In 1962, Horn raised the following problem: Let A and B be n-by-n Hermitian matrices with respective eigenvalues a_1,...,a_n and b_1,...,b_n. What can we say about the possible eigenvalues c_1,...,c_n of A + B?
The deterministic perspective is that the set of possible values for c_1,...,c_n are described by a collection of inequalities known as the Horn inequalities.
Free probability offers the following alternative perspective on the problem: if (A_n) and (B_n) are independent sequences of n-by-n random matrices with empirical spectra converging to probability measures mu and nu respectively, then the random empirical spectrum of A_n + B_n converges to the free convolution of mu and nu.
But how are these two perspectives related?
In this talk I will discuss approaches to free probability that bridge between the two perspectives. More broadly, I will discuss how the fundamental operations of free probability (such as free convolution, free compression etc) arise out of statistical physics/optimal transport arguments applied to corresponding finite representation theory objects (hives, Gelfand-Tsetlin patterns, characteristic polynomials, Horn inequalities etc).
This talk is based on joint work with Octavio Arizmendi (CIMAT, Mexico), Colin McSWiggen (Academia Sinica, Taiwan) and Joscha Prochno (Passau, Germany).
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