Abstract:Année de publication : 1993
The most general large $N$ eigenvalues distribution for the one matrix model is shown to consist of tree-like structures in the complex plane. For the $m=2$ critical point, such a solution describes the strong coupling phase of $2d$ quantum gravity ($c=0$ non-critical string). It is obtained by taking combinations of complex contours in the matrix integral, and the relative weight of the contours is identified with the non-perturbative ``$\theta$-parameter" that fixes uniquely the solution of the string equation (Painlev\'e I). This allows to recover by instanton methods results on the non-perturbative effects obtained by the Isomonodromic Deformation Method, and to construct for each $\theta$-vacuum the observables (the loop correlation functions) which satisfy the loop equations. The breakdown of analyticity of the large $N$ solution is related to the existence of poles for the loop operators.