Quantum random walks and minors of Hermitian Brownian motion (Q2903306)

The paper under review is concerned with the higher-dimensional extensions of the construction of Biane, who showed that a continuous time Markov process \((B_1(t), \sqrt{B_1(t)^2 + B_2(t)^2 + B_3(t)^2})\) built of independent Brownian motions \(B_1, B_2, B_3\) can be approximated by a discrete time noncommutative random walk. The extensions are formed via considering the universal enveloping algebra of the Lie algebra of the group \(GL_d(\mathbb{C})\) and of its block-diagonal subgroups. This leads to a family of noncommutative random walks approximating stochastic processes built of consecutive minors of matricial Brownian motions and allows the authors to analyse the Markovian properties of the latter. This gives in particular a new proof of the recent result of Adler, Nordenstam and Moerbeke, who proved that the process of eigenvalues of two consecutive minors of the Hermitian matricial Brownian motion is Markov, and the Markov property fails if one considers more than two minors.