On the computation of spectra in free probability

Computation of some examples of Brown's spectral measure in free probability

We use free probability techniques for computing spectra and Brown measures of some non hermitian operators in finite von Neumann algebras. Examples include u_n+u_oo where u_n and u_oo are the generators of Z_n and Z respectively, in the free product group Z_n*Z, or elliptic elements, of the form S_a+iS_b where S_a and S_b are free semi-circular elements of variance a and b. We give some pictorial evidence for connections with spectra of random matrices.

Spectra of infinite graphs via freeness with amalgamation

We use tools from free probability to study the spectra of Hermitian operators on infinite graphs. Special attention is devoted to universal covering trees of finite graphs. For operators on these graphs we derive a new variational formula for the spectral radius and provide new proofs of results due to Sunada and Aomoto using free probability. With the goal of extending the applicability of free probability techniques beyond universal covering trees, we introduce a new combinatorial product operation on graphs and show that, in the non-commutative probability context, it corresponds to the no…

矩阵集中不等式与自由概率 II：双侧界与应用

Matrix Concentration Inequalities and Free Probability II. Two-sided Bounds and Applications

本文针对矩阵集中不等式与自由概率研究，为谱边缘获得匹配下界以完善理论，其双侧界可解决经典方法无法处理的谱边缘首阶精确确定问题，并通过两个结果解释非齐次随机矩阵谱异常值相变，阐明多领域应用中的相变现象。

Free probability of type B and asymptotics of finite-rank perturbations of random matrices

We show that finite rank perturbations of certain random matrices fit in the framework of infinitesimal (type B) asymptotic freeness. This can be used to explain the appearance of free harmonic analysis (such as subordination functions appearing in additive free convolution) in computations of outlier eigenvalues in spectra of such matrices.

Polynomial convolutions and (finite) free probability

We introduce a finite version of free probability and show the link between recent results using polynomial convolutions and the traditional theory of free probability. One tool for accomplishing this is a seemingly new transformation that allows one to reduce computations in our new theory to computations using classically independent random variables. We then explore the idea of finite freeness and its implications. Lastly, we show applications of the new theory by deriving the finite versions of some well-known free distributions and then proving their associated limit laws directly. In the…