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A shortcut through the Coulomb gas method for spectral linear statistics on random matrices

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.