The semicircle law, free random variables, and entropy

By Fumio Hiai, Dénes Petz. 2006

739 citations recorded by the source corpus.

Subfields: Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5f219316f7980afd2ef3402c482005658e478698

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Citations in the corpus

Rectangular random matrices, related convolution (2005, background): “, [17]), so we will suppose that λ ∈ [0, 1).”
The stability–complexity relationship at age 40: a random matrix perspective (2015, positive): “Probably, the best tool to address this problem is represented by free probability (Hiai and Petz 2000).”
Random matrix theory in statistics: A review (2014, background): “The connection between RMT and free probability theory, another significant topic not discussed here, is explored in detail in Anderson et al. (2009), Hiai and Petz (2000), Mingo and Specicher (2006), Nica and Speicher (2006) and Edelman and Rao (2005).”
Extremal eigenvalues and eigenvectors of deformed Wigner matrices (2013, background): “More generally, the additive free convolution of two (probability) measures ω1 and ω2, usually denoted by ω1 ⊞ ω2, is defined as the distribution of the sum of two freely independent non-commutative random variables, having distributions ω1, ω2 respectively; we refer to [51, 35, 28, 2].”
Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles (2006, methodology): “10 of Hiai and Petz (2006) gives the same result for general β.”
Random matrix theory (2019, background): “2)- Voiculescu’s Entropy [2] : Definition of the Voiculescu free entropy.”
Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter (2002, methodology): “distributed entries. The measure (1.10) corresponds to the limit n,p → ∞, n/p → 1− A ∈ (0,1). In the theory of free probability, the measure (1.10) is also known as the free Poisson distribution, see [18] and the references cited therein. We prove Theorem 1.1 in section 2. The rest of the paper is d”
Strong asymptotic freeness for Wigner and Wishart matrices preprint (2005, background): “p(x1,...xr)|| a.s. (5.15) follows, as in Lemma 7.2 in [9], from the a.s. asymptotic freeness of the (X(i) n )i=1,...,r and supn kX (i) n k < ∞ a.s.. The ﬁrst point was proved by Hiai and Petz (see [10], [11]) and the second point follows from (5.14). Remark: If we only assume the convergence of p”