Longest increasing paths with gaps

Keywords: MSC 2010 Classification: 60K35, 60F15 combinatorial probability, longest increasing subsequences, longest increasing paths, last-passage percolation, Hammersley's process, Ulam's problem, BLIP (Bernoulli Longest Increasing Paths)

We consider a variant of the continuous and discrete Ulam-Hammersley problems: we study the maximal length of an increasing path through a Poisson point process (or a Bernoulli point process) with the restriction that there must be minimal gaps between abscissae and ordinates of successive points of the path.

For both cases (continuous and discrete) our approach rely on couplings with well-studied models: respectively the classical Ulam-Hammersley problem and last-passage percolation with geometric weights. Thanks to these couplings we obtain explicit limiting shapes in both settings. We also establish that, as in the classical Ulam-Hammersley problem, the fluctuations around the mean are given by the Tracy-Widom distribution.

Introduction

Motivated by the Ulam problem (which asks for the asymptotic behavior of the maximal length of an increasing subsequence in a uniform random permutation), Hammersley [START_REF] Hammersley | A few seedlings of research[END_REF] studied the problem of the maximal length L (x,t) of an increasing path in a Poisson process with intensity one in (0, x) × (0, t). He used subadditivity to prove the existence of a constant π/2 ≤ c ≤ e such that L (t,t) /t → c in probability and conjectured c = 2. The first probabilistic proof of c = 2 was obtained by Aldous-Diaconis [START_REF] Aldous | Hammersley's interacting particle process and longest increasing subsequences[END_REF], by exploiting the geometric construction of Hammersley. (We refer to [START_REF] Romik | The Surprising Mathematics of Longest Increasing Subsequences[END_REF] for a nice and modern introduction to the Ulam-Hammersley problem.) In this article we obtain the limiting behaviour of the maximal length of an increasing path in a Poisson process, if we impose minimal gaps between abscissae and ordinates of successive points in the path.

Our proof uses a coupling with the original Ulam-Hammersley problem, and therefore we make a strong use of Aldous-Diaconis' result. This coupling also allows us to use the celebrated result by Baik-Deift-Johansson regarding the fluctuations of L (x,t) around its mean. We obtain that, with the proper rescaling, the fluctuations of our problem around the mean are also given by the Tracy-Widom distribution.

It turns out that our strategy also applies to the discrete settings: we obtain explicit asymptotic results for the length of the longest increasing path with gaps through Bernoulli random points on the square lattice. We now state our results. Here we have L h (x,t) = 4, one of the maximizing paths is drawn in red.

x t ≥ h 1 ≥ h 2 ≥ h 1 ≥ h 1 ≥ h 2 ≥ h 2

Continuous settings

Let Ξ be a homogenous Poisson point process in (0, +∞) 2 with intensity 1. We write Ξ y,s = 0/1 for the absence/presence of a point of Ξ at (y, s), and we say that (y, s) ∈ Ξ if Ξ y,s = 1. Let h = (h 1 , h 2 ) be a pair of non negative real numbers. We introduce the strict order on (0, +∞) 2 defined by (y, s) h ≺ (y , s ) if and only if y + h 1 ≤ y , s + h 2 ≤ s .

For x, t > 0, we consider the random variable L h (x,t) given by the length of the longest increasing path in Ξ ∩ [0, x] × [0, t] with horizontal gaps h 1 and vertical gaps h 2 . Namely, L h (x,t) = max L; there are (y 1 , s 1 ), . . . , (y L , s L ) ∈ Ξ such that

0 < y 1 < • • • < y L < x, 0 < s 1 < • • • < s L < t, (y 1 , s 1 ) h ≺ (y 2 , s 2 ) h ≺ • • • h ≺ (y L , s L ).
In the case h = 0 := (0, 0), the random variable L (x,t) := L 0 (x,t) is just the length of the longest increasing path. It turns out that there exists a (random) coupling between L and L h . As an application, we will show the following identity: Theorem 1. For every x, t > 0, and every k ≥ 0,

P(L h (x,t) ≤ k) = P(L (x-h 1 k,t-h 2 k) ≤ k).
(In the above equation, we take the convention L (y,s) = 0 whenever y < 0 or s < 0.)

The asymptotic behavior of L (at,bt) for every a, b was obtained by Aldous-Diaconis [START_REF] Aldous | Hammersley's interacting particle process and longest increasing subsequences[END_REF]. (Identification of the limit actually dates back to [START_REF] Veršik | Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux[END_REF], different probabilistic proofs can be found in [START_REF] Seppäläinen | A microscopic model for the Burgers equation and longest increasing subsequences[END_REF][START_REF] Groeneboom | Hydrodynamical methods for analyzing longest increasing subsequences[END_REF].)

Theorem 2 (Aldous-Diaconis ([1], Th.5 )). Let a, b > 0. Then f (a, b) := lim t→+∞ L (at,bt) t = 2 √ ab. (1)
The convergence holds a.s. and in L 1 .

Theorem 1 allows us to extend the formula (1) to every pair of gaps :

Proposition 3. For every h 1 , h 2 ≥ 0, we have the following limit:

f h (a, b) := lim t→∞ L h (at,bt) t =        2(ah 2 + bh 1 ) -2 (ah 2 -bh 1 ) 2 + ab 4h 1 h 2 -1 if h 1 h 2 = 1/4, ab h 1 b + h 2 a if h 1 h 2 = 1/4.
(

) 2
The convergence holds a.s. and in L 1 .

(We have no probabilistic interpretation of the case h 1 h 2 = 1/4, but one can check that the right-hand side of ( 2) is continuous at every point of the line h 1 h 2 = 1/4.)

In some cases the above formula for f h (a, b) simplifies:

• If h 1 = h 2 = h, then f (h,h) (1, 1) = 2 1 + 2h . • If h 2 = 0 then f (h,0) (1, 1) = 2 h 2 + 1 -2h.
For h = (0, 0), the fluctuations of L (at,bt) around its mean have been determined by Baik-Deift-Johansson [START_REF] Baik | On the distribution of the length of the longest increasing subsequence of random permutations[END_REF].

Theorem 4 (Baik-Deift-Johansson [START_REF] Baik | On the distribution of the length of the longest increasing subsequence of random permutations[END_REF]). For every a, b > 0 and x ∈ R,we have

lim t→∞ P L (at,bt) -2 √ abt ( √ abt) 1/3 ≤ x = F T W (x),
where F T W is the distribution function of the Tracy-Widom distribution.

In fact the main result of [START_REF] Baik | On the distribution of the length of the longest increasing subsequence of random permutations[END_REF] is stated for the longest increasing subsequence in a uniform permutation. Theorem 4 follows by elementary poissonization arguments. This theorem can also been extended for every pair of gaps : Proposition 5. For every h 1 , h 2 , a, b ≥ 0 and x ∈ R, we have

lim t→∞ P L h (at,bt) -f h (a, b)t σ h (a, b)t 1/3 ≤ x = F T W (x), (3)
where F T W is the distribution function of the Tracy-Widom distribution and

σ h (a, b) = f h (a, b) 4/3 2 1/3 1 2(bh 1 + ah 2 ) + f h (a, b)(1 -4h 1 h 2 )
.

(In some cases the expression for σ h (a, b) simplifies, for instance σ (h,h) (1, 1) = (1 + 2h) -4/3 .) Thanks to the scale-invariance property of the Poisson process we also easily obtain asymptotic results in the case where gaps and intensity of the Poisson process depend on t (see Section 2.4).

Discrete settings

The same strategy allows us to obtain analogous results in the discrete settings. Let Ξ = (Ξ i,j ) i,j∈Z >0 be i.i.d. Bernoulli random variables with mean p. We also consider Ξ as a random set of integer points of the quarter-plane by saying that (i, j) is present in Ξ if Ξ i,j = 1.

Let h = (h 1 , h 2 ) be a pair of non-negative integers, we assume h = 0 = (0, 0). We introduce the strict order on (Z >0 ) 2 defined by

(i, j) h ≺ (i , j ) if and only if i + h 1 ≤ i , j + h 2 ≤ j .
We consider the random variable given by the length of the longest non-decreasing path from (1, 1) to (m, n) in Ξ with horizontal gaps h 1 and vertical gaps h 2 . Namely,

L h (m,n) = max L; there are (i 1 , j 1 ), . . . , (i L , j L ) ∈ Ξ such that 1 ≤ i 1 ≤ • • • ≤ i L ≤ m, 1 ≤ j 1 ≤ • • • ≤ j L ≤ m, (i 1 , j 1 ) h ≺ (i 2 , j 2 ) h ≺ • • • h ≺ (i L , j L ) .
This problem is close to what is sometimes called slope-constrained longest increasing subsequence (SCLIS) in the literature of algorithms [START_REF] Deorowicz | On some variants of the longest increasing subsequence problem[END_REF]. Two particular cases have received particular attention:

• If h = (1, 1), L (1,1)
(m,n) is the length of the longest increasing path in Ξ.

• If h = (1, 0), L (1,0)
(m,n) is the length of the longest non-decreasing path.

Both problems have been first studied by Seppäläinen (resp. in [START_REF] Seppäläinen | Increasing sequences of independent points on the planar lattice[END_REF] and [START_REF] Seppäläinen | Exact limiting shape for a simplified model of first-passage percolation on the plane[END_REF]). Similarly to our approach for the continuous settings, we will use a coupling between L h and a well-studied model: last passage percolation with geometric weights (which in turn is in correspondence with synchronous TASEP).

