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Almost sure weak convergence for the circular ensembles of Dyson.

Research output: Contribution to journalJournal article


Journal publication date2003
JournalStochastics: An International Journal of Probability and Stochastic Processes formerly Stochastics and Stochastics Reports
Number of pages9
Original languageEnglish


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→∞.