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Semicircle law and freeness for random matrices with symmetries or correlations. (English) Zbl 1095.82004

Consider real symmetric random matrices \(X_n=(\frac1{\sqrt{n}}\,a_n(p,q))\) where, besides the symmetry condition \(a_n(p,q)=a_n(q,p)\), each entry is an independent centered random variable with unit variance. E. P. Wigner [Ann. Math. (2) 62, 548–564 (1955; Zbl 0067.08403)] proved that under a growth condition on their moments, \[ \lim_{n\rightarrow\infty}\,\mathbb{E}\,\frac1n\,\text{Tr}_n(X_n^k)=\frac1{2\pi}\,\int_{-2}^2\,\sqrt{4-x^2}\,dx. \] That is, the moments of \(X_n\) approach those of the semicircle law.
In the paper under review, the authors prove that the result is still valid in cases where not all the entries above the diagonal are independent. Namely, they prove that if there is an equivalence relation among pairs \((p,q)\) and this relation satisfies certain technical conditions, then the condition of independence in Wigner’s theorem can be relaxed to independence of entries corresponding to non-equivalent pairs. The paper contains the proof of this result (which follows Wigner’s original proof) and a few applications, most notably the proof of a conjecture of Bellissard, that the density of states is semicircular for the higher dimensional version of the simplified effective Anderson model.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
81S25 Quantum stochastic calculus
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics

Citations:

Zbl 0067.08403
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