Electronic Journal of Probability

A Microscopic Model for the Burgers Equation and Longest Increasing Subsequences

Timo Seppäläinen

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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.

Article information

Source
Electron. J. Probab., Volume 1 (1996), paper no. 5, 51 pp.

Dates
Accepted: 6 March 1996
First available in Project Euclid: 25 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1453756468

Digital Object Identifier
doi:10.1214/EJP.v1-5

Mathematical Reviews number (MathSciNet)
MR1386297

Zentralblatt MATH identifier
0891.60093

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 35L65: Conservation laws 60C05: Combinatorial probability 82C22: Interacting particle systems [See also 60K35]

Keywords
Hydrodynamic scalinglimit Ulam's problem Hammersley's process nonlinear conservation law the Burgers equation the Laxformula

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Seppäläinen, Timo. A Microscopic Model for the Burgers Equation and Longest Increasing Subsequences. Electron. J. Probab. 1 (1996), paper no. 5, 51 pp. doi:10.1214/EJP.v1-5. https://projecteuclid.org/euclid.ejp/1453756468


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