## Extract from OP-SF NET

Topic #9 ------------- OP-SF NET 8.5 ------------ September 15, 2001
~~~~~~~~~~~~~
From: Steven Finch (sfinch@mathsoft.com)
Subject: Question on Painleve II numerics
[This appeared in opsftalk]
Here are four constants associated with the longest increasing subsequence
problem (Baik, Deift and Johansson):
mu=-1.77109, sigma=0.9018 (largest eigenvalue
of random GUE matrix)
mu'=3.6754, sigma'=0.7351 (second-largest e.v.)
of random GUE matrix)
http://front.math.ucdavis.edu/math.CO/9810105
http://front.math.ucdavis.edu/math.CO/9901118
These can be expressed as integrals involving a certain Painleve II ODE
solution that satisfies asymptotic boundary conditions.
The values came from Tracy and Widom:
http://front.math.ucdavis.edu/hep-th/9211141
who used asymptotic expansions of the Painleve II solution at both plus
and minus infinity to integrate forwards/backwards.
I am simply wondering if anyone has improved the estimates of these four
constants. Is Tracy-Widom's numerical analysis "state-of-the- art" for
this problem? Or can someone do better?
Thank you most kindly!
Steve Finch
MathSoft Engineering & Education, Inc.
101 Main St.
Cambridge, MA, USA 02142
http://www.mathsoft.com/asolve/sfinch.html

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