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Universal asymptotic statistics of maximal relative height in one-dimensional solid-on-solid models.

Grégory Schehr1, Satya N Majumdar

  • 1Theoretische Physik, Universität des Saarlandes, Saarbrücken, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 29, 2006
PubMed
Summary

We found a universal scaling form for the maximum height in 1D solid-on-solid models. This applies to large systems and is described by the Airy distribution function, confirmed by exact calculations and simulations.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Mathematical Physics

Background:

  • Studying the probability distribution of interface heights is crucial for understanding surface growth phenomena.
  • One-dimensional solid-on-solid models provide a tractable framework for investigating universal behaviors in fluctuating interfaces.

Purpose of the Study:

  • To determine the universal scaling form of the probability density function for the maximum relative height in finite-size 1D solid-on-solid models.
  • To verify theoretical predictions through exact analytical calculations and numerical simulations.

Main Methods:

  • Central limit arguments for large system sizes (L).
  • Exact computation using the transfer matrix technique for the anisotropic Ising model.
  • Numerical simulations to support analytical findings and investigate subleading corrections.

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Main Results:

  • A universal scaling form P(h(m), L) ~ (1/w(L)) * f(h(m)/w(L)) was identified for the maximum relative height, where f(x) is the Airy distribution function.
  • The subleading scaling function was found to be universal, given by the derivative of the Airy distribution function (f'(x)).
  • Exact analytical results were obtained for the anisotropic Ising model, validating the universal scaling form.

Conclusions:

  • The maximum relative height in a wide class of 1D solid-on-solid models exhibits universal behavior in the large-L limit.
  • The Airy distribution function and its derivative describe the universal scaling of the interface height distribution.
  • Theoretical predictions are robustly supported by both exact analytical methods and numerical simulations.