In recent years, the monotonic double Hurwitz number has been introduced as a natural combinatorial modification of the double Hurwitz number. Although monotonic double Hurwitz numbers share many structural properties with their classical counterparts such as piecewise polynomials, the quantitative properties of these two numbers are quite different. We consider real analogues of monotonic double Hurwitz numbers and study the asymptote of these real analogues. A key element is the interpretation of the actual tropical cover by arbitrary partitions as a symmetry group factorization that generalizes results from Guay-Paquet, Markwig, and Rau (Int. Math. Res. Not. IMRN, 2016 (1):258-293, 2016). Using the interpretation above, we consider three types of real analogues of monotonic double Hurwitz numbers: real monotonic double Hurwitz numbers associated with simple partitions, and real double Hurwitz numbers associated with arbitrary partitions. Heavy Hurwitz number and real mixed double Hurwitz number. Under certain conditions, find the lower bounds of these real analogues and obtain the logarithmic asymptote of the real monotonic double Hurwitz numbers for any split and real mixed multiple Hurwitz numbers. In particular, under the given conditions, real mixed double Hurwitz numbers are logarithmically equivalent to complex double Hurwitz numbers. Construct a family of true tropical cover and use them to show that true monotonic double Hurwitz numbers compared to simple splits are logarithmically equivalent to monotonic double Hurwitz numbers under certain conditions Indicates that This is consistent with the logarithmic equivalence of real double Hurwitz numbers and complex double Hurwitz numbers.
