32,126 research outputs found
Asymptotics of Plancherel measures for the infinite-dimensional unitary group
We study a two-dimensional family of probability measures on infinite
Gelfand-Tsetlin schemes induced by a distinguished family of extreme characters
of the infinite-dimensional unitary group. These measures are unitary group
analogs of the well-known Plancherel measures for symmetric groups. We show
that any measure from our family defines a determinantal point process, and we
prove that in appropriate scaling limits, such processes converge to two
different extensions of the discrete sine process as well as to the extended
Airy and Pearcey processes.Comment: 39 page
Representations of classical Lie groups and quantized free convolution
We study the decompositions into irreducible components of tensor products
and restrictions of irreducible representations of classical Lie groups as the
rank of the group goes to infinity. We prove the Law of Large Numbers for the
random counting measures describing the decomposition. This leads to two
operations on measures which are deformations of the notions of the free
convolution and the free projection. We further prove that if one replaces
counting measures with others coming from the work of Perelomov and Popov on
the higher order Casimir operators for classical groups, then the operations on
the measures turn into the free convolution and projection themselves.
We also explain the relation between our results and limit shape theorems for
uniformly random lozenge tilings with and without axial symmetry.Comment: 43 pages, 4 figures. v3: relation to the Markov-Krein correspondence
is updated and correcte
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