Let us recall formally this latter model. Let Ξ = Ξ i,j i,j∈Z >0 be i.i.d. geometric random variables with law

P(Ξ i,j = k) = p k (1 -p) for k ≥ 0 and let T (m,n) = max{ (i,j)∈P Ξ i,j ; P ∈ P m,n },
where P m,n denotes the set of paths from (1, 1) to (m, n) taking only North and East steps. The discrete analogous of Theorem 1 is the following: Theorem 6. Let h = (h 1 , h 2 ) = (0, 0). For every m, n ≥ 0, and every k ≥ 0,

P(L h (m,n) ≤ k) = P(T (m-h 1 k,n-h 2 k) ≤ k).
Remark 7. As we will see in the proof of Theorem 6, many relevant quantities regarding T are obtained by taking formally h = (0, 0) in the formulas for L h . As suggested by Theorem 6, the key difference is that the Bernoulli point process must replaced by a point process with geometric weights. Therefore we cannnot extend our methods and results to the study of L (0,0) . This last model can be seen as (directed) site percolation on the quarter-plane, for which the critical threshold remains unknown.

(m,n) = 4, L (3,2) (m,n) = 3.
In both pictures one of the maximizing paths is drawn in red.

Theorem 6 is related to previous results in literature. In the case h = (1, 1), a similar coupling was implicit in [START_REF] Rajesh | An exactly solvable anisotropic directed percolation model in three dimensions[END_REF] (see also [START_REF] Georgiou | Optimality regions and fluctuations for Bernoulli last passage models[END_REF][START_REF] Majumdar | Exact asymptotic results for the Bernoulli matching model of sequence alignment[END_REF]). Still in the case h = (1, 1) another coupling between with asynchronous TASEP (also called directed TASEP) was also given in [START_REF] Georgiou | Soft edge results for longest increasing paths on the planar lattice[END_REF]Sec.3], [START_REF] Priezzhev | Exact solution of the Bernoulli matching model of sequence alignment[END_REF].

The explicit formula for the limiting shape in last-passage percolation with geometric weights is originally due to Jockusch-Propp-Shor (in the context of synchronous TASEP).

Theorem 8 (Jockusch-Propp-Shor ( [START_REF] Jockusch | Random domino tilings and the arctic circle theorem[END_REF],Th.2), see also ([21],Th.2.2)). For every a, b > 0, p ∈ (0, 1)

g(a, b) := lim n→+∞ 1 n T ( an , bn ) = √ p 2 √ ab + (a + b) √ p 1 -p . (4)
Theorem 6 then allows us to deduce the limiting shape for L h . The limiting constant is less explicit than in the continuous settings. Proposition 9. Let h 1 , h 2 be two non-negative integers such that h = (h 1 , h 2 ) = (0, 0). For every a, b > 0, there exists a constant

g h (a, b) := lim n→+∞ 1 n L h ( an , bn ) ,
where the convergence holds a.s. and in L 1 . Moreover, we have

• If h 1 (h 2 -1)+1/p < a b < h 1 -1+1/p h 2 , g h (a, b) is the unique solution of equation g h (a, b) = g a -h 1 g h (a, b), b -h 2 g h (a, b) , ( 5
)
where g is defined by (4) ;

• If a b ≤ h 1 (h 2 -1)+1/p , g h (a, b) = a h 1 , • If a b ≥ h 1 -1+1/p h 2 , g h (a, b) = b h 2 .
The second and third cases correspond to a flat edge in the limiting shape. This differs from the continuous case.

We explicit here the solution of ( 5) in some cases:

• Increasing paths. For h = (1, 1) the above formula reduces to

g (1,1) (a, b) =      √ p 2 √ ab -(a + b) √ p 1 -p if p < min {a/b, b/a} , min {a, b} otherwise.
.

Thus we recover the asymptotic behavior of L

(1,1) ( an , bn ) which was obtained by Seppäläinen in [START_REF] Seppäläinen | Increasing sequences of independent points on the planar lattice[END_REF] using hydrodynamic limits of a given particle system (see also [13,

Sec.III], [2, Th.1.1.] ,[4, Th.2.2] for different proofs).
• Non-decreasing paths. If h = (1, 0) the above formula reduces to

g (1,0) (a, b) = 2 abp(1 -p) + (a -b)p if p < a/(a + b), a otherwise.
This was also first proved by Seppäläinen in a second article ( [START_REF] Seppäläinen | Exact limiting shape for a simplified model of first-passage percolation on the plane[END_REF], Th.1) (see again [START_REF] Basdevant | Discrete Hammersley's lines with sources and sinks[END_REF] for a different proof). In the more general case h = (h, 0) we obtain with (5) the following expression (we only write the formula for a = b = 1):

g (h,0) (1, 1) =        2(1 + h)p(1 -p) + 2 p(1 -p + h 2 p)(1 -p) h √ p + (1 -p + h 2 p)(1 -p) 2 if p < 1/(h + 1), 1/h otherwise.
• Symmetric case. If (a, b) = (1, 1) and h 1 = h 2 = h we can easily solve (5) and we get

g (h,h) (1, 1) = 2 √ p 1 + (2h -1) √ p .

Proofs in the continuous settings

We fix a pair h = (h 1 , h 2 ) of non-negative real numbers all along this section.

Preliminary results

We first justify that 1 t L h (at,bt) converges almost surely and in L 1 . Let us stress that for any x, x , t, t ≥ 0, we have the stochastic domination

L h (x+x ,t+t ) L h (x,t) + L h (x ,t ) -1,
where L h (x ,t ) has the same distribution as L h (x ,t ) but is independent of L h (x,t) . Indeed, if (i 1 , j 1 ), . . . , (i L , j L ) is a longest increasing path in Ξ with gaps h in (0, x) × (0, t) and (i 1 , j 1 ), . . . , (i L , j L ) a longest increasing path in (x, x + x ) × (t, t + t ) , then

(i 1 , j 1 ), . . . , (i L , j L ), (i 2 , j 2 ), . . . , (i L , j L )
is an increasing path with gaps h in (0, x + x ) × (0, t + t ). Thus, the family of random variables

L h (x,t) -1 x>0,t>0
is superadditive. Hence, Kingman's subadditive theory (see for example [17, Th.A2-A3]) implies the existence of a constant

f h (a, b) := lim t→∞ L h (at,bt) t ,
where the limit is a.s. and in L 1 .

Hammersley's lines and dilatation

A very useful way to handle the random variables L h (x,t) is the geometric interpretation of Hammersley's lines. In the classical case h = 0 this construction was first implicitly introduced by Hammersley [START_REF] Hammersley | A few seedlings of research[END_REF], a more explicit construction was given by Aldous-Diaconis in [START_REF] Aldous | Hammersley's interacting particle process and longest increasing subsequences[END_REF] (continuous settings) and by Seppäläinen [START_REF] Seppäläinen | Increasing sequences of independent points on the planar lattice[END_REF] (discrete settings). We now define Hammersley lines formally. These are a sequence H h 1 , H h 2 , . . . of broken lines in (0, +∞) 2 defined inductively as follows (an example is provided in Fig. 3).

x t h 1 h 2 H 1 H 2 H 3 H 4
The broken line H h 1 is the shortest path made of vertical and horizontal straight lines whose minimal points for 0 ≺ are exactly the minimal points of Ξ for 0

≺. The line H h

2 is defined as follows: we remove the points of

H h 1 + [0, h 1 ] × [0, h 2 ]
(hatched in gray in Fig. 3) and reiterate the procedure: H h 2 is the shortest path made of vertical and horizontal straight lines whose minimal points for 0 ≺ are exactly the minimal points of Ξ \

H h 1 + [0, h 1 ] × [0, h 2 ] for 0 ≺. Inductively we define H h 3 , H h 4 , .
. . in the same way.

Lemma 10. For each (x, t) ∈ (0, +∞) 2 , there are exactly L h (x,t) distinct Hammersley lines which intersect (0, x) × (0, t).

Proof of Lemma 10. Let denote by N (x,t) the number of distinct Hammersley lines which intersect (0, x) × (0, t).

Proof of L h (x,t) ≤ N (x,t) . Let P = (y 1 , s 1 ) ≺ • • • ≺ (y L (x,t) , s L (x,t
) ) be a maximizing path in Ξ for L (x,t) . For every ≤ N (x,t) , there is at most one point of P in the area

H h + [0, h 1 ] × [0, h 2 ]. Therefore L h (x,t) ≤ N (x,t) . Proof of L h (x,t) ≥ N (x,t) . Let H h 1 , H h 2 , .
. . , H h be given, we will construct an admissible path with points of Ξ (from top-right to bottom-left). We first take any point (y , s ) of

H h . Let Ĥ = H h -1 + (h 1 , h 2 )
be the translation of H h -1 by the gaps. By construction of H h the broken line Ĥ intersects (0, y ) × (0, s ). Since Ĥ takes only directions North/West, necessarily there is a point (y

-1 + h 1 , s -1 + h 2 ) ∈ Ĥ ∩ (0, y ) × (0, s ), with (y -1 , s -1 ) ∈ Ξ . H h -1 (y , s ) H h (y -1 , s -1 )
Ĥ Therefore, (y -1 , s -1 ) h ≺ (y , s ). By induction we construct an admissible path of points in Ξ.

The following Proposition was used in [START_REF] Aldous | Hammersley's interacting particle process and longest increasing subsequences[END_REF] in the case h = 0, it extends plainly to the general case. We want to make a coupling between random variables L (x,t) and L h (x ,t ) for some x ≥ x, t ≥ t. We fix a realization of Ξ in the quarter-plane, and denote by L (x,t) (Ξ), (x, t) ∈ (0, +∞) 2 the lengths of the longest paths corresponding to this realization.

We introduce the (random) function

φ h : (0, +∞) 2 → (0, +∞) 2 (y, s) → (y + h 1 L (y,s) -, s + h 2 L (y,s) -),
where L (y,s) -= lim ε→0 L (y-ε,s-ε) .

An example is drawn in Fig. 4. By construction, the image by φ h of every Hammersley line H 0 is a translation of H 0 (and the area between two consecutive Hammersley lines H 0 , H 0 +1 is also translated by φ h ).

