  Monotone Hurwitz numbers were introduced by the authors as a combinatorially
natural desymmetrization of the Hurwitz numbers studied in enumerative
algebraic geometry. Over the course of several papers, we developed the
structural theory of monotone Hurwitz numbers and demonstrated that it is in
many ways parallel to that of their classical counterparts. In this note, we
identify an important difference between the monotone and classical worlds:
fixed-genus generating functions for monotone double Hurwitz numbers are
absolutely summable, whereas those for classical double Hurwitz numbers are
not. This property is crucial for applications of monotone Hurwitz theory in
analysis. We quantify the growth rate of monotone Hurwitz numbers in fixed
genus by giving universal upper and lower bounds on the radii of convergence of
their generating functions.
