Matrix Models, Orthogonal Polynomials and Symmetric Freud weights

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• Peter Clarkson (University of Kent)
• Wednesday 07 August 2024, 15:00-16:00
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PDW - Emergent phenomena in nonlinear dispersive waves

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”.

This talk is part of the Isaac Newton Institute Seminar Series series.

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