Olivier Marchal

Université Jean Monnet · Institut Camille JORDAN

In this short article we propose a full large N asymptotic expansion of the probability that the m th power of a random unitary matrix of size N has all its eigenvalues in a given arc-interval centered in 1 when N is large. This corresponds to the asymptotic expansion of a Toeplitz determinant whose symbol is the indicator function of several inter...

We consider non-twisted meromorphic connections in $\mathfrak{sl}_2(\mathbb{C})$ and the associated symplectic Hamiltonian structure. In particular, we provide explicit expressions of the Lax pair in the geometric gauge supplementing previous results where explicit formulas have been obtained in the oper gauge. Expressing the geometric Lax matrices...

In this paper, we build the Hamiltonian system and the corresponding Lax pairs associated to a twisted connection in $\mathfrak{gl}_2(\mathbb{C})$ admitting an irregular and ramified pole at infinity of arbitrary degree, hence corresponding to the Painlev\'{e} $1$ hierarchy. We provide explicit formulas for these Lax pairs and Hamiltonians in terms...

In this paper, we study and build the Hamiltonian system attached to any $\mathfrak{gl}_2(\mathbb{C})$ rational connection with arbitrary number of non-ramified poles of arbitrary degrees. In particular, we propose the Lax pairs expressed in terms of irregular times associated to the poles and a map reducing the space of irregular times to isomonod...

In this short article we propose a full large $N$ asymptotic expansion of the probability that the $m^{\text{th}}$ power of a random unitary matrix of size $N$ has all its eigenvalues in a given arc-interval centered in $1$ when $N$ is large. This corresponds to the asymptotic expansion of a Toeplitz determinant whose symbol is the indicator functi...

We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of spectral Darboux coordinates on the moduli space of meromorphic $\mathfrak{sl}_2$-connections on $\mathbb{P}^1$...

We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e. the differential operator quantizing the algebraic equation defining the classical spectral curve consi...

Cet ouvrage présente la construction des estimateurs, des intervalles de confiance, des tests d’hypothèses et de la régression linaire associés aux modèles de statistiques paramétriques. Illustré d’exemples et de nombreux exercices intégralement corrigés, il permet une approche complète et cohérente des statistiques inférentielles. Il est principa...

In this paper, we show that it is always possible to deform a differential equation ∂_x Ψ(x) = L(x)Ψ(x) with L(x) ∈ sl_2(C)(x) by introducing a small formal parameter in such a way that it satisfies the Topological Type properties of Bergère, Borot and Eynard. This is obtained by including the former differential equation in an isomonodromic system...

Abstract We prove a monotonicity property of the Hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. We provide a probabilistic interpretation of our result by exploiting a connection between Hurwitz zeta function and the cumulants of the beta-exponential distribution.

In this article, we study the large $n$ asymptotic expansions of $n\times n$ Toeplitz determinants whose symbols are indicator functions of unions of arc-intervals of the unit circle. In particular, we use a Hermitian matrix model reformulation of the problem to provide a rigorous derivation of the general form of the large $n$ expansion when the s...

We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of spectral Darboux coordinates on the moduli space of meromorphic $\mathfrak{sl}_2$-connections on $\mathbb{P}^1$...

Study objectives: Our objectives were to determine in an obese population (body mass index > 35 kg/m²) the number of patients, after gastric bypass (GBP), who no longer met French Ministry of Health criteria for utilizing positive airway pressure (PAP), and the predictive factors of obstructive sleep apnea (OSA) improvement. Methods: Between Jun...

We prove a monotonicity property of the Hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. We provide a probabilistic interpretation of our result by exploiting a connection between Hurwitz zeta function and the cumulants of the exponential-beta distribution.

Introduction: Our objectives were to determine in an obese population (body mass index > 35 kg/m²) the number of patients, after gastric bypass (GBP), who no longer met French Ministry of Health criteria for utilizing positive airway pressure (PAP), and the predictive factors of obstructive sleep apnea (OSA) improvement. Methods: Between June 20...

We investigate the sub-Gaussian property for almost surely bounded random variables. If sub-Gaussianity per se is de facto ensured by the bounded support of said random variables, then exciting research avenues remain open. Among these questions is how to characterize the optimal sub-Gaussian proxy variance? Another question is how to characterize...

We investigate the sub-Gaussian property for almost surely bounded random variables. If sub-Gaussianity per se is de facto ensured by the bounded support of said random variables, then exciting research avenues remain open. Among these questions is how to characterize the optimal sub-Gaussian proxy variance? Another question is how to characterize...

