Existence of Boutroux curves, $g$-functions and spectral networks from Newton's polygon

Abstract

We prove the existence of an algebraic plane curve of equation $P(x,y)=0$, with prescribed asymptotic behaviors at punctures, and with the Boutroux property, namely, periods have vanishing real part, i.e, $\Re(\int_\gamma y dx)=0$ for every closed loop $\gamma$. This has applications in the Riemann-Hilbert problem, in random matrix theory, in spectral networks, in WKB analysis and Stokes phenomenon, in algebraic and enumerative geometry, and many applications in mathematical physics. From Newton's polygon we can define an affine space such that there exists always a Boutroux curve. This result is applied to random matrix and asymptotic theory, in which a key ingredient is called the $g$-function, the function $g(x)=\int_o^x Y dX$ is a $g$-function precisely if and only if the algebraic plane curve is a Boutroux curve.