44,655 research outputs found
Limit theorems for linear eigenvalue statistics of overlapping matrices
The paper proves several limit theorems for linear eigenvalue statistics of
overlapping Wigner and sample covariance matrices. It is shown that the
covariance of the limiting multivariate Gaussian distribution is diagonalized
by choosing the Chebyshev polynomials of the first kind as the basis for the
test function space. The covariance of linear statistics for the Chebyshev
polynomials of sufficiently high degree depends only on the first two moments
of the matrix entries. Proofs are based on a graph-theoretic interpretation of
the Chebyshev linear statistics as sums over non-backtracking cyclic pathsComment: 44 pages, 4 figures, some typos are corrected and proofs clarified.
Accepted to the Electronic Journal of Probabilit
From Jack polynomials to minimal model spectra
In this note, a deep connection between free field realisations of conformal
field theories and symmetric polynomials is presented. We give a brief
introduction into the necessary prerequisites of both free field realisations
and symmetric polynomials, in particular Jack symmetric polynomials. Then we
combine these two fields to classify the irreducible representations of the
minimal model vertex operator algebras as an illuminating example of the power
of these methods. While these results on the representation theory of the
minimal models are all known, this note exploits the full power of Jack
polynomials to present significant simplifications of the original proofs in
the literature.Comment: 14 pages, corrected typos and added comment on connections to the AGT
conjecture in introduction, version to appear in J. Phys.
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