Tau functions and their growth

The Szegö–Widom constant occurring in the asymptotic formula и truncated block Toeplitz determinants is a remarkable smooth function on the loop group of the complete linear group, with many applications to orthogonal polynomials and random matrix theory. In terms of the Riemann–Hilbert problem on the circle, this function was recently recognized as the tau function of solutions of soliton equations in the Segal–Wilson class. We extend the Toeplitz and Riemann–Hilbert set-up to cover all local holomorphic solutions. In particular, we prove that every local holomorphic (in $x$ and $t$) solution of the Korteweg–de Vries equation is the second logarithmic derivative of an entire function of the spatial variable $x$ and discuss the possible order of growth of this entire function along with similar results and conjectures for other soliton equations.