Abstract
In the last decade, spectral linear statistics on large dimensional random matrices have attracted significant attention. Within the physics community, a privileged role has been played by invariant matrix ensembles for which a two-dimensional Coulomb gas analogy is available. We present a critical revision of the Coulomb gas method in random matrix theory (RMT) borrowing language and tools from large deviations theory. This allows us to formalize an equivalent, but more effective and quicker route toward RMT free energy calculations. Moreover, we argue that this more modern viewpoint is likely to shed further light on the interesting issues of weak phase transitions and evaporation phenomena recently observed in RMT.
Highlights
We present a critical revision of the Coulomb gas method in random matrix theory (RMT) borrowing language and tools from large deviations theory
This article contains a critical revision of the two-dimensional (2D) Coulomb gas analogy in random matrix theory (RMT) and presents an alternative method to compute large deviation
A more modern viewpoint on the subject, based on concepts and tools borrowed from large deviations theory (LDT) seems worthwhile, as certain fundamental identities have been so far overlooked in the vast majority of previous works on this topic
Summary
This article contains a critical revision of the two-dimensional (2D) Coulomb gas analogy in random matrix theory (RMT) and presents an alternative method to compute large deviation. (i) To present an effective shortcut to the standard Coulomb gas technique for the evaluation of probabilities of linear statistics on invariant random matrix models. In addition to the shortcut, our investigations naturally prompt a corpus of new ideas on the issues of phase transitions and evaporation phenomena in Coulomb gas systems.
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