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A Microscopic Model for the Burgers Equation and Longest Increasing Subsequences

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1.	Title	Title of document	A Microscopic Model for the Burgers Equation and Longest Increasing Subsequences
2.	Creator	Author's name, affiliation, country	Timo Seppäläinen; Iowa State University
3.	Subject	Discipline(s)	Mathematics
3.	Subject	Keyword(s)	Hydrodynamic scalinglimit, Ulam's problem, Hammersley's process, nonlinear conservation law,the Burgers equation, the Laxformula
3.	Subject	Subject classification	Primary 60K35, Secondary 35L65, 60C05,82C22
4.	Description	Abstract	We introduce an interacting random process related to Ulam's problem, or finding the limit of the normalized longest increasing subsequence of a random permutation. The process describes the evolution of a configuration of sticks on the sites of the one-dimensional integer lattice. Our main result is a hydrodynamic scaling limit: The empirical stick profile converges to a weak solution of the inviscid Burgers equation under a scaling of lattice space and time. The stick process is also an alternative view of Hammersley's particle system that Aldous and Diaconis used to give a new solution to Ulam's problem. Along the way to the scaling limit we produce another independent solution to this question. The heart of the proof is that individual paths of the stochastic process evolve under a semigroup action which under the scaling turns into the corresponding action for the Burgers equation, known as the Lax formula. In a separate appendix we use the Lax formula to give an existence and uniqueness proof for scalar conservation laws with initial data given by a Radon measure.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	1996-03-06
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/5
10.	Identifier	Digital Object Identifier	10.1214/EJP.v1-5
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 1
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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