Connection problem for the tau-function of the Sine-Gordon reduction of Painlev\'{e}-III equation via the Riemann-Hilbert approach

TL;DR

This paper explicitly evaluates the constant pre-factor in the large x asymptotics of the Painlevé III tau-function using the Riemann-Hilbert approach, confirming a recent conjecture without relying on Virasoro conformal blocks.

Contribution

It provides a new Riemann-Hilbert based proof of the conjectural formula for the Painlevé III tau-function's asymptotic pre-factor, independent of Virasoro conformal blocks.

Findings

Explicit expression for the constant pre-factor in asymptotics

Confirmation of the conjectural formula by Lisovyy, Tykhyy, and the first co-author

Method based on Riemann-Hilbert approach

Abstract

We evaluate explicitly, in terms of the Cauchy data, the constant pre-factor in the large $x$ asymptotics of the Painlev\'e III tau-function. Our result proves the conjectural formula for this pre-factor obtained recently by O. Lisovyy, Y. Tykhyy, and the first co-author with the help of the recently discovered connection of the Painlev\'e tau-functions with the Virasoro conformal blocks. Our approach does not use this connection, and it is based on the Riemann-Hilbert method.