Jean Zinn-Justin

Matrix models of 2D quantum gravity are either exactly solvable for matter of central charge $ c\leq 1, $ or not understood. It would be useful to devise an approximate scheme which would be reasonable for the known cases and could be carried to the unsolved cases in order to achieve at least a qualitative understanding of the properties of the models. The double scaling limit is an indication that a change of the length scale induces a flow in the parameters of the theory, the size of the matrix and the coupling constants for matrix models, at constant long distances physics. We construct here these renormalization group equations at lowest orders in various cases to check that we reproduce qualitatively the properties of $ c\leq 1 $ models.

$O(N)$ invariant vector models have been shown to possess non-trivial scaling large $N$ limits, at least perturbatively within the loop expansion, a property they share with matrix models of 2D quantum gravity. In contrast with matrix models, however, vector models can be solved in arbitrary dimensions. We present here the analysis of field theory vector models in $d$ dimensions and discuss the nature and form of the critical behaviour. The double scaling limit corresponds for $d>1$ to a situation where a bound state of the $N$-component fundamental vector field $\phi$, associated with the $\phi^2$ composite operator, becomes massless, while the field $\phi$ itself remains massive. The limiting model can be described by an effective local interaction for the corresponding $O(N)$ invariant field. It has a physical interpretation as describing the statistical properties of a class of branched polymers.\par It is hoped that the $O(N)$ vector models, which can be investigated in their most general form, can serve as a test ground for new ideas about the behaviour of 2D quantum gravity coupled with $d>1$ matter.