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D. Shlyakhtenko

Publications and source records attributed to D. Shlyakhtenko.

At least 19 recordsLinked to original sources
arXiv2026

On the II$_{1}$ Factors of Fuchsian Groups

We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\mathbb{F}_{2}=\langle A,B\rangle$ is freely complemented in the group factor: $L(\mathbb{F}_{2})=W^{*}(w)*W^{*}(v)$ for some Haar unitary $v\in L(\mathbb{F}_{2})$ that is freely independent from $w$. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup $Γ\subset PSL_{2}(\mathbb{R})$, $L(Γ)$ is a free group factor, settling a conjecture of de la Harpe and Voiculescu. This result was obtained using OpenAI's ChatGPT Pro 6.0.

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arXiv2022

A Inequality for Non-Microstates Free Entropy Dimension for Crossed Products by Finite Abelian Groups

For certain generating sets of the subfactor pair $M\subset M\rtimes G$ where $G$ is a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Shreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of the form $M\rtimes(\mathbb{Z}/2\mathbb{Z})^{\oplus\infty}$ for a large class of algebras $M$.

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arXiv2016

Free transport for convex potentials

We construct non-commutative analogs of transport maps among free Gibbs state satisfying a certain convexity condition. Unlike previous constructions, our approach is non-perturbative in nature and thus can be used to construct transport maps between free Gibbs states associated to potentials which are far from quadratic, i.e., states which are far from the semicircle law. An essential technical ingredient in our approach is the extension of free stochastic analysis to non-commutative spaces of functions based on the Haagerup tensor product.

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arXiv2016

Von Neumann Algebras of Sofic Groups with $β_{1}^{(2)}=0$ are Strongly $1$-Bounded

We show that if $Γ$ is an infinite finitely generated finitely presented sofic group with zero first $L^{2}$ Betti number then the von Neumann algebra $L(Γ)$ is strongly $1$-bounded in the sense of Jung. In particular, $L(Γ)\not\cong L(Λ)$ if $Λ$ is any group with free entropy dimension $>1$, for example a free group. The key technical result is a short proof of an estimate of Jung using non-microstates entropy techniques.

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arXiv2015

Regularity of Polynomials in Free Variables

We show that the spectral measure of any non-commutative polynomial of a non-commutative $n$-tuple cannot have atoms if the free entropy dimension of that $n$-tuple is $n$ (see also work of Mai, Speicher, and Weber). Under stronger assumptions on the $n$-tuple, we prove that the spectral measure is not singular, and measures of intervals surrounding any point may not decay slower than polynomially as a function of the interval's length.

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arXiv2014

Free analysis and planar algebras

We study 2-cabled analogs of Voiculescu's trace and free Gibbs states on Jones planar algebras. These states are traces on a tower of graded algebras associated to a Jones planar algebra. Among our results is that, with a suitable definition, finiteness of free Fisher information for planar algebra traces implies that the associated tower of von Neumann algebras consists of factors, and that the standard invariant of the associated inclusion is exactly the original planar algebra. We also give conditions that imply that the associated von Neumann algebras are non-$Γ$ non-$L^2$ rigid factors.

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arXiv2013

Free Monotone Transport

By solving a free analog of the Monge-Ampère equation, we prove a non-commutative analog of Brenier's monotone transport theorem: if an $n$-tuple of self-adjoint non-commutative random variables $Z_{1},...,Z_{n}$ satisfies a regularity condition (its conjugate variables $ξ_{1},...,ξ_{n}$ should be analytic in $Z_{1},...,Z_{n}$ and $ξ_{j}$ should be close to $Z_{j}$ in a certain analytic norm), then there exist invertible non-commutative functions $F_{j}$ of an $n$-tuple of semicircular variables $S_{1},...,S_{n}$, so that $Z_{j}=F_{j}(S_{1},...,S_{n})$. Moreover, $F_{j}$ can be chosen to be monotone, in the sense that $F_{j}=\mathscr{D}_{j}g$ and $g$ is a non-commutative function with a positive definite Hessian. In particular, we can deduce that $C^{*}(Z_{1},...,Z_{n})\cong C^{*}(S_{1},...,S_{n})$ and $W^{*}(Z_{1},...,Z_{n})\cong L(\mathbb{F}(n))$. Thus our condition is a useful way to recognize when an $n$-tuple of operators generate a free group factor. We obtain as a consequence that the q-deformed free group factors $Γ_{q}(\mathbb{R}^{n})$ are isomorphic (for sufficiently small $q$, with bound depending on $n$) to free group factors. We also partially prove a conjecture of Voiculescu by showing that free Gibbs states which are small perturbations of a semicircle law generate free group factors. Lastly, we show that entrywise monotone transport maps for certain Gibbs measure on matrices are well-approximated by the matricial transport maps given by free monotone transport.

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arXiv2012

Loop models, random matrices and planar algebras

We define matrix models that converge to the generating functions of a wide variety of loop models with fugacity taken in sets with an accumulation point. The latter can also be seen as moments of a non-commutative law on a subfactor planar algebra. We apply this construction to compute the generating functions of the Potts model on a random planar map.

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arXiv2011

On operator-valued free convolution powers

We give an explicit realization of the $η$-convolution power of an $A$-valued distribution, as defined earlier by Anshelevich, Belinschi, Fevrier and Nica. If $η:A\to A$ is completely positive and $η\geq\operatorname{id}$, we give a short proof of positivity of the $η$-convolution power of a positive distribution. Conversely, if $η\not\geq\operatorname{id}$, and $s$ is large enough, we construct an $s$-tuple whose $A$-valued distribution is positive, but has non-positive $η$-convolution power.

