M Bergère

B Eynard

Universal scaling limits of matrix models, and (p, q) Liouville gravity

We show that near a point where the equilibrium density of eigenvalues of a matrix model behaves like y ∼ x p/q , the correlation functions of a random matrix, are, to leading order in the appropriate scaling, given by determinants of the universal (p, q)-minimal models kernels. Those (p, q) kernels are written in terms of functions solutions of a linear equation of order q, with polynomial coefficients of degree ≤ p. For example, near a regular edge y ∼ x 1/2 , the (1, 2) kernel is the Airy kernel. Those kernels are associated to the (p, q) minimal model, i.e. the (p, q) reduction of the KP hierarchy solution of the string equation. Here we consider only the 1-matrix model, for which q = 2.

Introduction

In this article, we shall consider "scaling limits" of matrix integrals.

We shall show, under certain assumptions, that scaling limits of matrix integrals are governed by some well known integrable systems. The fact that double scaling limits of matrix models are minimal models (p, q) of conformal field theories [START_REF] Di Francesco | Conformal field theory[END_REF], has been well known in the physics literature for a long time (see [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Cicuta | Phase transitions and random matrices[END_REF][START_REF]Two dimensional quantum gravity and random surfaces[END_REF] for review, and among others see [51,[START_REF] Va Kazakov | Critical properties of randomly triangulated planar random surfaces[END_REF][START_REF] Kazakov | The appearance of matter fields from quantum fluctuations of 2D-garvity[END_REF][START_REF] Kazakov | Bilocal regularization of models of random surfaces[END_REF][START_REF] Moore | Geometry of the string equations[END_REF][START_REF] Kostov | Conformal field theory techniques in random matrix models[END_REF][START_REF] Brézin | Exactly solvable field theories of closed strings[END_REF][START_REF] Douglas | Strings in less than one dimension[END_REF][START_REF] Gross | Nonperturbative two-dimensional quantum gravity[END_REF][START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF][START_REF] Gross | Nonperturbative two-dimensional quantum gravity[END_REF]), and here we merely summarize some results scattered in the physics literature, we present the main features of those universal limit laws, and provide a mathematical proof.

The idea of the proof works backwards: we show that (p, q) minimal models determinantal correlation functions satisfy the same recursion as the scaling limits of matrix models.

We shall consider only the 1-matrix model, whose corresponding limit integrable systems are the (p, 2) minimal models, reductions of KdV, and we hope to later generalize those results to multi-matrix models and general (p, q) limits, as claimed in many physics works [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Kazakov | The appearance of matter fields from quantum fluctuations of 2D-garvity[END_REF]51].

The main result, theorem 2.3, is that limit correlation functions are given by determinantal formulae of the (p, q) kernel. The equilibrium density of eigenvalues of a N ×N random hermitian matrix, generically behaves near the edge of the distribution, like:

ρ(x) ∼ x 1 2 .
(

It is well known that, after rescaling x by N 2/3 , the n-points correlation functions in the vicinity of the edge, are given by determinants of the Airy kernel which appears in Tracy-Widom law [START_REF] Tracy | Introduction to random matrices[END_REF] of extreme eigenvalues statistics:

ρ n (N -2/3 x 1 , . . . , N -2/3 x n ) ∼ N →∞ N 2n 3 det( KAiry (x i , x j )) (1 + O(N -1/3 )), [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF][START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF] and the Airy kernel is the Christoffel-Darboux kernel of the Airy function:

KAiry (x 1 , x 2 ) = Ai(x 1 )Ai ′ (x 2 ) -Ai ′ (x 1 )Ai(x 2 ) x 1 -x 2 .
(1-3)

Notice that the Airy function satisfies a 2nd order ODE, whose coefficients are polynomials of degree 1:

Ai ′′ (x) = x Ai(x), (1-4)
which can also be written as a 2 × 2 differential system:

d dx Ai(x) Ai ′ (x) = 0 1 x 0 Ai(x) Ai ′ (x)
.

(1-5)

p/(p+q)

x ρ( )

x x 1/2
x ỹ x p/q N N q/(p+q)

Figure 1: Generically, the equilibrium density of eigenvalues behaves like x 1/2 near endpoints. It may happen, after fine tuning the parameters, that it behaves like x p/q (q = 2 for 1-matrix model). The eigenvalue statistics at a scale x = x N q/(p+q) is governed by the universal (p, q) kernel law. The (p, q) kernel is an integrable kernel, associated to the (p, q) reduction of the KP hierarchy. For a regular endpoint (p, q) = (1, 2) it is the Airy kernel.

Higher (p, 2) laws

More generally consider a p/q singularity of the equilibrium density of eigenvalues: ρ(x) ∼ x p/q .

(1-6)

We shall consider only q = 2 and p = 2m + 1 in this article, but we recall that the physics literature claims that general y ∼ x p/q case can be treated the same way, and should correspond to (p, q) minimal models. We shall see, after rescaling x by N q/(p+q) , that the correlation functions in the vicinity of the edge, are given by determinants of the (p, q) kernel which appears in (p, q) minimal models of conformal field theory [START_REF] Di Francesco | Conformal field theory[END_REF].

ρ n (N -q p+q x 1 , . . . , N -q p+q x n ) ∼ N →∞

N n q p+q det( K(p,q) (x i , x j )) (1 + O(N -1/(p+q) )), [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF][START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF][START_REF] Belavin | On Correlation Numbers in 2D Minimal Gravity and Matrix Models[END_REF][START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF][START_REF] Belavin | On correlation functions in the perturbed minimal models M(2,2n+1)[END_REF][START_REF] Bertola | Boutroux curves with external field: equilibrium measures without a minimization problem[END_REF][START_REF] Bergère | Correlation functions of complex matrix models[END_REF] and the (p, 2) kernel is the Christoffel-Darboux kernel of the (p, 2) Baker-Akhiezer function: K(p,2) (x 1 , x 2 ) = ψ(x 1 ) ψ(x 2 ) -ψ(x 1 )ψ(x 2 )

x 1x 2 .

(1-8)

where the (ψ, ψ) functions satisfy a 2nd order ODE:

d dx ψ(x) ψ(x) = D (p,q) (x) ψ(x) ψ(x) (1-9)
where D (p,2) (x) is a 2 × 2 matrix with polynomial coefficients, such that the degree of det D is of degree at most p. Moreover this differential system, is associated to the Lax matrix of the (p, 2) reduction of the integrable KdV hierarchy, which means that the coefficients of the matrix D (p,q) (x), are themselves solution of some non-linear integrable differential equations.

The coefficients of D (p,q) (x) are differential polynomials of a function u(t) which satisfies (for p = 2m + 1, q = 2), the m + 1 th Gelfand-Dikii non linear equation [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF], of the form: u m+1 + u m-1 ü + . . . + u (2m) = t.

(1-10)

The time t measures the distance to the critical point.

For example for pure gravity (p, q) = (3, 2), D (3,2) (x) is a 2 × 2 matrix, with polynomial coefficients such that the degree of det D (3,2) (x) is 3:

D (3,2) (x, t) = u 2x -2u (x + 2u)(2x -2u) + ü -u (1-11)
and where u(t) is solution of the Painlevé I equation (1 st Gelfand-Dikii equation):

3u 2 (t) - ü(t) 2 = t. (1-12)
All this has been stated for a long time in the physics literature, and we shall just present it concisely and prove it.

Universality of eigenvalues statistics point of view

In this subsection, we summarize some well known facts about random matrices [START_REF] Mehta | Random Matrices[END_REF]70,[START_REF] Dyson | correlations between the eigenvales of a random matrix[END_REF][START_REF]Random Matrix Models and Their Applications[END_REF][START_REF] Van Moerbeke | Random Matrices and their applications[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Boutet De Monvel | On the statistical mechanics approach in the random matrix theory: Integrated density of states[END_REF][START_REF] Brézin | Universality of the correlations between eigenvalues of large random matrices[END_REF][START_REF] Johansson | On fluctuations of eigenvalues of random hermitian matrices[END_REF], and we fix the notations.

Consider a probability law of the form of the joint law of eigenvalues of a random hemitian type3 matrix:

dµ(λ 1 , . . . , λ N ) = 1 Z i<j (λ j -λ i ) 2 i e -N s V (λ i ) dλ i , (1-13)
where Z is the partition function

Z = dµ = dM e -N s tr V (M ) . (1-14)
Here s is a parameter, often called the temperature. we shall be interested in the large N limit, and possibly a limit s → s c , where Z has a singularity at s = s c . The name double scaling limit [START_REF] Kazakov | The appearance of matter fields from quantum fluctuations of 2D-garvity[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Va Kazakov | Critical properties of randomly triangulated planar random surfaces[END_REF][START_REF] Gross | Nonperturbative two-dimensional quantum gravity[END_REF][START_REF]Two dimensional quantum gravity and random surfaces[END_REF][START_REF] Douglas | Flow and instability in quantum gravity[END_REF][START_REF] Douglas | Strings in less than one dimension[END_REF][START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF][START_REF] Bleher | Double scaling limit in the matrix model: the Riemann-Hilbert approach[END_REF] means that we consider a regime where the limits s → s c and N → ∞ are related by a scaling relation

(s -s c ) N -α = O(1) (1-15)
where α is some appropriate exponent (α = 0 if Z is not singular). We are interested in computing expectation values of resolvents:

ωn (x 1 , . . . , x n ) = Tr 1 x 1 -M . . . Tr 1 x n -M = i 1 ,...,in 1 x 1 -λ i 1 . . . 1 x n -λ in (1-16) as well as in their cumulants ωn (x 1 , . . . , x n ) = Tr 1 x 1 -M . . . Tr 1 x n -M c . (1-17)
The density correlation functions ρ n (x 1 , . . . , x n ) can be easily deduced from them: densities are discontinuities of resolvents, and resolvents are Stieljes transforms of densities, for example for the 1-point function:

ω1 (x) = ρ 1 (x ′ )dx ′ x -x ′ , ρ 1 (x) = 1 2iπ (ω 1 (x -i0) -ω1 (x + i0)). (1-18)
Imagine, that, for s < s c , the potential V (x) is such that there is a large N expansion of the type:

ln Z = ∞ g=0 (N/s) 2-2g fg , (1-19)
and similarly:

ωn (x 1 , . . . , x n ) = ∞ g=0 (N/s) 2-2g-n ω(g) n (x 1 , . . . , x n ). (1-20)
First, let us emphasize that such an expansion does not exist for any potential V , it exists only if the integration contour for the λ i 's is a "steepest descent contour" for the potential V (i.e. a landpath and bridges path in the Riemann-Hilbert language of [START_REF] Bertola | Boutroux curves with external field: equilibrium measures without a minimization problem[END_REF][START_REF] Deift | Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory[END_REF]). For instance it was proved [START_REF] Guionnet | Combinatorial aspects of matrix models[END_REF] that such a large N expansion holds for s sufficiently small.

