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Universal distributions for growth processes in 1+1 dimensions and random matrices

Prahofer1, Spohn

  • 1Zentrum Mathematik and Physik Department, TU Munchen, D-80290 Munchen, Germany.

Physical Review Letters
|September 16, 2000
PubMed
Summary

We present a scaling theory for Kardar-Parisi-Zhang (KPZ) growth in one dimension. Our findings reveal three universal distributions for shape fluctuations, dependent on the macroscopic shape, derived from random matrix theory.

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Area of Science:

  • Surface growth phenomena
  • Statistical physics
  • Non-equilibrium systems

Background:

  • The Kardar-Parisi-Zhang (KPZ) equation describes the dynamics of interfaces in various physical systems.
  • Understanding universal scaling behaviors in one-dimensional growth is crucial for theoretical physics.
  • The polynuclear growth (PNG) model serves as a key discrete model for studying KPZ universality.

Purpose of the Study:

  • To develop a comprehensive scaling theory for one-dimensional Kardar-Parisi-Zhang (KPZ) growth.
  • To identify and characterize universal distributions governing shape fluctuations in the PNG model.
  • To establish a connection between KPZ universality and random matrix theory.

Main Methods:

  • Detailed analysis of the one-dimensional polynuclear growth (PNG) model.

Related Experiment Videos

  • Development of a scaling theory based on the identified universal distributions.
  • Computation of distribution functions using the partition function of Gaussian random matrices in a cosine potential.

    • Main Results:

    • Identification of three universal distribution functions for shape fluctuations.
    • Demonstration of the dependence of these distributions on the macroscopic shape of the growing interface.
    • Successful computation of these distributions through advanced random matrix techniques.

    Conclusions:

    • The developed scaling theory provides a unified framework for understanding KPZ growth in one dimension.
    • The identified universal distributions offer new insights into the statistical properties of fluctuating interfaces.
    • The application of random matrix theory proves effective in characterizing KPZ universality.