The main idea is that on the right picture of Figure 4 we re-sample new points in the regions which do not belong to the image of φ h , according to an independent Poisson process. New points of Ξ \ φ h (Ξ) are drawn in green. We have that L (x,t) (Ξ) = 3 = L h φ h (x,t) ( Ξ), as stated in Eq.( 6).

h 1 h 2 φ h H 0 1 φ(H 0 1 )=H h 1 φ h H 0 2 H 0 3 H 0 4 φ(H 0 2 )=H h 2 φ(H 0 3 )=H h 3 φ h (x, t) (x, t)
Lemma 12 (Dilatation). Let Ξ be the field of random points defined by

Ξy ,s = Ξ y,s if (y , s ) ∈ Image(φ h ) and φ h (y, s) = (y , s ), Ξy ,s = Y y ,s if (y , s ) / ∈ Image(φ h ),
where Y is a homogeneous Poisson process with intensity one, independent of Ξ. Then Ξ is also a homogeneous Poisson process with intensity one.

Proof of Lemma 12. The point process Ξ can be discovered by the following Markovian exploration of (0, +∞) 2 .

From bottom-left to top-right, both point processes Ξ and Ξ coincide up to H 0 1 . Then, conditional to H 0 1 , the points in the area φ h (H 0 1 ) + (0, h 1 ) × (0, h 2 ) (this is the first gray region in Fig. 4) are also distributed as an independent homogeneous Poisson process. This shows that, conditional to H 0 1 , Ξ is also a Poisson process up to φ h (H 0 1 ) + (0, h 1 ) × (0, h 2 ). But now, thanks to the Markovian property of Hammersley lines applied to H 0 1 we can reiterate the argument to show that conditional to H 0 2 , Ξ is a Poisson process up to φ h (H 0 2 ) + (0, h 1 ) × (0, h 2 ), and so on.

Proof of Theorem 1. We will first prove that almost surely, for every x, t we have

L (x,t) (Ξ) = L h (x ,t ) ( Ξ), (6)
where (x , t ) = φ h (x, t) i.e.

(x , t ) = (x + h 1 L (x,t) -(Ξ), t + h 2 L (x,t) -(Ξ)).
The quarter-plane (0, +∞) 2 is divided in two types of regions:

1. The region W defined by the interior of φ h ((0, +∞) 2 ) (represented in white in Fig. 4).

There are no points of Ξ in W.

The regions

G = φ h (H 0 ) + [0, h 1 ] × [0, h 2 ],
for ≥ 1 (represented in gray in Fig. 4).

Let P = (x 1 , s 1 ) 0 ≺ • • • 0 ≺ (y L (x,t) , s L (x,t
) ) be a maximizing path in Ξ. Because of the dilatation, the points of φ h (P ) have horizontal gaps h 1 and vertical gaps h 2 . Therefore the path φ h (P ) satisfies the gaps constraints and

L (x,t) (Ξ) ≤ L h (x ,t ) ( Ξ).
For the reverse inequality, we observe that because of the gaps constraint, an admissible path for the order h ≺ takes at most one point in each G . Since there are L (x,t) (Ξ) gray regions which intersect (0, x ) × (0, t ), this proves that L h (x ,t ) ( Ξ) ≤ L (x,t) (Ξ). Finally we have proved [START_REF] Georgiou | Soft edge results for longest increasing paths on the planar lattice[END_REF]. We now conclude the proof of the theorem. Let γ ≥ 0 be such that

γ = sup y ≥ 0, L (x-yh 1 ,t-yh 2 ) ≥ k + 1 (with sup ∅ = 0). If γ > 0 we have k + 1 = L (x-γh 1 ,t-γh 2 ) = L (x-γh 1 ,t-γh 2 ) -+ 1.

From (6)

k

+ 1 = L h (x-γh 1 +h 1 k,t-γh 2 +h 2 k) = L h (x-γh 1 +h 1 k,t-γh 2 +h 2 k) -+ 1.
By monotonicity of L we deduce that P(L h (x,t) < k + 1) = P(γ < k). On the other hand, by definition of γ,

P(γ < k) = P(L (x-h 1 k,t-h 2 k) < k + 1),
and Theorem 1 is proved.

Remark 13. There is a geometric interpretation of the coupling equality (6): the image of a Hammersley line under mapping φ h is a Hammersley line as well. More precisely, for every ,

φ h H 0 = H h ,
where on the left-hand side H 0 is defined with the points of Ξ and on the right-hand side, H h is defined with the points of Ξ.

Proof of Proposition 3. Recall the asymptotics known for the length of the longest increasing path:

f (a, b) := lim t→+∞ L (at,bt) t = 2 √ ab.
We fix (a, b) ∈ (0, +∞) 2 , let λ > 0 and t such that λt ∈ Z ≥0 . Using Theorem 1 we obtain

P(L h (at,bt) ≤ λt) = P(L (at-h 1 λt,bt-h 2 λt) ≤ λt) t→+∞ → 0 if λ < f (a -h 1 λ, b -h 2 λ) , 1 if λ > f (a -h 1 λ, b -h 2 λ) .
Therefore, 1 t L h (at,bt) converges in probability to the unique solution λ of the equation

λ = f (a -h 1 λ, b -h 2 λ) i.e. λ = 2 (a -h 1 λ)(b -h 2 λ). (7)
We easily check that if h 1 h 2 = 1/4 the solution of ( 7) is given by

λ = 2(ah 2 + bh 1 ) -2 (ah 2 -bh 1 ) 2 + ab 4h 1 h 2 -1 .
If h 1 h 2 = 1/4, then [START_REF] Georgiou | Optimality regions and fluctuations for Bernoulli last passage models[END_REF] reduces to λ = ab/(h 1 b + h 2 a).

Fluctuations of L h (at, bt)

Let us now explain how the combination of Baik-Deift-Johansson's result (Theorem 4) and Theorem 1 implies Proposition 5 for the fluctuations of L h (at, bt).

Proof of Proposition 5. Using the scaling invariance of a Poisson point process under transformations which preserve the volume, in the case without gaps constraint, the distribution of L (x,t) only depends on the value of the product xt. Thus, we can define a family of random variables (Z(s), s ≥ 0) such that Z(xt) d = L (x,t) for all x, t ≥ 0. Theorem 4 yields that ∀c ∈ R,

lim s→∞ P(Z(s 2 ) ≤ 2s + cs 1/3 ) = F T W (c),
where F T W is the distribution function of the Tracy-Widom distribution. Fix now a, b > 0 and let λ = f h (a, b). Using Theorem 1, we have that, for any t ≥ 0 and β ∈ R such that λt + βt 1/3 ∈ Z ≥0 ,

P(L h (at,bt) ≤ λt + βt 1/3 ) = P(L (t(a-h 1 λ)-βh 1 t 1/3 ,t(b-h 2 λ)-βh 2 t 1/3 ) ≤ λt + βt 1/3 ). ( 8
) = P(Z(s 2 ) ≤ λt + βt 1/3 )
with s ≥ 0 defined by

s 2 := (t(a -h 1 λ) -βh 1 t 1/3 )(t(b -h 2 λ) -βh 2 t 1/3 ) = t 2 (a -λh 1 )(b -λh 2 ) -t 4/3 β(h 1 b + h 2 a -2λh 1 h 2 ) + O(t) = t 2 λ 2 4 -t 4/3 β(h 1 b + h 2 a -2λh 1 h 2 ) + O(t),
where we use [START_REF] Georgiou | Optimality regions and fluctuations for Bernoulli last passage models[END_REF] in the last line. Inverting this equality gives

t = 2 λ s + βδs 1/3 + O(1) with δ := 2 4/3 h 1 b + h 2 a -2λh 1 h 2 λ 7/3 .
Plugging this expression of t in (8), we get

P(L h (at,bt) ≤ λt + βt 1/3 ) = P Z(s 2 ) ≤ 2s + β λδ + 2 1/3 λ 1/3 s 1/3 + O(1) . If we set σ h (a, b) = λδ + 2 1/3 λ 1/3
-1 and apply the above equation with β = cσ h (a, b), we obtain

lim t→∞ P(L h (at,bt) ≤ λt + cσ h (a, b)t 1/3 ) = lim s→∞ P(Z(s 2 ) ≤ 2s + cs 1/3 ) = F T W (c).
One can check that this definition of σ h (a, b) coincides with the one given in Proposition 5.

Case where h,λ depend on t

In this short section we show how to use the scale-invariance of the Poisson point process to derive asymptotics in the case where gaps and intensity of points depend on t. For the sake of simplicity we assume that vertical and horizontal gaps are identical. For every t, let h t = (h t , h t ) be a pair of gaps, λ t > 0 and denote by L h t ,λt (t,t) be the length of the longest increasing path with gaps h t when Ξ is a Poisson process with intensity λ t .

√ λ t t → +∞ then 1 √ λ t t L h t ,λt (at,bt) prob.
→ f (0,0) (a, b).

(ii) If c ∈ (0, +∞) and √ λ t t → +∞ then 1 √ λ t t L h t ,λt
(at,bt)

prob.

→ f (c,c) (a, b).

(iii) If c = +∞ and t/h t → +∞ then

h t t L h t ,λt (at,bt) prob.
→ min{a, b}.

Proof. We observe that by the scaling invariance of the Poisson process we have

L h t ,λt
(at,bt)

(d) = L h t √ λ t ,1 (at √ λt,bt √ λt) . Assume first that c = lim t→∞ h t √ λ t < +∞. Let ε ∈ (0, 1) and T such that c(1 -ε) ≤ h t √ λ t ≤ c(1+ε) for t ≥ T . Since L h
(at,bt) is a non-increasing function in h, we get, for t ≥ T , the stochastic domination:

1 t √ λ t L c(1+ε),1 (at √ λt,bt √ λt) 1 t √ λ t L h t ,λt (at,bt) 1 t √ λ t L c(1-ε),1 (at √ λt,bt √ λt)
Assuming that t √ λ t tends to infinity, the left-hand side tends to f (c(1+ε),c(1+ε)) (a, b) whereas the right-hand side tends to f (c(1-ε),c(1-ε)) (a, b). We conclude by continuity in c of the expression of f (c,c) (a, b).

