arXiv subjects

David Jekel

Publications and source records attributed to David Jekel.

44 records · Page 3Linked to original sources

arXiv2021

Property (T) and strong $1$-boundedness for von Neumann algebras

The notion of strong 1-boundedness for finite von Neumann algebras was introduced by Jung in arXiv:math/0510576 . This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all Property (T) von Neumann algebras with finite dimensional center and group von Neumann algebras of Property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung, and Shlyakhtenko. Our proofs are based on analysis of covering estimates of microstate spaces using an iteration technique in the spirit of Jung.

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arXiv2021

Tree convolution for probability distributions with unbounded support

We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in "An operad of non-commutative independences defined by trees" (Dissertationes Mathematicae, 2020, doi:10.4064/dm797-6-2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree $\mathcal{T}$ of the $N$-regular tree (with vertices labeled by alternating strings), we define the convolution $\boxplus_{\mathcal{T}}(μ_1,\dots,μ_N)$ for arbitrary probability measures $μ_1$, ..., $μ_N$ on $\mathbb{R}$ using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from doi:10.4064/dm797-6-2020. We prove a general limit theorem for iterated $\mathcal{T}$-free convolution similar to Bercovici and Pata's results in the free case in "Stable laws and domains of attraction in free probability" (Annals of Mathematics, 1999, doi:10.2307/121080), and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws.

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arXiv2020

A random matrix approach to absorption in free products

This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the first free entropy proof of Popa's famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer's results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if $\mathcal{M} = \mathcal{P} * \mathcal{Q}$, then $\mathcal{P}$ absorbs any subalgebra of $\mathcal{M}$ that intersects it diffusely and that has $1$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra $\mathcal{P} \leq \mathcal{M}$ to have this absorption property, it suffices for $\mathcal{M}$ to admit random matrix models that have exponential concentration of measure and that "simulate" the conditional expectation onto $\mathcal{P}$.

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arXiv2020

An Operad of Non-commutative Independences Defined by Trees

We study $N$-ary non-commutative notions of independence, which are given by trees and which generalize free, Boolean, and monotone independence. For every rooted subtree $\mathcal{T}$ of the $N$-regular tree, we define the $\mathcal{T}$-free product of $N$ non-commutative probability spaces and we define the $\mathcal{T}$-free additive convolution of $N$ non-commutative laws. These $N$-ary convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the operadic framework, the proof of convolution identities (such as the relation between free, monotone, and subordination convolutions studied by Lenczewski) can be reduced to combinatorial manipulations of trees. We also develop a theory of $\mathcal{T}$-free independence that closely parallels the free, Boolean, and monotone cases, provided that the root vertex has more than one neighbor. In particular, we study the case where the root vertex of $\mathcal{T}$ has $n$ children and each other vertex has $d$ children, and we relate the $\mathcal{T}$-free convolution powers to free and Boolean convolution powers and the Belinschi-Nica semigroup.

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arXiv2020

Conditional Expectation, Entropy, and Transport for Convex Gibbs Laws in Free Probability

Let $(X_1,\dots,X_m)$ be self-adjoint non-commutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,\dots,S_m)$ be a free semicircular family. We show that conditional expectations and conditional non-microstates free entropy given $X_1$, \dots, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the random matrix models associated to $V$. Then by studying conditional transport of measure for the matrix models, we construct an isomorphism $\mathrm{W}^*(X_1,\dots,X_m) \to \mathrm{W}^*(S_1,\dots,S_m)$ which maps $\mathrm{W}^*(X_1,\dots,X_k)$ to $\mathrm{W}^*(S_1,\dots,S_k)$ for each $k = 1, \dots, m$, and which also witnesses the Talagrand inequality for the law of $(X_1,\dots,X_m)$ relative to the law of $(S_1,\dots,S_m)$.

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arXiv2019

An Elementary Approach to Free Entropy Theory for Convex Potentials

We present an alternative approach to the theory of free Gibbs states with convex potentials. Instead of solving SDE's, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N(\mathbb{C})_{sa}^m$ to prove the following. Suppose $μ_N$ is a probability measure on on $M_N(\mathbb{C})_{sa}^m$ given by uniformly convex and semi-concave potentials $V_N$, and suppose that the sequence $DV_N$ is asymptotically approximable by trace polynomials. Then the moments of $μ_N$ converge to a non-commutative law $λ$. Moreover, the free entropies $χ(λ)$, $\underlineχ(λ)$, and $χ^*(λ)$ agree and equal the limit of the normalized classical entropies of $μ_N$.

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arXiv2017

Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological Algebra

We propose an algebraic framework for generalized graph Laplacians which unifies the study of resistor networks, the critical group, and the eigenvalues of the Laplacian and adjacency matrices. Given a graph with boundary $G$ together with a generalized Laplacian $L$ with entries in a commutative ring $R$, we define a generalized critical group $Υ_R(G,L)$. We relate $Υ_R(G,L)$ to spaces of harmonic functions on the network using the Hom, Tor, and Ext functors of homological algebra. We study how these algebraic objects transform under combinatorial operations on the network $(G,L)$, including harmonic morphisms, layer-stripping, duality, and symmetry. In particular, we use layer-stripping operations from the theory of resistor networks to systematize discrete harmonic continuation. This leads to an algebraic characterization of the graphs with boundary that can be completely layer-stripped, an algorithm for simplifying computation of $Υ_R(G,L)$, and upper bounds for the number of invariant factors in the critical group and the multiplicity of Laplacian eigenvalues in terms of geometric quantities.

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arXiv2016

Layering $\partial$-Graphs and Networks

We consider the inverse problem for countable, locally finite electrical networks with edge weights in an arbitrary field. The electrical inverse problem seeks to determine the weights of the edges knowing only the potential and current data of harmonic functions on a set of boundary nodes. Motivated by the results of Curtis-Ingerman-Morrow and de-Verdiere-Gitler-Vertigan and others, we formalize the idea of using layer-stripping and harmonic continuation to solve the inverse problem. Our strategy is to iteratively recover "vulnerable" edges near the boundary, then remove them by deletion or contraction. To recover the vulnerable edge, we set up a clever boundary value problem and solve it using discrete harmonic continuation. We define "scaffolds," a set of oriented edges that models the flow of information in harmonic continuation. We formulate a sufficient geometric condition ("recoverability by scaffolds") for the inverse problem to be solvable using the layer-stripping strategy. Recoverability by scaffolds is preserved under box products, harmonic subgraphs, covering graphs, and more generally under preimages by unramified harmonic morphisms. For critical circular planar graphs, we prove recoverability by scaffolds using the medial graph. We also connect the harmonic continuation process to Baez-Fong's compositional framework for networks and Lam-Pylyavskyy's electrical linear group. We use this to generalize results of Curtis-Ingerman-Morrow and de-Verdiere-Gitler-Vertigan relating the size of connections through the graph and the rank of submatrices of the response matrix. We give a symplectic characterization of the boundary behavior for networks and the electrical linear group, valid for fields other than $\mathbb{F}_2$. Many of our results also generalize to the nonlinear networks such as those of Johnson.

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