Dan Betea
email: dan.betea@gmail.com

Jérémie Bouttier
email: jeremie.bouttier@ipht.fr

Harriet Walsh
email: harriet.walsh@ens-lyon.fr

Multicritical random partitions

come

Introduction

Background. Schur measures, introduced by Okounkov [START_REF] Okounkov | Infinite wedge and random partitions[END_REF], are probability measures on integer partitions λ of the form

P(λ) = Z -1 s λ [θ 1 , θ 2 , . . . ]s λ [θ 1 , θ 2 , . . . ].
(1.1)

Here, the θ i , θ i are numbers such that Z = exp ∑ i≥1 θ i θ i i is well-defined, and s λ [θ 1 , θ 2 , . . . ] is the Schur symmetric function indexed by λ and evaluated at the specialization sending the i-th power sum p i to the value θ i , for all i ≥ 1. A more concrete expression is given by the Jacobi-Trudi identity s λ [θ 1 , θ 2 , . . . ] = det i,j h λ i -i+j [θ 1 , θ 2 , . . . ], the entries of the determinant being given by the generating series ∑ k≥0 h k [θ 1 , θ 2 , . . . ]z k = exp ∑ i≥1 θ i z i i . See [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF][START_REF] Stanley | Enumerative combinatorics[END_REF] for background on symmetric functions and specializations.

Example 1. For θ 1 = θ 1 = θ, and all other θ i , θ i set to zero, we obtain the poissonized Plancherel measure P(λ) = e -θ 2 θ |λ| f λ |λ|! 2 , discussed below. Here, f λ denotes the number of standard Young tableaux of shape λ.

Example 2. For θ 1 = θ 1 , θ 2 = θ 2 , and all other θ i , θ i set to zero, we get

P(λ) = e -θ 2 1 -θ 2 2 /2 ∑ µ=1 a 1 2 a 2 ∑ ν=1 b 1 2 b 2 χ λ (µ)χ λ (ν)θ a 1 +b 1 1 θ a 2 +b 2 2 2 a 2 +b 2 a 1 !a 2 !b 1 !b 2 ! (1.2)
where χ λ is the irreducible character of the symmetric group S |λ| indexed by λ and µ, ν are two-column partitions with |λ| = |µ| = |ν| using the notation of [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF]Ch. 1].

Schur measures and their generalizations appear in several combinatorial, probabilistic, and statistical mechanical models of mathematical and physical interest. For a brief list, see [START_REF] Okounkov | Infinite wedge and random partitions[END_REF][START_REF] Okounkov | Symmetric functions and random partitions[END_REF][START_REF] Borodin | Lectures on integrable probability[END_REF]] and references therein. One notable instance is the resolution of Ulam's problem on longest increasing subsequence of random permutations [START_REF] Baik | On the distribution of the length of the longest increasing subsequence of random permutations[END_REF]. Namely, if we consider the poissonized Plancherel measure in Example 1, then the Baik-Deift-Johansson theorem [START_REF] Baik | On the distribution of the length of the longest increasing subsequence of random permutations[END_REF] states that the first part λ 1 satisfies

lim θ→∞ P λ 1 -2θ θ 1/3 < s = F TW (s) (1.3)
with F TW (s) the Tracy-Widom GUE distribution [START_REF] Tracy | Level-spacing distributions and the Airy kernel[END_REF] from random matrix theory. By Schensted's theorem [START_REF] Schensted | Longest Increasing and Decreasing Subsequences[END_REF], λ 1 is equal in distribution to the longest increasing subsequence of a random permutation on S N , the symmetric group of N letters, where N in our case is a Poisson random variable N ∼ Poisson(θ 2 ). See [START_REF] Romik | The Surprising Mathematics of Longest Increasing Subsequences[END_REF] for more on this topic.

Main contribution.

We consider multicritical Schur measures, having as their salient feature an "edge" behavior different from (1.3). More precisely, for every n ≥ 2, we construct Schur measures for which the 1/3 fluctuation exponent is replaced by 1/(2n + 1) (we recover the poissonized Plancherel measure for n = 1). The limiting distribution then becomes a "higher-order analogue" of the Tracy-Widom distribution. It is a τfunction of a higher-order differential equation of the Painlevé II hierarchy [START_REF] Cafasso | Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes[END_REF] in the same way the Tracy-Widom distribution is for the "classical" Painlevé II equation [START_REF] Tracy | Level-spacing distributions and the Airy kernel[END_REF]. Our inspiration comes from the work of Le Doussal, Majumdar and Schehr [START_REF] Le Doussal | Multicritical edge statistics for the momenta of fermions in nonharmonic traps[END_REF], who found the same limiting distributions in the momenta statistics of fermions in nonharmonic traps. They also noted a coincidental connection with the multicritical unitary matrix models of Periwal and Shevitz [START_REF] Periwal | Exactly solvable unitary matrix models: Multicritical potentials and correlations[END_REF], which involve the Painlevé II hierarchy in their double scaling limit.

Our multicritical Schur measures explain the origin of this connection. On the one hand, as observed by Okounkov [START_REF] Okounkov | Infinite wedge and random partitions[END_REF], Schur measures admit a convenient description in terms of free fermions. Simple scaling arguments show that they have the same asymptotic edge behavior as the models considered in [START_REF] Le Doussal | Multicritical edge statistics for the momenta of fermions in nonharmonic traps[END_REF]. On the other hand, through a chain of classical identities that we will review, the distribution of λ 1 in a Schur measure can be expressed as the partition function of a unitary matrix model. For multicritical measures, we recover exactly the models of [START_REF] Periwal | Exactly solvable unitary matrix models: Multicritical potentials and correlations[END_REF]. Let us point out that there is a known connection between Ulam's problem and the Gross-Witten unitary matrix model, see [START_REF] Johansson | The longest increasing subsequence in a random permutation and a unitary random matrix model[END_REF] and references therein. We comment on the relation with our work in the conclusion.

Outline. In Section 2, we define the multicritical Schur measures and state our main theorems (Theorems 1 and 2) concerning their edge behavior. We compute limit shapes in Section 3. Section 4 reviews the connection between Schur measures and unitary matrix integrals. The proof of Theorems 1 and 2 is sketched in Section 5. Finally, concluding remarks are gathered in Section 6.

This is an extended abstract of the paper [START_REF] Betea | Multicritical Schur measures with edge scaling governed by higher-order analogues of the Tracy-Widom distribution[END_REF]. For brevity, we do not include a discussion of the physical interpretation in terms of fermions here, but we note that they manifest themselves via the determinantal point processes used in Section 5.

Multicritical Schur measures and their edge behavior

A partition λ may be characterized by the set S(λ) = {λ ii + 1 2 |i ≥ 1} ⊂ Z + 1 2 , see Figure 1 below. Note that the largest element of S(λ) is λ 1 -1 2 , and the smallest element of its complement is -(λ) + 1 2 , where (λ) denotes the length of λ (number of nonzero parts).

When λ is a distributed according to a Schur measure (1.1), it was shown by Okounkov [START_REF] Okounkov | Infinite wedge and random partitions[END_REF] that S(λ) is a determinantal point process, whose kernel admits an explicit expression (given in Section 5) in terms of the θ i , θ i parameters.

The study of the edge behavior-the statistics of the largest element(s) of S(λ), or of the smallest element(s) of its complement-is most conveniently done via a saddlepoint analysis [START_REF] Okounkov | Symmetric functions and random partitions[END_REF]. For generic parameters θ i , θ i (and, in particular, for the poissonized Plancherel measure), it is found that the edge behavior is characterized by the coalescence of two saddle points, which implies that the "action" has a double critical point, also known as "monkey saddle", explaining the 1/3 fluctuation exponent. Multicritical Schur measures are obtained by tuning the parameters in such a way that the action has a critical point of higher order.

For simplicity, we restrict to the case where θ i = θ i -ensuring that the probability (1.1) is indeed nonnegative-and where the set {i : θ i = 0} is finite and of fixed cardinal n ≥ 1. By symmetry reasons, the edge critical point is always of even order, and by tuning the θ i we expect 2n to be the maximal possible order. This is indeed the case.

Theorem 1 ("odd-even multicritical measure"). Let P oe n,θ denote the Schur measure (1.1) where we set θ i = (-1) i+1 (n-1)!(n+1)! (n-i)!(n+i)! θ for i = 1, . . . , n, and θ i = 0 for i > n. Then, we have:

lim θ→∞ P oe n,θ λ 1 -bθ (θ/d) 1 2n+1 < s = F(2n + 1; s), lim θ→∞ P oe n,θ (λ) -bθ (θ/ d) 1 3 < s = F(3; s) (2.1) with b = n+1 n , d = ( 2n n-1 ), b = n+1 n (2n)!! (2n-1)!! -1 , d = 2 2n-2 n ( 2n n-1 ) , F(3; s) = F TW (s) the Tracy-
Widom GUE distribution and F(2n + 1, s) its higher-order analogue defined at (2.6) below.

As we see, we obtain a nongeneric exponent 1/(2n + 1) for the fluctuations of λ 1 , but we still have the generic exponent 1/3 for the fluctuations of (λ). It is actually possible to have a more symmetric situation if, rather than taking θ 1 , . . . , θ n nonzero, we take θ 1 , θ 3 , . . . , θ 2n-1 nonzero.

