Electronic Journal of Probability

A Microscopic Model for the Burgers Equation and Longest Increasing Subsequences

Timo Seppäläinen

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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

This work is licensed under aCreative Commons Attribution 3.0 License.


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

Export citation


  • D. Aldous and P. Diaconis, Hammersley's interacting particle process and longest increasing subsequences, Probab. Theory Relat. Fields 103, (1995), 199–213.
  • R. Durrett, Probability: Theory and Examples, Wadsworth, Pacific Grove (1991).
  • M. Ekhaus and T. Sepp"al"ainen, Stochastic dynamics macroscopically governed by the porous medium equation for isothermal flow, Ann. Acad. Sci. Fenn. Ser. A I Math (to appear).
  • S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York (1985).
  • S. Feng, I. Iscoe, and T. Sepp"al"ainen, A class of stochastic evolutions that scale to the porous medium equation, J. Statist. Phys. ( to appear).
  • P. A. Ferrari, Shocks in one-dimensional processes with drift, Probability and Phase Transitions, ed. G. Grimmett, Kluwer Academic Publishers (1994).
  • J. M. Hammersley, A few seedlings of research, Proc. Sixth Berkeley Symp. Math. Stat. Probab. Vol. I, (1972), 345–394.
  • C. Kipnis, Central limit theorems for infinite series of queues and applications to simple exclusion, Ann. Probab. 14, (1986), 397–408.
  • P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10, (1957), 537–566.
  • P. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, (1973).
  • T. M. Liggett, Interacting Particle Systems, Springer-Verlag, New York, (1985).
  • P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London (1982).
  • F. Rezakhanlou, Hydrodynamic limit for attractive particle systems on $mmZ^d$, Comm. Math. Phys. 140, (1991), 417–448.
  • R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, (1970).
  • H. Rost, Non-equilibrium behaviour of a many particle process: Density profile and local equilibrium, Z. Wahrsch. Verw. Gebiete 58, (1981), 41–53.
  • J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, (1983).
  • Y. Suzuki and K. Uchiyama, Hydrodynamic limit for a spin system on a multidimensional lattice, Probab. Theory Relat. Fields 95, (1993), 47–74.