Related papers: Rosenthal type inequalities for free chaos
Using Voiculescu's notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical…
Noncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity…
In this paper we discuss the natural candidate for the one dimensional free Poincar\'e inequality. Two main strong points sustain this candidacy. One is the random matrix heuristic and the other the relations with the other free functional…
Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al, reductions based upon…
We present Rosenthal-type moment inequalities for matrix-valued U-statistics of order 2. As a corollary, we obtain new matrix concentration inequalities for U-statistics. One of our main technical tools, a version of the non-commutative…
We introduce notation Q(1) * ... * Q(n) * L(F_r)$ for von Neumann algebra II_1 factors where $r$ is allowed to be negative. This notation is defined by rescalings of free products of II_1 factors, and is proved to be consistent with known…
We show that the distribution of self-normalized sums of free self-adjoint random variables converges weakly to Wigner's semicircle law under appropriate conditions and estimate the rate of convergence in terms of the Kolmogorov distance.…
The notion of a $*$-law or $*$-distribution in free probability is also known as the quantifier-free type in Farah, Hart, and Sherman's model theoretic framework for tracial von Neumann algebras. However, the full type can also be…
We derive the exponential as well as power decreasing tail estimations for normed sums of centered independent identical distributed (or not) random variables on the Khintchine's form. We consider arbitrary, in particular, non-Rademacher's…
Beckner's inequality is a family of inequalities that interpolates the two fundamental functional inequalities, the logarithmic Sobolev and Poincar\'e's inequalities. It is parametrized by exponent $p\in (1,2]$ and it implies the…
This paper provides necessary and sufficient conditions for the existence of free boundaries in overdetermined value-problems (ODVP) for the Laplacian, and sufficient conditions for the bi-Laplacian, when the overdetermined boundary…
We derive novel concentration inequalities for the operator norm of the sum of self-adjoint operators that do not explicitly depend on the underlying dimension of the operator, but rather an intrinsic notion of it. Our analysis leads to…
We consider decoupling inequalities for random variables taking values in a Banach space $X$. We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be…
We give an alternate proof of one of the inequalities proved recently for martingales (=sums of martingale differences) in a non-commutative $L_p$-space, with $1<p<\infty$, by Q. Xu and the author. This new approach is restricted to $p$ an…
The main result of this paper is a generalization of Popa's free independence result for subalgebras of ultraproduct ${\rm II_1}$ factors [Po95] to the framework of ultraproduct von Neumann algebras $(M^\omega, \varphi^\omega)$ where $(M,…
We prove explicit finite-$N$ lower bounds for $\mathbb P(\bigcup_{k=1}^N A_k)$ when the $\sigma$-algebras generated by an event sequence satisfy quantitative $\varphi$- or $\alpha$-mixing bounds. The main $\varphi$-mixing estimate is…
In this article, we aim to define a Boolean entropy notion parallel to the framework of free entropy proposed by Voiculescu. Motivated by the work of Lenczewski and the work of C\'ebron & Gillers, we mainly investigated two random matrix…
The goal of the present paper is the study of some algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension $d - 1$. We prove that the inequality $d \leq \mathrm{reg}(\Delta) \cdot…
In this paper we describe the amalgamated free product of finite and semifinite hyperfinite von Neumann algebras over atomic type I subalgebras. To do this we extend the notions of free dimension and standard embeddings used in the related…
Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of L-functions in families with the same…