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We study random typical minimal factorizations of the $n$-cycle intotranspositions, which are factorizations of $(1, \ldots,n)$ as a product of$n-1$ transpositions. By viewing transpositions as chords of the unit disk andby reading them one after the other, one obtains a sequence of increasinglaminations of the unit disk (i.e. compact subsets of the unit disk made ofnon-intersecting chords).  When an order of $\sqrt{n}$ consecutive transpositions have been read, weestablish, roughly speaking, that a phase transition occurs and that theassociated laminations converge to a new one-parameter family of randomlaminations, constructed from excursions of specific L\'evy processes.  Our main tools involve coding random minimal factorizations by conditionedtwo-type Bienaym\'e--Galton--Watson trees. We establish in particular limittheorems for two-type BGW trees conditioned on having given numbers of verticesof both types, and with an offspring distribution depending on the conditioningsize. We believe that this could be of independent interest.

The geometry of random minimal factorizations of a long cycle via biconditioned bitype random trees