From now on, let us assume that we are in a situation where such an expansion exists. In that case, the coefficients ω(g) n and fg were computed in [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF][START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF][START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], and they are the "spectral invariants" of some spectral curve associated to V /s. The spectral curve ŷ(x), in that case, is the function ŷ

(x) = V ′ (x)/2 - ω(0) 1 (x)
, it is the large N density, also called equilibrium density ŷ(x) = iπρ eq (x) = iπρ (0) 1 (x): Theorem 1.1 (proved in [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF][START_REF] Chekhov | Hermitian matrix model free energy: Feynman graph technique for all genera[END_REF]) The coefficents fg and ω(g) n of the topological expansion of ln Z and ωn , are the spectral invariants (in the sense of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]) of the spectral curve:

ŷ(x) = iπ ρ eq (x) = 1 2 V ′ (x) - ω(0) 1 (x).
(1-21)

Remark 1.1 We refer the reader to [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] to see how to compute the spectral invariants of an arbitrary plane curve ŷ(x). We shall give an explicit example of computation of spectral invariants for formal matrix models below in section 1.2.1, see theorem 1.3.

Let us say that we shall not be really using any deep result of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] in this article, except the theorem 8.1. of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] (which is very easy to prove by recursion).

Here, as is well known in random matrix theory, ω(0) 1 (x) is an algebraic curve (hyperelliptical for the 1-matrix model), with typical square-root branchpoints at the endpoints of the distribution of eigenvalues, we shall write it:

ω(0) 1 (x) = 1 2 V ′ (x) -ŷ(x) (1-22)
where ŷ(x) is the square root of some polynomial ŷ2 = Polynomial(x).

(1-23) Generically, this s-dependent polynomial has only simple zeroes, and ŷ(x) has square root singularities, but for some appropriate choices of s = s c , the polynomial may have multiple zeroes, and we shall consider that, at s = s c , there is a zero of order 2m + 1 at x = 0:

s = s c -→ ŷ(x) ∼ x m+ 1 2 . (1-24)
When s is close to s c , we typically have:

ŷ ∼ m k=0 x k+ 1 2 c k (s -s c ) m-k m+1 (1 + O((s -s c ) 1 m+1 ) ) , (1-25)
which we write:

ŷ((s -s c ) 1 m+1 x) ∼ (s -s c ) 2m+1 2m+2 y(x) (1 + O((s -s c ) 1 m+1 ) , y(x) = m k=0 c k x k+ 1 2 .
(1-26) At s = s c we have ŷ ∼ x m+ 1 2 and at s = s c we have ŷ ∼ √ x.

Notice that a regular endpoint corresponds to m = 0, and in that case, s c can be chosen as any value of s.

Double scaling limit

In this article, we shall be interested in the behavior of ω(g) n when s is close to s c , and the x i 's are in the vicinity of a branchpoint. Theorem 8.1. of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], implies that after rescaling, we have (when 2 -2gn < 0): Theorem 1.2 (theorem 8.1. of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]): If m > 0:

fg ∼ (s -s c ) (2-2g) 2m+3 2m+2 f g , (1-27)
and if m ≥ 0:

ω(g) n ((s -s c ) 1 m+1 x 1 , . . . , (s -s c ) 1 m+1 x n ) ∼ (s -s c ) (2-2g-n) 2m+3 2m+2 -n m+1 ω (g) n (x 1 , . . . , x n ) (1 + O((s -s c ) 1 m+1
)), where the f g 's and ω (g) n 's are the spectral invariants of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] for the spectral curve y(x) appearing in eq. .

Remark 1.2 This theorem is very easy to prove by recursion on n and g from the definitions of spectral invariants in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]. A more detailed proof is also given in section 4.8.2 of [START_REF] Eynard | Algebraic methods in random matrices and enumerative geometry[END_REF].

Our goal in this article, is to show that the coefficients ω (g) n and f g can also be computed from determinantal formulae for the (p, 2) kernel appearing in the (p, 2) minimal model, this is our theorem 2.3.

Specific heat

If m > 0, we consider the resummation to leading order:

ln Z = g (N/s) 2-2g fg ∼ g (N/s c ) (2-2g) (s -s c ) (2-2g) 2m+3 2m+2 f g = F ((s -s c )N 2m+2 2m+3
).

(1-29) This shows that the double scaling limit is N → ∞, s → s c with a scaling:

t = (s -s c ) N 2m+2 2m+3 = O(1). (1-30)
This is a special case of the double scaling limit (ss c ) ∼ N -(p+q-1)/(p+q) for general (p, q). We defined the function

F (t) = ∞ g=0 t (2-2g) 2m+3 2m+2 f g . (1-31)
Consider its second derivative, often called specific heat:

u(t) = d 2 dt 2 F (t) = g u g t 1-g(2m+3) m+1
.

(1-32)

i.e.

u g = (1 -g) f g (2m + 3)(m + 2 -g(2m + 3)) (m + 1) 2 . (1-33)
We shall prove in this article, that, as claimed in many physics articles, this function satisfies the m+1 th Gelfand-Dikii non-linear equation. For instance if m = 1, it satisfies the Painlevé I equation:

3u 2 - 1 2 ü = t. (1-34)
Moreover, it is well known from general considerations in statistical physics, that the free energyln Z should be convex, i.e. u(t) should be negative for t > 0:

u(t) ≤ 0.
(1-35)

Remark 1.3

The case m = 0, needs some care. The correlation functions ω n 's, indeed correspond to the s → s c limits of ωn 's, i.e. a zoom x → (ss c )x near a regular branch point, but the free energy F = ln Z is not divergent at s = s c , and thus F (t) cannot be seen as the s → s c limit of ln Z. In that case m = 0, the 1st Gelfand-Dikii equation is not differential, it is simply

u(t) = - t 2 , (1-36)
and, if it made sense, it would correspond to a free energy diverging as F ∼ N 2 , but in fact the free energy F is not divergent, and one finds that f g = 0 for g ≥ 1.

Formal matrix models and combinatorics point of view

As it was discovered by Brezin-Itzykson-Parisi-Zuber [14], matrix integrals are (in the formal sense) generating functions for counting discrete surfaces of a given topology [START_REF] David | Planar diagrams, two-dimensional lattice gravity and surface models[END_REF][START_REF] Francesco | 2D gravity, matrix models and graph combinatorics[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF]. Consider the potential:

V (x) = x 2 2s - 1 s δV (x) , δV (x) = d+1 j=3 s j j x j (1-37)
Formal matrix integrals are defined as:

Z = formal dM e -N s Tr V (M ) def = ∞ k=0 N k k! s k H N dM ( tr δV (M)) k e -N s Tr M 2 2 . (1-38)
where dM is the Lebesgue measure on H N , normalized such that dM e -N s Tr M 2 2 = 1. In other words, we have exchanged the integral and the summation over k. Z is a formal power series in s, where each coefficient is a finite sum of polynomial expectation values of a Gaussian integral:

Z = 1 + ∞ j=1 s j
Ãj .

(1-39)

After taking the Log, we also have a formal power series:

ln Z = ∞ j=1 s j A j . (1-40)
It was noticed by t'Hooft [START_REF] Hooft | [END_REF] and then BIPZ [14], that, after dividing by N 2 , each coefficient A j is a polynomial in 1/N 2 , namely:

A j = gmax(j) g=0 N 2-2g A j,g . (1-41)
Then, one defines:

fg = ∞ j=1 s j+2-2g A j,g (1-42)
which is also a formal series in powers of s (one can easily prove that A j,g = 0 if j + 2 -2g < 0). In that case, the following large N topological expansion holds as an equality between formal series of s, order by order in s:

ln Z = ∞ g=0 (N/s) 2-2g fg .
(1-43)

We emphasize that this equality is not a large N asymptotic expansion, it is a small s asymptotic expansion, and order by order in s the sum over g is finite.

It was proved by BIPZ in 1978 [14], by a mere application of Wick's theorem, that fg is the generating function for maps of genus g:

fg = maps, genus g s #vertices #Aut s #triangles 3 s #quadrangles 4 . . . s #(d+1)-gons d+1
.

(1-44)

Similarly one may compute formal expectation values:

ωn (x 1 , . . . , x n ) = l 1 ,...,ln 1 x l 1 +1 1 . . . x ln+1 n tr M l 1 . . . tr M ln c,formal = tr 1 x 1 -M . . . tr 1 x n -M c,formal (1-45)
where c means cumulant, and formal means that we compute the integral by exchanging the order of the Taylor expansion of e -N s tr δV (M ) and the integral, as in eq. . ωn (x 1 , . . . , x n ) is thus a formal power series in s, whose coefficients are polynomial expectation values of a Gaussian integral. For each power of s, the coefficient in ωn is a polynomial in the 1/x i 's, and is a polynomial in 1/N. We write:

ωn (x 1 , . . . , x n ) = ∞ g=0 (N/s) 2-2g-n ω(g) n (x 1 , . . . , x n ) (1-46)
which is an equality between formal series of s (order by order the sum over g is finite). It was also proved by BIPZ [14], that ω(g) n is the generating function for maps of genus g, with n marked faces, and with 1 marked edge on each marked face:

ω(g) n = maps, genus g s #vertices #Aut s #triangles 3 s #quadrangles 4 . . . s #(d+1)-gons d+1 x l 1 +1 1 . . . x ln+1 n (1-47)
where l i is the length of the i th marked face. All this is now standard result in combinatorics of maps.

Spectral invariants and spectral curve

More recently, it was proved in [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF][START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], that the functions ω(g) n and fg are the spectral invariants (in the sense of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]) of the formal matrix model spectral curve E formal MM = (x, ŷ) defined parametrically by:

E formal MM = x(z) = α + γ(z + 1 z ) ŷ(z) = d k=1 u k (z k -z -k ) (1-48)
where the coefficients α, γ, u k are entirely determined by the following algebraic constraints:

x(z) -

d j=2 s j+1 x(z) j = d k=1 u k (z k + z -k ) , u 1 = s γ , u 0 = 0, (1-49)
which give an algebraic equation for α and γ, whose solution we choose such that α and γ 2 are formal power series of s starting with:

γ 2 = s + O(s 2 ) , α = O(s).
(1-50) (We give the example of quadrangulations below)

Then, from [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF][START_REF] Chekhov | Hermitian matrix model free energy: Feynman graph technique for all genera[END_REF] we have: [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF][START_REF] Chekhov | Hermitian matrix model free energy: Feynman graph technique for all genera[END_REF][START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]): ω(g) n and fg are the spectral invariants (in the sense of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]) of the formal matrix model spectral curve E formal MM = (x, ŷ).

