Título: Random Walks on Random Cayley Graph

Palestrante: Sam Olesker-Taylor, University of Bath
Data: 10/11/2021
Horário: 13:00hrs
Local: Transmissão online

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ID da reunião: 958 0581 3232

Resumo: We investigate mixing properties of RWs on random Cayley graphs of a finite group G with k ≫ 1 independent, uniformly random generators. Denote this Gₖ. Assume that 1 ≪ log k ≪ log |G|. Aldous and Diaconis (1985) conjectured that the RW exhibits cutoff for any group G whenever k ≫ log |G| and further that the time depends only on k and |G|. This was verified for Abelian groups by Dou and Hildebrand (1994, 1996). Their upper bound holds for all groups. We establish cutoff for the RW on Gₖ for all Abelian groups when 1 ≪ k ≲ log |G|, subject to some 'almost necessary' conditions. We also exhibit a non-Abelian matrix group which contradicts the second part of the AD conjecture. Lastly, we upper bound the mixing time of a RW on a nilpotent group by that of the RW on a corresponding Abelian group.