Cutoff profiles for quantum Lévy processes and quantum random transpositions

We consider a natural analogue of Brownian motion on free orthogonal quantum groups and prove that it exhibits a cuto at time N ln(N). Then, we study the induced classical process on the real line and compute its atoms and density. This enables us to nd the cuto prole, which involves free Poisson di...

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Bibliographic Details
Published inProbability theory and related fields Vol. 183; no. 3-4; pp. 1285 - 1327
Main Authors Freslon, Amaury, Teyssier, Lucas, Wang, Simeng
Format Journal Article
LanguageFrench
Published Springer Verlag 10.03.2022
Subjects
Mathematics
Operator Algebras
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Summary:We consider a natural analogue of Brownian motion on free orthogonal quantum groups and prove that it exhibits a cuto at time N ln(N). Then, we study the induced classical process on the real line and compute its atoms and density. This enables us to nd the cuto prole, which involves free Poisson distributions and the semicircle law. We prove similar results for quantum permutations and quantum random transpositions. Résumé Nous considérons un analogue naturel du mouvement brownien sur les groupes libres quantiques orthogonaux et montrons qu'il a une coupure au temps N ln(N). Nous étudions ensuite le processus classique induit sur la droite réelle et calculons ses atomes et sa densité. Cela nous permet de trouver le prol de coupure, qui fait intervenir des lois de Poisson libres et la loi du semi-cercle. Nous prouvons des résultats similaires pour les permutations quantiques et les transpositions aléatoires quantiques.
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-022-01121-4