Theorem 1.3 (proved in
The spectral invariants of that curve are defined as follows (see [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]):

ω(0) 2 (x(z 1 ), x(z 2 )) = 1 (z 1 -z 2 ) 2 x′ (z 1 )x ′ (z 2 ) - 1 (x(z 1 ) -x(z 2 )) 2 ,
(1-51)

and with J = {x(z 1 ), . . . , x(z n )}, we have recursively:

ω(g) n+1 (J, x(z n+1 )) = 1 2x ′ (z n+1 ) Res z→±1 z x′ (z) 2 dz z n+1 (x(z n+1 ) -x(z))ŷ(z) ω(g-1) n+2 (J, x(z), x(z)) + g h=0 ′ I⊂J ω(h) 1+|I| (I, x(z))ω (g-h) 1+n-|I| (J \ I, x(z)) (1 -52)
and for g ≥ 2:

fg = 1 2 -2g Res z→±1 ω(g) 1 (x(z)) Φ(z) dz (1-53)
where Φ′ (z) = ŷ(z) x′ (z), and for g = 1:

f1 = 1 24 ln (γ 2 ŷ′ (1)ŷ ′ (-1)) (1-54)
and for g = 0:

f0 = 1 2 j≥1 γ 2 j (u j+1 -u j-1 ) 2 + 2sγ j (-1) j (u 2j-1 -u 2j+1 ) + 3s 2 2 + s 2 ln γ 2 s .
(1 -55)

Example, for quadrangulations we have s 4 = 0 and all the other s k = 0. That gives:

E quadrangulations =                x(z) = γ(z + 1 z ) ŷ(z) = s γ (z -1 z ) -s 4 γ 3 (z 3 -z -3 ) γ 2 = 1 6s 4 (1 - √ 1 -12ss 4 ), u 0 = u 2 = α = 0, u 1 = s γ , u 3 = -s 4 γ 3 .
(1-56)

That gives:

f0 = 1 2 γ 2 2 (u 3 -u 1 ) 2 + γ 2 4 (u 1 ) 2 -2sγ(u 1 -u 3 ) + 2sγ 2 (u 3 ) + 3s 2 2 + s 2 ln γ 2 s , (1-57) f1 = 1 12 ln (2(2s -γ 2 )), (1-58) f2 = 178s 3 -465s 2 γ 2 + 420sγ 4 -130γ 6 6! s 2 4 (γ 2 -2s) 5 , (1-59)
and so on... Notice that at s = s c = 1 12s 4 , ŷ has a singular branch point ŷ ∼ (x -2γ) 3/2 , and when s → s c , fg diverges as (ss c ) 5 4 (2-2g) . Anticipating on what follows, we see that the exponent 5/4 = (p + q)/(p + q -1) indeed corresponds to the (p, q) = (3, 2) minimal model, of central charge c = 0, called pure gravity. In other words, the statistics of large quadrangulations is equivalent to the pure gravity (3, 2) conformal minimal model, i.e. Liouville field theory.

Limits of large maps

It is well known that the asymptotic large size behavior of a number of objects is related to the singularities of its generating series. Therefore, the number of maps with a large number of vertices (large maps), is governed by the singularities, i.e. the values of s c , such that ln Z is not analytical at s = s c . One should consider the singularity s c closest to the origin, i.e. |s c | minimal, and see how the fg and ω(g) n 's diverge at s → s c . Thus, the scaling limit s → s c of a formal matrix integral near a singularity s c , corresponds to the asymptotics of large discrete surfaces.

For instance, one easily sees from eq.(1-44), that the expectation value of the number of vertices for maps of genus g is:

< #vertices >= s d ds ln fg (1-60)
and thus large maps become dominant when s → s c a singularity of fg . Typically, if we have an algebraic singularity of the type fg ∼ (s cs) -αg f g , the expectation value of the number of vertices is:

< #vertices >∼ α g s c s c -s , (1-61)
and we see that (s cs), i.e. the distance to critical point, can be thought of as the "mesh size", so that the area (i.e. number of vertices times mesh size) remains finite in the limit.

Another way to say that, is imagine that fg has an algebraic singularity of type

fg ∼ (s c -s) -αg f g (1-62)
and notice that

(1 -s/s c ) -αg = ∞ v=0 -α g v (-s/s c ) v = ∞ v=0 Γ(v + α g ) v! Γ(α g ) (s/s c ) v (1-63)
This means that the (possibly weighted) number of maps of genus g with v vertices, behaves for large v as:

f g s -αg-v c Γ(v + α g ) v! Γ(α g ) ∼ f g s -αg-v c v αg-1 Γ(α g ) (1-64)
where we used the large v Stirling asymptotic formula for the Γ-function. Similarly, the s → s c asymptotics of ω(g) n (x 1 , . . . , x n ) give the enumeration of large maps with n marked faces, and if we also rescale x i → (s c -s) αn,g xi , we ca also consider large maps with large marked faces.

Therefore, we see that the enumeration of large maps, is asymptotically given by the knowledge of: -the exponents α n,g (and α g = α 0,g ), -the critical point s c -and the prefactor f g .

• It turns out that the critical point s c is independent of n and g, and it can be easily found from the resolvent ω (0) 1 , it is not universal, it is strongly model dependent.

• The exponent α n,g turns out to be proportional to (2 -2gn):

α n,g = (2 -2g -n) (1 -γ string /2) (1-65)
where γ string is a universal exponent, it depends only on (p, q), and it is one of the exponents computed by the famous KPZ formula. Here, we shall see that it is:

γ string = -2 p + q -1
.

(1-66)

• The last thing to compute, is the prefactor f g , or ω

(g) n .
Here, in this article, we prove in theorem 2.3 the long claim statement that this prefactor is the same as the one computed directly from conformal field theory technics, with the Liouville theory coupled to matter reprensented by a minimal model (p, q) of central charge c = 1 -6(pq) 2 /pq (notice that the (3, 2) model has c = 0 and thus is called pure Liouville gravity). In particular, we show that the generating function of the coefficients f g , satisfies the Gelfand-Dikii non linear equation, see eq.(1-31).

Singularities of spectral invariants

One can easily convince oneself that the algebraic equations eq.(1-49) obeyed by α and γ, are singular whenever ŷ′ (1) = 0 or ŷ′ (-1) = 0, and then from theorem 1.3, one can see that the fg 's and ω(g) n 's diverge whenever ŷ′ (1) = 0 or ŷ′ (-1) = 0, i.e. whenever ŷ doesn't behave as a square root branchpoint.

Let us assume that we fix the parameters s k and s = s c such that:

ŷ(z) ∼ z→1 (x(z) -x(1)) m+ 1 2 . (1-67)
This can be obtained for instance if we choose:

V ′ (x) = (x -α -2) m (T m+1 (x -α) -T m (x -α)) s c = (-α -2) m (T ′ m+1 (-α) -T ′ m (-α)) T m+1 (-α) = T m (-α) (1-68)
where T m (z + z -1 ) = z m + z -m is the Tchebychev's polynomial of degree m. In that case we have at s = s c :

x(z) = z + 1 z ŷ(z) = (z -1) 2m+1 -( 1 z -1) 2m+1 (1-69)
When s is close to s c but not exactly equal to s c , we have like in eq.(1-26):

ŷ((s -s c ) 1 m+1 x) ∼ (s -s c ) 2m+1 2m+2 y(x) (1 + O((s -s c ) 1 m+1 ) , y(x) = m k=0 c k x k+ 1 2 .
(1-70) At s = s c we have ŷ ∼ xm+ 1 2 and at s = s c we have ŷ ∼ xx(1). The value of m and the coefficients c k depend on which limit of large maps we are interested in. Indeed we may fine-tune the coefficients s j , in order to favor one value of m or another.

Again, theorem 8.1. of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] implies that:

Theorem 1.4 (theorem 8.1. of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]): If m > 0:

fg ∼ (s -s c ) (2-2g) 2m+3 2m+2 f g , (1-71)
and if m ≥ 0:

ω(g) n ((s -s c ) 1 m+1 x 1 , . . . , (s -s c ) 1 m+1 x n ) ∼ (s -s c ) (2-2g-n) 2m+3 2m+2 -n m+1 ω (g) n (x 1 , . . . , x n ) (1 + O((s -s c ) 1 m+1 )), (1-72)
where the f g 's and ω (g) n 's are the spectral invariants of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] for the spectral curve y(x) appearing in eq. .

It was argued and highly debated, that this limit should be equivalent to the Liouville gravity conformal field theory, coupled to some matter field given by a conformal minimal model (p, q) of central charge c = 1 -6 (p-q) 2 pq . Intuitively, discrete surfaces made of a very large number of small polygons, should give a good approximation of smooth Riemann surfaces... It was indeed proved that the critical exponents -α g = (2 -2g) 2m+3 2m+2 are the same (given by KPZ formula [51,[START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF][START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF]) as those of the Liouville conformal field theory, but it is only recently that it became possible to compute explicitly partition functions and correlation functions on both sides: on the matrix model side (in particular in the double scaling limit), and in the conformal theory side.

On the Liouville conformal theory side, recent progress was obtained following Zamolodchikov, Belavin, Hosomichi, Ribault, Teschner, ... [START_REF] Belavin | On Correlation Numbers in 2D Minimal Gravity and Matrix Models[END_REF][START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF][START_REF] Belavin | On correlation functions in the perturbed minimal models M(2,2n+1)[END_REF][START_REF] Teschner | From Liouville Theory to the Quantum Geometry of Riemann Surfaces Proceedings of the ICMP 2003[END_REF][START_REF] Ribault | H(3)+ correlators from Liouville theory[END_REF][START_REF] Hosomichi | Solution of the H3+ model on a disc[END_REF].

On the matrix model side, recent progress was obtained in [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF], and formalized as a special case of the symplectic invariants of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], which allow to compute all correlation functions of all genus.

From here, we can repeat all what was said in section 1.1, after theorem 1.1.

In this article, we shall show how to apply the spectral invariants method of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], for the double scaling limit of matrix models which is expected to coincide with Liouville theory.

