For sums of G-independent semicircle variables, the operator norm is bounded by 2√ω(G) and is exactly the spectral radius of the Cayley graph of the associated trace monoid.
Strong convergence for tensor GUE random matrices
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Haagerup and Thorbj{\o}rnsen proved that iid GUEs converge strongly to free semicircular elements as the dimension grows to infinity. Motivated by considerations from quantum physics -- in particular, understanding nearest neighbor interactions in quantum spin systems -- we consider iid GUE acting on multipartite state spaces, with a mixing component on some sites and identity on the remaining sites. We show that under proper assumptions on the dimension of the sites, strong asymptotic freeness still holds. Our proof relies on an interpolation technology recently introduced by Bandeira, Boedihardjo and van Handel.
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Khintchine inequalities, trace monoids and Tur\'an-type problems
For sums of G-independent semicircle variables, the operator norm is bounded by 2√ω(G) and is exactly the spectral radius of the Cayley graph of the associated trace monoid.