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Symmetric partitions and pairings

Ferenc Oravecz — 2000

Colloquium Mathematicae

The lattice of partitions and the sublattice of non-crossing partitions of a finite set are important objects in combinatorics. In this paper another sublattice of the partitions is investigated, which is formed by the symmetric partitions. The measure whose nth moment is given by the number of non-crossing symmetric partitions of n elements is determined explicitly to be the "symmetric" analogue of the free Poisson law.

On the powers of Voiculescu's circular element

Ferenc Oravecz — 2001

Studia Mathematica

The main result of the paper is that for a circular element c in a C*-probability space, ( c , c n * ) is an R-diagonal pair in the sense of Nica and Speicher for every n = 1,2,... The coefficients of the R-series are found to be the generalized Catalan numbers of parameter n-1.

The Leray measure of nodal sets for random eigenfunctions on the torus

Ferenc OraveczZeév RudnickIgor Wigman — 2008

Annales de l’institut Fourier

We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in d 2 dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity 𝒩 . The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant,...

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