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Almost sure weak convergence for the circular ensembles of Dyson.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

<mark>Journal publication date</mark>2003
<mark>Journal</mark>Stochastics: An International Journal of Probability and Stochastic Processes formerly Stochastics and Stochastics Reports
Issue number6
Number of pages9
Pages (from-to)425-433
Publication StatusPublished
<mark>Original language</mark>English


The circular ensembles of Dyson satisfy isoperimetric inequalities and concentration of measure phenomena for large particle numbers analogous to the isoperimetric inequality for surface measure on the sphere in Euclidean space of high dimension. This leads to a geometrical proof of a result of Johansson [Bull. Sci. Math. (2) 112, (1988), 257-304] that the empirical distribution of energy levels under such ensembles converge weakly almost surely to normalized arclength on the unit circle as n→∞.