Freely Independent Random Variables with Non-Atomic Distributions

Abstract

We examine the distributions of non-commutative polynomials of non-atomic, freely independent random variables. In particular, we obtain an analogue of the Strong Atiyah Conjecture for free groups thus proving that the measure of each atom of any $n\times n$ matricial polynomial of non-atomic, freely independent random variables is an integer multiple of $n^{-1}$. In addition, we show that the Cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real-analytic except at a finite number of points.

References in corpus (1)

• A Random Matrix Approach to the Lack of Projections in C*_red(F_2)

Cited by in corpus (15)

• On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices
• The free field: realization via unbounded operators and Atiyah property
• Hölder Continuity of Cumulative Distribution Functions for Noncommutative Polynomials under Finite Free Fisher Information
• The free field: zero divisors, Atiyah property and realizations via unbounded operators
• The Rank Theorem and $L^2$-invariants in Free Entropy: Global Upper Bounds
• Absence of algebraic relations and of zero divisors under the assumption of full non-microstates free entropy dimension
• Preservation of algebraicity in free probability
• Regularity of distributions of Wigner integrals
• Absence of algebraic relations and of zero divisors under the assumption of finite non-microstates free Fisher information
• Regularity of Polynomials in Free Variables
• The atoms of the free additive convolution of two operator-valued distributions
• Universality of free random variables: atoms for non-commutative rational functions
• Invariant projections for operators that are free over the diagonal
• Computing the noncommutative inner rank by means of operator-valued free probability theory
• Pseudo-Polynomial Time Algorithm for Computing Moments of Polynomials in Free Semicircular Elements