Assume now that c = lim t→∞ h t √ λ t = +∞. First, by definition of gaps, L h t ,λ (at,bt) ≤ min{a, b}× t/h t a.s. This gives the upper bound in (iii). For the lower bound, let A > 0 and T such that h t √ λ t ≥ A for t ≥ T . For t ≥ T , λ t ≥ (A/h t ) 2 , thus using the monotonicity of a Poisson point process with respect to its intensity, we have

h t t L h t ,λt (at,bt) h t t L h t ,(A/ht) 2 (at,bt) . (9)
Using (ii) with λt = (A/h t ) 2 , we get, if t/h t → +∞, the following convergence in probability:

lim t→∞ h t At L h t ,(A/ht) 2 (t,t) = f (A,A) (a, b).
Observe that lim A→+∞ Af (A,A) (a, b) = a+b-|a-b| 2 = min{a, b}, so we obtain the lower bound by letting A tend to infinity in [START_REF] Hammersley | A few seedlings of research[END_REF].

3 Proofs in the discrete settings

Proof of Theorem 6

The main task of this Section is to prove the coupling between L h and T which leads to the identity of Theorem 6. For the sake of clarity we first exhibit a coupling between L h and L (1,1) . Lemma 15. Let h = (h 1 , h 2 ) with (h 1 , h 2 ) = (0, 0). For every m, n ≥ 0, and every k ≥ 0, (i,j) , the four Hammersley lines are drawn in blue.

P(L h (m,n) ≤ k) = P(L (1,1) (m-(h 1 -1)k,n-(h 2 -1)k) ≤ k). m n H 1 H 2 H 3 H 4

Proof of Lemma 15: the case

h 1 > 0, h 2 > 0
In the discrete settings and if h 1 > 0, h 2 > 0, the proof of Lemma 15 is almost identical to that of Theorem 1. We only explain how to change the definitions of the Hammersley lines and the function φ h . The definition of the Hammersley lines is identical to the continuous case (an example is provided in Fig. 5): the broken line H h 1 is the shortest path made of vertical and horizontal straight lines whose minimal points for 0 ≺ are exactly the minimal points of Ξ for 0 ≺. The line H h 2 is defined as follows: we remove the points of H h 1 + {0, 1, . . . , h 1 -1} × {0, 1, . . . , h 2 -1} and reiterate the procedure: H h 2 is the shortest path made of vertical and horizontal straight lines whose minimal points for 0 ≺ are exactly the minimal points of Ξ \ H h 1 + {0, 1, . . . , h 1 -1} × {0, 1, . . . , h 2 -1} for 0 ≺. Inductively we define H h 3 , H h 4 , . . . in the same way. The function φ h has to be replaced by its discrete counterpart:

φ h : (Z >0 ) 2 → (Z >0 ) 2 (m, n) → (m + (h 1 -1)L (1,1) (m-1,n-1) , n + (h 2 -1)L (1,1) (m-1,n-1) ). (10)
We define a new set of points Ξ by

Ξi ,j = Ξ i,j if (i , j ) ∈ Image(φ h ) and φ h (i, j) = (i , j ), Ξi ,j = Y i ,j if (i , j ) / ∈ Image(φ h ),
where (Y i ,j ) i,j≥1 are independent Bernoulli random variables with mean p (see an example in Figure 6). In the same manner as in the continuous settings, we prove that ( Ξi,j ) i,j≥1 are i.i.d Bernoulli random variables with mean p and for every m, n ≥ 1 we have

L (1,1) (m,n) (Ξ) = L h (m ,n ) ( Ξ), where (m , n ) = φ h (m, n) = (m + (h 1 -1)L (1,1) (m-1,n-1) , n + (h 2 -1)L (1,1)
(m-1,n-1) ). We deduce then Lemma 15 for h 1 h 2 > 0 in the same manner as in Theorem 1. ) and h = (3, 2). The points of the quarter-plane indicated by small gray squares are not in the image of φ h . The values of Ξ at these points are independent of Ξ.

Proof of Lemma 15: the case

h 1 > 0, h 2 = 0
As in the previous section, we can exhibit a coupling between L (h 1 ,0) and L (1,1) which shows that Lemma 15 also holds for h = (h 1 , 0). However, some change must be made compared to the case h 1 h 2 > 0 since the function φ h defined in [START_REF] Jockusch | Random domino tilings and the arctic circle theorem[END_REF] is no more a dilation.

To make the exposition clearer, it is more convenient to explain the coupling between the model with gap (h, 0) with the one with gap (h, 1). Thus, let us consider a Bernoulli field Ξ on (Z >0 ) 2 and construct the associated random variables L (h,1) (m,n) . Define the function ψ h by

ψ h : (Z >0 ) 2 → (Z >0 ) 2 (m, n) → (m, n -L (h,1) (m-1,n-1) ).
Contrary to φ h , the function ψ h is surjective but no more injective. More precisely, for any (m, n ) ∈ (Z >0 ) 2 , there exist n ≥ 1 and k ≥ 0 such that

(ψ h ) -1 (m, n ) = {(m, n), (m, n + 1), . . . , (m, n + k)}.
We define now the new set of points Ξ by

Ξm,n = Ξ m,n+k where (ψ h ) -1 (m, n ) = {(m, n), (m, n + 1), . . . , (m, n + k)}.
Again, one can prove that the random variables Ξ = ( Ξi,j ) i,j≥1 are i.i.d. Bernoulli random variables with mean p and for every m, n ≥ 1 we have

L (h,1) (m,n) (Ξ) = L (h,0) (m ,n ) ( Ξ), where (m, n ) = ψ h (m, n) = (m, n -L (h,1) (m-1,n-1)
). Then, with the same argument as in the proof of Theorem 1, we get that, for every m, n ∈ Z ≥0 , and every k ∈ Z ≥0 ,

P(L (h,0) (m,n) ≤ k) = P(L (h,1) (m,n+k) ≤ k) = P(L (1,1) (m-(h-1)k,n+k) ≤ k).

Proof of Theorem 6: coupling with T

We conclude the proof of Theorem 6 with our last coupling between L (1,1) (m,n) and T (m ,n ) , for some (m , n ). As already said, this coupling already appeared in [START_REF] Rajesh | An exactly solvable anisotropic directed percolation model in three dimensions[END_REF][START_REF] Georgiou | Optimality regions and fluctuations for Bernoulli last passage models[END_REF][START_REF] Majumdar | Exact asymptotic results for the Bernoulli matching model of sequence alignment[END_REF]. There is a point in Ξ at (i , j ) i.f.f. there is one in Ξ at its highest antecedent by ψ h .

ψ h ψ h
Let us consider a Bernoulli field Ξ on (Z >0 ) 2 and construct the associated random variables L

(1,1) (m,n) associated to the gaps (1, 1). Formally, in the case h = (0, 0), the function φ h defined in [START_REF] Jockusch | Random domino tilings and the arctic circle theorem[END_REF] becomes

φ 0 : (Z >0 ) 2 → (Z >0 ) 2 (m, n) → (m -L (m-1,n-1) , n -L (m-1,n-1) ). (11)
As in the previous case h = (h, 0), the function φ 0 is surjective but not injective. More precisely, for any (m , n ) ∈ (Z >0 ) 2 , there exist n, m ≥ 1 and k ≥ 0 such that

(φ 0 ) -1 (m , n ) = {(m, n), (m + 1, n + 1), . . . , (m + k, n + k)}.
We first define a new collection of random variables Ξ = { Ξi,j , i, j ≥ 1} ∈ {0, 1} (Z >0 ) 2 by

Ξm,n = Ξ m,n = 1 if (m, n) is a minimal point of some H (1,1)
Ξm,n = 0 otherwise, and we define now the family of random variables Ξ = { Ξi,j , i, j

≥ 1} ∈ (Z ≥0 ) (Z >0 ) 2 by Ξm ,n = (m,n)∈(φ 0 ) -1 (m ,n ) Ξm,n .
For every m , n we have

Ξ(m ,n ) = k if Ξ(m,n) = Ξ(m+1,n+1) = • • • = Ξ(m+k-1,n+k-1) = 1, Ξ(m+k,n+k) = 0,
which occurs with probability p k (1p). Therefore we can show that Ξ is a family of i.i.d. geometric random variables:

P( Ξm,n = k) = p k (1 -p) for k ≥ 0.
For m, n ≥ 1, recall the notation 1) . Right:

T (m,n) ( Ξ) = max{
The associated realization of last-passage percolation with geometric weights. We have Ξ3,1 = Ξ3,1 + Ξ4,2 + Ξ5,3 + Ξ6,4 = 1 + 1 + 1 + 0 = 3.

where (m , n

) = φ 0 (m, n) = (m -L (1,1) (m-1,n-1) , n -L (1,1) (m-1,n-1)
). We deduce that for every m, n ≥ 0, and every k ≥ 0,

P(T (m,n) ≤ k) = P(L (1,1) (m+k,n+k) ≤ k),
which is in fact equivalent to Eq.(4.1) in [START_REF] Georgiou | Optimality regions and fluctuations for Bernoulli last passage models[END_REF]. Combining this equality with Lemma 15 yields Theorem 6.

Proof of the limiting shape: Proposition 9

Proof of Proposition 9. We fix (a, b) ∈ (0, +∞) 2 , let λ > 0 and n such that λn ∈ Z ≥0 and λ ≤ λ 0 := min(a/h 1 , b/h 2 ). This last condition implies in particular that (anh 1 λn) and (bnh 2 λn) are non negative. Note also that, due to the gap constraint, we have L h (an,bn) ≤ λ 0 n a.s. Combining Theorem 6 and Theorem 8, we have P(L h (an,bn) ≤ λn) = P(T (an-h 1 λn,bn-h 2 λn) ≤ λn)

n→+∞ → 0 if λ < g (a -h 1 λ, b -h 2 λ) , 1 if λ > g (a -h 1 λ, b -h 2 λ)
where g is defined in (4). Therefore, 1 n L h (an,bn) converges in probability to

g h (a, b) := sup{λ ≤ λ 0 , λ < g (a -h 1 λ, b -h 2 λ)}.
Note that in (4), g is only defined on (R * + ) 2 but one can extend g on (R + ) 2 by continuity so that g (ah 1 λ 0 , bh 2 λ 0 ) is well defined. Two cases can occur:

• either λ 0 ≥ g (a -h 1 λ 0 , b -h 2 λ 0 ) ,
and the equation

λ = g (a -h 1 λ, b -h 2 λ) , (12)
has a solution which is necessarily unique since the right hand side of [START_REF] Johansson | Discrete orthogonal polynomial ensembles and the Plancherel measure[END_REF] decreases with respect to λ. Then 1 n L h (an,bn) converges to this unique solution.

