Invariant beta ensembles and the Gauss-Wigner crossover

Area of Science:

• Mathematics
• Probability Theory
• Random Matrix Theory

Background:

• Beta ensembles are crucial in random matrix theory.
• Understanding the behavior of random matrices, especially their limiting distributions, is a key area of research.
• The connection between Gaussian and Wigner distributions is of significant interest.

Purpose of the Study:

• To define a novel diffusive matrix model.
• To provide an explicit construction for beta ensembles of random matrices.
• To interpolate between Gaussian and Wigner distributions for specific beta values.

Main Methods:

• Development of a new diffusive matrix model.
• Analysis of the model's convergence properties.
• Mathematical computation of interpolating limit distributions.

Main Results:

• The model converges to the β-Dyson Brownian motion for β in [0,2].
• It offers an explicit construction of orthogonal and unitary invariant beta ensembles.
• Smooth interpolation between Gaussian and Wigner semicircle distributions is achieved for small β.
• A one-parameter family of interpolating limit distributions is explicitly computed.
• Finite-size corrections to the semicircle distribution are calculated.

Conclusions:

• The new model provides a unified framework for studying beta ensembles.
• It bridges the gap between different random matrix distributions.
• The explicit computations enable deeper analysis of random matrix properties.