FW: Painleve II numerics

["Daniel Lozier" <lozier@nist.gov>]

From: opsftalk@nist.gov [mailto:opsftalk@nist.gov]On Behalf Of Steven
Finch
Sent: Tuesday, August 21, 2001 1:06 PM
To: dlozier@nist.gov
Subject: Painleve II numerics

-----Original Message-----
Sender: Steven Finch <sfinch@mathsoft.com>
Subject: Painleve II numerics


Hello!

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

	
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