@article{7512,
  abstract     = {We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically.},
  author       = {Erdös, László and Krüger, Torben H and Nemish, Yuriy},
  issn         = {10960783},
  journal      = {Journal of Functional Analysis},
  number       = {12},
  publisher    = {Elsevier},
  title        = {{Local laws for polynomials of Wigner matrices}},
  doi          = {10.1016/j.jfa.2020.108507},
  volume       = {278},
  year         = {2020},
}

@article{956,
  abstract     = {We study a class of ergodic quantum Markov semigroups on finite-dimensional unital C⁎-algebras. These semigroups have a unique stationary state σ, and we are concerned with those that satisfy a quantum detailed balance condition with respect to σ. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to σ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the 2-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical 2-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance.},
  author       = {Carlen, Eric and Maas, Jan},
  issn         = {00221236},
  journal      = {Journal of Functional Analysis},
  number       = {5},
  pages        = {1810 -- 1869},
  publisher    = {Academic Press},
  title        = {{Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance}},
  doi          = {10.1016/j.jfa.2017.05.003},
  volume       = {273},
  year         = {2017},
}