- 95-403 Timo Seppalainen
- A Microscopic Model for the Burgers Equation
and Longest Increasing Subsequences
Sep 1, 95
(auto. generated ps),
of related papers
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 an alternative view of
Hammersley's particle system recently used by Aldous
and Diaconis to give a new solution to Ulam's problem.
Along the way to the scaling limit we also
produce a solution to this
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. This semigroup
for nonlinear scalar
conservation laws in one space variable is developed
in a separate appendix, where we give an existence result
and a uniqueness criterion for solutions of
such equations with initial
data given by a Radon measure.