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The right tail exponent of the Tracy–Widom β distribution

Laure DumazBálint Virág — 2013

Annales de l'I.H.P. Probabilités et statistiques

The Tracy–Widom β distribution is the large dimensional limit of the top eigenvalue of β random matrix ensembles. We use the stochastic Airy operator representation to show that as a the tail of the Tracy–Widom distribution satisfies P ( 𝑇𝑊 β g t ; a ) = a - ( 3 / 4 ) β + o ( 1 ) exp - 2 3 β a 3 / 2 .

Amenability of linear-activity automaton groups

Gideon AmirOmer AngelBálint Virág — 2013

Journal of the European Mathematical Society

We prove that every linear-activity automaton group is amenable. The proof is based on showing that a random walk on a specially constructed degree 1 automaton group – the mother group – has asymptotic entropy 0. Our result answers an open question by Nekrashevych in the Kourovka notebook, and gives a partial answer to a question of Sidki.

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