• Or

λ 0 < g (a -h 1 λ 0 , b -h 2 λ 0 ) ,
and in this case, 1 n L h (an,bn) converges λ 0 = min(a/h 1 , b/h 2 ).

Using the expression of g given in (4), one can check that λ 0 ≥ g (ah 1 λ 0 , bh 2 λ 0 ) i.f.f.

h 1 (h 2 -1)+1/p ≤ a b ≤ h 1 -1+1/p h 2 3.3 Fluctuations of L h (an,bn)
Johansson [START_REF] Johansson | Shape fluctuations and random matrices[END_REF] has computed the fluctuations of T ( an , bn ) around its mean:

Theorem 16 (Cube root fluctuations (Johansson [11], Theorem 1.2)). For every a, b > 0 and x ∈ R, we have

lim n→∞ P T ( an , bn ) -ng(a, b) σ(a, b)n 1/3 ≤ x = F T W (x),
where F T W is the distribution function of the Tracy-Widom distribution, and

σ(a, b) = p 1/6 1 -p (ab) -1/6 ( √ a + pb) 2/3 ( √ b + √ pa) 2/3 .
In another paper Johansson [START_REF] Johansson | Discrete orthogonal polynomial ensembles and the Plancherel measure[END_REF]Th.5.3] has also obtained Tracy-Widom fluctuations for longest increasing paths in the case h = (1, 0). The authors of [START_REF] Majumdar | Exact asymptotic results for the Bernoulli matching model of sequence alignment[END_REF] state a close result for the fluctuations of L (an,bn) around its mean (see also Section 4 in [START_REF] Priezzhev | Exact solution of the Bernoulli matching model of sequence alignment[END_REF]). However, we have not been able to fill the gap between their result and the convergence of rescaled fluctuations.

From Theorem 16, it is not obvious to obtain a result as neat as Proposition 5 for every direction (a, b). The proof of Proposiion 5 relies on the scaling invariance property of the Poisson process: the law of L (x,t) only depends on the value of xt. There is of course no analogous for fields of Bernoulli random points. However, one can still show that for any a, b ≥ 0 and h, the fluctuations of L h ( an , bn ) are also of order n 1/3 (outside the flat edges of the limiting shape). Before stating our result about the fluctuations of L h ( an , bn ) , we must first prove a technical lemma.

Lemma 17. Let h be a gap constraint. For all a, b > 0 such that g h (a, b) < min{a/h 1 , b/h 2 }, there exists a unique couple (α, β) of positive numbers such that g h (a, b) = g(α, β). Moreover, (α, β) is solution to the system

α + h 1 g(α, β) = a, β + h 2 g(α, β) = b.
Proof. By symmetry, we can assume that g

h (a, b) < a/h 1 ≤ b/h 2 . Recall that in this case, g h (a, b) is the unique λ solution of λ = g (a -h 1 λ, b -h 2 λ) , (13)
and we necessarily have g 0, b -h 2 a h 1 < a h 1 . Assume that there exists a solution (α, β) of the system

α + h 1 g(α, β) = a, β + h 2 g(α, β) = b.
Putting g(α, β) in ( 13), we see that g(α, β) satisfies this equality and thus g(α, β) = g h (a, b) < a/h 1 . In particular, we necessarily have α, β > 0. It remains to prove that the system as indeed a (unique) solution. Noticing that g(α, β) = βg(α/β, 1) and setting γ = α/β, we see now that the system is equivalent to

β(γ + h 1 g(γ, 1)) = a, β(1 + h 2 g(γ, 1)) = b. (14)
In particular, we have b(γ + h 1 g(γ, 1)) = a(1 + h 2 g(γ, 1)).

Hence, if a/h 1 = b/h 2 , we get γ = a/b. In the other case : a/h 1 < b/h 2 , we get

g(γ, 1) = a -bγ bh 1 -ah 2 .
The left hand side is increasing with γ whereas the right hand side decreases. Thus, there exists a unique solution γ > 0 if and only if g(0, 1) < a bh 1ah 2 which coincides with the condition g 0, b -h 2 a h 1 < a h 1 stated above. Finally, using ( 14), we see that the existence and unicity of γ implies the existence and unicity of (α, β). and such that g h (a, b) = g(α, β) =: λ. Using ( 16), we obtain P(T (αn,βn) ≤ λn + yn 1/3 ) = P(L h (n(α+h 1 λ)+h 1 yn 1/3 ,n(β+h 2 λ)+h 2 yn 1/3 ) ≤ λn + yn 1/3 ). = P(L h (an+h 1 yn 1/3 ,bn+h 2 yn 1/3 ) ≤ λn + yn 1/3 ). We obtain the lower bound in the same way, setting

Ñ := n + h 2 b yn 1/3
and γ the function such that Ñ γ( Ñ ) = n + h 1 a yn 1/3 .

Due to the condition a/h 1 ≤ b/h 2 , we now have γ( Ñ ) ≥ 1 for any y ≥ 0. The case x ≤ 0 is also obtained with similar arguments. Finally, note that in the particular case a/h 1 = b/h 2 , we have α/a = β/b and we obtain the corollary.

Aknowledgements. The authors are glad to acknowledge N.Georgiou for helpful comments regarding the literature around the problem of fluctuations.

Figure 1 :

 1 Figure 1: A realization of L h (x,t) (points of Ξ are represented with •).Here we have L h (x,t) = 4, one of the maximizing paths is drawn in red.

Figure 2 :

 2 Figure 2: Realizations of L (2,1) (m,n) (left) and L (3,2) (m,n) (right) for (m, n) = (10, 8) and the same sampling of Ξ (points of Ξ are represented with •). Here we have L (2,1)

Figure 3 :

 3 Figure 3: An example of Hammersley lines for the same realization of Ξ as that of Fig.1. The four Hammersley lines are drawn in blue.

Figure 4 :

 4 Figure 4: An example of the function φ h . Left: A sample of Ξ. Right:The same realization after the dilatation φ h . The gray areas correspond to regions which are not in the image of φ h . New points of Ξ \ φ h (Ξ) are drawn in green. We have that L (x,t) (Ξ) = 3 = L h φ h (x,t) ( Ξ), as stated in Eq.(6).

Theorem 14 .

 14 Let c t = h t √ λ t and assume that c := lim t→∞ c t exists in [0, +∞]. Then, (i) If c = 0 and

Figure 5 :

 5 Figure 5: An example with h = (2, 1) for the same realization of Ξ as that of Fig.2. In every unit square (i, j) we write the value of L (2,1)

Figure 6 :

 6 Figure 6: An example of the function φ h for (m, n) = (8, 6) and h = (3, 2). The points of the quarter-plane indicated by small gray squares are not in the image of φ h . The values of Ξ at these points are independent of Ξ.

Figure 7 :

 7 Figure 7: An example of the function ψ h and the definition of set of points Ξ for h = (1, 0).There is a point in Ξ at (i , j ) i.f.f. there is one in Ξ at its highest antecedent by ψ h .

(Figure 8 :

 8 Figure 8: An example of the function φ 0 . Left: a realization of Hammersley lines H(1,1) . Right:

Proposition 18 . 3 . 3 . 20 .

 183320 Let h = (h 1 , h 2 ) be a gap constraint and let a, b > 0 be such that g h (a, b) < min{a/h 1 , b/h 2 }. Let us define (α, β) as in Lemma 17 such thatg h (a, b) = g(α, β). Set W h an,bn = L h ( an , bn )g h (a, b)n σ(α, β)n 1/Assuming for example that a/h 1 ≤ b/h 2 , we have, for all x ≥ 0,F T W bx β ≤ lim inf n→∞ P W h an,bn ≤ x ≤ lim sup n→∞ P W h an,bn ≤ x ≤ F T W ax α F T W -ax α ≤ lim inf n→∞ P W h an,bn ≤ -x ≤ lim sup n→∞ P W h an,bn ≤ -x ≤ F T W -bx β .In one particular direction the LHS and RHS of above inequalities coincide:Corollary 19. Let h = (h 1 , h 2 ) be a gap constraint and let a, b > 0 be such that a/h 1 = b/h 2 . Thenlim n→∞ P L h (an,bn)g h (a, b)n σ h (a, b)n 1/3 ≤ x = F T W (x),(15)whereσ h (a, b) = σ(α, β) αβ ab .Again, we have a simple expression for h = (h, h) and a = b = 1:σ (h,h) (1, 1) = (1p) 1/3 p 1/6 (1 + (2h -1) √ p) 4/Remark Proposition 18 states that, ouside the flat edge of the limiting shape, the fluctuations of L h (an,bn) are of order n 1/3 . Inside the flat edge (except in the critical direction), one can easily show that P(|L h ( an , bn )g h (a, b)n| ≥ 1) tends to 0.Proof. A change of indexes in Theorem 6 yieldsP(T (m,n) ≤ k) = P(L h (m+h 1 k,n+h 2 k) ≤ k).(16)Let a, b > 0 such that g h (a, b) < min{a/h 1 , b/h 2 } and, according to Lemma 17, take α, β > 0 solution of the system α + h 1 g(α, β) = a, β + h 1 g(α, β) = b

)Using ( 18 )

 18 Assume now that a/h 1 ≤ b/h 2 . SetN := n + h 1 a yn 1/3 so that n = N -h 1 a yN 1/3 + o(N 1/3 )and define the function γ such thatN γ(N ) = n + h 2 b yn 1/3 , observe that because of a/h 1 ≤ b/h 2 we have N γ(N ) ≤ n + h 1 a yn 1/3 = N.(18)With this notation,[START_REF] Romik | The Surprising Mathematics of Longest Increasing Subsequences[END_REF] becomesP(L h (aN,bγ(N )N ) ≤ λN + y(1 -h 1 λ a )N 1/3 + o(N 1/3)) = P(T (αn,βn) ≤ λn + yn 1/3 ). we have that L h (aN,bN γ(N )) ≤ L h (aN,bN ) and therefore by putting x = yα/a we have for any x ≥ 0,P(L h (aN,bN ) ≤ λN + xN 1/3 + o(N 1/3 )) ≤ P(L h (aN,bN γ(N )) ≤ λN + xN 1/3 + o(N 1/3 )) ≤ P(T (αn,βn) ≤ λn + xa α n 1/3). Using Johansson's result, we obtain, for x ≥ 0, lim sup N →∞ P(W h (aN,bN ) ≤ x) ≤ lim n→∞ P(T (αn,βn) ≤ λn + xσ(α, β) a α n 1/3 ) = F T W ( xa α ).

