The non-perturbative corrections to the free energy of the two-matrix model are expressed in terms of its spectral curve. The eigenvalue instantons are associated with the vanishing cycles of the curve. Compared with the world-sheet theory, the results for the (p,q) critical points lead to intriguing identifications between different Liouville and matter boundary conditions in non-critical string theories. De nombreux travaux récents étudient les limites continues dárbres aléatoires discrets. Ces arbres discrets peuvent être définis soit de manière combinatoire (arbre choisi au hasard parmi tous les arbres à n sommets dún certain type), soit de manière probabiliste (en donnant la loi du “nombre dénfants” de chaque sommet de lárbre). Un passage à la limite où le nombre de sommets de lárbre tend vers línfini et où simultanément la longueur de chaque arête tend vers 0, conduit à des arbres aléatoires continus, dont le prototype est le Continuum Random Tree (CRT) introduit par Aldous. Léxposé décrira la manière dont ces arbres sont codés et en quel sens ils sont limites des arbres discrets. On donnera aussi diverses propriétés géométriques et fractales de ces arbres continus. Si le temps le permet, on introduira le modèle appelé ISE (Integrated Super-Brownian Excursion) qui combine la structure de branchement du CRT avec un déplacement brownien (les ïndividus” dont le CRT décrit la généalogie se déplacent dans léspace de manière brownienne). Les travaux récents de Slade et de ses co-auteurs ont montré que lÍSE intervient dans lásymptotique de différents modèles de mécanique statistique (arbres sur réseaux, percolation, etc.). There are many situations in physics where all the hallmarks of a phase transition are present but for which the idealized notions of analyticity are difficult to apply.

A theory of nonequilibrium statistical mechanics that is based on stochastic dynamics can deal with this through study of the spectral properties of the underlying stochastic matrix. We have shown the relation between near-degeneracy (of the associated eigenvalues) of the slowest modes of this matrix and the occurrence of one or more stable or metastable modes. This can be applied to ordinary metastable states, e.g., supercooled water, and to more subtle situations like spin glasses (although the latter application is far from complete). An extension of these ideas is used to quantify the coarse graining process that lies at the foundations of statistical mechanics. A recently completed next-to-leading-order program to calculate neutrino cross sections, including power-suppressed mass correction terms, has been applied to evaluate the Paschos-Wolfenstein relation, in order to quantitatively assess the validity and significance of the NuTeV anomaly. In particular, we study the shift of sin² q_W obtained in calculations with a new generation of PDF sets that allow for an asymmetry between the strange and the anti-strange quark sea distributions, enabled by recent neutrino dimuon data from CCFR and NuTeV, as compared to the previous s = s-bar parton distribution functions like CTEQ6M. The extracted value of sin² q_W is closely correlated with the strangeness asymmetry momentum integral. We also consider isospin violating effects that have recently been explored by the MRST group. The results of our study suggest that the new dimuon data, the Weinberg angle measurement, and other data sets used in global QCD parton structure analysis can all be consistent within the Standard Model. .

Brookhaven National Laboratory – Nuclear theory