On the lower bounds for real double Hurwitz numbers

Abstract

The real double Hurwitz number counts ramified covers of Riemann sphere with compatible involutions satisfying certain ramification data. J. Rau established a lower bound for these numbers in [19] using tropical covers with odd multiplicity. We improve Rau's lower bound by adding some tropical covers with even multiplicity. As an application, we prove the logarithmic equivalence of real and classical Hurwitz numbers without the existence assumption of Rau's lower bound.