Theorem 2 ("odd multicritical measure"). Let P o n,θ denote the Schur measure (1.1) where we set θ 2i-1 = (-1) i+1 (n-1)!n! (2i-1)(n-i)!(n+i-1)! θ for i = 1, . . . , n, and all other θ i to zero. Then, P o n,θ is invariant under the conjugation of partitions λ → λ , and we have:

lim θ→∞ P o n,θ λ 1 -bθ (θ/d) 1 2n+1 < s = lim θ→∞ P o n,θ (λ) -bθ (θ/d) 1 2n+1 < s = F(2n + 1; s) (2.2) with b = 2 4n-1 n( 2n n ) 2 , d = (2n-1)!! (2n-2)!! ,
∑ i i k θ i = δ k,0 bθ 2 + (2n)!δ k,2n dθ, k = 0, 2, . . . , 2n -2, 2n. (2.3)
When n = 1, both measures reduce to the poissonized Plancherel measure, and we recover the convergence in distribution (1.3). As soon as n ≥ 2, they involve specializations which are not Schur positive, but the measures are nevertheless probability measures. The distributions F(2n + 1; s) appearing in Theorems 1 and 2 have been previously encountered in [START_REF] Le Doussal | Multicritical edge statistics for the momenta of fermions in nonharmonic traps[END_REF][START_REF] Cafasso | Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes[END_REF], and we now give their definition in a self-contained way. First, we recall that, if K is an integral operator with kernel K(x, y) acting on L 2 (X) (X is an open interval in what follows), it acts on functions f ∈ L 2 (X) via "matrix multiplication" (K f )(x) = X K(x, y) f (y)dy. For such operators which are trace-class one can define the Fredholm determinant of 1 -K (1 the identity operator) on L 2 (X) by

det(1 -K) L 2 (X) = ∑ m≥0 (-1) m m! X • • • X det 1≤i,j≤m [K(x i , x j )]dx 1 • • • dx m (2.4)
where there are m integrals in the m-th summand (and the case m = 0 yields 1). Consider first the following generalized (order 2n + 1) Airy function:

Ai 2n+1 (x) = iR+δ exp (-1) n-1 ζ 2n+1 2n + 1 -xζ dζ 2πi (2.5)
where δ > 0 is small and the contour is up-oriented. A(x) = (-1) n-1 xA(x) and that Ai 3 is the usual Airy Ai function. Then F(2n + 1; s) is the following Fredholm determinant

F(2n + 1; s) = det(1 -A 2n+1 ) L 2 (s,∞) (2.6)
where A 2n+1 is the higher order Airy kernel given by

A 2n+1 (x, y) = iR-δ dω 2πi iR+δ dζ 2πi exp (-1) n-1 ζ 2n+1 2n+1 -xζ exp (-1) n-1 ω 2n+1 2n+1 -yω 1 ζ -ω = ∞ 0 Ai 2n+1 (x + t)Ai 2n+1 (y + t)dt = (-1) n-1 ∑ 2n-1 i=0 (-1) i Ai (i) 2n+1 (x)Ai (2m-1-i) 2n+1 (y) x -y (2.7) (both contours above are up-oriented). Let us note that A 3 (x, y) = Ai 3 (x)Ai 3 (y)-Ai 3 (x)Ai 3 (y)
x-y is the usual Airy kernel and F(3; s) = F TW (s) the Tracy-Widom GUE distribution [START_REF] Tracy | Level-spacing distributions and the Airy kernel[END_REF].

In the x = y case, the third equality should be taken in the l'Hôpital limit sense.

Limit shapes

In this section we describe limit shapes for the multicritical P o n,θ -and P oe n,θ -distributed random partitions of Theorems 1 and 2. This section is descriptive; stating precise results is harder not least because of topological and analytical considerations beyond the scope and space afforded by this note; see [START_REF] Romik | The Surprising Mathematics of Longest Increasing Subsequences[END_REF] for precise statements in the n = 1 case. To begin, recall that the Young diagram of a partition can be represented in "Russian convention" as the graph of a piecewise linear function composed of slope ±1 segments, which we call its profile. See Figure 1.

If λ is distributed according to the measures P oe n,θ or P o n,θ of Theorems 1 and 2, and if we rescale by a factor 1/ √ θ in both directions, then the profile converges as θ → ∞ to the graph of a deterministic 1-Lipschitz function, denoted Ω. We have Ω = 1 -2ρ, where ρ is the limiting density profile of the set S(λ).

The limiting density profiles may be computed exactly. Let us denote them as follows:

ρ o/oe (u) = lim θ→∞ ∑ λ:θu∈S(λ)
P o/oe n,θ (λ).

ρ oe (u) = 1 π arccos 1 -1 2 2n n-1 1 n (b -u 1 n , u ∈ -b, b (3.2) 1 2 n = 1 n = 2 n = 3 n = 4 n = 5 Ω o (u) u ρ o (u) -2 -1 1 2
Figure 3: Limit shape and density profile of P o n,θ -distributed random partitions, for n = 1, . . . , 5. Notice the symmetry with respect to the vertical axis. and ρ oe (u) = 1, u < -b, ρ oe (u) = 0, u > b. The limit profile-depicted in Figure 2

-is Ω oe (u) = b + u -b [1 -2ρ(v)] dv.
A similar profile, for n=2, was recently observed in tight-binding fermions [START_REF] Bocini | Non-probabilistic fermionic limit shapes[END_REF].

In the o case and for b = 2 4n-1 n( 2n n )

2 we have:

ρ o (u) = χ(u) π , χ(u) 0 (2 sin φ) 2n-1 dφ = (-1) n+1 2n-1 n ) u, u ∈ [-b, b] (3.3) continued to ρ o (u) = 1, u < -b, ρ o (u) = 0, u > b.
The limit shape, symmetric under the vertical axis and shown in Figure 3, is

Ω o (u) = b + u -b [1 -2ρ(v)] dv.
Both Ω o and Ω oe are extensions of the Vershik-Kerov-Logan-Shepp limit curve-see e.g. [START_REF] Romik | The Surprising Mathematics of Longest Increasing Subsequences[END_REF]-to multicritical random partitions; indeed they become the former if n = 1.

Toeplitz determinants and unitary matrix integrals

In this section we review the connection between Schur measures and unitary matrix integrals, and we relate our multicritical measures to the integrals studied in [START_REF] Periwal | Exactly solvable unitary matrix models: Multicritical potentials and correlations[END_REF]. For simplicity, we assume that the parameters θ i , θ i of (1.1) are such that θ i = θ i for all i, and θ i = 0 for i large enough. We introduce the polynomials V and Ṽ defined by

V(z) = ∑ i≥1 θ i z i i , Ṽ(z + z -1 ) = V(z) + V(z -1 ). (4.1)
In physical parlance Ṽ, modulo a multiplicative constant, is often called the potential.

Example 5. If V(z) = θ 1 z + θ 2 2 z 2 + θ 3 z 3 we have Ṽ(x) = -θ 2 + (θ 1 -θ 3 )x + θ 2 2 x 2 + θ 3 3 x 3 .
Proposition 6. For λ distributed as above, we have:

e ∑ i θ 2 i /i • P( (λ) ≤ ) = det 1≤i,j≤ [ f j-i ] = E U∈U ( ) exp tr Ṽ(U + U * ) (4.2)
where the middle Toeplitz determinant has symbol ∑ k∈Z f k z k = exp Ṽ(z + z -1 ), and E U∈U ( ) is the expectation with respect to the Haar measure over the unitary group U ( ).

Proof. The left-hand side is equal to 2 which, by Gessel's identity [START_REF] Gessel | Symmetric functions and P-recursiveness[END_REF]Thm. 16], is equal to the middle Toeplitz determinant. The second equality is Heine's identity-see e.g. [START_REF] Diaconis | Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture[END_REF].

∑ (λ)≤ (s λ [θ 1 , θ 2 , . . .])
We also have the following similar identity regarding λ 1 .

Proposition 7. For λ distributed as above, we have:

e ∑ i θ 2 i /i • P(λ 1 ≤ ) = det 1≤i,j≤ [g j-i ] = E U∈U ( ) exp tr (-Ṽ(-U -U * )) (4.3)
where the middle Toeplitz determinant has symbol

∑ k∈Z g k z k = exp(-Ṽ(-z -z -1 )).
It is a straightforward consequence of Proposition 6 and the following: Lemma 8. If λ is distributed according to the Schur measure (1.1), then the conjugate partition λ is distributed according to the Schur measure of parameters θi = (-1) i-1 θ i , θi = (-1) i-1 θ i .

Proof. This follows from the relation s λ [θ 1 , θ 2 , . . .] = s λ [ θ1 , θ2 , . . .] that results from the classical involution ω on the algebra of symmetric function mapping the power sum p i to (-1) i-1 p i and the Schur function s λ to s λ .

Another consequence of the above lemma is the fact, mentioned in Theorem 2, that P o is invariant under conjugation.

When we specialize Proposition 7 to the multicritical measures P oe n,θ of Theorem 1, then the right-hand side of (4.3) matches, up to a change of variable U → -U, the multicritical unitary matrix integrals of Periwal and Shevitz [START_REF] Periwal | Exactly solvable unitary matrix models: Multicritical potentials and correlations[END_REF]. Indeed, the derivative V k (z) given on p. 737 of op. cit. is proportional to V (z) for k = n in our present notations, and the proportionality constant can be reabsorbed in θ. It seems that the case of an odd multicritical potential was not considered in their paper.

Sketch of proof

Let us sketch the proof of Theorems 1 and 2. We present the argument for the P o n,θ measure as it is slightly simpler, and make comments at the end on the difference with the P oe n,θ measure.

We use the fact, already mentioned at the beginning of Section 2, that S(λ) is a determinantal point process. This means that, fixing m and k 1 , . . . , k m ∈ Z + 1 2 , we have

P o ({k 1 , . . . , k m } ∈ S(λ)) = det 1≤i,j≤m K(k i , k j ) (5.1)
where, by [START_REF] Okounkov | Infinite wedge and random partitions[END_REF], the discrete ( 2 operator) kernel K equals (for some small > 0)

K(k, ) = 1 (2πi) 2 |w|=1-|z|=1+ e V(z)-V(z -1 ) e V(w)-V(w -1 ) dzdw z k+1/2 w -+1/2 (z -w) (5.2)
with V as in (4.1). Combinatorially, the above integral is just coefficient extraction: we look at the coefficient of z k /w in the respective generating series (since > 0, 1 z-w should be expanded as ∑ i≥0 w i z i+1 ). Moreover, inclusion-exclusion gives that the gap probability P o (λ 1 ≤ l) is equal to the discrete Fredholm determinant det(1 -K) 2 {l+1/2,l+3/2,... } .

