94 research outputs found
On the Vershik-Kerov Conjecture Concerning the Shannon-McMillan-Breiman Theorem for the Plancherel Family of Measures on the Space of Young Diagrams
Vershik and Kerov conjectured in 1985 that dimensions of irreducible
representations of finite symmetric groups, after appropriate normalization,
converge to a constant with respect to the Plancherel family of measures on the
space of Young diagrams. The statement of the Vershik-Kerov conjecture can be
seen as an analogue of the Shannon-McMillan-Breiman Theorem for the
non-stationary Markov process of the growth of a Young diagram. The limiting
constant is then interpreted as the entropy of the Plancherel measure. The main
result of the paper is the proof of the Vershik-Kerov conjecture. The argument
is based on the methods of Borodin, Okounkov and Olshanski.Comment: To appear in GAFA. Referee's suggestions incorporated: in particular,
a new subsection 4.2 explains in greater detail the convergence of the
integral (15). Misprints corrected, references update
Rigidity of Determinantal Point Processes with the Airy, the Bessel and the Gamma Kernel
A point process is said to be rigid if for any bounded domain in the phase
space, the number of particles in the domain is almost surely determined by the
restriction of the configuration to the complement of our bounded domain. The
main result of this paper is that determinantal point processes with the Airy,
the Bessel and the Gamma kernels are rigid. The proof follows the scheme of
Ghosh [6], Ghosh and Peres [7]: the main step is the construction of a sequence
of additive statistics with variance going to zero.Comment: 10 page
Quasi-Symmetries of Determinantal Point Processes
The main result of this paper is that determinantal point processes on the
real line corresponding to projection operators with integrable kernels are
quasi-invariant, in the continuous case, under the group of diffeomorphisms
with compact support (Theorem 1.4); in the discrete case, under the group of
all finite permutations of the phase space (Theorem 1.6). The Radon-Nikodym
derivative is computed explicitly and is given by a regularized multiplicative
functional. Theorem 1.4 applies, in particular, to the sine-process, as well as
to determinantal point processes with the Bessel and the Airy kernels; Theorem
1.6 to the discrete sine-process and the Gamma kernel process. The paper
answers a question of Grigori Olshanski.Comment: The argument on regularization of multiplicative functionals has been
simplified. Section 4 has become shorter. Subsections 2.9, 2.13, 2.14, 2.15
have been added: in particular, formula (43) simplifies the argument for
unbounded function
On the Rate of Convergence in the Central Limit Theorem for Linear Statistics of Gaussian, Laguerre, and Jacobi Ensembles
Under the Kolmogorov--Smirnov metric, an upper bound on the rate of
convergence to the Gaussian distribution is obtained for linear statistics of
the matrix ensembles in the case of the Gaussian, Laguerre, and Jacobi weights.
The main lemma gives an estimate for the characteristic functions of the linear
statistics; this estimate is uniform over the growing interval. The proof of
the lemma relies on the Riemann--Hilbert approach.Comment: 45 pages, 5 figures. Final version. Cleared up exposition, added new
section "Outline of proof and discussion", fixed minor typo
The explicit formulae for scaling limits in the ergodic decomposition of infinite Pickrell measures
The main result of this paper, Theorem 1.1, gives explicit formulae for the
kernels of the ergodic decomposition measures for infinite Pickrell measures on
spaces of infinite complex matrices. The kernels are obtained as the scaling
limits of Christoffel-Uvarov deformations of Jacobi orthogonal polynomial
ensembles.Comment: 45 page
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