  Proposition 11 (spatial Markov property for Hammersley's lines). Conditional on the -th Hammersley line H

h , Ξ y,s , (y, s) 0 H h is independent of Ξ y,s , (y, s) 0 ≺ H h , and distributed as a homogeneous Poisson process with intensity one. Here (y, s) 0 H h (resp. 0 ≺) means that (y, s) 0 (y , s ) (resp. 0 ≺) for at least one point (y , s ) in H h . In particular, conditional on H h , the line H h +1 is independent of H h 1 , H h 2 , . . . , H h -1 . Proof. Let H h 1 , H h 2 , . . . , H h -1 be given. By construction of Hammersley lines, the fact that (y, s) belongs to H h or not only depends on Ξ in the rectangle [0, y] × [0, s].

Work partially supported by ANR PPPP and ANR GRAAL.


Richard Sempere

Kawamura

come

Comparative distributions of dicarboxylic acids and related polar compounds in snow, rain and aerosols from

urban atmosphere


Ph Menini

B Durand

G Larrieu

A Chapelle

A Lecestre

L Mazenq

Ch Talhi

Micro and nanotechnology evolution towards vertical Silicon nanowires for ultra-sensitive gas detection

Most commercial sensors, using metal oxide as sensitive layer, exhibit sensitivity in the ppm range with low selectivity. A part of our previous work is focused on developing new gas sensor array in a small chip based on microhotplates with four silicon membranes and various nanoscaled sensing materials on top deposited by inkjet printing. Moreover, to enhance significantly their performances, these layers operate at various temperatures in pulsed operating mode which drastically increases the possibility to discriminate mixtures of gases thank to appropriate data analysis. The main drawbacks of these structures concern the complexity of the process, the limit of detection above ppm and the power consumption as well.

The one-dimensional nanostructures, such as nanowires (NWs) [START_REF] Ramgir | Nanowire-Based Sensors[END_REF] or the nanotubes [START_REF] Bondavalli | Carbon nanotubes based transistors as gas sensors: State of the art and critical review[END_REF] are very promising for new generations of sensors with very low power consumption needed and potential sub-ppm detection. This is due to their very high form factor (surface / volume ratio), but also by the ability to be functionalized. The silicon NWs, popularized by Lieber [START_REF] Cui | Nanowire Nanosensors for Highly Sensitive and Selective Detection of Biological and Chemical Species[END_REF], are particularly attractive due to their high sensitivity, biocompatibility and diversity of possible applications (gas [START_REF] Cao | Silicon Nanowire-Based Devices for Gas-Phase Sensing[END_REF] chemicals [START_REF] Chen | Silicon-based nanoelectronic fieldeffect pH sensor with local gate control[END_REF]

biological [6] sensors).

There are two main approaches to structure NWs: a "bottom-up" approach (rising) and "top-down" (down) [START_REF] Chen | Nanowire-based gas sensors[END_REF]. The first method involves the growth of NWs by synthesis from elementary constituents by chemical reaction (polymerization) or physico-chemical assembly (crystal growth). The main limitation of this approach is the integration of NWs obtained in devices, due to difficulties in positioning the one hand, and to contact them electrically on the other hand. In the case of the "top-down" approach, the nanowire is formed by masking and then etching of bulk material. The key point of this method is the solution of nano-masking patterns, which often requires expensive equipment (electron beam lithography).

NWs based gas sensors have already been discussed in the literature and various materials have been studied, such as SnO2 [START_REF] Choi | Novel fabrication of an SnO 2 nanowire gas sensor with high sensitivity[END_REF], CuO [START_REF] Kim | CuO nanowire gas sensors for air quality control in automotive cabin[END_REF], Si [START_REF] Wan | Silicon nanowire sensor for gas detection fabricated by nanoimprint on SU8/SiO2/PMMA trilayer[END_REF] and InAs [START_REF] Offermans | Gas Detection with Vertical InAs Nanowire Arrays[END_REF]. Whenever in a configuration known as "resistor", which consists in measuring the variation of electrical resistance of NW induced by variation of surround atmosphere. Currently, the best performance for a gas sensor based on silicon NWs (without any functionalization), from the viewpoint of the detection NH3 and NO2 was obtained by using horizontal NWs with a diameter of 130 nm [START_REF] Wan | Silicon nanowire sensor for gas detection fabricated by nanoimprint on SU8/SiO2/PMMA trilayer[END_REF]. The detection limit is closed to 0.5 ppm, and the sensitivity around 27%. The InAs NWs also demonstrate good sensitivity to NO2 (at 115 ppb), but with a relatively long response time (10 min) [START_REF] Offermans | Gas Detection with Vertical InAs Nanowire Arrays[END_REF].

The objective of our present studies is to develop a new generation of highly sensitive gas sensors based on nanostructuring silicon as vertical nanowires well controlled with reproducible manner by a "top-down" approach. These nanowires are electrically contacted by a metallic suspended bridge. Thus a large surface NWs is available, and can react with the surrounding gas. This approach allows networks of NWs defined by optical photolithography. The number, diameter, spacing and position of NWs are perfectly controlled and thus we obtain reproducible devices. The sensors were then electrically characterized and mounted to be tested under different gas atmospheres (CO, NH3 and NO2).

The first best results showed a strong sensitivity to low concentration of NO2 (30% at 50 ppb) with very low limit of detection (few ppb) and low sensitivity to both other gases.


Henri Gouin
email: henri.gouin@univ.u-3mrs.fr

Sergey Gavrilyuk
email: sergey.gavrilyuk@univ.u-3mrs.fr

Dissipative Two-Fluid Models

From Hamilton's principle of stationary action, we derive governing equations of two-fluid mixtures and extend the model to the dissipative case without chemical reactions. For both conservative and dissipative cases, an algebraic identity connecting equations of momentum, mass, energy and entropy is obtained by extending the Gibbs identity in dynamics. The obtained system is hyperbolic for small relative velocity of the phases.

Introduction

The knowledge of governing equations for fluid mixtures is scientifically and industrially an important challenge. Many authors derived the governing system by using axioms of balance of mass, momentum, energy and second law of thermodynamics [START_REF] Bowen | Theory of mixtures[END_REF]. The mixtures were considered as a collection of different media co-existing in the physical space. For example, the balance law of momentum is given in the form :

∂ ∂t D ρ α u α dD + ∂D ρ α u α u αn dσ = ∂D T α n dσ + D b α dD, α = 1, 2 (1)
where D is a fixed volume, ∂D is its boundary, n is the unit normal to ∂D, ρ α are the densities of components, u α are the associated velocities, u αn are the normal part of the velocities at ∂D, T α are the stress tensors and b α are the volume forces associated with internal forces and interaction between components. The principle of material frame-indifference requires that T α and b α depend on the thermodynamic parameters of the mixture and on the relative velocity w = u 2u 1 . To include the added mass effect into consideration, one should be supposed that T α and b α depend also on accelerations of phases. The structure of this dependence which is an important source of interaction, is not clear: should it be frame-indifferent or simply Galilean invariant? The method of balance laws [START_REF] Bowen | Theory of mixtures[END_REF] does not give a definite answer to this question. It exists a different approach based on Hamilton's principle [START_REF] Bedford | Recent advances. Theories of immiscible and structured mixtures[END_REF]- [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF] which is used for construction of conservative (non-dissipative) mathematical models of continuous media with complex internal structure. The terms including interaction between different components of the mixtures do not require constitutive postulates difficult to interpret experimentally. They come from the direct knowledge of a unique potential for the mixture. According to [START_REF] Nigmatulin | Fundamentals of mechanics of heterogeneous mixtures[END_REF], we call a homogeneous mixture if each component of the mixture occupies the whole volume of the physical space, and a heterogeneous mixture if each component occupies only a part of the mixture volume. In this paper, we consider only homogeneous binary mixtures, but the method can be extended to the case of heterogeneous mixtures [START_REF] Gavrilyuk | Mathematical and Numerical Modeling of Twophase Compressible Flows with Micro-Inertia[END_REF]. The plan for the article is as follows :

In section 2, we formulate an extended form of Hamilton's principle of stationary action allowing us to obtain the governing equations of motion. The Lagrangian is the difference between the kinetic energy depending on the reference frame and a thermodynamic potential which is a Galilean invariant. The equations of motion introduce two new vector fields different from the velocity fields by taking into account the relative velocity of the components. They play the same role as the velocity field does in the case of a single fluid.

In section 3, the governing equations are extended to the dissipative case without chemical reactions. An algebraic identity connecting equations of momentum, equations of mass, energy equation and equations of entropy is obtained. This identity can be considered as the dynamic form of the Gibbs identity.

In section 4, we justify the compatibility of the governing equations with the second law of thermodynamics.

