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