We prove that the scaling limits of the matrix model correlation functions, i.e. the generating functions counting discrete surfaces, is indeed the (p, 2) reduction of KdV satisfying string equation, i.e. the minimal model (p, 2).

Minimal models

There exists several equivalent definitions of minimal models coupled to gravity. They correspond to representations of the conformal group in 2 dimensions. They are classified by two integers (p, q), and their central charge is:

c = 1 -6 (p -q) 2 pq (2-1)
Some of them have received special names:

• (1, 2) = Airy, c = -2 (related to Tracy-Widom law [66]) • (3, 2) = pure gravity, c = 0 • (5, 2) = Lee-Yang edge singularity, c = -22 5 • (4, 3) = Ising, c = 1 2 • (6, 5) = Potts-3, c = 4 5
Minimal models can also be viewed as finite reductions of the Kadamtsev-Petviashvili (KP) integrable hierarchy of partial differential equations.

The case q = 2 is a little bit simpler to address, and is a reduction of the Korteweg de Vries (KdV) hierarchy.

The KdV hierarchy, and the minimal models (p, 2) have generated a huge amount of works, and have been presented in many different (but equivalent) formulations. For instance in terms of a string equation for differential operators, in terms of a Lax pair, in terms of commuting hamiltonians, in terms of Schrödinger equation, in terms of Hirota equations, in terms of isomonodromic systems, in terms of Riemann Hilbert problems, in terms of tau functions, in terms of Grasman manifolds, in terms of Yang-Baxter equations, ...etc, see [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF] for a comprehensive lecture.

All those formulations are equivalent, and let us recall some of the well known features of the (p, 2) reduction of KdV (see [START_REF] Di Francesco | Conformal field theory[END_REF][START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF]), presented in a way convenient for our purposes.

String equation

The KdV minimal model (p, 2) with p = 2m + 1 can be formulated in terms of two differential operators P , Q of respective orders p and 2, satisfying the string equation:

[P, Q] = 1 N Id (2-2) Q = d 2 -2u(t) , P = d p -p u d p-2 + . . . , d = 1 N d dt (2-3) 1
N is a scaling parameter, which we can send to zero to get the "classical limit".

The general solution of the string equation eq.(2-2) is of the form:

P = m j=0 t j (Q j+1/2 ) + , t m = 1 (2-4)
where (Q j+1/2 ) + is the unique differential operator of order 2j + 1, such that:

order[((Q j+1/2 ) + ) 2 -Q 2j+1 ] ≤ 2j. (2-5)
For example:

(Q 1/2 ) + = d , (Q 3/2 ) + = d 3 -3ud - 3 u 2 , (2-6) (Q 5/2 ) + = d 5 -5ud 3 - 15 u 2 d 2 - 25ü 4 d - 45u 2 2 d - 15 8
...

u - 45u u 2 . (2-7)
It is a classical result (see [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF]) that it satisfies:

[(Q j-1/2 ) + , Q] = 1 N d dt (R j (u(t))) (2-8)
where the right hand side is a function (a differential operator of order 0), and the coefficients R j (u) are the Gelfand-Dikii differential polynomials [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF]. They can be obtained by the recursion:

R 0 = 2 , Ṙj+1 = -2u Ṙj -uR j + 1 4 N 2 ...

Rj .

(2-9)

The first few of them are:

R 0 = 2 R 1 = -2u R 2 = 3 u 2 -1 2 N 2 ü R 3 = -5u 3 + 5 2N 2 uü + 5 4N 2 u2 - 1 8N 4 .... u . . . (2-10)
and in general:

R j (u) = 2 (-1) j (2j -1)!! j! u j + . . . - 2 (2N) 2j-2 u (2j-2) .
(2-11)

After substitution of eq.(2-4) into the string equation eq.(2-2), the property eq.(2-8) gives a non-linear differential equation for the function u(t):

m j=0 t j R j+1 (u) = t.
(2-12) Since R 0 = 2, we see that we can identify t with t = -2t -1 .

• For instance for Airy p = 1, this gives:

-2u = t.
(2-13)

• For instance for pure gravity p = 3, this is the Painlevé I equation:

3 u 2 - 1 2N 2 ü -2t 0 u = t.
(2-14)

• For instance for Lee-Yang p = 5, we have:

-5u 3 + 5 2N 2 uü - 1 4N 2 u2 - 1 8N 4 .... u + t 1 (3 u 2 - 1 2N 2 ü) -2t 0 u = t.
(2-15)

Tau function

We define the Tau-function τ (t, t 0 , . . . , t m ) and its log, the free energy function F (t, t 0 , . . . , t m ) = ln τ (t, t 0 , . . . , t m ) such that:

N -2 F = u. (2-16)
The Tau-function has many other properties, which can be found in textbooks and classical works on the subject [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF][START_REF] Krichever | The τ -function of the universal Whitham hierarchy, matrix models and topological field theories[END_REF][START_REF] Krichever | The τ -function of the universal Whitham hierarchy, matrix models and topological field theories[END_REF], but which are beyond the scope of the present article.

Lax pair

Consider the following matrices:

R(x, t) = 0 1 x + 2u(t) 0 , (2-17)
and for any integer k:

D k (x, t) = A k B k C k -A k , (2-18)
where

A k (x, t), B k (x, t), C k (x, t) are polynomials of respective degree k -1, k, k + 1 in
x, which are determined by:

B k (x, t) = k j=0 x k-j R j (u) , A k = - 1 2N Ḃk , C k = (x + 2u) B k + 1 N Ȧk .
(2-19) The recursion relation eq.(2-9) implies that B k satisfies the equation:

2 uB k + 2(x + 2u) Ḃk - 1 2N 2 ... Bk = -2 Ṙk+1 (u) (2-20)
and we see that the matrix D k (x, t) satisfies:

1 N ∂ ∂t D k (x, t) + [D k (x, t), R(x, t)] = - 2 N Ṙk+1 (u) 0 0 1 0 , (2-21)
the right hand side is independent of x, and is proportional to ∂ ∂x R(x, t).

Lax equation

If we consider u solution of the string equation eq.(2-12), then, the matrix:

D(x, t) = m j=0 t j D j (x, t) , t m = 1 (2-22)
satisfies the Lax equation:

1 N ∂ ∂t D(x, t) + [D(x, t), R(x, t)] = - 2 N ∂ ∂x R(x, t) (2-23)
which can also be written as a commutation relation:

2 N ∂ ∂x + D(x, t), R(x, t) - 1 N ∂ ∂t = 0 (2-24)
This relation means that the operator 2

N ∂ ∂x + D(x, t) is a Lax operator [2].

The differential system

The Lax equation eq.(2-24) is the compatibility condition, which says that the following two differential systems have a common solution Ψ(x, t):

1 N d dx Ψ(x, t) = - 1 2 D(x, t) Ψ(x, t) , 1 N d dt Ψ(x, t) = R(x, t) Ψ(x, t) (2-25)
and Ψ(x, t) is a matrix such that:

Ψ(x, t) = ψ φ ψ φ , det Ψ = 1.
(2-26)

In particular we have the Schrödinger equation for ψ:

1 N 2 ψ(x, t) = (x + 2u(t)) ψ(x, t) (2-27)
where t can be interpreted as the space variable, and x the energy. x is called the spectral parameter.

Correlators

Consider the Christoffel-Darboux kernel associated to the system D(x):

K(x 1 , x 2 ) = ψ(x 1 ) φ(x 2 ) -ψ(x 1 )φ(x 2 ) x 1 -x 2 (2-28)
Definition 2.1 We define the connected correlation functions by the "determinantal formulae":

W 1 (x) = lim x ′ →x K(x, x ′ ) - 1 x -x ′ = ψ ′ (x) φ(x) -ψ′ (x)φ(x) (2-29)
and for n ≥ 2:

W n (x 1 , . . . , x n ) = - δ n,2 (x 1 -x 2 ) 2 -(-1) n σ=cyles n i=1 K(x σ(i) , x σ(i+1) ) (2-30)
For example:

W 3 (x 1 , x 2 , x 3 ) = K(x 1 , x 2 )K(x 2 , x 3 )K(x 3 , x 1 ) + K(x 1 , x 3 )K(x 3 , x 2 )K(x 2 , x 1 ). (2-31)
Although we have not written it explicitly, the kernel K and the correlators W n depend on t.

The non-connected correlation functions are defined by: where J = {x 1 , . . . , x n } and the sum runs over all partitions of J into k non-empty disjoint subsets. In other words, the connected W n 's are the cumulants of the nonconnected ones.

W n, n.c. (x 1 , . . . , x n ) = k J 1 ∪J 2 ∪...∪J k =J k i=1 W |J i | (J i ),
For instance:

W 2, n.c. (x 1 , x 2 ) = W 2 (x 1 , x 2 ) + W 1 (x 1 )W 1 (x 2 ), (2-33) W 3, n.c. (x 1 , x 2 , x 3 ) = W 3 (x 1 , x 2 , x 3 ) + W 1 (x 1 )W 2 (x 2 , x 3 ) + W 1 (x 2 )W 2 (x 1 , x 3 ) +W 1 (x 3 )W 2 (x 1 , x 2 ) + W 1 (x 1 )W 1 (x 2 )W 1 (x 3 ). (2-34)
The formula eq.(2-30) is called "determinantal formula", because for the non-connected correlation functions we have:

W n, n.c. (x 1 , . . . , x n ) = ′ det(K(x i , x j )), (2-35)
where det ′ means that when we compute the determinant as a sum over permutations of products (-1) σ i K(x i , x σ(i) ), then if σ(i) = i we replace K(x i , x i ) by W 1 (x i ), and if σ(i) = j and σ(j) = i, we replace K(x i , x j )K(x j , x i ) by -W 2 (x i , x j ), see [START_REF] Bergère | Correlation functions of complex matrix models[END_REF].