In section 5, Fick's law is derived. We prove that Fick's law is not yet a linear phenomenological law but a direct consequence of governing equations and Stokes drag force hypothesis.

In Section 6, we check the properties of hyperbolicity of the governing system for small relative velocity of phases.

As a convention, in the following we shall use asterisk ′′ ⋆ ′′ to denote conjugate mappings or covectors (vector lines); subscripts α = 1, 2 indicate the parameters of the α th component; the symbol I indicates the identity; ∇ means the gradient operator-line; ∇ ⋆ means the gradient operator-column; a ⋆ b means the scalar product of vectors a, b (the vector line is multiplied by the vector column); a b ⋆ means the tensor product of vectors a, b (the vector column is multiplied by the vector line); A a means the product of the mapping A by a vector a ; b ⋆ A means the covector c ⋆ defined by the rule c ⋆ = (A ⋆ b) ⋆ ; div A denotes the divergence of a linear transformation A which is a covector defined as follows: for any vector a, div (A a) = (divA) a + tr A ∂a ∂x .

Governing equations in conservative case

In paper [START_REF] Gavrilyuk | Hyperbolic models of homogeneous two-fluid mixtures[END_REF], we considered a pure mechanical case (without entropy). Now we consider the general case. We take the Lagrangian of the binary system in the following form:

L = 2 α=1 1 2 ρ α u 2 α -ρ α Ω α -W (ρ 1 , ρ 2 , s 1 , s 2 , w) (2)
where in the whole paper the summation is over fluid components (α = 1, 2) and ρ α are the densities of components, s α are the specific entropies, u α are the velocities, w = u 2u 1 is the relative velocity, Ω α are the external force potentials, W is a potential per unit volume of the mixture. The dependance of W with respect to the relative velocity is analog to take into account the added mass effect in heterogeneous two-fluid theory as it was done by Geurst [START_REF] Geurst | Variational principles and two-fluid hydrodynamics of bubbly liquid/gas mixtures[END_REF]. The fact that W depends on two entropies is classically adopted in the literature [START_REF] Bedford | Recent advances. Theories of immiscible and structured mixtures[END_REF][START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF][START_REF] Müller | Thermodynamics, Interaction of Mechanics and Mathematics Series[END_REF][START_REF] Müller | Rational Extended Thermodynamics[END_REF]. The potential W is related with the internal energy U of the mixture through the transformation

U = W - ∂W ∂w w (3)
so that the total energy of the system is [START_REF] Gavrilyuk | Hyperbolic models of homogeneous two-fluid mixtures[END_REF]:

ε = 2 α=1 1 2 ρ α u 2 α + ρ α Ω α + U
Let us note that to define the internal energy of one-velocity media it is useful to consider a moving coordinate system in which the elementary volume of the continuum is at rest. The total energy of the continuum with respect to this system is called the internal energy of the medium. For a two-velocity medium, there is no reference frame in which any motion could be disregarded. This is the reason why the standard definition of internal energy is dependent on the relative motion of components.

The formula (3) implies that the internal energy U is a Galilean invariant. The dependence of U (or W ) on w is an important property of multicomponent fluid mixtures.

Let x be the Eulerian coordinates, t be the time, X α be the Lagrangian coordinates of each component. The mass and the entropy conservation laws in the Eulerian coordinates are:

∂ρ α ∂t + div (ρ α u α ) = 0 , ∂ ∂t (ρ α s α ) + div (ρ α s α u α ) = 0 . (4)
In the Lagrangian coordinates, equations ( 4) are equivalent to:

ρ α det F α = ρ α0 (X α ) , s α = s α0 (X α ), (5)
where

F α = ∂x ∂X α (6)
is the deformation gradient at X α ; ρ α0 (X α ) and s α0 (X α ) do not depend on t.

The relation between the Eulerian and Lagrangian coordinates is given by the local diffeomorphism x = φ α (X α , t), where φ α (X α , t) is the solution of the Cauchy problem :

dφ α dt = u (φ α , t) , φ α (X α , 0) = X α Let X α = ψ α (x, t) be its inverse mapping ( φ α • ψ α = I).
We define the virtual motion of the mixture such that [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF][START_REF] Serrin | Mathematical principles of classical fluid mechanics in[END_REF]:

x = Φ α (X α , t, ε α ) , X α = Ψ α (x, t, ε α ) , Φ α • Ψ α = I Φ α (X α , t, 0) = φ α (X α , t) , Ψ α (x, t, 0) = ψ α (x, t),
where ε α belong to a vicinity of zero. The Lagrangian and Eulerian virtual displacements are defined respectively as :

δX α = ∂ Ψ α ∂ε α (x, t, ε α ) | εα=0 , δ α x = ∂Φ α ∂ε α (X α , t, ε α ) | εα=0 . (7)
The definitions (6-7) imply the following relation between δX α and δ α x [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF]:

δ α x = -F α δX α (8)
The variations of u α (t, x), ρ α (t, x) and s α (t, x) are deduced from ( 6) -( 7) and from the definition of the Lagrangian coordinates X α :

d α X α dt = 0 , d α dt = ∂ ∂t + u ⋆ α ∇ ⋆ .
We obtain in Appendix A the values of δu α (x, t), δρ α (x, t) and δs α (x, t), where δf (t, x) means the variation of f when t, x are fixed and div α (δX α ) means the divergence with respect to the coordinates X α . We note that in [START_REF] Gavrilyuk | Hyperbolic models of homogeneous two-fluid mixtures[END_REF] we used different but equivalent expressions for these variations. Using the definition (2) of the Lagrangian L as a function of ρ α , u α , s α , we introduce the following quantities:

                           R α ≡ ∂L ∂ρ α = 1 2 u 2 α - ∂W ∂ρ α -Ω α , K ⋆ α ≡ 1 ρ α ∂L ∂u α = u ⋆ α - (-1) α ρ α ∂W ∂w , ρ α θ α ≡ - ∂L ∂s α = ∂W ∂s α (9)
We note that a best set of independent variables is : ρ α , j α = ρ α u α , ρ α s α . However, in this case, the corresponding derivation should be given in four-dimensional space [START_REF] Gavrilyuk | A new form of governing equations of fluids arising from Hamilton's principle[END_REF]. For the sake of simplicity we use the first set of independent variables. The last formula defines the thermodynamic temperature θ α of each component which is now a dynamical quantity depending on the relative velocity of the components.

Let ω = D × [t 1 , t 2 ] be the domain in the four-dimensional space (x, t) and ω α be its image in the (X α , t)-space. Here [t 1 , t 2 ] is a time interval and D is a fixed domain. We consider Hamilton's principle in the form,

δ α a ≡ δ α ω L dω = 0
under constraints (4) where δ α a are variations of a associated with the variation of

X α = Ψ α (x, t, ε α ). It means that δ α a = da dε α | εα = 0.
We have to emphasis on the fact that the domain D is fixed in the physical space. This particularity is related to the impossibility to have material volume in general motion of the mixture. Taking into account formulae [START_REF] Müller | Thermodynamics, Interaction of Mechanics and Mathematics Series[END_REF], variations in Appendix A and definitions (9), we get

δ α a = ω R α δρ α + ρ α K ⋆ α δu α -ρ α θ α δs α dω = ωα R α div α (ρ α0 δX α ) -ρ α0 K ⋆ α F α ∂ ∂t (δX α ) -ρ α0 θ α ∂s α0 ∂X α δX α dω α .
In the last expression all quantities are considered as functions of (X α , t). Hence,

δ α a = ωα ρ α0 - ∂R α ∂X α + ∂ ∂t (K ⋆ α F α ) -θ α ∂s α0 ∂X α δX α dω α + ωα Div α (ρ α0 G) dω α = 0
where G = (R α δX α , -K ⋆ α F α δX α ) and Div α is the divergence operator in the 4-dimensional space ω α associated with (X α , t). All the functions are assumed to be smooth enough in the domain ω α and δX α = 0 on ∂ω α . Then, we obtain the equations of motion for each component in Lagrangian coordinates:

∂ ∂t (K ⋆ α F α ) - ∂R α ∂X α -θ α ∂s α0 ∂X α = 0 (10)
Taking into account the identity d α F α dt -∂u α ∂x F α = 0, we rewrite [START_REF] Serrin | Mathematical principles of classical fluid mechanics in[END_REF] in Eulerian coordinates in the form :

d α K ⋆ α dt + K ⋆ α ∂u α ∂x = ∂R α ∂x + θ α ∂s α ∂x (11)
If u 1 = u 2 = u , then K α = u and ( 10) is equivalent to

du dt + ∇ ⋆ (h + Ω) = θ ∇ ⋆ s , d dt = ∂ ∂t + u ⋆ ∇ ⋆
where is the enthalpy and Ω is an external potential [START_REF] Serrin | Mathematical principles of classical fluid mechanics in[END_REF]. Conservations of the total momentum and the total energy are a consequence of the governing equations ( 4), ( 11):

2 α=1 ∂ρ α K ⋆ α ∂t + div ρ α u α K ⋆ α + ρ α ∂W ∂ρ α -W I + ρ α ∂Ω α ∂x = 0 (12) 2 α=1 ∂ ∂t ρ α 1 2 u 2 α + Ω α + U + div ρ α u α (K ⋆ α u α -R α ) -ρ α ∂Ω α ∂t = 0
(13) The covector K ⋆ α is an essential quantity; indeed, ρ α K α (but not ρ α u α ) is the impulse for the α th component of the mixture. Morever, for adiabatic motions, the definition of potential flows for two-component mixtures is associated with rot K α = 0 and not with rot u α = 0. For potential motion, equation [START_REF] Gavrilyuk | A new form of governing equations of fluids arising from Hamilton's principle[END_REF] yields additional conservation laws [START_REF] Gavrilyuk | Hyperbolic models of homogeneous two-fluid mixtures[END_REF][START_REF] Gavrilyuk | A new form of governing equations of fluids arising from Hamilton's principle[END_REF]:

∂K ⋆ α ∂t + ∇ (K ⋆ α u α -R α ) = 0
In the particular case of bubbly liquids, Geurst was the first to carry out a term analog to K α [START_REF] Geurst | Variational principles and two-fluid hydrodynamics of bubbly liquid/gas mixtures[END_REF]. Moreover, if rot K α = 0, the system of governing equations is not conservative in terms of K α , ρ α and s α (the number of conservation laws admitted by the system is less than the number of unknown variables); but, nevertheless the system can be rewritten in conservative form if we add the gradient tensor F α as unknown variable [START_REF] Gavrilyuk | Hyperbolic models of homogeneous two-fluid mixtures[END_REF].

