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Statistical self-similarity of one-dimensional growth processes

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  • Prähofer, Michael
  • Spohn, Herbert

Abstract

For one-dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times. We argue that for self-similar growth from a single seed the universal distribution is the Tracy–Widom distribution from the theory of random matrices and that for the growth from a flat substrate it is some other, only numerically determined distribution. In particular, for the polynuclear growth model in the droplet geometry the height maps onto the longest increasing subsequence of a random permutation, from which the height distribution is identified as the Tracy–Widom distribution.

Suggested Citation

  • Prähofer, Michael & Spohn, Herbert, 2000. "Statistical self-similarity of one-dimensional growth processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 279(1), pages 342-352.
    • Handle: RePEc:eee:phsmap:v:279:y:2000:i:1:p:342-352
    DOI: 10.1016/S0378-4371(99)00517-8
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    Cited by:

    1. Spohn, Herbert, 2006. "Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 369(1), pages 71-99.
    2. Wio, H.S. & Deza, J.I. & Sánchez, A.D. & García-García, R. & Gallego, R. & Revelli, J.A. & Deza, R.R., 2022. "The nonequilibrium potential today: A short review," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).

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