The Hurwitz–quadrangulation correspondence conjecture for Painlevé I

Let $b_g$ be the constants in the asymptotic expansion of the ramified-covering numbers $h_{g,n;\emptyset}$, and let $b'_g$ be the corresponding constants for the numbers $Q_{g,n}$ of quadrangulations. Let $F(y)$ be the free energy defined by

$$F(y) = \Gamma(-5/2)b_0y^{5/2}-b_1\ln y+\sum_{g\geq 2}\Gamma\bigl(5(g-1)/2\bigr)b_g y^{5(1-g)/2}.$$

Hurwitz–quadrangulation correspondence conjecture. The constants satisfy

$$b'_g=2^{\frac32(g-1)+1}\cdot b_g.$$

Equivalently, the function $u(y)=F”(y)$ satisfies the Painlevé I equation

$$\frac16u”(y)+u(y)^2=2y.$$

This conjecture asserts that the free energy arising from ramified coverings of the sphere agrees, after the stated rescaling, with the free energy of the quadrangulation model. The paper reports numerical evidence for the relation and its equivalent Painlevé I description; its status is open.