A Microscopic Model for the Burgers Equation and Longest Increasing Subsequences
Timo Seppäläinen, Iowa State University
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.
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