In the multicritical regime we look for numbers β and θ 1 , θ 3 , . . . , θ 2n-1 satisfying

∑ i=1,3,...,2n-1 i k θ i = -δ k,0 β 2 , k = 0, 2, . . . , 2n -2 (5.3)
and solve for each of them in terms of θ 1 = θ. The solution is, up to an overall factor, given in the statement of the theorem:

θ 1 = θ, β = 2 4n-1 n ( 2n n ) 2 θ, θ 2i-1 = (-1) i+1 (n -1)!n! (2i -1)(n -i)!(n + i -1)!
θ, i = 2, 3, . . . , n. (5.4)

We call b = 2 4n-1 n -1 ( 2n n ) -2 = β/θ. The correlation kernel becomes

K(k, ) = 1 (2πi) 2 |w|=1-|z|=1+
e θ[S 0 (z)-S 0 (w)] dzdw z k+1/2 w -+1/2 (zw)

(5.5)

with S 0 (z) = n ∑ i=1 (-1) i+1 (n-1)!n! (2i-1)(n-i)!(n+i-1)! (z 2i-1 -z 1-2i ) 2i-1
. The equations (5.3) ensure that

(z∂ z ) i [S 0 (z) -b log z] z=1 = 0, 1 ≤ i ≤ 2n (5.6)
meaning z = 1 is a critical point of order 2n. The same is true for z = -1. Notice that the relation (5.6) is automatically satisfied for even i by the symmetry relation S 0 (z) + S 0 (z -1 ) = 0; the specific choice of coefficients ensures that it also holds for odd i between 1 and 2n -1.

We now analyze the scaling regime

θ → ∞, k = bθ + x(θd) 1 2n+1 , = bθ + y(θd) 1 2n+1
(5.7) with d = (2n-1)!! (2n-2)!! . In this regime, the integral (5.5) will be dominated by the vicinity of the critical point z = w = 1 (if we considered instead the regime k, ≈ -bθ, then the critical point z = w = -1 would dominate). We perform the change of variable

z = 1 + ζ(dθ -1 ) 1 2n+1 , w = 1 + ω(dθ -1 )
1 2n+1 (5.8) where ζ and ω are to be integrated over iR + δ and iRδ respectively. The quantity θS 0 (z)k ln z which appears exponentiated in the integral may be approximated as

S (2n+1) (1) (2n + 1)! ζ 2n+1 d -xζ + O 1 θ 1/(2n+1) = (-1) n+1 ζ 2n+1 2n + 1 -xζ + O 1 θ 1/(2n+1)
(5.9)

and we estimate -θS 0 (w) + ln w similarly. Plugging these estimates into (5.5), we recognize the double integral representation (2.7) for A 2n+1 (x, y). Modulo standard arguments (dominated convergence, tail bounds...) to justify the approximation, we obtain

(d -1 θ) 1 2n+1 K bθ + x(dθ) 1 2n+1 , bθ + y(dθ) 1 2n+1 → A 2n+1 (x, y) as θ → ∞.
(5.10)

To finish the proof, we show that K(k, ) has exponential decay which then shows the discrete Fredholm determinant P o n,θ (λ 1 ≤ l) = det(1 -K) 2 {l+1/2,l+3/2,... } converges to the continuous one det(1 -A 2n+1 ) L 2 (s,∞) = F(2n + 1; s) when l = bθ + s(dθ) 1 2n+1 . In the odd+even multicritical case, the analysis of the scaling regime (5.7) is the same. However, we lose symmetry under conjugation. This means that V(z) and hence the function S 0 (z) appearing in (2.7) are not odd functions of z anymore. At the point z = -1, which is relevant for studying the asymptotics of (λ), we find that S 0 (z) + b ln z has a generic double critical point which, by standard calculations, leads to the second equality in (2.1).

Concluding remarks

In this paper we have introduced Schur measures displaying the same multicritical edge behavior as the fermionic models considered by Le Doussal, Majumdar and Schehr [START_REF] Le Doussal | Multicritical edge statistics for the momenta of fermions in nonharmonic traps[END_REF]. We also computed limit shapes and explained how our measures map exactly to the Periwal-Shevitz multicritical unitary matrix models [START_REF] Periwal | Exactly solvable unitary matrix models: Multicritical potentials and correlations[END_REF]. This gives a combinatorial explanation to the coincidence noted in [START_REF] Le Doussal | Multicritical edge statistics for the momenta of fermions in nonharmonic traps[END_REF].

The approach of Periwal and Shevitz relies on the method of orthogonal polynomials. Through this approach, one obtains a different expression for the higher order distributions F(2n + 1; s) in terms of solutions of the Painlevé II hierarchy. It is shown in [START_REF] Cafasso | Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes[END_REF]-see also Appendix G of the arXiv version of [START_REF] Le Doussal | Multicritical edge statistics for the momenta of fermions in nonharmonic traps[END_REF]-that it is indeed equal to the Fredholm determinant (2.6). Multicriticality of a similar flavor was also observed at the spectrum edge of Hermitian random matrix ensembles by Claeys, Its and Krasovsky [START_REF] Claeys | Higher-order analogues of the Tracy-Widom distribution and the Painlevé II hierarchy[END_REF].

For n = 1, our measures reduce to the poissonized Plancherel measure, while on the unitary random matrix side we obtain a model first studied by Gross and Witten, see e.g. [START_REF] Johansson | The longest increasing subsequence in a random permutation and a unitary random matrix model[END_REF] and references therein. Our work shows that the connection observed by Johansson in [START_REF] Johansson | The longest increasing subsequence in a random permutation and a unitary random matrix model[END_REF] extends to higher orders n ≥ 2 of multicriticality, even though the formulation in terms of longest increasing subsequences seems more elusive.

Main result of this note (odd case). Let us summarize, in one place, the results of Theorem 2 on one hand and of Propositions 6 and 7 on the other. 2Fix n ≥ 1 and let λ be P o n,θ -distributed (1.1) with θ 2i-1 = (-1) i+1 (n-1)!n! (2i-1)(n-i)!(n+i-1)! θ for i = 1, . . . , n and θ > 0. Let b = 2 4n-1 n( 2n n )

2 ; d = (2n-1)!! (2n-2)!! ; V(z) = ∑ n i=1 θ 2i-1 z 2i-1 2i-1 ; Ṽ(z + z -1 ) = V(z) + V(z -1 ); and ∑ k∈Z f k z k = exp[V(z) + V(z -1 )].

Example 4 .

 4 For n = 2, P oe n,θ has the form given in Example 2 with θ 1 = θ, θ 2 = -θ 4 , while P o n,θ has θ 1 = θ, θ 3 = -θ 9 as nonzero parameters.

Figure 1 : 5 ρFigure 2 :

 152 Figure 1: The profile (thick blue) and the set S(λ) (dots) for λ = (4, 2, 1).

(3. 1 )

 1 In the oe case we have, with b = n+1 n , b = n+1 n (2n)!! (2n-1)!! -1 :

  Then the quantitiesP o n,θ (λ 1 ≤ ), P o n,θ ( (λ) ≤ ), det 1≤i,j≤ [ f j-i ]are all equal, and equal to the Fredholm determinant det(1 -K) (K as in (5.2)) on { + 1/2, + 3/2, . . . }. Asymptotically, they all equal the distribution F(2n + 1; s) in (2.6) when = bθ + s(θd)

	e ∑ n i=1 θ 2 2i-1 /(2i-1)	e ∑ n i=1 θ 2 2i-1 /(2i-1)	(6.1)

, E U∈U ( ) exp tr Ṽ(U + U * ) 1 2n+1 and θ → ∞.

Comparing with [10, Eq. (5)], we chose different integration conventions for the same function. Their expression is different for n even and comes from the change of variables z = -ζ. Otherwise said, the contours of [10, Eq. (5)] are such that (z

2n+1 ) < 0 whereas ours have ((-1) n-1 ζ 2n+1 ) < 0.

An analogous result could be stated for Theorem 1; we omit it for brevity.

Acknowledgements

We would like to thank S. Bocini, G. Chapuy, T. Claeys, A. Kuijlaars, P. Le Doussal, A. Occelli, G. Schehr and J.-M. Stéphan for support and conversations regarding this project.

supported by FWO Flanders project EOS 30889451. . J.B. is partially supported by the project ANR-18-CE40-0033 "Dimers". . H.W. is supported by the grant ERC-2016-STG 716083 "CombiTop".


Claude Godrèche

Non stationarity of renewal processes with power-law tails

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Non stationarity of renewal processes with power-law tails

Introduction

Consider the class of models defined as the succession of a fluctuating number of time, or space, independent and identically distributed (iid) intervals conditioned by the value of their sum. These intervals describe for example the lapses of time between some recurrent events, or spin domains for particular one-dimensional magnetic systems. The correlation function for such tied-down renewal processes has been recently investigated in two references [1,2], with different methods and different results. The aim of this note is to clarify this issue.

Reference [1] revisits the tidsi model, a truncated version of a microscopic onedimensional spin model with long-range interactions, dubbed the inverse distance squared Ising (idsi) model [3,4,5]. This tidsi model, which has been investigated in a series of papers in recent years [6,7,8], is made of a fluctuating number of domains, filling up the total size L of the system. A configuration is thus entirely specified by the number and sizes of these domains. The Boltzmann weight associated to the realisation { 1 , 2 , . . . , n , n} of such a configuration reads [1] p( 1 , 2 , . . . , n , n; L) =

y n g( 1 ) . . . g( n )δ n i=1 i , L Z(L) , (1.1)
where the denominator

Z(L) = n≥1 1... n y n g( 1 ) . . . g( n )δ n i=1 i , L (1.2)
ensures the normalisation and δ(•, •) is the Kronecker delta. In (1.1), y denotes the fugacity and g( ) is given by

g( ) = 1 1+θ , (1.3)
where = 1, 2, . . ., and the tail index θ is positive. In [1,6,7] the phase diagram is analysed according to the values of the fugacity y and the index θ, which are the two parameters of the model. For y = y c , such that

y c = 1 g( ) = 1 ζ(1 + θ) ,
where ζ(•) is the Riemann zeta function, the system is critical, separating a paramagnetic phase from a ferromagnetic one.

. . . . . . . . .

