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Random Matrices with Log-Range Correlations

Banff International Research Station for Mathematical Innovation and Discovery

Moving beyond the Wigner matrix paradigm, there is a vast literature on "band matrices" -- random matrices with entries that are not identically distributed.  In almost all cases, however, it is still assumed that the entries are independent (modulo the Hermitian condition), in order to prove convergence of the empirical eigenvalue distribution. One case where independence has been relaxed is block matrices.  If $X_n$ is a random Hermitian matrix with blocks of size $m\times m$ that are independent, then <i>operator valued free probability</i> may be used to analyze the limit empirical spectral distribution (as discovered by Shlyakhtenko, and continued in the work of Bryc, Oraby, Rashidi Far and Speicher).  This analysis, so far, can only handle the case that $m$ is constant, or, dually, $m$ is proportional to $n$. In this talk, I will discuss recent joint work with my former student David Zimmermann, where we handle a much more general intermediate case.  We show that, if the matrix $X_n$ can be partitioned into independent ``blocks'' (not necessarily rectangular) each of size $o(\log n)$, then the empirical spectral distribution converges to its mean in probability.  (The limit is usually <i> not </i> semicircular.)  The main tool we use is a mollified logarithmic Sobolev inequality, which I will discuss.

Mathematics; Functional analysis; Probability theory and stochastic processes; Operator theory/algebras

video/mp4

Author affiliation: University of California, San Diego

2017-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61622

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/