Zero-temperature limit of quantum weighted Hurwitz numbers

TL;DR

This paper investigates the zero-temperature limit of quantum weighted Hurwitz numbers, revealing their connection to Belyi curves and classical Hurwitz numbers through asymptotic analysis of the partition function.

Contribution

It provides the leading and next-order asymptotic terms for the partition function and quantum Hurwitz numbers as temperature approaches zero, linking quantum and classical enumerations.

Findings

Leading term reproduces classical Hurwitz numbers for Belyi curves

Quantum generating functions converge to classical ones at zero temperature

Asymptotic analysis connects quantum weights to classical enumerations

Abstract

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.