Governing equations in the dissipative case and dynamic Gibbs identity

The conservative fluid mixture model presented in section 2 is relevant to the first gradient theory [START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus[END_REF]: the forces applied to the continuous medium are divided into volume forces and surface forces. In fluid mixture flows it is reasonable to neglect the surface friction forces compared to galilean invariant algebraic volume forces.

The virtual work δT α of dissipative forces applied to the α th component is in the form δT α = f ⋆ α δ α x. For the same virtual displacement of two components,

δx = δ α x 1 = δ α x 2 , the total virtual work of dissipative forces is δT = 2 α=1 f ⋆ α δx.
For a solid displacement, the work δT is equal to zero and consequently

2 α=1 f ⋆ α = 0.
We specify later the behavior of forces f ⋆ α . Let us introduce the quantities M α , B α , S and E such that:

M ⋆ α = ρ α d α K ⋆ α dt + ρ α K ⋆ α ∂u α ∂x -ρ α ∂R α ∂x -ρ α θ α ∂s α ∂x -f ⋆ α B α = ∂ρ α ∂t + div(ρ α u α ) S = 2 α=1 ρ α θ α d α s α dt + f ⋆ α u α E = 2 α=1 ∂ ∂t ρ α ( 1 2 u 2 α + Ω α ) + U + div ρ α u α (K ⋆ α u α -R α ) -ρ α
∂Ω α ∂t We prove in Appendix B the following property:

Theorem: For any motion of the mixture, we have the identity

E - 2 α=1 M ⋆ α u α + (K ⋆ α u α -R α ) B α -S ≡ 0
This relation is the most general expression of the Gibbs identity in dynamics. Analogous identities were obtained earlier for thermocapillary mixtures [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF] and bubbly liquids [START_REF] Gavrilyuk | Media with equations of state that depends on derivatives[END_REF]. For each component of the mixture, equation of momentum and equation of mass are in the form

M ⋆ α = 0, B α = 0 (14)
The Gibbs identity implies S = E. Hence, the equation of the entropy S = 0,

2 α=1 ρ α θ α d α s α dt + f ⋆ α u α = 0 (15)
is equivalent to the equation of the energy E = 0. We note also that the equations M ⋆ α = 0 and B α = 0 imply conservation of the total momentum of the mixture M = 0 (see ( 12))

The second law of thermodynamics

In conservative case, the system with two different entropies is closed by [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF]. In dissipative case we need additional arguments to obtain equations for each entropy s α which could replace equations (4). We take these equations in the form :

ρ α θ α d α s α dt + f α ⋆ (u α -u) + q α = 0 (16)
which must be compatible with [START_REF] Godunov | An interesting class of quasilinear systems[END_REF]. Here ρ u = q α = 0. The last relation means that we have only internal heat exchanges between components. Consequently, if

2 α=1 f ⋆ α θ α (u α -u) + q α θ α ≤ 0, (17)
we obtain the entropy inequality [8, 9]

2 α=1 ρ α d α s α dt ≥ 0
Let us note that relation [START_REF] Boillat | Non-linear hyperbolic fields and waves in[END_REF] is verified if

f 1 = k ( u 2 -u θ 2 - u 1 -u θ 1 ) , f 2 -f 1 , k > 0 and q 1 = κ ( 1 θ 2 - 1 θ 1 ), q 1 = -q 2 , κ > 0
The fact that the inverse temperatures (coldness) appear in the closure relations play an important role in other applications [START_REF] Ruggeri | Relativistic extended thermodynamics in[END_REF].

Fick's law as a consequence of the governing equations

The governing equations for each component are :

M ⋆ α ≡ ρ α d α K ⋆ α dt + ρ α K ⋆ α ∂u α ∂x -ρ α ∂R α ∂x α -ρ α θ α ∂s α ∂x -f ⋆ α = 0
For slow isothermal motions (θ 1 = θ 2 = θ 0 = const), we can rewrite these equations in the following approximate form:

M ⋆ α ≃ ρ α ∂ ∂x ∂W ∂ρ α -ρ α θ 0 ∂s α ∂x -f ⋆ α = 0 or M ⋆ α ≃ ρ α ∂µ α ∂x -f ⋆ α = 0
where µ α = ∂W ∂ρ α θ 0 s α is the chemical potential for the α th phase. Considering the difference M ⋆ 2 -M ⋆ 1 we obtain :

∇µ = f ⋆ 2 ρ 2 - f ⋆ 1 ρ 1 ≡ ρ f ⋆ ρ 1 ρ 2 (18)
where f ⋆ =f ⋆ 1 and µ = µ 2µ 1 . Equation ( 18) is the general form of Fick's law. So, Fick's law is not a linear phenomenological law but a direct consequence of equations of motion and the Stokes drag hypothesis which was previously noticed by Bowen with an other model [START_REF] Bowen | Theory of mixtures[END_REF].

Hyperbolicity of the two-fluid mixture model

The hyperbolicity of governing equations is very important because it implies wellposedness of the Cauchy problem. In mechanical case (when we neglect the equations of entropies) we are back to our previous study [START_REF] Gavrilyuk | Hyperbolic models of homogeneous two-fluid mixtures[END_REF][START_REF] Gavrilyuk | A new form of governing equations of fluids arising from Hamilton's principle[END_REF]. The only difference is the right-hand side algebraic terms f α due to the Stokes-like drag forces. Obviously, they do not affect the hyperbolicity analysis. The potential W is then a function of ρ 1 , ρ 2 and w = |w|. The Lagrangian is (with

Ω α = 0) L = 2 α=1 1 2 ρ α u 2 α -W (ρ 1 , ρ 2 , w)
We gave a sufficient condition of the hyperbolicity of system (4), [START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus[END_REF] in the multidimensional irrotational case where rot K α = 0. Recall the main results we obtained in this case: after a change of variables, system ((4),( 12)) takes the form

∂ ∂t ∂G ∂σ α -div ∂ ∂σ α 2 β=1 σ β j β = 0, (19)
∂ ∂t ∂G ∂j α -div ∂ ∂j α 2 β=1 σ β j β = 0. (20)
where

G(σ 1 , σ 2 , j 1 , j 2 ) L(ρ 1 , ρ 2 , j 1 , j 2 ) - 2 α=1 σ α ρ α , with σ α = ∂L ∂ρ α
The function G is a partial Legendre transformation of L (ρ 1 , ρ 2 , j 1 , j 2 ) with respect to the variables ρ α :

∂G ∂σ α = -ρ α , ∂G ∂j α = K * α ,
and the system (19), (20) can be rewritten in a symmetric form [START_REF] Godunov | An interesting class of quasilinear systems[END_REF][START_REF] Friedrichs | Systems of conservation laws with a convex extension[END_REF][START_REF] Boillat | Non-linear hyperbolic fields and waves in[END_REF]]

A ∂u ∂t + B i ∂u ∂x i = 0 , A = A * , B i = (B i ) * , i = 1, 2, 3 (21)
where

u * = (σ 1 , σ 2 , j * 1 , j * 2 ) , A = ∂ 2 G ∂u 2
and the matrices B i can be obtained from (19), (20). If A is positive definite, system (21) is hyperbolic. We proved in [START_REF] Gavrilyuk | Hyperbolic models of homogeneous two-fluid mixtures[END_REF][START_REF] Gavrilyuk | A new form of governing equations of fluids arising from Hamilton's principle[END_REF] that the conditions

∂ 2 W ∂w 2 < 0 , ∂ 2 W ∂ρ 2 1 > 0, ∂ 2 W ∂ρ 2 1 ∂ 2 W ∂ρ 2 2 - ∂ 2 W ∂ρ 1 ∂ρ 2 2 > 0 (22)
guarantee the hyperbolicity of our system for small relative velocity of phases. Due to relation (3), the inequalities (22) mean the convexity of the internal energy U that corresponds to a natural condition of stability. Finally, we established that the stability implies the hyperbolicity of the governing equations for small relative velocity w, provided that rot K α = 0 (condition always fulfilled for one-dimensional flows).

Appendix B.

The proof of the Gibbs identity is obtained by summing the following algebraic identities af : For dissipative terms, a.

f ⋆ 1 u 1 + f ⋆ 2 u 2 -f ⋆ 1 u 1 -f ⋆ 2 u 2 ≡ 0 For the external potentials Ω α , b. ∂ ∂t ρ α Ω α + div (ρ α Ω α u α ) -ρ α ∂Ω α ∂x u α -B α Ω α -ρ α ∂Ω α ∂t ≡ 0
For the velocity fields u α , c. ∂ ∂t

1 2 ρ α u 2 α + div ρ α u α (u 2 α - 1 2 u 2 α ) -B α u 2 α - 1

2 α=1 ρ α u α is the total momentum, ρ 2 α=1 ρ α and 2 α=1

 222

Acknowledgments

We are grateful to Professor Tommaso Ruggeri for helpful discussions.

Appendix A.

The definition of the Lagrangian coordinates X α implies ∂X α ∂t + ∂X α ∂x u α = 0. Taking the derivative with respect to ε α at zero, we obtain the following equation :

Equation ( 5) yields :

Using the Euler-Jacobi identity, δ(det F α ) = det F α (x, t) tr F -1 α δF α and δF α (x, t) = -F α (x, t) δF -1 (x, t) F α (x, t) , δF -1 (x, t) = ∂δX α ∂x , we deduce :

Substituting ( 24) into (23) we obtain :