Modèles de matrices aléatoires à N grand, groupe de renormalisation, solutions exactes et universalité
Résumé
Random matrix models can be related to a great number of problems : nuclei, atoms in chaotic regimes, mesoscopic physics, string theory, or quantum 2-dimensional gravity with matter fields (i.e. non-geometrical degrees of freedom) with a central charge c. When dealing with quantum gravity, we consider the vicinity of critical points, where a double-scaling limit enables us to recover a continuous quantum gravity theory. The central charge of its matter fields can be derived from the critical exponents of the matrix model by the KPZ (Knizhnik, Polyakov, Zamolodchikov) formula. This formula, however, is not valid for c>1, and, up to now, no random matrix model has been solved, which, in the continuous limit, would allow us to recover a quantum gravity plus matter model with central charge c>1. Our purpose has been, first [G. Bonnet, F. David, Nucl. Phys. B552 (1999) 511-528], to understand better this c=1 barrier by studying the renormalization group flows of random matrix models under a shift on the size of the matrix. We have improved the renormalization group techniques by using loop equations to reduce drastically the number of operators appearing in the renormalization process: we have been able to obtain a renormalized action in the form of the initial action, apart from additional couplings allowing several planar surfaces to be glued to each other in one point. Thanks to this renormalization group process, we could improve the precision of the renormalization group method and have a $0.016\%$ precision on the position of the critical point of a $tr \Phi^4$ one-matrix model instead of a previous $0.9\%$, found by Higuchi et al.. We also showed that critical exponents also converge, which is not always true when one does not reduce enough the number of operators appearing in the renormalization process. Other interesting results are that we recovered (in this simple one-matrix model), the qualitative renormalization flows and the 2 fixed points which were predicted in this case by F. David's conjecture. This conjecture also predicts that, when $c \rightarrow 1$, these fixed points coalesce, then, for ``c>1'', disappear. Thus, according to the conjecture, it is impossible to model c>1 2-dimensional quantum gravity plus matter systems by a random matrix model. Finally, using our renormalization group techniques, we also found good approximations of the critical points and flows for the 2-matrix Ising model. This model is a $c=(1 \over 2)$ matrix model. In our second and third works [G. Bonnet, Phys. Lett. B459 (1999) 575, B. Eynard, G. Bonnet Phys. Lett. B 453 (1999) 273], we concentrated on solving the general Potts-q models on planar random surfaces, using the loop equations method. These are models where the q matrices are coupled to each other, thus making the resolution of the model difficult. We showed that, though there does not generally (for 2 or more matrix models) appear to be a closed set of loop equations; in the Potts-3 case, it is indeed the case. By taking advantage of the invariance by circular permutation of traces, we could relate all the expectation values of the operators of the Potts-3 model on planar random surfaces to only 4 unknown operators. These can in turn be determined by a one-cut assumption, and we finally obtain an order five algebraic equation for the resolvent, as well as the critical points and exponents of the model. As for the Potts-q models, a more general method enables us to relate the expectation values of traces of odd powers of one of the matrices to the expectation values of even powers of the matrix. Then, we obtain the equation for the resolvent, which general solution is an elliptic function. However, in the rational case $q=2-2 cos ((l \over r) \pi)$, we obtain an explicit algebraic equation for the resolvent. This equation is of order r-1 if r+l is even and of order 2 r -1 if r+l is odd. We also derive, in the general case, the critical exponents of the model. Finally, in a last part (this paper is still being written), we concentrated on a rather different subject : Brezin and Deo have shown symmetric 2-cut random matrix problems problems have an unexpected (-1)N term, depending on the size N of the matrix, in the 2-point correlation function. We generalize their results to non-symmetric cuts. We obtain the general 2-point correlation function, the corresponding orthogonal polynomials, and the expression of the recursion relation between them, along with the expression of K(x,y).
Les modèles de matrices aléatoires, d'abord introduits en physique pour décrire les statistiques de niveaux d'énergie en physique nucléaire, ont par la suite trouvé des applications dans des domaines extrêmement variés, du chaos quantique et de la physique mésoscopique, à la chromodynamique quantique, la théorie des cordes et la gravité quantique via les modèles de surfaces aléatoires. Bien que certains modèles de matrices soient bien compris, il s'agit principalement des cas particuliers de matrices couplées en chaîne, correspondant à des théories de gravité quantique ou des théories des cordes de charge centrale conforme inférieure ou égale à un. Ainsi, tout un pan des modèles de matrices aléatoires, les modèles de matrices de charge centrale c>1, nous échappe. J'ai cherché, au cours de mon travail de thèse, à mieux comprendre et à résoudre ces modèles. La méthode de groupe de renormalisation nous a permis, par l'étude de l'évolution des flots en fonction de la charge centrale conforme, de mieux comprendre le lien entre celle-ci et le comportement des modèles de matrices [G. Bonnet, F. David, Nucl. Phys. B552 (1999) 511-528, hep-th/9811216]. Par la méthode des équations de boucles, nous avons résolu [G. Bonnet, Phys. Lett. B 459 (1999) 575, hep-th/9904058; B. Eynard, G. Bonnet Phys. Lett. B 453 (1999) 273, hep-th/9906130] des modèles de matrices couplées deux à deux : les modèles de Potts-q sur réseau aléatoire. Cette résolution ouvre la voie à celle d'une classe de modèles plus vaste que les simples modèles de matrices couplées en chaîne. Enfin, bien que dans notre étude nous nous soyons intéressés principalement à la limite planaire, où la taille N des matrices tend vers l'infini, nous avons aussi étudié l'effet, sous-dominant dans la fonction de partition du modèle, de la discrétisation des valeurs propres. Nous avons montré [G. Bonnet, F. David, B. Eynard, cond-mat/0003324] que, dans le cas d'un modèle où le support des valeurs propres est non-connexe, il n'y a pas de développement topologique en puissances de N. L'influence de la discrétisation des valeurs propres devient alors d'ordre dominant dans les fonctions de corrélation à deux points ou au-delà. Nous allons commencer ici par la signification physique des modèles de matrices, puis nous parlerons des techniques classiques de résolution, enfin, nous décrirons les résultats que nous avons obtenus au cours de cette thèse.