On the lower bounds for real double Hurwitz numbers

As the real counterpart of double Hurwitz number, the real double Hurwitz number depends on the distribution of real branch points. We consider the problem of asymptotic growth of real and complex double Hurwitz numbers. We provide a lower bound for real double Hurwitz numbers based on the tropical computation of real double Hurwitz numbers. By using this lower bound and J. Rau’s result ( Math Ann 375: 895-915, 2019), we prove the logarithmic equivalence of real and complex Hurwitz numbers.