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Deformations of the Tracy-Widom distribution
O Bohigas1, J X de Carvalho, M P Pato
1LPTMS, CNRS, Université Paris-Sud, UMR 8626, Orsay Cedex F-91405, France.
Summary
This study explores transformations of the Tracy-Widom distribution in random matrix theory. Introducing disorder or removing eigenvalues leads to new distributions, revealing competition and continuous transitions.
Area of Science:
- Mathematics
- Physics
- Statistics
Background:
- The Tracy-Widom distribution models the largest eigenvalue in random matrix theory.
- Understanding eigenvalue distributions is crucial in various scientific fields.
Purpose of the Study:
- To investigate transformations of the Tracy-Widom distribution under modified random matrix models.
- To analyze the impact of external randomness and eigenvalue removal on eigenvalue distributions.
Main Methods:
- Introduction of disorder into Gaussian ensembles via an external source.
- Random removal of a fraction of correlated eigenvalues from random matrices.
- Application of Fredholm determinant formalism.
Main Results:
- A competition between the Tracy-Widom and normal distributions emerges with disorder, dependent on disorder spread.
- A continuous transition from the Tracy-Widom to the Weibull distribution is observed when eigenvalues are randomly removed.
- The Weibull distribution characterizes extreme values of uncorrelated sequences.
Conclusions:
- The Tracy-Widom distribution is not rigid and can be transformed through specific modifications to random matrix ensembles.
- These findings extend the understanding of eigenvalue statistics in random matrices and their relationship to extreme value theory.
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