On the connection problem for Painlev\'e I

Understand

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.