Coalescence: emergence of the map-Airy law

Maps are planar graphs presented together with an embedding in the plane, and as such, they model the topology of many geometric arrangements. This talk concerns itself with the statistical properties of random maps, and focusses on connectivity issues. The analysis that we introduce is largely based on a method of ``coalescing saddle points''. We exhibit here a new class of ``universal'' phenomena that are of the exponential-cubic type $\exp ix^3$, corresponding to nonstandard distributions that involve the Airy function. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Joint work with Ph. Flajolet, G. Schaeffer and M. Soria. (Malgr\'e les apparences, l'expos\'e sera donn\'e en fran\c cais.)