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y j j q b 1 f s A j n p w z R z r K S T 9 S n V y m 2 c a 3 5 d y / A 8 O l j z o = < / l a t e x i t > Figure 1. A tied-down renewal process is defined as the succession of a fluctuating number of time (or space) iid intervals between points (which can be events, domain walls, . . . .) conditioned to filling up the total size L of the system. The quantity of interest pm(x, r, L) is the probability that m points fall between the two arbitrary positions x and x + r. The correlation function C(x, r, L) is defined in (3.1).

The main result of [1] is the expression of the critical correlation function between two spins at x and x + r, for θ < 1, in the regime of short separations (1 r L), for a system with periodic boundary conditions. As stated in [1], since the system is translation invariant, this expression is independent of the position x of the first spin and only depends on the separation r. The expression derived in [1] is

C(r, L) ≈ 1 - 1 θ r L 1-θ . (1.4)
The same question, in more generality, was posed in [2] and solved by exact methods, for open boundary conditions (see figure 1). Restricting to the regime of short separations between x and x + r (1 r x ∼ L), the following result holds, when θ < 1,

C(x, r, L) ≈ 1 - sin πθ π(1 -θ) x L 1 - x L θ-1 r L 1-θ , (1.5)
giving clear evidence for the role of the position x, that is, for the non stationarity of the process. The question therefore naturally arises of understanding the relationship, if any, between the two expressions (1.4) and (1.5). Averaging C(x, r, L) in (1.5) on x yields

C(r, L) ≈ 1 - sin πθ π(1 -θ) Γ(θ) 2 Γ(2θ) r L 1-θ , (1.6)
to be compared to (1.4). As will be explained later, averaging (1.5) on x is the correct way to take into account periodic boundary conditions, in case these are imposed on the system. As we shall see, the central point, beyond the existence of a discrepancy between the two expressions (1.4) and (1.6), is the difference in nature of the methods used to derive them. While in [1] invariance by translation is invoked at the very start in order to justify the use of a stationary formalism where the dependence in x is discarded, in contrast, in [2] the dependence in x is essential when deriving (1.5), before any possible tracing of this expression upon x is performed, resulting in (1.6).

We shall explain below why the method used in [1] in order to derive (1.4)-the socalled independent interval approximation-is not exact, as claimed in this reference, and can only predict the scaling behaviour C(r, L) -1 ∼ (r/L) 1-θ .

The class of tied-down renewal processes

Before discussing this issue, we first show that the tidsi model defined by (1.1), (1.2), (1.3), when taken at criticality, belongs to the class of tied-down renewal processes considered in [2,9] (independently of any considerations on the boundary conditions).

For this class of processes, a configuration is again specified by a fluctuating number of intervals (e.g., domains) τ 1 , τ 2 , . . . which are iid random variables with common probability distribution f ( ) = Prob(τ = ).

(

A realisation { 1 , 2 , . . . , n , n} of such a configuration, where the number of intervals takes the value n and the random variables τ i take the values i , has weight

p( 1 , 2 , . . . , n , n; L) = f ( 1 ) . . . f ( n )δ n i=1 i , L Z(L) , (2.2)
where

Z(L) = n≥1 1... n f ( 1 ) . . . f ( n )δ n i=1 i , L . (2.3)
These processes are renewal processes because the intervals are iid random variables, and they are tied down because these intervals are conditioned to sum up to a given value L, generalizing the tied-down random walk, starting from the origin and conditioned to end at the origin at a given time [10,9] (see section 4). (For the tieddown random walk, intervals are temporal, while in the rest of this note all intervals are spatial.) The definitions (2.1), (2.2) and (2.3) easily generalize to the case where intervals are continuous random variables.

It is now evident that, at criticality, i.e., if y = y c , (1.1) reduces to (2.2). Indeed, the product y c g( ) is normalised and can be identified with the probability distribution f ( ), where in this specific case we have

f ( ) = 1 ζ(1 + θ) 1 1+θ .
(2.4)

Therefore, as already stated in [2,9], the class of tied-down renewal processes defined by (2.1), (2.2) and (2.3) above, encompasses as a particular case the tidsi model considered in [1,6,7,8], at criticality, with the consequence that all results found in [2,9] can be immediately transposed to the latter. This applies, in the present case, to (1.5) and (1.6).

The fact, demonstrated above, that, at criticality, the tidsi model belongs to the class of tied-down renewal processes was contradicted in [1]. As emphasised in this reference: 'A joint distribution similar to equation (10) was studied in the context of the tied-down renewal process [10], with the important difference that in the latter case the fugacity y was taken to be exactly 1. As we will see later, in our tidsi model where y can vary, there is a mixed-order phase transition at a critical value y = y c (which need not be 1)' ‡. This assertion does not hold true. The fact that f ( ) in (2.2) is a normalised probability distribution was missed in [1]. This corresponds to taking y = y c in (1.1), and not y = 1 as incorrectly stated in [1].

Translation invariance and non stationarity

We now analyse the nature of the methods leading respectively to (1.4) and (1.6). In section 3.1, we first define the correlation function C(x, r, L) for an open system, we then explain the consequences of imposing periodic boundary conditions. The intermediate section 3.2 gives some preparatory material on renewal theory needed for section 3.3, where we give a concise overview of the method used in [1] in order to obtain (1.4), then conclude on its validity.

Correlation function for a finite system

Let us start with the open system (0, L), depicted in figure 1. The correlation function is defined as

C(x, r, L) = (-1) N (x,x+r) = m≥0 (-1) m p m (x, r, L), (3.1)
where N (x, x+r), the number of points (events, domain walls, . . . ) comprised between two arbitrary positions x and x + r, has probability distribution denoted as

p m (x, r, L) = Prob(N (x, x + r) = m).
As shown in [2], for a broad distribution f ( ) with tail index θ < 1, if x, r, L are all simultaneously large, the correlation function C(x, r, L) is dominated by p 0 (x, r, L), the probability that the interval (x, x + r) does not contain any point. From now on we shall focus on this quantity. We assume that the first interval begins at site 0.

If periodic boundary conditions are imposed on the system, any configuration with weight (2.2) can be represented by (a fluctuating number of) intervals on a circle, where the two points 0 and L are identified. Since the position of this point on the circle is arbitrary, p 0 (x, r, L) should be averaged on x. Likewise (1.6) ensues from (1.5).

Another point should be checked beforehand. In the case of the tidsi model, intervals represent spin domains. Therefore if the system is periodic, the number of domains between 0 and L should be even. It turns out that resuming the derivations made in [2] with n even does not change the expression (1.5), hence, after averaging on x, does not change (1.6) either.

Note finally that if L is sent to infinity, the conditioning disappears. So p 0 (x, r, L → ∞) = p 0 (x, r), which is the probability that the interval (x, x + r) does not contain any point for an ordinary renewal process.

Elements of renewal theory

It is useful to recall some results of renewal theory before coming to the next section. Consider first an ordinary renewal process, i.e., without tied-down conditioning. The following dichotomy holds.

(i) If the distribution of intervals f ( ) has a finite first moment

τ = ∞ 0 d f ( ), (3.2)
e.g., when f ( ) is narrow, with finite moments, or broad with a power-law tail of index θ > 1, then the process reaches a stationary regime, where p 0 (x, r) → p 0,stat (r), in the limit x → ∞ (see e.g., [11]). In Laplace space

L r p 0,stat (r) = p0,stat (s) = 1 s - 1 -f (s) τ s 2 . (3.3) (ii)
The situation is different when f ( ) is a broad distribution with tail index θ < 1.

Then τ is no longer finite and p 0 (x, r) keeps a dependence in x even at large values of this variable. The process is genuinely non stationary [11,12]. Now, let us come back to tied-down renewal processes (conditioned by the sum of intervals equal to L). The system is now finite. Let us denote by τ the first moment of the marginal distribution f ( |L) of a single interval, obtained by tracing the full distribution p( 1 , 2 • • • , n , n; L) on all i but one. Alternatively, τ is equal to the product of L by the mean inverse number of intervals in (0, L).

(i) If τ is finite, then τ is asymptotically equal to τ at large L.

(ii) If τ is infinite, then [9] τ ≈ πc sin πθ L 1-θ , (3.4)
where c is the tail coefficient of the distribution f ( ) ≈ c/ 1+θ . For the tidsi model c is equal to y c (see (2.4)). The prediction (3.4) is checked below on the example of the tied-down random walk, which corresponds to taking θ = 1/2 (see figure 2).

Derivation of (1.4) in the independent interval approximation

We now come to the derivation of (1.4) given in [1] when θ < 1. It proceeds as follows.

(i) The system is taken infinite, stationary and the distribution of sizes of domains f ( ) is assumed to have a finite first moment τ . Therefore the stationary probability that an interval of size r does not contain any point is, in Laplace space, given by (3.3), which expresses p0,stat (s) in terms of f (s) and τ .

(ii) In this formalism, f ( ) is thought as being f ( |L), but nevertheless approximated by f ( ), if

τ ≈ y c 1 -θ L 1-θ . (3.5)
(iii) The expansion of the Laplace transform of f ( ) with respect to , f (s) ≈ 1 + y c Γ(-θ)s θ , (3.6) is carried into the expression (3.3) of p0,stat (s). This gives p0,stat (s) ≈

1 s + y c Γ(-θ) τ s θ-2 ,
where again τ is replaced by τ given by (3.5), yielding finally [1] p 0,stat (r) ≈ 1 -

1 θ r L 1-θ , (3.7)
which is (1.4), since the correlation function is dominated by p 0,stat (r). This heuristic line of reasoning-the so-called independent interval approximation (IIA)-would predict the correct asymptotic stationary regime if f ( ) had a finite first moment. But, in the present situation of a broad distribution with index θ < 1, it does not lead to exact results because it does not treat the genuine non stationarity of the process adequately. In particular the various assumptions on which it is based do not form a coherent whole, as we now comment.

On the one hand, using (3.3) requires τ to be finite. On the other hand, using the expansion (3.6) only makes sense if τ is infinite. Taking τ in place of τ does not circumvent this contradiction because the finiteness of τ is an intrinsic property of the distribution f ( ), independent of the size of the system-i.e., holding even for an infinite system. In contrast, τ is a property of the finite system and depends on L.