For instance W 3, n.c. is the sum of 6 terms coming from the 6 permutations:

W 3, n.c. (x 1 , x 2 , x 3 ) = ′ det   K(x 1 , x 1 ) K(x 1 , x 2 ) K(x 1 , x 3 ) K(x 2 , x 1 ) K(x 2 , x 2 ) K(x 2 , x 3 ) K(x 3 , x 1 ) K(x 3 , x 2 ) K(x 3 , x 3 )   = W 1 (x 1 )W 1 (x 2 )W 1 (x 3 ) + W 1 (x 1 )W 2 (x 2 , x 3 ) + W 1 (x 2 )W 2 (x 1 , x 3 ) +W 1 (x 3 )W 2 (x 1 , x 2 ) + K(x 1 , x 2 )K(x 2 , x 3 )K(x 3 , x 1 ) +K(x 1 , x 3 )K(x 3 , x 2 )K(x 2 , x 1 ) (2-36)
It was proved in [START_REF] Bergère | Determinantal formulae and loop equations[END_REF], that the correlators W n satisfy an infinite set of equations, called loop equations, and equivalent to Virasoro constraints for the τ function. The loop equation simply states that the following quantity: Theorem 2.1 Loop equations (proved in [START_REF] Bergère | Determinantal formulae and loop equations[END_REF]):

P n (x; x 1 , . . . , x n ) = W n+2, n.c. (x, x, x 1 , . . . , x n ) + n j=1 ∂ ∂x j W n (x, x 1 , . . . , x j-1 , x j+1 , . . . , x n ) -W n (x 1 , . . . , x n ) x -x j (2 -37)
is a polynomial of the variable x.

For example, one can easily check that:

P 0 (x) = W 2 (x, x) + W 1 (x) 2 = -det D(x, t). (2-38) Notice that 1 N ∂ ∂t det D(x, t) = 2B(x, t) = 2 m j=0 t j B j (x, t).
(2-39)

Classical limit

The classical limit is the large N limit, or equivalently, it is also the large t limit. Intuitively, in the classical limit, P and Q commute, and they can be represented without differential operators. In this limit d → z can be represented as a number, and operators Q = d 2 -2u and P are replaced by functions of z and t. Therefore, in analogy with Q = d 2 -2u(t), and P = d p + . . ., let us define two functions x(z, t) and y(z, t):

x(z, t) = z 2 -2u 0 (t) , y(z, t) = z p + . . . .

(2-51)

In the classical limit, we replace the string equation [P, Q] = N -1 with a Poisson bracket:

{y, x} = 1 = ∂y ∂z ∂x ∂t - ∂y ∂t ∂x ∂z (2-52)
whose general solution is:

x(z, t) = z 2 -2u 0 (t) , y(z, t) = m j=0 t j z 2j+1 (1 - 2u 0 (t) z 2 ) j+1/2 + , (2-53)
where () + means the positive part in the large z Laurent series expansion. Explicitly we get:

y(z, t) = m j=0 j l=0 t j z 2j+1-2l (-u 0 /2) l (2j + 1)! j! (j -l)! l! (2j + 1 -2l)! . (2-54)
The string equation {y, x} = 1 reduces to:

u0 y ′ (0) = -1 2 , (2-55)
i.e.

m j=0 t j u0 (-u 0 /2) j (2j + 1)! (j!) 2 = - 1 2 (2-56)
which can be integrated with respect to t and gives a polynomial equation for u 0 (t):

P(u 0 ) = m j=0 t j (-u 0 /2) j+1 (2j + 1)! j! (j + 1)! = t 4 (2-57)
which is clearly the classical limit of eq. [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF][START_REF] Belavin | On Correlation Numbers in 2D Minimal Gravity and Matrix Models[END_REF][START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF][START_REF] Belavin | On correlation functions in the perturbed minimal models M(2,2n+1)[END_REF][START_REF] Bertola | Boutroux curves with external field: equilibrium measures without a minimization problem[END_REF][START_REF] Bergère | Correlation functions of complex matrix models[END_REF][START_REF] Bergère | Determinantal formulae and loop equations[END_REF][START_REF]Random Matrix Models and Their Applications[END_REF][START_REF] Bleher | Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model[END_REF][START_REF] Bleher | Double scaling limit in the matrix model: the Riemann-Hilbert approach[END_REF][START_REF] Bleher | Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations[END_REF]. In other words, the non-linear differential equation eq. [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF][START_REF] Belavin | On Correlation Numbers in 2D Minimal Gravity and Matrix Models[END_REF][START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF][START_REF] Belavin | On correlation functions in the perturbed minimal models M(2,2n+1)[END_REF][START_REF] Bertola | Boutroux curves with external field: equilibrium measures without a minimization problem[END_REF][START_REF] Bergère | Correlation functions of complex matrix models[END_REF][START_REF] Bergère | Determinantal formulae and loop equations[END_REF][START_REF]Random Matrix Models and Their Applications[END_REF][START_REF] Bleher | Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model[END_REF][START_REF] Bleher | Double scaling limit in the matrix model: the Riemann-Hilbert approach[END_REF][START_REF] Bleher | Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations[END_REF] for u(t), becomes an algebraic equation for u 0 (t).

For example, for pure gravity m = 1 we have the classical limit of eq.(2-14):

4P(u 0 ) = 3 u 2 0 -2t 0 u 0 = t.
(2-58)

Topological expansion

We now have the polynomial equation eq.(2-57):

P(u 0 ) = t/4 (2-59)
which implies: -60) and in general, any derivative of u 0 with respect to t can be written as a rational function of u 0 . Since u 0 (t) satisfies the string equation eq. [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF][START_REF] Belavin | On Correlation Numbers in 2D Minimal Gravity and Matrix Models[END_REF][START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF][START_REF] Belavin | On correlation functions in the perturbed minimal models M(2,2n+1)[END_REF][START_REF] Bertola | Boutroux curves with external field: equilibrium measures without a minimization problem[END_REF][START_REF] Bergère | Correlation functions of complex matrix models[END_REF][START_REF] Bergère | Determinantal formulae and loop equations[END_REF][START_REF]Random Matrix Models and Their Applications[END_REF][START_REF] Bleher | Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model[END_REF][START_REF] Bleher | Double scaling limit in the matrix model: the Riemann-Hilbert approach[END_REF][START_REF] Bleher | Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations[END_REF] at N = ∞, the full solution u(t) to the string equation eq. [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF][START_REF] Belavin | On Correlation Numbers in 2D Minimal Gravity and Matrix Models[END_REF][START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF][START_REF] Belavin | On correlation functions in the perturbed minimal models M(2,2n+1)[END_REF][START_REF] Bertola | Boutroux curves with external field: equilibrium measures without a minimization problem[END_REF][START_REF] Bergère | Correlation functions of complex matrix models[END_REF][START_REF] Bergère | Determinantal formulae and loop equations[END_REF][START_REF]Random Matrix Models and Their Applications[END_REF][START_REF] Bleher | Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model[END_REF][START_REF] Bleher | Double scaling limit in the matrix model: the Riemann-Hilbert approach[END_REF][START_REF] Bleher | Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations[END_REF], can be expanded as an N -2 power series:

u0 = 1 4P ′ (u 0 ) , ü0 = -P ′′ (u 0 ) 16 (P ′ (u 0 )) 3 , . . . ( 2
u(t) = u 0 + k N -2k u k (t) (2-61)
where all coefficients u k are rational functions of u 0 (their denominator is a power of P ′ (u 0 )).

For example for pure gravity m = 1, the Painlevé equation eq. [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF][START_REF] Belavin | On Correlation Numbers in 2D Minimal Gravity and Matrix Models[END_REF][START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF][START_REF] Belavin | On correlation functions in the perturbed minimal models M(2,2n+1)[END_REF][START_REF] Bertola | Boutroux curves with external field: equilibrium measures without a minimization problem[END_REF][START_REF] Bergère | Correlation functions of complex matrix models[END_REF][START_REF] Bergère | Determinantal formulae and loop equations[END_REF][START_REF]Random Matrix Models and Their Applications[END_REF][START_REF] Bleher | Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model[END_REF][START_REF] Bleher | Double scaling limit in the matrix model: the Riemann-Hilbert approach[END_REF][START_REF] Bleher | Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations[END_REF][START_REF] Boutet De Monvel | On the statistical mechanics approach in the random matrix theory: Integrated density of states[END_REF](14) implies that to the first few orders we have:

u(t) = u 0 - 3 N 2 (6u 0 -2t 0 ) -4 + O(N -4 ). (2-62)
And the Free energy F (t) such that u = 1 N 2 F , also has a 1/N 2 expansion:

ln τ = F = ∞ g=0 N 2-2g F g (u 0 ) , Fg = u g . (2-63)
Also, since the coefficients of the differential system D(x, t) depend on u(t), the matrix D(x, t) has a 1/N 2 expansion:

D(x, t) = g N -2g D (g) (x, t) (2-64)
To leading order we have:

D (0) (x, t) = 0 B(x, u 0 ) (x + 2u 0 ) B(x, u 0 ) 0 (2-65) B(x, u 0 ) = 2 m j=0 j k=0 t j x j-k u k 0 (-1) k (2k -1)!! k! (2-66)
Notice that:

z B(z 2 -2u 0 , u 0 ) = y(z, t). (2-67)
The classical spectral curve is given by the eigenvalues of D (0) (x, t)), i.e. the values of y such that det (y -D (0) (x, t)) = 0, i.e., if we parametrize x as x = z 2 -2u 0 , we have:

y = ± y(z, t) (2-68)
where y(z, t) is the function defined in eq. . This explains why we call the function y(z, t) the classical spectral curve.

Written in a parametric form where u 0 = u 0 (t), the classical spectral curve is thus:

E (2m+1,2) = x(z) = z 2 -2u 0 y(z) = j l t j z 2j+1-2l (-u 0 /2) l (2j+1)! j! (j-l)! l! (2j+1-2l)! (2-69)
It is important to notice that it is a genus 0 hyperelliptical curve, which is equivalent to saying that it can be parametrized by a complex variable z (higher genus would be parametrized by a variable z living on a Riemann surface), and which is equivalent to saying that the polynomial y 2 , written as a polynomial in x, has only one simple zero, located at x = -2u 0 , all the other zeroes are double zeroes.

BKW expansion

Similarly, we can look for a BKW asymptotic solution of the solutions ψ(x, t) of the differential system. It takes the form:

ψ(x, t) ∼ e N R x -2u 0 ydx √ 2 (-x -2u 0 ) 1 4 1 + k N -k ψ k (x, u 0 ) (2-70) ψ(x, t) ∼ e N R x -2u 0 ydx (x + 2u 0 ) 1 4 1 + k N -k ψk (x, u 0 ) (2-71)
and we recall that z = (x + 2u 0 )

1 2
. the BKW expansion of the other solutions φ and φ, are obtained by changing the sign of the square root z → -z.

