scholarly journals Martin boundaries of the duals of free unitary quantum groups

2019 ◽  
Vol 155 (6) ◽  
pp. 1171-1193
Author(s):  
Sara Malacarne ◽  
Sergey Neshveyev
Keyword(s):  
Random Walk ◽  
Quantum Group ◽  
Quantum Groups ◽  
Free Group ◽  
Cayley Tree ◽  
Martin Boundary ◽  
Finite Range ◽  
Quantum Random Walk ◽  
Martin Boundaries ◽  
Topological Boundary

Given a free unitary quantum group $G=A_{u}(F)$, with $F$ not a unitary $2\times 2$ matrix, we show that the Martin boundary of the dual of $G$ with respect to any $G$-${\hat{G}}$-invariant, irreducible, finite-range quantum random walk coincides with the topological boundary defined by Vaes and Vander Vennet. This can be thought of as a quantum analogue of the fact that the Martin boundary of a free group coincides with the space of ends of its Cayley tree.

scholarly journals Probabilistic boundaries of finite extensions of quantum groups

2017 ◽  
Vol 20 (04) ◽  
pp. 1750026 ◽  
Author(s):  
Sara Malacarne ◽  
Sergey Neshveyev
Keyword(s):  
Random Walk ◽  
Harmonic Function ◽  
Quantum Group ◽  
Quantum Groups ◽  
Tensor Categories ◽  
Martin Boundaries ◽  
Exact Sequences

Given a discrete quantum group [Formula: see text] with a finite normal quantum subgroup [Formula: see text], we show that any positive, possibly unbounded, harmonic function on [Formula: see text] with respect to an irreducible invariant random walk is [Formula: see text]-invariant. This implies that, under suitable assumptions, the Poisson and Martin boundaries of [Formula: see text] coincide with those of [Formula: see text]. A similar result is also proved in the setting of exact sequences of C[Formula: see text]-tensor categories. As an immediate application, we conclude that the boundaries of the duals of the group-theoretical easy quantum groups are classical.

scholarly journals RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

2021 ◽  
pp. 1-37
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Riesz Transform ◽  
Wreath Products ◽  
Compact Quantum Group ◽  
Riesz Transforms ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Free Products ◽  
Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

scholarly journals Quantum random walk with Rb atoms

2009 ◽  
Vol 185 ◽  
pp. 012026 ◽  
Author(s):  
Kia Manouchehri ◽  
Jingbo Wang
Keyword(s):  
Random Walk ◽  
Quantum Random Walk

The quantum random walk within the Schrödinger-Langevin approach

1997 ◽  
Vol 71 (2-3) ◽  
pp. 187-194 ◽  
Author(s):  
Manuel O. Cáceres ◽  
Ana K. Chattah
Keyword(s):  
Random Walk ◽  
Quantum Random Walk ◽  
Langevin Approach

scholarly journals Controlling quantum random walk with a step-dependent coin

2018 ◽  
Vol 20 (8) ◽  
pp. 083028 ◽  
Author(s):  
S Panahiyan ◽  
S Fritzsche
Keyword(s):  
Random Walk ◽  
Quantum Random Walk

scholarly journals From Quantum Groups to Groups

2013 ◽  
Vol 65 (5) ◽  
pp. 1073-1094 ◽  
Author(s):  
Mehrdad Kalantar ◽  
Matthias Neufang
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Locally Compact ◽  
Large Classes ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Recent Developments ◽  
Compact Quantum Groups ◽  
Quantum Version ◽  
Quantum Point

AbstractIn this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign to each locally compact quantum group 𝔾 a locally compact group 𝔾˜ that is the quantum version of point-masses and is an invariant for the latter. We show that “quantum point-masses” can be identified with several other locally compact groups that can be naturally assigned to the quantum group 𝔾. This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of 𝔾 are encoded by 𝔾˜; the latter, despite being a simpler object, can carry very important information about 𝔾.

scholarly journals Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

2014 ◽  
Vol 57 (4) ◽  
pp. 708-720 ◽  
Author(s):  
Michael Brannan
Keyword(s):  
Strong Convergence ◽  
Quantum Group ◽  
Quantum Groups ◽  
Convergence Theorem ◽  
Convergence Result ◽  
Operator Norm ◽  
Matrix Coefficient ◽  
Strong Convergence Theorem ◽  
Convergence Results ◽  
Noncommutative Polynomials

AbstractIt is known that the normalized standard generators of the free orthogonal quantum groupO+Nconverge in distribution to a free semicircular system as N → ∞. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators ofO+Nconverges asN→ ∞ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-knownL2-L∞norm equivalence for noncommutative polynomials in free semicircular systems.

scholarly journals Martin boundary of a killed random walk on a quadrant

2010 ◽  
Vol 38 (3) ◽  
pp. 1106-1142 ◽  
Author(s):  
Irina Ignatiouk-Robert ◽  
Christophe Loree
Keyword(s):  
Random Walk ◽  
Martin Boundary
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