Universal exponents and tail estimates in the enumeration of planar maps

It has been observed that for most classes of planar maps, the number of maps of size n grows asymptotically like c · n−5/2γn, for suitable positive constants c and γ. It has also been observed that, if dk is the limit probability that the root vertex in a random map has degree k, then again for most classes of maps the tail of the distribution is asymptotically of the form dk ∼ c·kq as k →∞, for positive constants c, q with q < 1. We provide a rationale for this universal behaviour in terms of analytic conditions on the associated generating functions. The fact that generating functions for maps satisfy as a rule a quadratic equation with one catalytic variable, allows us to identify a critical condition implying the shape of the above-mentioned asymptotic estimates. We verify this condition on several well-known families of planar maps.