We have the following Lemma:

Lemma 2.1 Each ψ k (x, u 0 ) and ψk (x, u 0 ) is a polynomial of 1/z.

proof:

The proof uses the Schrödinger equation eq.(2-27):

1 N 2 ψ(x, t) = (x + 2u(t)) ψ(x, t).
(2-72)

Let us write:

ψ(x, t) = f (x, t) e R t dt ′ f (x,t ′ ) . (2-73)
The Schrödinger equation implies that:

N 2 (x + 2u(t)) f 2 (x, t) = 1 2 f (x, t) f (x, t) - 1 4 ḟ (x, t) 2 + 1, (2-74)
and after differentiating once more with respect to t, we obtain a third order linear equation for f :

(x + 2u(t)) ḟ(x, t) + u(t) f (x, t) = 1 2 N 2 ... f (x, t).
(2-75)

To leading order we have u(t) = u 0 (t), and recall that u(t) has a 1/N 2 expansion, therefore, one easily sees that:

f (x, t) = -1 N x + 2u 0 (t) 1 + k N -2k f k (x, t) , (2-76)
and by an easy recursion, we see that each f k (x, t) is a polynomial in 1/z with z = x + 2u 0 (t). Then, notice that the Poisson equation eq.(2-52) implies:

∂ y ∂t x = - 1 x ′ (z) = - 1 2z = - 1 2 √ x + 2u 0 (2-77)
And therefore:

∂ x ydx ∂t x = -z. (2-78)
This implies that:

t 1 f (x, t) = N x ydx + k≥1 N 1-2k g k (x, t) (2-79)
and where all coefficients g k (x, t) are polynomials of 1/z.

Since ψ(x, t) = f (x, t) e R t dt ′ f (x,t ′ )
, we find that ψ(x, t) is of the form:

ψ(x, t) ∼ e N R x ydx (x + 2u 0 ) 1 4 1 + k N -k ψ k (x, u 0 ) (2-80)
where each ψ k (x, t) is a polynomial in 1/z.

The proof for ψ(x, t) works in a similar manner.

This lemma implies that the kernel also have a 1/N expansion:

K(z 1 , z 2 ) = e N R z 1 z 2 ydx 2 √ z 1 z 2 (z 1 -z 2 ) 1 + k N -k K k (z 1 , z 2 ) , (2-81)
where each K k (z 1 , z 2 ) is a polynomial in 1/z 1 and in 1/z 2 .

This implies that the correlators also have a 1/N expansion:

Lemma 2.2 W n (x 1 , . . . , x n ) = g N 2-2g-n W (g) n (x 1 , . . . , x n ) (2-82)
where each W

n is a rational function of the z i = √ x i + 2u 0 , with poles only at z i = 0, except W (0) 2 and W (0) 1 which are:

W (0) 1 = y(z, t) (2-83) W (0) 2 = 1 4z 1 z 2 1 (z 1 -z 2 ) 2 - 1 (z 2 1 -z 2 2 ) 2 = 1 4z 1 z 2 (z 1 + z 2 ) 2 .
(

The important point, is that each W

n has no other pole than z i = 0, in particular, has no pole at the other zeroes of y(z, t). proof:

Notice that in the products i K(z σ(i),σ(i+1) ), all the exponentials cancel, and the result is, order by order in N -k , a rational fraction of the z i 's having poles at z i = 0, or at z i = z j . Except for W (0) 1 and W (0) 2 , the poles at z i = z j are simple poles, and it is easy to see that in the sum over permutations, the residues cancel, therefore, each W (g) n is a rational function of the z i 's having poles only at z i = 0. The cases of W 2 and W 1 need to be treated separately, and are easy.

The fact that W n has a 1/N 2 expansion instead of 1/N comes from a simple symmetry argument. In the expression of W n , changing ψ → φ and ψ → φ, can also be obtained as changing the order of the x i 's, and since we take a symmetric sum, only the terms which are invariant under the exchange ψ → φ and ψ → φ contribute to W n . Exchanging the two solutions ψ → φ and ψ → φ, is also equivalent to changing N → -N, and therefore W n has a given parity in N.

Symplectic invariants

It was found in [START_REF] Bergère | Determinantal formulae and loop equations[END_REF], that the correlators obtained from the determinantal formulae eq.(2-29), eq.(2-30) of a Christoffel-Darboux kernel K of type eq.(2-28), do satisfy loop equations, i.e. for any n and g, and J = {x 1 , . . . , x n }, the following quantity:

P (g) n (x; J) = g h=0 I⊂J W (h) 1+|I| (x, I)W (g-h)
1+n-|I| (x, J/I)

+ n j=1 ∂ ∂x j W (g) n (x, J/{x j }) -W (g) n (x j , J/{x j }) x -x j (2-85)
is a polynomial in x. This property, as was proved in [START_REF] Bergère | Determinantal formulae and loop equations[END_REF], is a direct consequence of eq.(2-28) and eq.(2-29), eq.(2-30).

Moreover we know from section 2.10, that W (g) n (x(z 1 ), . . . , x(z n )) the following differential form:

W (g) n (z 1 , . . . , z n ) = W (g) n (x(z 1 ), . . . , x(z n )) x ′ (z 1 ) . . . x ′ (z n ) + δ n,2 δ g,0 x ′ (z 1 )x ′ (z 2 ) (x(z 1 ) -x(z 2 )) 2 (2-86)
is a symmetric rational function of all its variables, and if 2g + n -2 > 0, due to lemma 2.2, it has poles only at z i = 0, and

W (0) 2 (z 1 , z 2 ) = 1 (z 1 -z 2 ) 2
(2-87)

It was found in [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF], that the unique solution of loop equations eq.(2-85) which has a topological expansion for which the W (g) n 's have the poles given by lemma 2.2, can be obtained by the following recursion relation:

Theorem 2.2 W (g) n+1 (z 1 , . . . , z n , z n+1 ) = -1 4 Res z→0 dz (z 2 n+1 -z 2 ) y(z) W (g-1)
n+2 (z, -z, J)

+ g h=0 ′ I⊂J W (h) 1+|I| (z, I)W (g-h) 1+n-|I| (-z, J/I) (2-88)
where J = {z 1 , . . . , z n }, and h ′ I , means that we exclude the terms (h, I) = (0, ∅) and (h, I) = (g, J).

proof:

The proof proceeds exactly like in [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF]. Write the Cauchy residue formula:

W (g) n+1 (z 1 , . . . , z n , z n+1 ) = Res z→z n+1 dz z -z n+1 W (g) n+1 (z 1 , . . . , z n , z) (2-89)
and move the integration contour, to enclose all the other poles, i.e. only z = 0, and thus:

W (g) n+1 (z 1 , . . . , z n , z n+1 ) = Res z→0 dz z n+1 -z W (g) n+1 (z 1 , . . . , z n , z) = Res z→0 x ′ (z) dz z n+1 -z W (g) n+1 (x(z 1 ), . . . , x(z n ), x(z)) (2 -90)
Then, insert in the right hand side eq.(2-85):

-2W (0) 1 (x)W (g) n+1 (x 1 , . . . , x n , x) = g h=0 ′ I⊂J W (h) 1+|I| (x, I)W (g-h) 1+n-|I| (x, J/I) + n j=1 ∂ ∂x j W (g) n (x, J/{x j }) -W (g) n (x j , J/{x j }) x -x j -P (g) n (x; x 1 , . . . , x n ) (2-91)
and notice that the polynomial P The recursion relation eq.(2-88) is precisely the definition of the symplectic invariant's correlators defined in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]. In [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], it is explained how to associate an infinite family of W (g) n 's, to any spectral curve defined by a pair of functions (x(z), y(z)). Examples:

• eq.(2-88) gives:

W (0) 3 (z 1 , z 2 , z 3 ) = 1 2 y ′ (0) 1 z 2 1 z 2 2 z 2 3
(2-92)

• eq.(2-88) gives:

W (1) 1 (z 1 ) = -1 4 Res z→0 dz (z 2 1 -z 2 ) y(z) 1 4z 2 = -1 32
d 2 dz 2 z (z 2 1 -z 2 ) y(z) z=0 = y ′′′ (0) 48 y ′ (0) 2 1 z 2 1 - 1 16 y ′ (0) 1 z 4 1
(2-93) 2.12 Double scaling limit and (p, 2) kernel By comparison with theorem 1.2, we conclude that: Theorem 2.3 the s → s c double scaling limit of (possibly formal) matrix integrals correlation functions W (g) n are the determinantal formula correlation functions of the (p, 2) kernel:

ω (g) n (x 1 , . . . , x n ) = W (g) n (x 1 , . . . , x n ). (2-94)
where

ω (g) n (x 1 , . . . , x n ) = lim s→sc (s -s c ) (2g+n-2) 2m+3 2m+2 + n m+1 ω(g) n ((s -s c ) 1 m+1 x 1 , . . . , (s -s c ) 1 m+1 x n ) (2-95)
Since φ n (x) is discontinuous across the integration contour of dx ′ , the matrix Ψ n (x) is also discontinuous, and has jumps of the form:

Ψ n + (x) = Ψ n -(x) 1 2iπ 0 1 . (2-104)
Therefore Ψ n (x) satisfies an isomonodromic Riemann-Hilbert problem (the jump matrix, called the monodromy, is independent of n and of V (x)).

A general method was invented [START_REF] Deift | Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory[END_REF] to find large N asymptotic solution of isomonodromic Riemann-Hilbert problems. In the case where we approach a (2m + 1, 2) singularity, the method of Deift& co [START_REF] Deift | Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory[END_REF] requires to have an ansatz for a parametrix asymptotics for Ψ n (x) in the vicinity of the singularity.

We claim that the correct parametrix for the (2m + 1, 2) singularity, is the matrix of Baker-Akhiezer functions of eq.(2-26) for the (2m + 1, 2) minimal model:

Ψ n ((s -s c ) 1 m+1 x) ∼ Ψ(x) (1 + O((s -s c ) 1 m+1 )).
(2-105)

This should be checked by the steepest descent Riemann-Hilbert method of [START_REF] Deift | Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory[END_REF].

