Non-homogenous disks in the chain of matrices

We investigate the generating functions of multi-colored discrete disks with non-homogenous boundary conditions in the context of the Hermitian multi-matrix model where the matrices are coupled in an open chain. We show that the study of the spectral curve of the matrix model allows one to solve a set of loop equations to get a recursive formula computing mixed trace correlation functions to leading order in the large matrix limit.Comment: 25 pages, 4 figure

Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies

We compute the generating functions of a O(n) model (loop gas model) on a random lattice of any topology. On the disc and the cylinder, they were already known, and here we compute all the other topologies. We find that the generating functions (and the correlation functions of the lattice) obey the topological recursion, as usual in matrix models, i.e they are given by the symplectic invariants of their spectral curve.Comment: pdflatex, 89 pages, 12 labelled figures (15 figures at all), minor correction

Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach

We solve the loop equations of the $\beta$-ensemble model analogously to the solution found for the Hermitian matrices $\beta=1$. For \beta=1$, the solution was expressed using the algebraic spectral curve of equation$y^2=U(x)$. For arbitrary$\beta$, the spectral curve converts into a Schr\"odinger equation$((\hbar\partial)^2-U(x))\psi(x)=0$with$\hbar\propto (\sqrt\beta-1/\sqrt\beta)/N$. This paper is similar to the sister paper~I, in particular, all the main ingredients specific for the algebraic solution of the problem remain the same, but here we present the second approach to finding a solution of loop equations using sectorwise definition of resolvents. Being technically more involved, it allows defining consistently the B-cycle structure of the obtained quantum algebraic curve (a D-module of the form$y^2-U(x)$, where$[y,x]=\hbar$) and to construct explicitly the correlation functions and the corresponding symplectic invariants$F_h$, or the terms of the free energy, in 1/N^2$-expansion at arbitrary $\hbar$. The set of "flat" coordinates comprises the potential times $t_k$ and the occupation numbers \widetilde{\epsilon}_\alpha$. We define and investigate the properties of the A- and B-cycles, forms of 1st, 2nd and 3rd kind, and the Riemann bilinear identities. We use these identities to find explicitly the singular part of$\mathcal F_0$that depends exclusively on$\widetilde{\epsilon}_\alpha$.Comment: 58 pages, 7 figure

Large deviations of the maximal eigenvalue of random matrices

We present detailed computations of the 'at least finite' terms (three dominant orders) of the free energy in a one-cut matrix model with a hard edge a, in beta-ensembles, with any polynomial potential. beta is a positive number, so not restricted to the standard values beta = 1 (hermitian matrices), beta = 1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This model allows to study the statistic of the maximum eigenvalue of random matrices. We compute the large deviation function to the left of the expected maximum. We specialize our results to the gaussian beta-ensembles and check them numerically. Our method is based on general results and procedures already developed in the literature to solve the Pastur equations (also called "loop equations"). It allows to compute the left tail of the analog of Tracy-Widom laws for any beta, including the constant term.Comment: 62 pages, 4 figures, pdflatex ; v2 bibliography corrected ; v3 typos corrected and preprint added ; v4 few more numbers adde

Second and Third Order Observables of the Two-Matrix Model

In this paper we complement our recent result on the explicit formula for the planar limit of the free energy of the two-matrix model by computing the second and third order observables of the model in terms of canonical structures of the underlying genus g spectral curve. In particular we provide explicit formulas for any three-loop correlator of the model. Some explicit examples are worked out.Comment: 22 pages, v2 with added references and minor correction

Les valeurs mobilières détenues par les Français en mars 2005.

placements financiers, détenteurs de titres, actionnaires, titres, actions, obligations, titres d’OPCVM, épargne salariale.

Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

We develop techniques to compute multi-instanton corrections to the 1/N expansion in matrix models described by orthogonal polynomials. These techniques are based on finding trans-series solutions, i.e. formal solutions with exponentially small corrections, to the recursion relations characterizing the free energy. We illustrate this method in the Hermitian, quartic matrix model, and we provide a detailed description of the instanton corrections in the Gross-Witten-Wadia (GWW) unitary matrix model. Moreover, we use Borel resummation techniques and results from the theory of resurgent functions to relate the formal multi-instanton series to the nonperturbative definition of the matrix model. We study this relation in the case of the GWW model and its double-scaling limit, providing in this way a nice illustration of various mechanisms connecting the resummation of perturbative series to nonperturbative results, like the cancellation of nonperturbative ambiguities. Finally, we argue that trans-series solutions are also relevant in the context of topological string theory. In particular, we point out that in topological string models with both a matrix model and a large N gauge theory description, the nonperturbative, holographic definition involves a sum over the multi-instanton sectors of the matrix modelComment: 50 pages, 12 figures, comments and references added, small correction

Geometric Transition as a Change of Polarization

Taking the results of hep-th/0702110 we study the Dijkgraaf-Vafa open/closed topological string duality by considering the wavefunction behavior of the partition function. We find that the geometric transition associated with the duality can be seen as a change of polarization.Comment: 20 page

Matrices coupled in a chain. I. Eigenvalue correlations

The general correlation function for the eigenvalues of $p$ complex hermitian matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson.Comment: ftex eynmeh.tex, 2 files, 8 pages Submitted to: J. Phys.

Bridging tools to better understand environmental performances and raw materials supply of traction batteries in the future EU fleet

Sustainable and smart mobility and associated energy systems are key to decarbonise the EU and develop a clean, resource efficient, circular and carbon-neutral future. To achieve the 2030 and 2050 targets, technological and societal changes are needed. This transition will inevitably change the composition of the future EU fleet, with an increasing share of electric vehicles (xEVs). To assess the potential contribution of lithium-ion traction batteries (LIBs) in decreasing the environmental burdens of EU mobility, several aspects should be included. Even though environmental assessments of batteries along their life-cycle have been already conducted using life-cycle assessment, a single tool does not likely provide a complete overview of such a complex system. Complementary information is provided by material flow analysis and criticality assessment, with emphasis on supply risk. Bridging complementary aspects can better support decision-making, especially when different strategies are simultaneously tackled. The results point out that the future life-cycle GWP of traction LIBs will likely improve, mainly due to more environmental-friendly energy mix and improved recycling. Even though second-use will postpone available materials for recycling, both these end-of-life strategies allow keeping the values of materials in the circular economy, with recycling also contributing to mitigate the supply risk of Lithium and Nickel