In conclusion, the IIA used in [1] only predicts the power 1 -θ of r/L in (1.4) or (3.7). The correctness of the prefactor 1/θ in front of (r/L) 1-θ is illusory. Actually, this prefactor should be modified because the expression (3.5) for τ is inaccurate and should be replaced by (3.4). By so doing, the prefactor 1/θ in (1.4) or (3.7) is changed to sin πθ/[πθ(1 -θ)], which does not give the correct expression for the correlation function either. Another shortfall of the method is that it is unable to predict the non-stationary correlation function C(x, r, L) for an open system §.

The tied-down random walk as a benchmark

We can illustrate and validate the results seen above, such as (1.5), (1.6) for the correlation function and (3.4) for τ , on the case of tied-down processes with θ = 1/2.

The tied-down random walk is the simplest representative of this class, with the advantage of being intuitively understandable [10,9]. The walk (with steps ±1) is said to be tied down because it is conditioned to return to the origin at a given time, L. The weight of a configuration of the walk is given by (2.2), (2.3). The distribution f ( ) in (2.2) is the probability that the first return to zero of the random walk starting at the origin occurs at time . Note that both and L are even . Asymptotically, for large, this distribution has a power-law tail with index θ equal to 1/2, and tail [13]. This result was questioned by the present Author since it contradicted (1.5) (in particular by missing its x dependence) [14]. In the published version [1], the boundary conditions were changed to periodic, the derivation of (1.4) was left unchanged.

parameter c = 1/ √ 2π, f ( ) ≈ 1 √ 2π 3/2 . ( 4
This induces an additional factor 2 in the denominator of the continuum expression (4.1). The denominator Z(L) in (2.3) is the probability of return of the walk to the origin at time L. It has the explicit expression

Z(L) = 1 2 L L L/2 ≈ 2 πL .
It can be shown that [9]

τ = 1 Z(L) , (4.2)
hence, asymptotically,

τ ≈ πL 2 , (4.3)
which is precisely (3.4) with c = 1/ √ 2π and θ = 1/2. Furthermore, for the Brownian bridge, which is the continuum limit of the tieddown random walk, the analytical form of the full correlation function (of the sign of the position of the walker) is known [ This compact expression encompasses all regimes of interest. In particular, in the regime of short separations (1 r x ∼ L), it reads

2] C(x, r, L) = 1 - 2 π arccos x(L -x -r) (x + r)(L -x) . ( 4
C(x, r, L) ≈ 1 - 2 π x L 1 - x L -1/2 r L 1/2 , (4.5)
which is (1.5), with θ = 1/2. Averaging this expression on x yields

C(r, L) ≈ 1 -2 r L 1/2 , (4.6)
which is (1.6) with θ = 1/2. Actually, the average of (4.4) on x has the exact form

C(r, L) = 1 - r L 1/2 2 = 1 -2 r L 1/2 + r L , (4.7)
where the first two terms in the right side reproduce (4.6), for r/L small, and the last term gives the explicit correction to this expression. Finally, we can complement our study by performing numerical simulations of the tied-down random walk.

Figure 2 depicts a comparison between the measured mean interval τ , its theoretical prediction (4.2), and its asymptotic expression (4.3), showing excellent agreement. We also measured p 0 (x, r, L), for 0 < x < L-r, on 10 5 samples of the tieddown random walk, with L = 20 000 and r = 2000. For x > Lr, p 0 (x, r, L) vanishes because there is always the point at L inside the interval. The result is depicted in figure 3 together with the analytical expression (4.4) of the correlation function for the Brownian bridge. Comparison between the two sets of data demonstrates the adequacy between the two processes, and confirms also that p 0 (x, r, L) accounts faithfully for the full correlation function.

Conclusion

Tied-down renewal processes generated by a power-law distribution of intervals with tail index θ < 1, of which the critical tidsi model (a spin model with long-range order) is a particular example, are genuinely non stationary. This holds for the infinite system as well as for the finite one, when the intervals are conditioned by the value of their sum. For such processes, the independent interval approximation put forward in [1] for the computation of the spin-spin correlation function is not the proper approach, even if periodic boundary conditions are on the system, entailing translation invariance. In the case at hand, translation invariance only implies tracing the nonstationary correlation function C(x, r, L) on x.

. 1 )

 1 § The first version of [1] already predicted (1.4) for the critical correlation function of the same model with open boundary conditions

Figure 2 .

 2 Figure 2. Mean time interval τ between two successive returns to the origin of a tied-down random walk, against √ L, where L is the time of ultimate return to the origin, L = 2, 4, . . . , 40 (see text). Comparison of the exact expression (4.2) (green dots), simulation (blue empty circles), and asymptotic prediction (4.3) (continuous line).

Figure 3 .

 3 Figure 3. Probability for an interval of size r = 2000 to be empty as a function of x/L, for a tied-down random walk of L = 20 000 steps, measured on 10 5 samples (red dots). This probability is compared to the analytical expression of the correlation function of the sign of the position for Brownian bridge (continuous curve).

‡ In the first sentence quoted above, equation(10) refers to (2.2), and reference[10] to[2].


Eric Bertin

Emergence of simple characteristics for heterogeneous complex social agents

Keywords: Sociophysics, statistical physics, heterogeneity, phase transition

Models of interacting social agents often represent agents as very simple entities having a small number of degrees of freedom, as exemplified by binary opinion models for instance. Understanding how such simple individual characteristics may emerge from potentially much more complex agents is thus a natural question. It has been proposed recently in [E. Bertin, P. Jensen, C. R. Phys. 20, 329 (2019)] that some types of interactions among agents with many internal degrees of freedom may lead to a 'simplification' of agents, which are then effectively described by a small number of internal degrees of freedom. Here, we generalize the model to account for agents intrinsic heterogeneity. We find two different simplification regimes, one dominated by interactions, where agents become simple and identical as in the homogeneous model, and one where agents remain strongly heterogeneous although effectively having simple characteristics.

Introduction

When modeling complex systems, statistical physicists often posit that the interacting entities they consider have simple individual properties, and that the possibly more complex behavior observed at a collective level, when considering many such simple entities, is simply the result of interactions between entities [START_REF] Castellano | Statistical physics of social dynamics[END_REF][START_REF]Complex systems[END_REF][START_REF] Barrat | Dynamical Processes on Complex Networks[END_REF][START_REF] Bouchaud | Crises and collective socio-economic phenomena: simple models and challenges[END_REF]. The emergence of macroscopically different properties from simpler interacting entities, often through a collective symmetry breaking mechanism, has been emphasized long ago by P. W. Anderson in his seminal paper "More is Different" [START_REF] Anderson | More is different[END_REF] as one of the key mechanisms at play to account for the wealth of different objects or behaviors found in the real world. In this paper, Anderson actually presents the statistical physics approach as a generic and somewhat iterative method, to deal with the emerging complexity in a multilevel way, the outcome of a given level of analysis being the building blocks (i.e., the interacting entities) of the next level. The key role played by symmetry breaking phenomena in the emerging complexity at collective scale has been widely acknowledged in many different contexts (see, e.g., [START_REF] Chaikin | Principles of condensed matter physics[END_REF][START_REF] Bellac | Quantum and statistical field theory[END_REF][START_REF]Complex systems[END_REF][START_REF] Marchetti | Hydrodynamics of soft active matter[END_REF] among many others). Interestingly, a perhaps less explicit suggestion of Anderson's paper is also to consider statistical physics models where the interacting entities are not simple objects with a handful of characteristics, but are already rather complex objects with many internal degrees of freedom. Although it has been formulated almost fifty years ago, this suggestion looks very timely by now.

In the last decades, statistical physics has gone beyond the equilibrium paradigm based on molecular entities, and has indeed started to consider assemblies of macroscopic and potentially complex objects like grains of sand [START_REF] De Gennes | Granular matter: a tentative view[END_REF][START_REF] Puglisi | Transport and fluctuations in granular fluids[END_REF], or active particles modeling for instance some types of bacteria, or self-phoretic colloids [START_REF] Marchetti | Hydrodynamics of soft active matter[END_REF]. However, in these examples, macroscopic particles may in practice still be considered as simple particles, as their many internal degrees of freedom can be subsumed into a small number of effective parameters encoding their macroscopic, non-equilibrium character, like dissipation coefficients [START_REF] De Gennes | Granular matter: a tentative view[END_REF][START_REF] Puglisi | Transport and fluctuations in granular fluids[END_REF] or self-propulsion forces [START_REF] Marchetti | Hydrodynamics of soft active matter[END_REF]. To find genuine examples of assemblies of complex entities in a theoretical context, one may rather turn to the field of population dynamics and evolution in theoretical biology, where for instance large populations of individuals characterized by a complicated genome evolve under some evolutionary rules [START_REF] Drossel | Biological evolution and statistical physics[END_REF][START_REF] Sella | The application of statistical physics to evolutionary biology[END_REF].

When considering models of social systems, it would be necessary at first sight to take into account the intrinsic complexity of human beings [START_REF] Castellano | Statistical physics of social dynamics[END_REF]. However, this complexity is too extreme to be captured by any type of statistical physics models, so that statistical physicists have often considered very simple models of social agents, retaining a small number of characteristics, for instance a binary [START_REF] Sznajd-Weron | Opinion evolution in closed community[END_REF] or continuous [START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF] opinion. A natural question is then to understand how more complex agents could in some situations reduce their intrinsic complexity to effectively appear as simple. This question can already be addressed within the framework of statistical physics, because there is no need to model the full complexity of human beings to address at least some aspects of this issue. A simple tentative answer has been given in [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF], by illustrating on a toy model how the simplification of agents with many internal degrees of freedom may result from interactions among agents. This point of view is qualitatively consistent with the idea advocated by some sociologists that 'the whole is less than the parts' [START_REF] Latour | The whole is always smaller than its parts, a digital test of gabriel tardes monads[END_REF][START_REF] Jensen | The politics of physicists' social models[END_REF], in the sense that, roughly speaking, human beings may leave aside part of their complexity to build a group. It has been argued in [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF] that this point of view can be reconciled with the seemingly antagonist viewpoint of statistical physicists according to which 'the whole is more than the sum of the part', due to collective phenomena and symmetry breakings.

