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Hari Bercovici

Publications and source records attributed to Hari Bercovici.

At least 19 recordsLinked to original sources
arXiv2024

On the support of free convolutions

We extend to arbitrary measures results of Bao, Erd\"os, Schnelli, Moreillon, and Ji on the connectedness of the supports of additive convolutions of measures on \mathbb{R} and of free multiplicative convolutions of measures on \mathbb{R}_+. More precisely, the convolution of two measures with connected supports also has connected support. The result holds without any absolute continuity or bounded support hypotheses on the measures being convolved. We also show that the results of Moreillon and Schnelli concerning the number of components of the support of a free additive convolution hold for arbitrary measures with bounded supports. Finally, we provide an approach to the corresponding results in the case of free multiplicative convolutions of probability measures on the unit circle.

math.OA↗
arXiv2022

The Brown measure of a sum of two free nonselfadjoint random variables, one of which is R-diagonal

Suppose that $X_{1}$ and $X_{2}$ are two $*$-free (generally unbounded) random variables with Brown measures $\mu_{X_{1}}$ and $\mu_{X_{2}}$, respectively. Using properties of classical free additive convolutions, we develop a method for calculating $\mu_{X_{1}+X_{2}}$when $X_{2}$ is $R$-diagonal. This method determines a density relative to Lebesgue measure on an open set whose closure contains the support of $\mu_{X_{1}+X_{2}}$. Effective calculations are possible in important cases. Biane and Lehner were the first to make significant progress on the problem we consider, even in some cases in which neither $X_{1}$ nor $X_{2}$ is $R$-diagonal. Our examples overlap with theirs, but we emphasize the use of subordination functions. When $X_{2}$ is circular, $\mu_{X_{1}+X_{2}}$ was studied earlier using two different approaches, one involving Hamilton-Jacobi equations, and another using standard free probability techniques. Our work extends the second approach.

math.PR↗
arXiv2022

Upgrading subordination properties in free probability theory

The existence of Voiculescu's subordination functions in the context of non-tracial operator-valued C*-probability spaces has been established using analytic function theory methods. We use a matrix construction to show that the subordination functions thus obtained also satisfy an appropriately modified form of subordination for conditional expectations.

math.OA↗
arXiv2022

On the convergence of Denjoy-Wolff points

If $φ$ is an analytic function from the unit disk $\mathbb{D}$ to itself, and $φ$ is not a conformal automorphism, we denote by $λ_φ$ its Denjoy-Wolff point, that is, the limit of the iterates $φ(φ(\cdotsφ(0)\cdots))$. A result of Heins shows that, given a sequence $(φ_{n})_{n\in\mathbb{N}}$ of such analytic functions that convergence pointwise to $φ$, it follows that $\lim_{n\to\infty}λ_{φ_{n}}=λ_φ$. This allows us to improve results about the contnuous extensions of the subordination functions that arise in the study of free convolutions. We also offer an alternate proof of the result of Heins.

math.DS↗
arXiv2022

Regularity for free multiplicative convolution on the unit circle

It is shown that the free multiplicative convolution of two nondegenerate probability measures on the unit circle has no continuous singular part relative to arclength measure. Analogous results have long been known for free additive convolutions on the line and free multiplicative convolution on the positive half-line.

math.OA↗
arXiv2022

Superconvergence in free probability limit theorems for arbitrary triangular arrays

It is known that limit theorems for triangular arrays with identically distributed rows yields convergence of densities rather than just convergence in distribution. We show that this superconvergence result holds -- at least at points at which the limit density is nonzero -- even if the rows of the array are not identically distributed.

math.PR↗
arXiv2021

Superconvergence and regularity of densities in free probability

The superconvergence phenomenon is shown for products of free, identically distributed random variables. We also show that a certain Holder regularity, first demonstrated by Biane for the density of a free additive convolution with a semicircular law, extends to free additive and multiplicative convolutions with arbitrary freely infinitely divisible laws and to free convolution semigroups.

math.FA↗
arXiv2020

The atoms of the free additive convolution of two operator-valued distributions

Suppose that $X\_{1}$ and $X\_{2}$ are two selfadjoint random variables that are freely independent over an operator algebra $\mathcal{B}$. We describe the possible operator atoms of the distribution of $X\_{1}+X\_{2}$ and, using linearization, we determine the possible eigenvalues of an arbitrary polynomial $p(X\_{1},X\_{2})$ in case $\mathcal{B}=\mathbb{C}$.