In this paper, we show that it is always possible to deform a differential equation $\partial_x \Psi(x) = L(x) \Psi(x)$ with $L(x) \in \mathfrak{sl}_2(\mathbb{C})(x)$ by introducing a small formal parameter $\hbar$ in such a way that it satisfies the Topological Type properties of Berg\`ere, Borot and Eynard. This is obtained by including the forme...

Dans cet ouvrage inspiré de son expérience d’enseignant, l’auteur présente, pour un public non spécialiste des mathématiques, les bases des statistiques appliquées modernes ainsi que leur implémentation dans le logiciel libre R. L’ouvrage aborde ainsi l’aspect descriptif des statistiques (représentations graphiques, moyenne, écarts-types empiriques...

Textbook for a fundamental probability class via measure theory.

In this article we prove that Lax pairs associated with the $\hbar$-dependent all six Painlev\'e equations satisfy the topological type property proposed by Berg\`ere, Borot and Eynard for any generic choice of the monodromy parameters. Consequently we show that one can reconstruct the formal $\hbar$-expansion of the isomonodromic $\tau$-function a...

The goal of this "Habilitation \`a diriger des recherches" is to present two different applications, namely computations of certain partition functions in probability and applications to integrable systems, of the topological recursion developed by B. Eynard and N. Orantin in 2007. Since its creation, the range of applications of the topological re...

The goal of this ``Habilitation à diriger des recherches'' is to present two different applications, namely computations of certain partition functions in probability and applications to integrable systems, of the topological recursion developed by B. Eynard and N. Orantin in 2007. Since its creation, the range of applications of the topological re...

The purpose of this article is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a $\hbar$-difference equation: $\Psi(x+\hbar)=\left(e^{\hbar\frac{d}{dx}}\right) \Psi(x)=L(x;\hbar)\Psi(x)$ with $L(x;\hbar)\in GL_2( (\mathbb{C}(x))[\hbar])$. In particular, we extend the notion of determinantal formula...

To any differential system $d\Psi=\Phi\Psi$ where $\Psi$ belongs to a Lie group (a fiber of a principal bundle) and $\Phi$ is a Lie algebra $\Lieg$ valued 1-form on a Riemann surface $\curve$, is associated an infinite sequence of ``correlators" $W_n$ that are symmetric $n$-forms on $\curve^n$. The goal of this article is to prove that these corre...

We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, thus proving upper bounds recently conjectured by Elder (2016). We provide different proof techniques for the symmetrical (around its mean) case and the non-symmetrical case. The technique in the latter case relies on studying the ordinary differential equation satis...

The objective of the study was To compare patients’ outcomes of young people with cystic fibrosis (CF) before and after transfer from child to adult CF center.

Introduction Le syndrome d’apnees hypopnees obstructives du sommeil (SAHOS) est un probleme majeur de sante publique. Son facteur de risque principal est l’obesite. L’objectif de notre etude etait d’evaluer l’evolution (clinique/polygraphique) et les facteurs predictifs d’amelioration du SAHOS modere a severe, apres court-circuit gastrique de patie...

Starting from a $d\times d$ rational Lax pair system of the form $\hbar \partial_x \Psi= L\Psi$ and $\hbar \partial_t \Psi=R\Psi$ we prove that, under certain assumptions (genus $0$ spectral curve and additional conditions on $R$ and $L$), the system satisfies the "topological type property". A consequence is that the formal $\hbar$-WKB expansion o...

Background: Dermoscopy improves diagnostic accuracy in melanoma, as shown by several meta-analyses. Although it is used by general practitioners (GPs) in Australia, Canada and Italy, no published data on this topic are available in France. Objectives: To review the opinions and use of dermoscopy by GPs in France and to understand their practice...

This short note is the result of a French "Hippocampe internship" that aims at introducing the world of research to young undergraduate French students. The problem studied is the following: imagine yourself locked in a cage barred with $n$ different locks. You are given a keyring with $N \geq n$ keys containing the $n$ keys that open the locks. In...

The purpose of this article is to study the eigenvalues $u_1^{\, t}=e^{it\theta_1},\dots,u_N^{\,t}=e^{it\theta_N}$ of $U^t$ where $U$ is a large $N\times N$ random unitary matrix and $t>0$. In particular we are interested in the typical times $t$ for which all the eigenvalues are simultaneously close to $1$ in different ways thus corresponding to r...

