Topological recursion for irregular spectral curves

We study topological recursion on the irregular spectral curve xy² - xy + 1 = 0, which produces a weighted count of dessins d'enfant. This analysis is then applied to topological recursion on the spectral curve xy² = 1, which takes the place of the Airy curve x = y² to describe asymptotic behaviour of enumerative problems associated to irregular spectral curves. In particular, we calculate all one-point invariants of the spectral curve xy² = 1 via a new three-term recursion for the number of dessins d'enfant with one face.