Abstract

Orthogonal polynomials associated with symmetric Freud weights which arise in the context of Hermitian matrix models and random symmetric matrix ensembles, in particular the sextic Freud weight \[ w(x)=\exp\left\{-N \!\left(g_6x^{6+ g_4x}4+g_2x^2 \right)\right\},\eqno(1)\] with $N$, $g_6$, $g_4$ and $g_2$ parameters. In the 1990s the behaviour of the recurrence coefficients in the three-term recurrence relation associated with these orthogonal polynomials for the weight (1) was described as being ``chaotic” and more recently the ``chaotic phase” has been interpreted as a dispersive shock.  In this talk I will discuss  properties of the recurrence coefficients in the three-term recurrence relation associated with these orthogonal polynomials associated with the sextic Freud weight (1). In particular  For this weight,  the recurrence coefficients satisfy a fourth-order discrete equation which is the second member of the first discrete Painlev\’e hierarchy, and also known as the ``string equation”.