Kontsevich's integral

In this section, we also propose a combinatorical interpretation of the coefficients of the f g 's and ω The spectral curve eq.(2-69) of the (2m + 1, 2) minimal model

E (2m+1,2) = x(z) = z 2 -2u 0 y(z) = j l t j z 2j+1-2l (-u 0 /2) l (2j+1)! j! (j-l)! l! (2j+1-2l)! (3-1)
is of the same form as the Kontsevich integral's spectral curve (see in [START_REF] Eynard | Recursion between Mumford volumes of moduli spaces[END_REF], or see below), and thus it has the same correlators and spectral invariants F g as those of the Kontsevich integral. Therefore, the correlators and F g 's of the minimal model (2m + 1, 2) can be written as integrals of tautological classes on the moduli spaces of Riemann surfaces. This can be viewed as a proof that the double scaling limits of matrix models, i.e. the limit of large maps generating function, indeed coincides with topological gravity, as claimed by Witten [START_REF] Witten | Two-dimensional gravity and intersection theory on moduli space[END_REF][START_REF] Dijkgraaf | Intersection theory, integrable hierarchies and topological field theory[END_REF] and then proved by Kontsevich [START_REF] Kontsevich | Intersection theory on the moduli space of curves and the matrix Airy function[END_REF]. The Kontsevich integral [START_REF] Kontsevich | Intersection theory on the moduli space of curves and the matrix Airy function[END_REF]:

Z K (Λ) = dMe -N Tr M 3 3 -M Λ 2 = e P g N 2-2g Fg , τ k = 1 N Tr Λ -k (3-2)
(to simplify we assume τ 1 = 0 here) is the generating function for intersection numbers of cotangent line bundles at marked points of Riemann surfaces of genus g:

F g = W (g) 0 = P i d i =3g-3 i τ d i i d i ! i ψ d i i Mg , ψ i = c 1 (L i ) (3-3)
where ψ i is the Chern class of the cotangent line bundle at point i, and where τi = (2i -1)!! τ 2i+1 [START_REF] Belavin | On Correlation Numbers in 2D Minimal Gravity and Matrix Models[END_REF][START_REF] Belavin | Four-point function in Super Liouville Gravity[END_REF] and more generally, correlation functions of the Kontsevich integral give access [START_REF] Eynard | Recursion between Mumford volumes of moduli spaces[END_REF] to integrals of Mumford κ characteristic classes [START_REF] Mumford | Towards an enumerative geometry of the moduli space of curves[END_REF]:

W (g) n (z 1 , . . . , z n ) = 2 -dg,n (τ 3 -2) 2-2g-n d 0 +d 1 +...+dn=dg,n d 0 k=1 1 k! b 1 +...+b k =d 0 ,b i >0 n i=1 2d i + 1! d i ! dz i z 2d i +2 i k l=1 τb l < k l=1 κ b l n i=1 ψ d i i > g,n (3 -5)
The class κ 0 is the Euler class, and 2πκ 1 is the curvature form of the Weil-Petersson symplectic metrics. The dual times τk are closely related to the τ k 's, see the relation in [START_REF] Eynard | Recursion between Mumford volumes of moduli spaces[END_REF] or eq.(3-9) below. It was shown in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF][START_REF] Eynard | Recursion between Mumford volumes of moduli spaces[END_REF], that Kontsevich's integral's W (g) 0

= F g 's and correlators W (g) n can be computed as the symplectic invariants F g = F g (E K ) of a spectral curve:

E K = x(z) = z 2 y(z) = z -1 2 j τ j+2 z j (3-6)
We see that the minimal model E (p,2) spectral curve eq.(2-69) can be identified with Kontsevich integral's spectral curve E K , under the identification of times:

δ k,0 - 1 2 τ 2k+3 = k! (2k + 1)! l t l+k (-u 0 /2) l (2l + 2k + 1)! l! (l + k)! (3-7)
In particular 1 -

1 2 τ 3 = y ′ (0) = 1 -2 u0 (3-8)
The dual times τk are given by their generating function g(r) = k τk r k and g(r) = ln (1g(r)) with:

1 -g(r) = -2 u0 k≥0 r k l t l+k (-u 0 /2) l (2l + 2k + 1)! l! (l + k)! = e -g(r) (3-9) g(r) = k τk r k (3-10) I.e. 1 -g(r) = -2 u0 j j l=0 t j r j-l (-u 0 /2) l (2j + 1)! l! j! = -2 u0 j (2j + 1)! j! t j r j j l=0 1 l! (-u 0 /2r) l = -2 u0 j (2j + 1)! j! t j r j e -u 0 /2r + (3 -11)

Derivatives

The general method to compute derivatives of F g and W (g) n 's with respect to any parameter entering the spectral curve is explained in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF].

Here, our spectral curve E (p,2) depends on the parameters t j 's and t = -2t -1 .

[36] says that we first have to study the variation of y(z)x ′ (z) under variation of any such parameter, and we write it: ∂y(z) ∂t j

x ′ (z) -∂x(z) ∂t j y ′ (z) = Λ ′ j (z) (4-1)

Here, we find for j ≥ 0:

Λ j (z) = -2 (2j + 1)! j! u 0 l z 2j+1-2l (-u 0 /2) l (jl)! (l + 1)! (2j + 1 -2l)! (4-2)

and for j = -1:

Λ -1 (z) = -2 u0 y(z) = - u0 z m j=0 t j Λ ′ j (z) (4-3)
The theorem 5.1 of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], then shows that those functions are such that: ∂W This implies:

∂ 3 F 0 ∂t j 1 ∂t j 2 ∂t j 3 = Res ∞ Res ∞ Res ∞ W (0) 3 (z 1 , z 2 , z 3 ) Λ j 1 (z 1 ) Λ j 2 (z 2 ) Λ j 3 (z 3 ) = -1 2y ′ (0) Λ ′ j 1 (0) Λ ′ j 2 (0) Λ ′ j 3 (0) (4 -8)
Notice that Λ ′ j (0) = (2j + 1)! j! (j + 1)! (-2u 0 ) j+1 2 -2j , Λ ′ -1 (0) = 1 (4-9)

In particular this implies that:

∂ 3 F 0 ∂t 3 = -1 2y ′ (0) = u0 (t) (4-10)
and thus, as expected we recover: The recursion relation eq.(2-88) gives:

W (1)
1 (z) = W In other words, W

n is homogeneous of degree 2 -2gn.

5 The (2m, 1) minimal model

Another kind of universal limit of matrix models may arise when two connected components of the eigenvalues support merge, typically the equilibrium density of eigenvalues behaves as: ŷ ∼ x 2m .

(5-1)

The case m = 1 was treated in [START_REF] Bleher | Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model[END_REF][START_REF] Bleher | Double scaling limit in the matrix model: the Riemann-Hilbert approach[END_REF][START_REF] Brower | Symmetry breaking in the double-well hermitian matrix models[END_REF] The results concerning general m, were described without proof in [START_REF] Bleher | Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations[END_REF]. The universal limit is given by the (2m, 1) reduction of the mKdV hierarchy. All the results can be proven in a way very similar to the (2m + 1, 2) case, and here we just summarize the results the results stated in [START_REF] Bleher | Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations[END_REF]. Figure 2: When two cuts merge, the equilibrium density of eigenvalues behaves like x 2m near the merging endpoints. The universal eigenvalues statistics is given by determinants of the (2m, 1) kernel, associated to the integrable mKdV hierarchy.

asymptotics of the ω(g) n 's and fg 's are indeed those obtained from conformal field theory. Our proof is based on the fact that the limits ω (g) n 's of matrix models correlators, are the spectral invariants of the limit spectral curve, and the fact that the determinantal correlators of the (2m + 1, 2) minimal model kernel are also the spectral invariants of the same spectral curve.

We recall that those correlators can be interpreted in the Kontsevich integral's framework, and have a combinatorial interpretation as intersection numbers of some tautological classes on the moduli spaces of Riemann surfaces.

We claim that the same method can be applied to other sorts of universal limits, in particular the merging of two cuts like in [START_REF] Bleher | Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations[END_REF], and hopefully, we can work out the same kind of proof for multi-matrix models, whose universal limits should be the (p, q) minimal models with arbitrary p and q. Unfortunately, one of the key points should be the equivalent of [START_REF] Bergère | Determinantal formulae and loop equations[END_REF] (i.e. the fact that determinantal correlators obey loop equations), but this is not proved yet for differential systems of order q > 2.

  1.0.1 Example: (1, 2) law: Airy kernel and Tracy-Widom law

  and doesn't contribute to the residue. All what remains is eq.(2-88).

n

  's, based on the comparison with Kontsevich integral.

  1 , . . . , z n , z z+1 ) Λ j (z n+1 ) x ′ (z n+1 ) k F g ∂t j 1 . . . ∂t j k = Res ∞ . . . Res ∞ W (g) k (z 1 , . . . , z k ) Λ j 1 (z 1 ) . . . Λ j k (z k ) (4-6) 1 , z 2 , z 3 ) x ′ (z 1 )x ′ (z 2 )x ′ (z 3

∂ 2 F 0 ∂t 2 = u 0

 20

t

  l+1 (-u 0 /2) l (2l + 3)! l!(l + 1)! (4 -13)We have:(2 -2gn) W (g) n (z 1 , . . . , z n ) 1 , . . . , z n , z z+1 ) Φ(z n+1 )

Hermitian matrices correspond to real eigenvalues and positive measure dµ, but it is customary to generalize random matrices to normal matrices having their eigenvalues on some contours in the complex plane, and the measure dµ can be complex. The loop equations are the same for all those models, they are independent of the integration contour, and thus, they can all be treated in the same framework.

Acknowledgments

We would like to thank A. Belavin, N. Orantin, J. Teschner, C. Tracy, S. Venakides, for useful and fruitful discussions on this subject. This work is partly supported by the ANR project Grandes Matrices Aléatoires ANR-08-BLAN-0311-01, the European Science Foundation through the Misgam program, and the Quebec government with the FQRNT.

Example: Airy kernel

Let us write the (1, 2) model, i.e. m = 0. We have:

the string equation is:

The Lax pair is:

The differential system is:

i.e. ψ ′′ = N 2 (xt)ψ whose solution is the Airy function [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]:

and the other independent solution is the "Bairy" function [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]:

and thus the kernel is the famous Airy kernel [START_REF] Tracy | Level-spacing distributions and the Airy kernel[END_REF]:

The Airy kernel plays a very important role in many problems, in particular in the universal laws of extreme values, related to the Tracy-Widom law [START_REF] Tracy | Introduction to random matrices[END_REF]. The τ function is simply:

(2-49)

For the Airy system, the polynomial of theorem 2.1 is simply:

is the double scaling limit of matrix integrals correlators, and W (g) n (x 1 , . . . , x n ) is the g th term in the BKW expansion of the determinantal correlator of the (2m + 1, 2) minimal model (def 2.1).

And similarly if m > 0:

where fg are the free energies of the matrix model and F g are the free energies of the (2m + 1, 2) minimal model:

(2-97)

The double scaling limit is:

Therefore, we have proved that, as announced, the double scaling limit of matrix models, is given by the Liouville minimal models (2m + 1, 2) coupled to gravity.

