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

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.
• 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.