Yet, the model introduced in [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF] considered an assembly of identical agents, while agents heterogeneity may be expected to be an important characteristics of human beings. In this note, we extend the model of [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF] by including heterogeneity between agents. We show, using methods inspired by the physics of glasses, that two different types of agents simplification can occur in this model, one driven by interactions as in [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF], and the other one driven by heterogeneity.

Model

We introduce a model of complex agents generalizing the one introduced in [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF] by now considering heterogeneous agents. The model is composed of N interacting agents having an internal state described by a configuration C i ∈ {1, ..., H}, with i = 1, . . . , N and where the number H of configurations is large. We write for later convenience H in the form H = n M 0 , where n 0 is a fixed integer number, and M is assumed to be large. This is typically the case if the configuration C i is composed of M degrees of freedom, each of which taking n 0 possible values. Each agent is endowed with a characteristic that can be either present or absent depending on the configuration C i . Intuitively, this characteristic could be a preference for a specific kind of music or tempo for the members of a vocal ensemble [START_REF] Jensen | The politics of physicists' social models[END_REF] for instance, or more generally be related to a binary opinion [START_REF] Sznajd-Weron | Opinion evolution in closed community[END_REF]. This feature is conveniently encoded by a variable S i (C i ) ∈ {0, 1}: the characteristic is present when S i (C i ) = 1, and absent when S i (C i ) = 0. The general idea is that the characteristic would be present only in a small number of internal states, so that it would typically remain unobserved except if there is a strong probability bias towards the few configurations for which S i (C i ) = 1. To implement this idea in practice in the model, we assume that the characteristic is present in a single configuration, that we label C i = 1.

We now need to define the dynamics of agents. Following standard practice in the modeling of social agents [START_REF] Phan | Social interactions in economic theory: an insight from statistical mechanics[END_REF], we assume that the dynamics is driven by an individual utility function u i = u i (C i |C j =i ) that accounts both for individual preferences and for interactions with agents. An agent i stochastically changes configurations according to the following rule. Given the current configuration C i , the new configuration C i is randomly chosen with a probability rate given by the logit rule,

W (C |C) = 1 1 + e -∆u i /T
(1)

with ∆u i = u i (C i ) -u i (C i
) the variation of utility generated by the change of configuration (note that configurations C j of the other agents j = i are kept fixed). The parameter T plays a role similar to temperature in statistical physics, and characterizes the degree of stochasticity in the decision rule.

Our goal is to model heterogeneous agents that interact through their characteristic S i (their internal state is otherwise invisible to other agents). With this aim in mind, we choose the following form of the utility function of agent i,

u i (C i |C j =i ) = U i (C i ) + K N j( =i) S i (C i )S j (C j ) , (2)
where U i (C i ) is the intrinsic (or idiosyncratic) utility of configuration C i for agent i, and K is the coupling constant characterizing the interaction with the other agents. Note the 1/N scaling of the interaction term, typical of fully connected models where all particles or agents interact with each other in a similar way. This interaction terms was already present in the homogeneous model of Ref. [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF]. We then model the heterogeneity of agents as a quenched randomness of the intrinsic utilities (which were absent from the model of [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF]). More precisely, for all i = 1, . . . , N and

C i = 1, . . . , H, the intrinsic utility U i (C i ) is randomly drawn from a Gaussian distribution ρ(U ), ρ(U ) = 1 √ πM J 2 e -U 2 /M J 2 , (3)
where we recall that M is defined by

H = n M 0 . The utilities U i (C i ) do not change in time.
The utility variation ∆u i can be reformutated as the variation ∆E = ∆u i of a global observable E that plays a role similar to the energy in physics (up to a change of sign). Note that contrary to u i , the pseudo-energy E does not depend on the agent i.

Here, the function E takes the form

E(C 1 , . . . , C N ) = N i=1 U i (C i ) + K 2N i,j(i =j) S i (C i )S j (C j ) . (4)
The quantity E is thus different from the total utility i u i , due to the factor 1 2 in the interaction term. Note also that the present model shares similarities with the Random Energy Model [START_REF] Derrida | Random-energy model: Limit of a family of disordered models[END_REF][START_REF] Derrida | Random-energy model: An exactly solvable model of disordered systems[END_REF][START_REF] Bouchaud | Universality classes for extreme-value statistics[END_REF], as well as with the Ising model [START_REF] Bellac | Quantum and statistical field theory[END_REF], the Potts model [START_REF] Wu | The potts model[END_REF] or the Blume-Emery-Griffiths spin-1 model [START_REF] Blume | Ising model for the λ transition and phase separation in he 3 -he 4[END_REF]. However, it also exhibits important differences with each of these models.

Given the property ∆u i = ∆E, the dynamics defined by the transition rate Eq. ( 1) obeys the detailed balance property in terms of the equilibrium distribution

P (C 1 , . . . , C N ) = 1 Z e βE(C 1 ,...,C N ) , (5)
with β ≡ 1/T , and where Z is a normalization constant. Now that we determined the equilibrium distribution of the model, our goal is to investigate its phase diagram to assess the effect of the competition between agents heterogeneity (due to their quenched intrinsic utility) and collective effects that could arise from interactions. As recalled in the introduction, the homogeneous version of the model, studied in [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF], exhibits a transition driven by interactions between a phase where agents essentially visit all their internal states and thus have no strongly preferred configurations, and an ordered state where all agents 'standardize' in the same configuration C 1 such that S i = 1 (i.e., the characteristic of the agent becomes visible). In the present generalization of the model, we wish to explore whether the standardized state survives the heterogeneity of agents intrinsic utility. Indeed, the internal state C 1 such that S i = 1 may have a lower intrinsic utility U i (C 1 ) than other more favored configurations of agent i, and this may prevent a common standardization of all agents in the same state.

To investigate this issue, we introduce the order parameter q defined as

q = 1 N N i=1 S i . (6)
If the number N of agents is large, the pseudo-energy E can be expressed in terms of the order parameter q as

E = N i=1 U i + 1 2 N Kq 2 . ( 7
)
We now wish to determine the distribution P (q) of the order parameter q. This distribution is obtained by summing the joint distribution P (C 1 , . . . , C N ) over all configurations (C 1 , . . . , C N ) having a given value of q, namely

P (q) = C 1 ,...,C N P (C 1 , . . . , C N ) δ q(C 1 , . . . , C N ) -q (8)
with δ the Kronecker delta. For a given value of q, there are qN agents in configuration C i = 1 and (1 -q)N agents in any of the other n M 0 -1 configurations. We denote by S q,N a subset with qN elements of the set [1, N ]. With these notations, P (q) can be written as

P (q) = 1 Z S q,N   i∈S q,N e βU i (1)     i / ∈S q,N n M 0 C i =2 e βU i (C i )   e 1 2 βN Kq 2 , (9)
where in the second product, the index i is implicitly restricted to the interval [1, N ]. The distribution P (q) defined in Eq. ( 8) actually depends on the specific realization of the random utilities U i (C i ), so that P (q) should in principle be eventually averaged over these random utilities. However, to make calculations easier, we do not compute explicitly the average over U i (C i ), but rather use heuristic arguments to evaluate the typical values of the random quantities appearing in Eq. [START_REF] De Gennes | Granular matter: a tentative view[END_REF]. Such an estimate is expected to be sufficient to determine the leading exponential behavior P (q) ∼ e -N f (q) of the distribution P (q). The first product between brackets in Eq. ( 9) can be rewritten as the exponential of i∈S q,N βU i (1). The latter sum is (up to the factor β) a sum of independent and identically distributed Gaussian random variables drawn from the distribution Eq. ( 3). The sum is thus also Gaussian distributed, with zero mean and variance 1 2 qN m(βJ) 2 . It follows that

i∈S q,N e βU i (1) ∼ e O( √ N ) , (10)
and this product can thus be neglected (i.e., replaced by 1) when looking for the behavior of P (q) at exponential order in N .

The key point in order to determine P (q) explicitly is now to evaluate the typical value Z typ of the sum

Z i = n M 0 C i =2 e βU i (C i ) , (11)
where we recall that each U i (C i ) is an independent random variable drawn from the distribution ρ(U ) given in Eq. ( 3). Once this estimate is known, the distribution P (q) can be approximated as

P (q) ∼ 1 Z N qN Z (1-q)N typ e 1 2 βN Kq 2 . ( 12
)
The quantity Z i has the same form as the partition function of the Random Energy Model (REM) [START_REF] Derrida | Random-energy model: Limit of a family of disordered models[END_REF][START_REF] Derrida | Random-energy model: An exactly solvable model of disordered systems[END_REF], and we can thus borrow some methods from the REM to evaluate it. A standard approach to study the REM is to evaluate the density of states of a typical realization of the disorder. It is convenient at this stage to define the rescaled variable u = U/M . Denoting as n(u) the density of state of a given realization, the average n(u) over the disorder is given by n

(u) = n M 0 ρ(u) ∼ e M (ln n 0 -u 2 /J 2 ) ( 13
)
to exponential order in M , having defined ρ(u) = M ρ(M u). Hence if ln n 0u 2 /J 2 > 0, corresponding to |u| < u 0 = J √ ln n 0 , the average density of state n(u) is exponentially large with M , and for a typical sample n(u) ≈ n(u) . By contrast, when |u| > u 0 , the average density of state is exponentially small with M , meaning that in most realizations there are actually no states with |u| > u 0 (the exponentially small value of the average density of states comes from very rare and atypical realizations having a few states in this range). Therefore, to describe a typical realization, one can in practice consider that the density of state is equal to zero for |u| > u 0 , and equal to n(u) for |u| < u 0 . We can thus evaluate Z i as

