Special solutions of a discrete Painlev\'e equation for quantum minimal surfaces

Abstract

We consider solutions of a discrete Painlev\'e equation arising from a construction of quantum minimal surfaces by Arnlind, Hoppe and Kontsevich, and in earlier work of Cornalba and Taylor on static membranes. While the discrete equation admits a continuum limit to the continuous Painlev\'e I equation, we find that it has the same space of initial values as the Painlev\'e V equation with certain specific parameter values. We further explicitly show how each iteration of this discrete Painlev\'e I equation corresponds to a certain composition of B\"acklund transformations for Painlev\'e V, as was first remarked in work by Tokihiro, Grammaticos and Ramani. In addition, we show that some explicit special function solutions of Painlev\'e V, written in terms of modified Bessel functions, yield the unique positive solution of the initial value problem required for quantum minimal surfaces.