Upper bound of some character ratios and large genus asymptotic behavior of Hurwitz numbers

In [14] we found the large genus asymptotics of Hurwitz numbers for the Riemann sphere with a fixed number of general profiles and some (2,1^{d-2}) profiles. In this paper, motivated from [3], we generalize these results to Hurwitz numbers of an arbitrary compact Riemann surface with a fixed number of general profiles and some (r,1^{d-r}) profiles.