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Title:
Global Fluctuations of Linear Statistics of beta Ensembles
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Abstract:
Dyson in the 1960’s observed that the eigenvalues of random matrices behave like a two-dimensional Coulomb gas in equilibrium under an external potential. Classical random matrices are usually labelled by a discrete index beta=1,2,4, which in this analogy plays the role of the inverse temperature. In many applications, however, it is interesting to look at random matrix models where beta can assume any positive real value; these models are usually referred to as `beta ensembles’. Central Limit Theorems for linear statistics of eigenvalues of random matrices (for both classical and beta ensembles) have been studied for almost twenty years. Not many results, however, are available for the correction to the limiting behaviour of the probability distributions of linear statistics. Not much is know about finite matrix dimensions too. These corrections have important applications to Quantum Transport, Quantum Chaos, Quantum Field Theory and Combinatorics. In this talk we will discuss recent results on the fluctuation of linear statistics for both Hermitian and non-Hermitian beta ensembles
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