Z i = C i e βU i (C i ) ≈ M πJ u 0 -u 0 du e M (ln n 0 -u 2 /J 2 +βu) ≡ Z typ . ( 14
)
The integral in Eq. ( 14) can be evaluated by a saddle-point calculation in the large M limit. Defining g(u) = ln n 0 -u 2 /J 2 + βu, Z typ is given for large M by Z typ ∼ e M gmax , where g max is the maximum value of g(u) over the interval [-u 0 , u 0 ]. Let us first look for the maximum of g(u) over the whole real axis. Defining u * such that g (u * ) = 0, we find u * = 1 2 βJ 2 . Recalling that β = 1/T , and defining

T g = J 2 √ ln n 0 , (15)
we find that -u 0 < u * < u 0 for T > T g , whereas u * ≥ u 0 for T ≤ T g . Hence g max = g(u * ) for T > T g , and g max = g(u 0 ) for T ≤ T g , taking into account the fact that g(u) is monotonously increasing for u < u * . We thus obtain

Z typ ≈ e M (ln n 0 + 1 4 β 2 J 2 ) if T > T g , e M βJ √ ln n 0 if T ≤ T g . (16)
Using Eqs. ( 12) and ( 16), we obtain after expanding the factorials thanks to the Stirling formula that P (q) takes a large deviation form P (q) ∼ e N f (q) . In physical terms, f (q) may be thought of as (the opposite of) a free energy. The explicit expression of f (q) depends on the temperature range. For T > T g , f (q) reads

f (q) = -q ln q -(1 -q) ln(1 -q) + (1 -q)M ln n 0 + J 2 4T 2 + 1 2T Kq 2 , (17)
whereas for T ≤ T g ,

f (q) = -q ln q -(1 -q) ln(1 -q) + (1 -q)M J T ln n 0 + 1 2T Kq 2 . ( 18
)
Note that we did not take into account in f (q) the contribution coming from the normalization factor Z, as this would simply add a constant to f (q). Inspired by Ref. [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF], where interesting results were obtained for a coupling constant K ∼ M , we assume in what follows that

K = kM , (19)
and take the reduced constant k as the relevant control parameter in the model (on top of temperature T ). For large M , the expression of f (q) then simplifies to

f (q) = M T (1 -q) T ln n 0 + J 2 4T + 1 2 kq 2 if T > T g , M T (1 -q) J √ ln n 0 + 1 2 kq 2 if T ≤ T g . (20)
We first observe that in this large-M approximation, f (q) is a convex function of q for all values of temperature T , so that the maximum of f (q) over the interval 0 ≤ q ≤ 1 is either f (0) or f (1). The most probable state is found to be q = 0 for k < k c (T ) and q = 1 for k > k c (T ), where the critical line k c (T ) is defined as (with

k 0 = 2J √ ln n 0 ) k c (T ) = k 0 2 T Tg + Tg T if T > T g , k 0 if T ≤ T g . (21)
The curve k c (T ) thus separates the (k, T ) phase diagram into two regions, a region with q = 0 at low coupling and a region with q = 1 at high coupling.

The corresponding phase diagram is plotted in Fig. 1. Note that for J = 0 (i.e., in the absence of disorder), the glassy region in the phase diagram disappears, and one recovers the phase transition at T c = k/(2 ln n 0 ) between a high-temperature phase with q = 0 and a low-temperature phase with q = 1 found in the homogeneous model [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF]. Besides, we have also seen that a change of behavior occurs at T = T g . For T > T g , the agents dynamically visit a large number of configurations, while for T < T g their dynamics becomes essentially frozen, and only few configurations have a significant probability to be visited. In other words, for T < T g agents become 'stuck' in a small number of configurations having the highest utility. In the context of the Random Energy Model for glasses, the temperature T g corresponds to the glass transition.

Hence there are actually three different regions in the phase diagram shown in Fig. 1. For T > T g and k < k c (T ), agents have no preferred configurations and visit many different configurations over time. For T < T g and k < k c (T g ), each agent spends a lot of time in a small set of preferred configurations. In other words agents look simple, but they remain different one from the other. This regime is dominated by agents heterogeneity, and there is on average no macroscopic overlap between agents configurations (q = 0). In the last region k > k c (T ), the coupling between agents dominates over agents heterogeneity, and all agents are essentially in the same configuration, leading to a strong overlap (q = 1) and to the emergence of a common characteristic, a phenomenon that has been called standardization in [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF]. For k < k c (T ), agents configurations do not overlap (q = 0), while for k > k c (T ) agents become standardized by interactions (q = 1). The q = 0 area is subdivided into high-and low-temperature regions.

For T > T g , agents may be in any of their internal configurations, while for T < T g , they are dynamically blocked in the few configurations with the highest intrinsic utility. In this latter case, agents appear simple but remain heterogeneous.

Conclusion

We have extended here the model proposed in [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF], where agents have many internal configurations, but can select a specific configuration thanks to interactions, leading to simplified, or standardized agents. While Ref. [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF] focused on homogeneous agents, we have in the present work extended the model to account for agents heterogeneity, introduced through random idiosyncratic utilities associated with each configurations of each agent. Heterogeneity is introduced in a minimal way, directly inspired from the Random Energy Model for glasses [START_REF] Derrida | Random-energy model: Limit of a family of disordered models[END_REF]. Including heterogeneity in the model leads to the onset of a new phase at low temperature and small coupling, when heterogeneity dominates over interactions among agents. In physical terms, this new phase shares similarity with a glass phase. The main difference here is that we consider not a single glassy system as in physics, but a large assembly of interacting agents, each of which experiencing an internal glass transition.

In this glassy phase, agents are stuck in a small number of possible internal configurations, those with maximal idiosyncratic utility. In spite of the individual simplification of agents, the assembly remains heterogeneous, because all agents are in different configurations and do not share common characteristics. Increasing the coupling strength, interactions eventually dominate over heterogeneity, effectively leading to simple agents all occupying the same configuration and sharing the same characteristic. The latter phase is similar to the low temperature phase found in the homogeneous model of [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF], where agents have been called standardized. Finally, at high temperature, agents keep their internal complexity and do not significantly feel interactions or heterogeneity, again similarly to the homogeneous model [START_REF] Bertin | In social complex systems, the whole can be more or less than (the sum of) the parts[END_REF].

A further step in the study of this model would to study possible collective effects that could arise once agents are simplified. This has been done in the homogeneous model by assuming that standardized agents could be in one of two distinct states, corresponding for instance to two different opinions (or to a spin in physical terms). Interactions between these binary degrees of freedom may then lead, at low temperature, to a collectively ordered state. It would be interesting to see in more details how agents heterogeneity could possibly modify this simple picture. In addition, the role of a lower connectivity (in the present work, agents interact with all other agents) would clearly deserve to be investigated, as connectivity is known in many situations to modify critical properties [START_REF] Bellac | Quantum and statistical field theory[END_REF].

Figure 1 :

 1 Figure 1: Phase diagram of the model in the (k, T )-plane (reduced coupling constant k = K/M versus temperature), showing the three different regions separated by the critical line k c (T ) (full line) and the 'glass' temperature T g (dashed vertical line). For k < k c (T ), agents configurations do not overlap (q = 0), while for k > k c (T ) agents become standardized by interactions (q = 1). The q = 0 area is subdivided into high-and low-temperature regions. For T > T g , agents may be in any of their internal configurations, while for T < T g , they are dynamically blocked in the few configurations with the highest intrinsic utility. In this latter case, agents appear simple but remain heterogeneous.

Preprint submitted to Elsevier December 7, 2020

Acknowledgment

The author is grateful to Pablo Jensen for a critical reading of the manuscript and interesting comments.


Jean Guillard

Daniel Gerdeaux

Jean-Marc Chautru

The use of geostatistics for abundance estimation by echo integration in lakes: the example of Lake Annecy

The use of geosta tistics for abundance estimation by echo integration in lakes: the example of Lake Annecy

Introduction

Estimation of fish biomass in lakes allows us to expand our knowledge of the ichthyological resources of such environments and, above all, enables us to follow stock fluctuations over a period of time (Forbes and Nakken, 1972;[START_REF] Marchai | La détection acoustique dans l'étude des peuplements piscaires[END_REF]. One must not only obtain an estimate, but also establish a confidence limit. This problem has been addressed frequently in recent work [START_REF] Gohin | Geostatistics applied to fish distribution as derived from acoustic surveys[END_REF][START_REF] Laloe | Contribution à l'étude de la variance d'estima teur de biomasse de poissons obtenus par échointégration[END_REF][START_REF] Burczynski | Application of dual-beam acoustic survey to limnetic populations of juve nile sockeye salmon (O ncorhynchus nerka)[END_REF]). An approach using classical statistical methods often results in very large confidence limits owing to wide sampling variance [START_REF] Williamson | Cluster sampling estimation of the variance of abundance estimates derived from quantitative echo sounder surveys[END_REF][START_REF] Francis | Variability in hydroacoustic biomass estimates[END_REF]. Those methods which take into account the spa tial distribution of sampling locations should result in estimates with smaller confidence limits, except in rare cases where no spatial correlation exists. Geostatistics allows us to analyse this type of data and to determine an unbiased estimator [START_REF] Matheron | The theory of regionalized variables and its applications[END_REF]Journel and Huijbregts, 1978;[START_REF] Delhomme | Spatial variability and uncertainty in groundwater flow parameters: a geostatistical approach[END_REF].

Materials and methods

Materials

Lake Annecy is situated in the northern region of the French Alps at an elevation of 446.5 metres. It is 13.7 km long with a surface area of 27 k nr and consists of two basins separated by a narrow strait. Its maximum depth is 65 m. The detectable biomass consists essen tially of six species: perch (Perea fluviatilis), roach (Rutilus rutilus), coregonids (Coregonus lavaretus), trout (Salmo trutta), char (Salvelinus alpinus), and pike (Esox lucius). The echo-sounding survey was conducted at night with a boat 8 m long, sailing at a constant speed of 10 km/h. The acoustic equipment used consisted of a Simrad EY-M sounder (70 kHz, TVG = 20 log R) fitted with a transducer with a full beam angle of 11°. The signal received, converted into frequency, and the trig ger pulse were digitally recorded on tape. Echo in tegration was performed by an A GENOR echo integra tor [START_REF] Person | Sys tème d'échointégration numérique pour l'évaluation des stocks "AG EN O R[END_REF].