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arXiv2010

Free probability, Planar algebras, Subfactors and Random Matrices

To a planar algebra P in the sense of Jones we associate a natural non- commutative ring, which can be viewed as the ring of non-commutative polynomials in several indeterminates, invariant under a symmetry encoded by P. We show that this ring carries a natural structure of a non-commutative probability space. Non-commutative laws on this space turn out to describe random matrix ensembles possessing special sym- metries. As application, we give a canonical construction of a subfactor and its symmetric enveloping algebra associated to a given planar algebra P. This talk is based on joint work with A. Guionnet and V. Jones.

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arXiv2009

A semi-finite algebra associated to a planar algebra

We canonically associate to any planar algebra two type II_{\infty} factors M_{+} and M_{-}. The subfactors constructed previously by the authors in a previous paper are isomorphic to compressions of M_{+} and M_{-} to finite projections. We show that each \mathfrak{M}_{\pm} is isomorphic to an amalgamated free product of type I von Neumann algebras with amalgamation over a fixed discrete type I von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that M_{+} \cong M_{-} is the amplification a free group factor on a finite number of generators. As an application, we show that the factors M_{j} constructed in our previous paper are isomorphic to interpolated free group factors L(\mathbb{F}(r_{j})), r_{j}=1+2δ^{-2j}(δ-1)I, where δ^{2} is the index of the planar algebra and I is its global index. Other applications include computations of laws of Jones-Wenzl projections.

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arXiv2009

Free probability of type B: analytic interpretation and applications

In this paper we give an analytic interpretation of free convolution of type B, introduced by Biane, Goodman and Nica, and provide a new formula for its computation. This formula allows us to show that free additive convolution of type B is essentially a re-casting of conditionally free convolution. We put in evidence several aspects of this operation, the most significant being its apparition as an 'intertwiner' between derivation and free convolution of type A. We also show connections between several limit theorems in type A and type B free probability. Moreover, we show that the analytical picture fits very well with the idea of considering type B random variables as infinitesimal deformations to ordinary non-commutative random variables.

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arXiv2008

Random matrices, free probability, planar algebras and subfactors

Using a family of graded algebra structures on a planar algebra and a family of traces coming from random matrix theory, we obtain a tower of non-commutative probability spaces, naturally associated to a given planar algebra. The associated von Neumann algebras are II$_{1}$ factors whose inclusions realize the given planar algebra as a system of higher relative commutants. We thus give an alternative proof to a result of Popa that every planar algebra can be realized by a subfactor.

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arXiv2008

Lower estimates on microstates free entropy dimension

By proving that certain free stochastic differential equations have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain $n$-tuples $X_{1},...,X_{n}$: we show that Abstract. By proving that certain free stochastic differential equations with analytic coefficients have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain n-tuples X_{1},...,X_{n}. In particular, we show that δ_{0}(X_{1},...,X_{n})\geq\dim_{M\bar{\otimes}M^{o}}V where M=W^{*}(X_{1},...,X_{n}) and V=\{(\partial(X_{1}),...,\partial(X_{n})):\partial\in\mathcal{C}\} is the set of values of derivations A=\mathbb{C}[X_{1},... X_{n}]\to A\otimes A with the property that \partial^{*}\partial(A)\subset A. We show that for q sufficiently small (depending on n) and X_{1},...,X_{n} a q-semicircular family, δ_{0}(X_{1},...,X_{n})>1. In particular, for small q, q-deformed free group factors have no Cartan subalgebras. An essential tool in our analysis is a free analog of an inequality between Wasserstein distance and Fisher information introduced by Otto and Villani (and also studied in the free case by Biane and Voiculescu).

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arXiv2007

Free diffusions and Matrix models with strictly convex interaction

We study solutions to the free stochastic differential equation $dX_t = dS_t - \half DV(X_t)dt$, where $V$ is a locally convex polynomial potential in $m$ non-commuting variables. We show that for self-adjoint $V$, the law $μ_V$ of a stationary solution is the limit law of a random matrix model, in which an $m$-tuple of self-adjoint matrices are chosen according to the law $\exp(-N \textrm{Tr}(V(A_1,...,A_m)))dA_1... dA_m$. We show that if $V=V_β$ depends on complex parameters $β_1,...,β_k$, then the law $μ_V$ is analytic in $β$ at least for those $β$ for which $V_β$ is locally convex. In particular, this gives information on the region of convergence of the generating function for planar maps. We show that the solution $dX_t$ has nice convergence properties with respect to the operator norm. This allows us to derive several properties of $C^*$ and $W^*$ algebras generated by an $m$-tuple with law $μ_V$. Among them is lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II$_1$ factor. We show that the microstates free entropy $χ(τ_V)$ is finite. A corollary of these results is the fact that the support of the law of any self-adjoint polynomial in $X_1,...,X_n$ under the law $μ_V$ is connected, vastly generalizing the case of a single random matrix.

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arXiv2007

On Classical Analogues of Free Entropy Dimension

We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on $\mathbb{R}^n$. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.

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arXiv2005

A Free Analogue of Shannon's Problem on Monotonicity of Entropy

We prove a free probability analog of a result of Artstein-Bally-Barthez-Naor. In particualar we prove that if X_{1},X_{2},... are freely independent identically distributed random variables, then the free entropy chi(X_{1}+...+X_{n}/\sqrt{n}) is monotone increasing for all n. Our proof also leads to a slight simplification of the original argument in the classical case.

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