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

Understand

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.