Sampling method

The survey was systematically run in sections over the entire surface of the lake using a classical procedure [START_REF] Marchai | Première estimation de la population piscicole du lac Léman par échointégration[END_REF]. Between each transect, the boat followed the shore for about six minutes. So, the distance between any two transects was never ex actly the same, and we can estimate that this distance is randomly around 500 m. We consider that we made a stratification of the lake in a regular grid and place each transect randomly along the grid knots. This grid was arbitrarily chosen as a function of the time to carry out a complete survey in one night (Fig. 1).

A copper ball 32 mm in diameter, attached by three

o O oo =0 1 km O-,

O0

Figure 1. Echo-integration results of Lake Annecy. Left: location of hydroacoustics transects. Right: plots of echo-integration values, for two-minute se quences. Each circle is proportional to the echointegration units for the layer 0 -2 5 m.

wires at a distance of 7 m from the transducer was used to calibrate the measuring equipment. The duration of echo-integration sequences was first established at two minutes for eight 5-m layers of water (from the surface to 40-m depth), and two 10-m layers (from 40 to 60 m). To facilitate data processing the integration layers were combined into two depth strata (0-25 and 25-60 m). The study objectives did not re quire us to develop biomass estimates. Therefore results were expressed in echo-integration units (V2 x rrT2). Later on, the duration of sequences was decreased to 30, then to 10 seconds, to fit the variograms and for kriging.

Overview of the theory of regionalized variables

The calculations were carried out within the probabilis tic framework of the theory of regionalized variables [START_REF] Matheron | The theory of regionalized variables and its applications[END_REF], where each experimental datum is considered as one realization of a random variable.

Our objective was to estimate the average value of any additive variable (e.g., biomass density) in a given field, starting from a linear combination of samples. In each case it is necessary, in practice, to adjust a model starting with the experimental data. Since the variogram is the tool currently used in this field, it is useful to define its most important properties. The variogram y(h) is defined by the expression:

y(h) = 0.5 E {[z(x+h)-z(x)]2}
where z(x) is the ob served value at point x (Fig. 2).

The sill and range only exist in the case where the field studied is sufficiently homogeneous. If not, one will be able to observe a continuous growth of the variogram (the linear shape of Figure 2). It should be noted that the variogram is directiondependent. Different behaviour in various directions therefore demonstrates and quantifies the anisotropy of the phenomenon studied. Because of this, one must take care not to treat as a whole, any samples which are regularized using different criteria (supports), in partic ular on long segments in different directions. The regu larized function Zv(x) with a support v is the average over the volume v of the random function Z(x) defined on a point support. Thus, to avoid serious incoherences, the echo-integration sequences along the east-west transects (sections) should be studied separately from those along the cross-transects, approximately n o rthsouth.

SILL VARIANCE OF THE SAMPLE! LINEAR MODEL SPHERICAL MODEL 2 < Ù1 o o cr < > RANGE DISTANCE BEYOND WHICH 2 POINTS ARE UNCORRELATED NUGGET EFFECT] DISTANCE H
For more information, the reader may refer to Matheron (1971), [START_REF] David | Geostatistical ore reserve estimation[END_REF], and Journel and Huijbregts (1978).

Results

Plotting of the echo-integration results shows the heter ogeneity of the spatial distribution of the fish (Fig. 1). The coastal areas contain the major portion of the bio mass. Note the existence of a gradient of decreasing density from the shore towards the pelagic zone. The majority of the fish are located at the thermocline, 12 to 15 m deep. Observation of these gradients demon strates the spatial correlations between samples.

Variogram study

The calculations were carried out separately for each of the two parts of the lake, the "upper lake" and the "lower lake" . Each of these zones is assumed to be homogeneous, in other words, having a unique struc ture. Since only the "upper lake" was sufficiently sam pled to allow precise calculations, the following results are applicable to this zone only.

Starting from echo-integration sequences of two min utes' duration (63 locations), it was impossible to fit the variogram, and thus to determine a structure logically. The size of each sequence (about 330 m) perhaps places too much emphasis on the regularization of the phe nomenon, masking the structures. Therefore, we had to increase the number of sampling locations and work on echo-integration sequences of 30 seconds. The number of samples for the transects of the "upper lake" was thus 202; there were, accordingly, enough pairs of data points in the principal east-west direction to fit the variogram (Fig. 3).

The arrangement of points corresponding to the first steps of the variogram leads to uncertainty regarding the ordinate at the origin: was it zero or not?; in other words, was there or was there not a nugget effect? We finally decreased the duration of sequences to 10 sec onds to obtain enough points at short distances. With this additional information, we were able to select a theoretical linear model which permitted adjustment of the variogram in the principal east-west direction and which incorporated a fairly big nugget effect (Fig. 4).

As we have already pointed out, the nugget effect includes both inevitable measurement errors (precision of the apparatus, stability of adjustments, etc.), and microstructures which are not detectable on the scale of the study. In our particular case, we can say that the behaviour of a fish depended more on its immediate neighbours than on those 30 metres away. The primary spatial structure must therefore be on the order of a metre. Because our scale of study was in tens of metres, this structure would not be detectable, and it was not surprising to notice a significant nugget effect. The nug get effect entered into the calculations as a purely ran dom component.

It should be noted that the particular case of the nugget effect alone (the variogram still equal to its sill) corresponds to the independence of the random var iables introduced at each point. The more significant the nugget effect, the closer we get to conditions where classical statistical methods apply.

Incidentally, because of the heterogeneity and small dimensions of the field studied, it is not surprising to observe a linear variogram, i.e., without a sill or range.

One can see, moreover, in Figure 1 some anisotropy of the fish biomass, with a north-south elongation, which would be useful to quantify. To do this, we have to calculate and fit the variogram in many directions. Two types of data are available: on the one hand, the samples along the east-west transects theoretically al low us to obtain a discontinuous north-south vario gram, but the transects are too far apart to enable us to adjust it appropriately in the vicinity of the origin. On the other hand, the samples of the cross-transects allow us to calculate the regularized north-south variogram which is comparable to that in the east-west direction in Figure 3. However, because there are so few samples and they are concentrated toward the edge of the lake, the points on the variogram have little statistical signif icance and there is a risk of bias. One can nevetheless perform the calculation and adjust the variogram based on the nugget effect and a linear model (Fig. 5). By comparing the observed slopes in Figures 3 and5, after standardization by the variance, we can estimate the order of magnitude of the coefficient of anisotropy to be about 2.5. Since this value is compatible with natural observations, we will hypothesize that it is cor rect and use it in subsequent calculations.

Estimation by kriging

The variogram models chosen (Fig. 3) were introduced into the kriging calculations (computer program BLUE-PACK 3D, Centre de Géostatistique, Fontainebleau). We observed an estimator of the average biomass (in units of echo integration) for the whole of each zone sampled and for each link in the network.

With each estimation, there is an associated estima tion variance which allows us to define a confidence limit for each result obtained (Fig. 6); for example, the average of the density in the "upper lake" is 27 Rapports et Procès-Verbaux 413 659 ± 2 x 62. The estimation of total stock is made by multiplying the average value for each zone by the surface area (similarly for the confidence limit).

Discussion

An estimate of the biomass present in the lake of ±18 % has been obtained. If we could obtain an even narrower confidence interval, it would make our comparison tests all the more powerful.

Calculation of the data and adjustment of the va riogram must be performed with enough points to avoid contradicting the theoretical model. In our particular case, we have had to establish certain hypotheses, since the configuration of our sampling plan did not take everything into consideration. Our first hypothesis was the structural homogeneity of the chosen units. These zones were selected based on our knowledge of the environment (bathymetry, fish behaviour, etc.), and experience of local professional and sports fishermen. The sampling routine, following different directions, does not allow us to take into account simultaneously all of the sequences. To mitigate this problem, the se quences must be sufficiently short that the regulariza tion effect is negligible. The samples can therefore be considered as points on the scale of the lake in all directions and can be studied together as a whole. An other solution would consist of sampling in only one direction, east-west for example, but bringing the tran sects as close together as possible and keeping the se quences relatively short, yet not necessarily treating them as point values.

In general, the problems encountered during this study could be eliminated, or at least greatly reduced, if the following sampling procedure were followed. Sam ples should at first be taken along closely spaced tran sects in order to obtain an accurate variogram. These transects could begin at the shore of the lake and end in the middle; there should be on the order of 10 to have enough points in the direction perpendicular to the tran sects, and the distance between the two courses which are farthest apart should be greater than the assumed range of the variogram. One should try within reason to duplicate this sampling procedure in another zone in order to test the homogeneity hypothesis. This prelimi nary work should be carried out some time before the main echo-integration survey. It will then be possible to define the range of the phenomenon and to calculate the estimation variance for many sampling plans. De pending on the desired precision, one can select the plan best suited to one's needs and constraints, and finally perform the survey by selecting an adequate sampling procedure for each zone.

For stock estimation, there will always be an error on the surface of the lake which is not precisely known; but this type of error is common to all estimation methods, and can perhaps be calculated with the help of approxi mation formulas proposed by [START_REF] Matheron | The theory of regionalized variables and its applications[END_REF].

A procedure of this type will result in a satisfactory sampling plan which enables us to obtain a reliable estimation of the biomass. If narrower confidence limits can be obtained, it will be easier to show stock fluctu ations.

Figure 2 .

 2 Figure 2. The variogram: definition of terms.

Figure 3 .

 3 Figure 3. E ast-w est variogram, with 30-second sequences. Possible models: y(H): 150000 x (1.5 x (FI/150) -0.5 x x (H/150)3 + 210. y(H): 150000 + 210 x H.

Figure 4

 4 Figure 4. E ast-w est variogram, with 10-second sequences. Possible models: y(H ): 750000 + 340 x H.

Figure 5

 5 Figure 5. N orth-south variogram, with 30-second sequences. Possible models: y (H ): 2100000 + 5 000XH .

Figure 6 .

 6 Figure6. Contour map for kriging biomass densities (estima tion in echo-integration units) in the "upper lake". Superposi tion (hachures) of standard deviation for local estimation.