxygen species. [START_REF] Zhao | Programmed Death of Injured Pseudomonas aeruginosa on Mechano-Bactericidal Surfaces[END_REF] As an example of the first mechanism, it has been reported that cicada wings have bactericidal activity only against Gram-negative bacteria due to a specific nanopillar topography responsible for the stretching and further rupture of the adherent bacterial wall between two adjacent nanopillars. [START_REF] Linklater | Mechanobactericidal actions of nanostructured surfaces[END_REF][START_REF] Tripathy | Natural and bioinspired nanostructured bactericidal surfaces[END_REF] It can be hypothesized that the contact-killing property of the oriented ChNC films is due to similar mechanisms, thanks to the spike-like nanotopography (20-30 up to 40 nm in height).

This leads to a thickening of the bacterial cell wall to resist the rupture and also increases its rigidity for all the ChNC-based oriented films that have the same nanotopography features. The second mechanism may also play an important role in the ChNC-terminated oriented films, which has been proposed to occur on strongly adhesive nanostructured, [START_REF] Linklater | Mechanobactericidal actions of nanostructured surfaces[END_REF][START_REF] Nowlin | Adhesion-dependent rupturing of Saccharomyces cerevisiae on biological antimicrobial nanostructured surfaces[END_REF] or strongly positive surfaces. [START_REF] Gottenbos | Antimicrobial effects of positively charged surfaces on adhering Gram-positive and Gram-negative bacteria[END_REF] The higher charge of the ChNC-oriented films leads to stronger and multiple interactions between the positively charged spikes and the negatively charged bacterial cell wall. Thus, the ChNC-terminated oriented films are likely to cause more stretching and further rupture of the cell wall when bacteria are forced to move away from their attachment location compared to the other films. As a result, they have the strongest contact-killing property (with between 44-50% damaged S. aureus and E. coli), whereas the other oriented films have only between 11-14% damaged S. aureus. Finally, the steeper spike-like character and the higher positive charge of the oriented ChNC film are both likely to cause the antibacterial property of this surface by leading to the cell wall puncture induced either by the compression of bacterial cells on the spikes and/or by a shear force applied to highly electrostatically adherent cells (Figure 9).

Figure 2 .

 2 Figure 2. Thickness and surface characterization of ChNC LbL films. (a) Evolution of the

Figure 3 .GGG

 3 Figure 3. The contact-killing activity of the oriented ChNC films. Antibacterial properties of

Figure 4 :

 4 Figure 4: The antibiofilm property of ChNC LbL films. S. aureus biofilm on ChNC films and

  O r ie n t e d C h N C N o n -o r ie n t e d C h N and uncoated glass (with p-value ≤ 0.0001) and 10% damaged bacteria (similar to uncoated glass) (Figure 5c-d).

Figure 5 .

 5 Figure 5. The crucial role of the ending layer on the antibacterial activity of the oriented ChNC

Figure 6 :

 6 Figure 6: Characterization of adherent S. aureus on the ChNC-based films vs glass after 24 h

Figure 7 :

 7 Figure 7: Nanotopography and chemical characterization of the ChNC-based films. (a) 3D AFM 292 images (1 × 1 µm 2 ) of the topography of ChNC films with their respective cross-section profiles. (b) 293

Figure 8 : 19 1. 4 . 4

 81944 Figure 8: Surface charge density of ChNC-based films. Adhesion forces, measured by chemical

Figure 9 :

 9 Figure 9: Contact-killing mechanism of the oriented ChNC film. A schematic representation of

nm in height) (Figure7aand Supplementary Information FigureS8b).

Acknowledgments

The authors thank the CarMAc of ICS https://www.ics-cnrs.unistra.fr/plateforme-carmac.html, the Spectroscopy and Microscopy Service Facility (SMI) of LCPME www.lcpme.ul.cnrs.fr/equipements/smi/, where the AFM and infrared experiments were performed and the NanoBio-ICMG Platform (UAR 2607, Grenoble) for granting access to the Electron Microscopy facility. M.H.I. thanks the Higher Education Commission (HEC) Pakistan for his Ph.D. scholarship. Fonds Régional de Coopération pour la Recherche of Région Grand Est ERMES project

cell wall by the positively charged spikes under the shear force due to bacterial motility or movement of the liquid environment. This hypothesis is supported by the morphology and mechanical properties of bacterial cell walls. The fully bio-based nature of ChNC-based nanocoatings and their brush-based fabrication method make them a suitable and affordable candidate as a sustainable antibacterial film for (bio)materials.

CRediT authorship contribution statement

Muhammad

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Supplementary data to this article can be found online at.

Data availability statement.

Data available upon request.