Zero-temperature limit of quantum weighted Hurwitz numbers

The partition function for quantum weighted double Hurwitz numbers can be interpreted in terms of the energy distribution of a quantum Bose gas with vanishing fugacity. We compute the leading term of the partition function and the quantum weighted Hurwitz numbers in the zero temperature limit $T \rightarrow 0$, as well as the next order corrections. The leading term is shown to reproduce the case of uniformly weighted Hurwitz numbers of Belyi curves. In particular, the KP or Toda $\tau$-function serving as generating function for the quantum Hurwitz numbers is shown in the limit to give the one for Belyi curves and, with suitable scaling, the same holds true for the partition function, the weights and the expectations of Hurwitz numbers.