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SUMMARY:Matrix Models\, Orthogonal Polynomials and Symmetric Freud weights
  - Peter Clarkson (University of Kent)
DTSTART:20240807T140000Z
DTEND:20240807T150000Z
UID:TALK219364@talks.cam.ac.uk
DESCRIPTION:Orthogonal polynomials associated with symmetric Freud weights
  which arise in the context of Hermitian matrix models and random symmetri
 c matrix ensembles\, in particular the sextic Freud weight\n\\[ w(x)=\\exp
 \\left\\{-N \\!\\left(g_6x^6+ g_4x^4+g_2x^2 \\right)\\right\\}\,\\eqno(1)\
 \]\nwith $N$\, $g_6$\, $g_4$ and $g_2$ parameters.\nIn the 1990s the behav
 iour of the recurrence coefficients in the three-term recurrence relation 
 associated with these orthogonal polynomials for the weight (1) was descri
 bed as being ``chaotic" and more recently the ``chaotic phase" has been in
 terpreted as a dispersive shock.&nbsp\;\nIn this talk I will discuss&nbsp\
 ; properties of the recurrence coefficients in the three-term recurrence r
 elation associated with these orthogonal polynomials associated with the s
 extic Freud weight (1). In particular&nbsp\;\nFor this weight\,&nbsp\; 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".
LOCATION:External
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