On the connection problem for Painlev\'e I

TL;DR

This paper investigates how the tau function of Painlevé I depends on monodromy data, computing connection constants that relate asymptotic behaviors using dilogarithms of cluster coordinates.

Contribution

It provides explicit formulas for connection constants of the Painlevé I tau function in terms of cluster coordinates and dilogarithms, linking monodromy data to asymptotics.

Findings

Computed connection constants for Painlevé I tau function asymptotics.

Expressed connection constants using dilogarithms of cluster coordinates.

Linked monodromy data to asymptotic behaviors of the tau function.

Abstract

We study the dependence of the tau function of Painlev\'e I equation on the generalized monodromy of the associated linear problem. In particular, we compute connection constants relating the tau function asymptotics on five canonical rays at infinity. The result is expressed in terms of dilogarithms of cluster type coordinates on the space of Stokes data.