1,161 research outputs found
Martin Boundary Theory of some Quantum Random Walks
In this paper we define a general setting for Martin boundary theory
associated to quantum random walks, and prove a general representation theorem.
We show that in the dual of a simply connected Lie subgroup of U(n), the
extremal Martin boundary is homeomorphic to a sphere. Then, we investigate
restriction of quantum random walks to Abelian subalgebras of group algebras,
and establish a Ney-Spitzer theorem for an elementary random walk on the fusion
algebra of SU(n), generalizing a previous result of Biane. We also consider the
restriction of a quantum random walk on introduced by Izumi to two
natural Abelian subalgebras, and relate the underlying Markov chains by
classical probabilistic processes. This result generalizes a result of Biane.Comment: 29 page
New scaling of Itzykson-Zuber integrals
We study asymptotics of the Itzykson-Zuber integrals in the scaling when one
of the matrices has a small rank compared to the full rank. We show that the
result is basically the same as in the case when one of the matrices has a
fixed rank. In this way we extend the recent results of Guionnet and Maida who
showed that for a latter scaling the Itzykson-Zuber integral is given in terms
of the Voiculescu's R-transform of the full rank matrix
Free products of sofic groups with amalgamation over monotileably amenable groups
We show that free products of sofic groups with amalgamation over
monotileably amenable subgroups are sofic. Consequently, so are HNN extensions
of sofic groups relative to homomorphisms of monotileably amenable subgroups.
We also show that families of independent uniformly distributed permutation
matrices and certain families of non-random permutation matrices (essentially,
those coming from quasi--actions of a sofic group) are asymptotically *-free as
the matrix size grows without bound.Comment: 16 pages. Version 2 has a shorter proof of Lemma 2.2, thanks to Ion
Nechita. Version 3 corrects a mistake. The previous Lemma 3.1 was incorrect.
Version 3 has a new proof of the main result, but it is (apparently) weaker
than in Version 2. Note that the paper's title also changed accordingly.
Version 4 corrects a minor mistake around equation (4
The strong asymptotic freeness of Haar and deterministic matrices
In this paper, we are interested in sequences of q-tuple of N-by-N random
matrices having a strong limiting distribution (i.e. given any non-commutative
polynomial in the matrices and their conjugate transpose, its normalized trace
and its norm converge). We start with such a sequence having this property, and
we show that this property pertains if the q-tuple is enlarged with independent
unitary Haar distributed random matrices. Besides, the limit of norms and
traces in non-commutative polynomials in the enlarged family can be computed
with reduced free product construction. This extends results of one author (C.
M.) and of Haagerup and Thorbjornsen. We also show that a p-tuple of
independent orthogonal and symplectic Haar matrices have a strong limiting
distribution, extending a recent result of Schultz.Comment: 12 pages. Accepted for publication to Annales Scientifique de l'EN
Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
We revisit the work of the first named author and using simpler algebraic
arguments we calculate integrals of polynomial functions with respect to the
Haar measure on the unitary group U(d). The previous result provided exact
formulas only for 2d bigger than the degree of the integrated polynomial and we
show that these formulas remain valid for all values of d. Also, we consider
the integrals of polynomial functions on the orthogonal group O(d) and the
symplectic group Sp(d). We obtain an exact character expansion and the
asymptotic behavior for large d. Thus we can show the asymptotic freeness of
Haar-distributed orthogonal and symplectic random matrices, as well as the
convergence of integrals of the Itzykson-Zuber type
Low entropy output states for products of random unitary channels
In this paper, we study the behaviour of the output of pure entangled states
after being transformed by a product of conjugate random unitary channels. This
study is motivated by the counterexamples by Hastings and Hayden-Winter to the
additivity problems. In particular, we study in depth the difference of
behaviour between random unitary channels and generic random channels. In the
case where the number of unitary operators is fixed, we compute the limiting
eigenvalues of the output states. In the case where the number of unitary
operators grows linearly with the dimension of the input space, we show that
the eigenvalue distribution converges to a limiting shape that we characterize
with free probability tools. In order to perform the required computations, we
need a systematic way of dealing with moment problems for random matrices whose
blocks are i.i.d. Haar distributed unitary operators. This is achieved by
extending the graphical Weingarten calculus introduced in Collins and Nechita
(2010)
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