The goal of this article is to prove that the determinantal formulas of the Painlev\'e $2$ system identify with the correlation functions computed from the topological recursion on their spectral curve for an arbitrary non-zero monodromy parameter. The result is established for a WKB expansion of two different Lax pairs associated to the Painlev\'e...

The goal of this article is to rederive the connection between the Painlev\'e $5$ integrable system and the universal eigenvalues correlation functions of double-scaled hermitian matrix models, through the topological recursion method. More specifically we prove, \textbf{to all orders}, that the WKB asymptotic expansions of the $\tau$-function as w...

The purpose of the article is to provide partial proofs of two conjectures given by Witte and Forrester in "Moments of the gaussian beta ensembles and the large $N$ expansion of the densities" with the use of the topological recursion adapted for general $\beta$ in the Gaussian case. In particular, the paper uses a version at coinciding points givi...

The purpose of the article is to provide partial proofs for two conjectures given by Witte and Forrester in “Moments of the Gaussian β Ensembles and the large N expansion of the densities” ([1]) with the use of the topological recursion adapted for general β Gaussian case. In particular, the paper uses a version at coinciding points that provides a...

In this paper, we study in detail the modified topological recursion of the one-matrix model for arbitrary β in the one-cut case. We show that, for polynomial potentials, the recursion can be computed as a sum of residues. However, the main difference with the Hermitian matrix model is that the residues cannot be set at the branch points of the spe...

In this note, we prove that the free energies F_g constructed from the Eynard-Orantin topological recursion applied to the curve mirror to C^3 reproduce the Faber-Pandharipande formula for genus g Gromov-Witten invariants of C^3. This completes the proof of the remodeling conjecture for C^3.

We write the loop equations for the $\beta$ two-matrix model, and we propose a topological recursion algorithm to solve them, order by order in a small parameter. We find that to leading order, the spectral curve is a "quantum" spectral curve, i.e. it is given by a differential operator (instead of an algebraic equation for the hermitian case). Her...

In this article, we study in detail the modified topological recursion of the one matrix model for arbitrary $\beta$ in the one cut case. We show that for polynomial potentials, the recursion can be computed as a sum of residues. However the main difference with the hermitian matrix model is that the residues cannot be set at the branchpoints of th...

We construct the solution of the loop equations of the β-ensemble model in a form analogous to the solution in the case of the Hermitian matrices β = 1. The solution for β = 1 is expressed in terms of the algebraic spectral curve given by y2 = U(x). The spectral curve for arbitrary β converts into the Schrödinger equation (ħ∂)2 − U(x) ψ(x) = 0, whe...

This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of integrable systems. The variety of these applications shows why matrix models are important from a mathematical point o...

This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of integrable systems. The variety of these applications shows why matrix models are important from a mathematical point o...

In Part I [“A matrix model for the topological string. I: Deriving the matrix model”, Preprint, arXiv:1003.1737], we presented a matrix model reproducing the topological string partition function on an arbitrary given toric Calabi-Yau manifold. Here, we compute the spectral curve of our matrix model and thus provide a matrix model derivation of the...

We construct a matrix model that reproduces the topological string partition function on arbitrary toric Calabi-Yau 3-folds. This demonstrates, in accord with the BKMP "remodeling the B-model" conjecture, that Gromov-Witten invariants of any toric Calabi-Yau 3-fold can be computed in terms of the spectral invariants of a spectral curve. Moreover, i...

In this article, we show that the double scaling limit correlation functions of a random matrix model when two cuts merge with degeneracy $2m$ (i.e. when $y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the determinantal formulae defined by conformal $(2m,1)$ models. Our approach follows the one developed by Berg\`{e}re and E...

In this article, we solve the loop equations of the \beta-random matrix model, in a way similar to what was found for the case of hermitian matrices \beta=1. For \beta=1, the solution was expressed in terms of algebraic geometry properties of an algebraic spectral curve of equation y^2=U(x). For arbitrary \beta, the spectral curve is no longer alge...

In this article, we define a non-commutative deformation of the "symplectic invariants" of an algebraic hyperelliptical plane curve. The necessary condition for our definition to make sense is a Bethe ansatz. The commutative limit reduces to the symplectic invariants, i.e. algebraic geometry, and thus we define non-commutative deformations of some...

We consider the Itzykson-Zuber-Eynard-Mehta two-matrix model and prove that the partition function is an isomonodromic tau function in a sense that generalizes Jimbo-Miwa-Ueno's. In order to achieve the generalization we need to define a notion of tau-function for isomonodromic systems where the ad-regularity of the leading coefficient is not a nec...