math.OA↗
arXiv2020

The enumeration of extreme rigid honeycombs

Rigid tree honeycombs were introduced by Knutson, Tao, and Woodward and they were shown by Dykema, Collins, Timotin, and the authors to be sums of extreme rigid honeycombs, with uniquely determined summands up to permutations. Two extreme rigid honeycombs are essentially the same if they have proportional exit multiplicities and, up to this identification, there are countably many equivalence classes of such honeycombs. We describe two ways to approach the enumeration of these equivalence classes. The first method produces a (finite) list of all rigid tree honeycombs of fixed weight by looking at the locking patterns that can be obtained from a certain quadratic Diophantine equation. The second method constructs arbitrary rigid tree honeycombs from rigid overlays of two rigid tree honeycombs with strictly smaller weights. This allows, in principle, for an inductive construction of all rigid tree honeycombs starting with those of unit weight. We also show that some rigid overlays of two rigid tree honeycombs give rise to an infinite sequence of rigid tree honeycombs of increasing complexity but with a fixed number of nonzero exit multiplicities. This last result involves a new inflation/deflation construction that also produces other infinite sequences of rigid tree honeycombs.

math.CO↗
arXiv2018

On the outlying eigenvalues of a polynomial in large independent random matrices

Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A\_N,B\_N)$, where $A\_N$ and $B\_N$ are independent Hermitian random matrices and the distribution of $B\_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A\_N$ and $B\_N$ converge almost surely to deterministic probability measures $μ$ and $ν$, respectively. In addition, the eigenvalues of $A\_N$ and $B\_N$ are assumed to converge uniformly almost surely to the support of $μ$ and $ν,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A\_N$. It is known that almost surely the empirical distribution of the eigenvalues of $P(A\_N,B\_N)$ converges to a certain deterministic probability measure $η$ (sometimes denoted $η=P^\square(μ,ν)$) and, when there are no spikes, the eigenvalues of $P(A\_N,B\_N)$ converge uniformly almost surely to the support of $η$. When spikes are present, we show that the eigenvalues of $P(A\_N,B\_N)$ still converge uniformly to the support of $η$, with the possible exception of certain isolated outliers whose location can be determined in terms of $μ,ν,P$, and the spikes of $A\_N$. We establish a similar result when $B\_N$ is replaced by a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extend known facts from the special case in which $P(X,Y)=X+Y$.

math.OA↗
arXiv2018

Analytic subordination for bi-free convolution

In this paper we study some analytic properties of bi-free additive convolution, both scalar and operator-valued. We show that using properties of Voiculescu's subordination functions associated to free additive convolution of operator-valued distributions, simpler formulas for bi-free convolutions can be derived. We use these formulas in order to prove a result about atoms of bi-free additive convolutions.

math.OA↗
arXiv2016

Eta-diagonal distributions and infinite divisibility for R-diagonals

The class of R-diagonal *-distributions is fairly well understood in free probability. In this class, we consider the concept of infinite divisibility with respect to the operation $\boxplus$ of free additive convolution. We exploit the relation between free probability and the parallel (and simpler) world of Boolean probability. It is natural to introduce the concept of an eta-diagonal distribution that is the Boolean counterpart of an R-diagonal distribution. We establish a number of properties of eta-diagonal distributions, then we examine the canonical bijection relating eta-diagonal distributions to infinitely divisible R-diagonal ones. The overall result is a parametrization of an arbitrary $\boxplus$-infinitely divisible R-diagonal distribution that can arise in a C*-probability space, by a pair of compactly supported Borel probability measures on $[ 0, \infty )$. Among the applications of this parametrization, we prove that the set of $\boxplus$-infinitely divisible R-diagonal distributions is closed under the operation $\boxtimes$ of free multiplicative convolution.

math.OA↗
arXiv2015

Superconvergence to freely infinitely divisible distributions

The phenomenon of superconvergence is proved for all freely infinitely divisible distributions. Precisely, suppose that the partial sums of a sequence of free identically distributed, infinitesimal random variables converge in distribution to a nondegenerate freely infinitely divisible law. Then the distribution of the sum becomes Lebesgue absolutely continuous with a continuous density in finite time, and this density can be approximated by that of the limit law uniformly, as well as in all $L^{p}$-norms for $p>1$, on the real line except possibly in the neighborhood of one point. Applications include the global superconvergence to freely stable laws and that to free compound Poisson laws over the whole real line.

math.PR↗
arXiv2015

Intersection theory and the Horn inequalities for invariant subspaces

We provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C_{0}, of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where `inequality' is replaced by `divisibility'. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. The result is true even for operators acting on nonseparable Hilbert spaces. For such operators the usual Horn inequalities are supplemented so as to apply to all the Jordan blocks in the model.

math.FA↗
arXiv2015

Outliers in the spectrum of large deformed unitarily invariant models

In this paper we characterize the possible outliers in the spectrum of large deformed unitarily invariant additive and multiplicative models, as well as the eigenvectors corresponding to them. We allow both the non-deformed unitarily invariant model and the perturbation matrix to have non-trivial limiting spectral measures and spiked outliers in their spectrum. We uncover a remarkable new phenomenon: a single spike can generate asymptotically several outliers in the spectrum of the deformed model. The free subordination functions play a key role in this analysis.

math.PR↗
arXiv2013

On series of free $R$-diagonal operators

For a series of free $R$-diagonal operators, we prove an analogue of the three series theorem. We show that a series of free $R$-diagonal operators converges almost uniformly if and if two numerical series converge.

math.FA↗