Calculus of variations arbitrary delta free#

Calculus of variations arbitrary delta free#

At the random matrix level, this reflects the fact that the minor of a GUE matrix is again a GUE matrix (up to a renormalizing constant).Īs first observed by Bercovici and Voiculescu and developed further by Nica and Speicher, among other authors, the notion of a free convolution power of can be extended to non-integer, thus giving the notion of a “fractional free convolution power”. For instance, if is the Wigner semicircular measure, then. The reason for this notation is that there is a surprising description of this measure when is a natural number, namely it is the free convolution of copies of, pushed forward by the dilation map. When is large and the matrix is a random matrix with empirical spectral distribution converging to some compactly supported probability measure on the real line, then under suitable hypotheses (e.g., unitary conjugation invariance of the random matrix ensemble ), a “concentration of measure” effect occurs, with the spectral distribution of the minors for for any fixed converging to a specific measure that depends only on and. These eigenvalues are related to each other by the Cauchy interlacing inequalitiesįor, and are often arranged in a triangular array known as a Gelfand-Tsetlin pattern, as discussed in these previous blog posts.

For me, this project (which we started during the 2018 IPAM program on quantitative linear algebra) was motivated by a desire to understand the behavior of the minor process applied to a large random Hermitian matrix, in which one takes the successive upper left minors of and computes their eigenvalues in non-decreasing order. Dimitri Shlyakhtenko and I have uploaded to the arXiv our paper Fractional free convolution powers.