Parametrics of orthogonal polynomials and Baker-Akhiezer functions

Many approaches of matrix models use some orthogonal polynomials (see Mehta [START_REF] Mehta | Random Matrices[END_REF]):

which we prefer to make orthonormal:

as well as their Hilbert transforms:

(2-101)

The matrix

satisfies a 2 × 2 differential system D n (x) with polynomial coefficients of degree at most deg

It is easy to see that tr D n (x) = 0 and det Ψ n (x) = h n-1 /h n is constant.

In particular:

and so on ...

Homogeneity relation

Theorem 4.7 of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] gives another relation which we can apply here: the homogeneity equation. Let Φ(z) such that

and thus:

The (2m, 1) minimal model

Again, we shall write a Lax pair D(x, t) and R(x, t) where D(x, t) is polynomial in x of degree 2m, and R(x, t) is polynomial of degree 1, and such that:

We choose:

and the matrix D(x, t) is of the form:

with:

where

They can be found by recursion:

where Rk (u) and Řk (u) are the modified Gelfand-Dikii differential polynomials:

For example: (5-10)

The Baker-Akhiezer functions Ψ(x, t) = ψ(x, t) φ(x, t) ψ(x, t) φ(x, t) , [START_REF] Belavin | On correlation functions in the perturbed minimal models M(2,2n+1)[END_REF][START_REF] Bertola | Boutroux curves with external field: equilibrium measures without a minimization problem[END_REF][START_REF] Bergère | Correlation functions of complex matrix models[END_REF][START_REF] Bergère | Determinantal formulae and loop equations[END_REF][START_REF]Random Matrix Models and Their Applications[END_REF][START_REF] Bleher | Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model[END_REF][START_REF] Bleher | Double scaling limit in the matrix model: the Riemann-Hilbert approach[END_REF] are given by the common solutions of the two compatible systems:

(5-12)

It was claimed in [START_REF] Bleher | Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations[END_REF] that the parametrics asymptotics of orthogonal polynomials near the singularity are given by:

)πǫ) ψ(x) + . . . (5 -13) and where ǫ/(1ǫ) is the ratio of the number of eigenvales in the 2 cuts which merge at the singularity (ǫ = 1/2 is the symmetric case).

The Christoffel-Darboux kernel K(x 1 , x 2 ) is the same as eq.(2-28):

x 1x 2 [START_REF] Belavin | On correlation functions in the perturbed minimal models M(2,2n+1)[END_REF][START_REF] Bertola | Boutroux curves with external field: equilibrium measures without a minimization problem[END_REF][START_REF] Bergère | Correlation functions of complex matrix models[END_REF][START_REF] Bergère | Determinantal formulae and loop equations[END_REF][START_REF]Random Matrix Models and Their Applications[END_REF][START_REF] Bleher | Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model[END_REF][START_REF] Bleher | Double scaling limit in the matrix model: the Riemann-Hilbert approach[END_REF][START_REF] Bleher | Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations[END_REF][START_REF] Boutet De Monvel | On the statistical mechanics approach in the random matrix theory: Integrated density of states[END_REF](14) and the correlators are obtained by the same determinantal formulae eq. .

And again the claim is that the determinantal correlators of the (2m, 1) minimal model, are the limits of matrix models correlators.

Conclusion

In this article, we have summarized some properties of scaling limits of matrix models (formal or not), known for a long time. We have provided a mathematical proof that the


Jérôme Härri
email: haerri@kit.edu

Moritz Killat
email: killat@kit.edu

Tessa Tielert
email: tielert@kit.edu

Jens Mittag
email: mittag@kit.edu

Hannes Hartenstein
email: hartenstein@kit.edu

DEMO: Simulation-as-a-Service for ITS Applications

I. INTRODUCTION

The development of Intelligent Transportation Systems (ITS) applications requires the evaluation of its impact on vehicular traffic. For logistics, flexibility, and financial reasons, evaluations using simulations are usually preferred over field tests as a first step toward deployment. Simulations offer a large modularity in the evaluated ITS applications and full flexibility in the chosen scenarios. Considering the large scale typically involved in ITS scenarios, simulations also require high computational capabilities including maintenance and technical expertise potentially resulting in significant financial investments on local simulation infrastructures. Moreover, such infrastructures are usually not efficiently used, in a sense that they are overused when simulations are conducted and underused when not.

Cloud computing is a novel paradigm that offers a differentiation of the usage of a resource or a service and its physical location on remote High Capacity Computing (HCC) platforms. Typical benefits from this approach are for instance an improved resource sharing between remote users, userfriendly web-based access to and configuration of a service, efficient maintenance and easier extensibility. Due to its high demand in processing power, the large-scale simulation of ITS applications is likely to benefit from cloud computing.

In this work, we therefore propose a simulation-as-a-service approach to evaluate ITS applications. We segment a simulation process in two building blocks: a web-interface/server front-end used by users to configure the remote simulations of their ITS application, and a back-end consisting of a controller and a HCC platform to conduct the remote simulations (see Fig. 1). The controller is in charge of configuring the HCC platform as well as distributing and scheduling the simulations on it. HCC platforms usually provide users with only a restricted control on the simulation environment and a limited set of available libraries. To mitigate these drawbacks, we make use of the virtualization support of recent HCC platforms and employ Kernel-based Virtual Machines (KVMs). Each virtual machine (VM) is a fully configured simulation environment that is transparent to the computing platform. Our objective is to illustrate how simulations requiring a large number of executions could be configured on a web-interface and remotely run in parallel on HCC platforms, for example on the High Performance Computing (HPC) platform of the Steinbuch Center for Computing (SCC) [START_REF]The Steinbuch Center for Computing (SCC[END_REF] at the Karlsruhe Institute of Technology (KIT). Our approach may be beneficial to research entities that could federate their simulation capabilities, but also companies providing professional simulators that could offer not only an ownership of a license but also a right-ofuse of their simulators on their simulation platform.

II. FRONT-END

The objective of the front-end is to provide a user-friendly web-based configuration of the simulation study, as well as an interface between the user and the simulation infrastructure. A web-interface gathers the inputs from users regardless of their location or used device, while the web-server saves the scenarios and other data transmitted by the users, connects to the controller, transmits the simulation scenarios to the controller and retrieves and stores the simulation results. The current version of the front-end provides the following services:

• upload of simulation scenarios • specification of the number of simulation executions • provision of a link to download the aggregated results The front-end technology is based on the Eclipse framework and on the Eclipse Rich Ajax Platform (RAP) [START_REF]The Eclipse Rich Ajax Platform (RAP)[END_REF]. It has been chosen as it allows the development of AJAX-enabled applications based on the Eclipse Rich Client Platform (RCP) and its comprehensive flexible component-based development model and convenient user-interfaces.

As depicted in Fig. 2, a major benefit of the Eclipse framework and the tight coupling between Eclipse RAP and RCP is the OSGi standard for the development of modular componentbased applications, and the Eclipse Modeling Framework (EMF) for its (de-)serialization mechanism and efficient data models composition and validation. Another major benefit, also illustrated in Fig. 2, is RAP's single sourcing capability, which lets the development of rich clients as well as web clients from a single source base. For example, the web frontend implemented in this demonstrator could be converted to an Eclipse RCP application with minor code modifications. III. BACK-END The back-end is the core of the simulation-as-a-service approach to be demonstrated. It consists of a controller connected to a HCC platform. As illustrated in Fig. 1, the back-end may be physically separated from the front-end, but the controller may also be physically separated from the HCC platform.

Once the controller receives a simulation request from the front-end, it performs the following tasks:

• It configures the scenarios for the HCC platform.

• It distributes the jobs to the required and available cores.

• It gathers and returns the results to the front-end. The HCC where simulations are remotely run is the HPC of the SCC [START_REF]The Steinbuch Center for Computing (SCC[END_REF], which offers us a computing capability of approx. 800 GFLOPS on average. Yet, our concept does not restrict the connection to a single HCC platform.

The back-end also employs Kernel-based Virtual Machines (KVM) [START_REF]Kernel-based Virtual Machine[END_REF], a virtualization solution for Linux that makes an extensive use of hardware virtualization support in recent x86 and x86-64 processors. This approach offers the possibility to run fully configurable Windows or Linux images adapted to the simulation requirements on the HCC platforms, which allows us to benefit from their high computation capacities regardless of the required simulation environment. Fig. 3 illustrates the conceptual schema of the back-end.

Although the back-end concept uses KVM, it is not restricted to it as virtualizations based on Xen [START_REF]The Xen hypervisor[END_REF], such as Amazon EC2 [START_REF]Amazon Elastic Compute Cloud (EC2)[END_REF], could also be controlled by the controller.

IV. DEMONSTRATION Our demonstration will consist of three phases: First, we will show the configuration of a simulation scenario for an ITS application called Green Light Optimal Speed Advisory (GLOSA) 1 in an urban environment. Second, we will connect 1 GLOSA is an application making use of signal transition timings transmitted by traffic lights to approaching vehicles in order to provide the optimal approaching speed avoiding a stop at traffic lights. to the web-interface, request the number of simulation executions, and upload the configured scenario to the controller. Finally, the requested multiple instances of the scenario will be run in parallel on multiple cores on the remote computing platform. As GLOSA is simulated using the Windows-based VISSIM traffic simulator [START_REF]VISSIM Traffic Simulator[END_REF], we will run Windows as KVM guest platforms and visualize a subset of the running KVMs with a remote desktop connection. Fig. 4 depicts what attendees will be able to see.

Fig. 1 .

 1 Fig. 1. Simulation-as-a-Service -Front-end and Back-end Concepts.

Fig. 2 .

 2 Fig. 2. Rich Ajax Platform (RAP) development highlights. OSGi offers modular component-based developments and EMF modular data model compositions and validations. RAP provides AJAX-based web-applications but it may also generate RCP applications with minor code modifications.

Fig. 3 .

 3 Fig. 3. Back-end concept, composed of a controller and a computing platform. In the demonstrator, Kernel-based Virtual Machines (KVM) run Windows images.

Fig. 4 .

 4 Fig. 4. Illustration of the demonstrator -a GLOSA simulation campaign is configured on the front-end and remotely dispatched on multiple KVMs running in parallel in the back-end.

ACKNOWLEDGMENTS

We would also like to thank Dr. Michael Ortgiese and Dr. Thomas Benz of PTV AG for their help and support in this project.

support of the Ministry of Science, Research and the Arts of Baden-Württemberg (Az: Zu ), the Klaus Tschira Stiftung, the INIT GmbH and the PTV AG for the research group on Traffic Telematics. T. Tielert acknowledges the support of Deutsche Forschungsgemeinschaft within the Research Training Group GRK 1194 Selforganizing Sensor-Actuator-Networks.