Related papers: Free Stein Kernel and Moments maps
In this work, we describe new constructions of free Stein kernels. Firstly, in dimension one, we propose a free analog to the construction of Stein kernels using moment maps as the one proposed by Fathi. This will be possible for a class of…
We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge-Amp\`ere equation. As a consequence, we show how regularity bounds on these maps control the rate of convergence in the classical…
Stein kernels are a way of comparing probability distributions, defined via integration by parts formulas. We provide two constructions of Stein kernels in free probability. One is given by an explicit formula, and the other via free…
We combine the notion of free Stein kernel and the free Malliavin calculus to provide quantitative bounds under the free (quadratic) Wasserstein distance in the multivariate semicircular approximations for self-adjoint vector-valued…
We prove a quantitative Fourth Moment Theorem for Wigner integrals of any order with symmetric kernels, generalizing an earlier result from Kemp et al. (2012). The proof relies on free stochastic analysis and uses a new biproduct formula…
We introduce a free version of the Stein kernel, relative to a semicircular law. We use it to obtain a free counterpart of the HSI inequality of Ledoux, Peccatti and Nourdin, which is an improvement of the free logarithmic Sobolev…
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…
In arXiv:1304.0630, it was shown that convex, almost everywhere continuous functions coordinatize a broad class of probability measures on $\mathbb{R}^n$ by the map $U \mapsto (\nabla U)_{\#} e^{-U} dx$. We consider whether there is a…
We develop a multidimensional Stein methodology for non-degenerate self-decomposable random vectors in $\mathbb{R}^d$ having finite first moment. Building on previous univariate findings, we solve an integro-partial differential Stein…
Given an isotropic probability measure $\mu$ on ${\mathbb R}^d$ with ${\rm d}\mu \left( x \right) = {\left( {\varrho \left( x \right)} \right)^{ - \alpha }}{\rm d}x$, where $\alpha > d + 1$ and $\varrho :{{\mathbb R}^d} \to \left( {0, +…
A quantitative "fourth moment theorem" is provided for any self-adjoint element in a homogeneous Wigner chaos: the Wasserstein distance is controlled by the distance from the fourth moment to two. The proof uses the free counterpart of the…
In this paper we propose and study a class of simple, nonparametric, yet interpretable measures of association between two random variables $X$ and $Y$ taking values in general topological spaces. These nonparametric measures -- defined…
We give a necessary and sufficient condition on the cost function so that the map solution of Monge's optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and…
We establish a class of sufficient conditions, ensuring that a sequence of multiple integrals with respect to a free Poisson measure converges to a semicircular limit. We use this result to construct a set of explicit counterexamples,…
We investigate stability of invariant measures of diffusion processes with respect to $L^p$ distances on the coefficients, under an assumption of log-concavity. The method is a variant of a technique introduced by Crippa and De Lellis to…
A simple mapping procedure is presented by which classical orbits and path integrals for the motion of a point particle in flat space can be transformed directly into those in curved space with torsion. Our procedure evolved from…
The antiferromagnetic Heisenberg model is studied on a two-dimensional bipartite quasiperiodic lattice. The distribution of local staggered magnetic moments is determined on finite square approximants with up to 1393 sites, using the…
We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg…
We formulate quantum mechanics in spacetimes with real-order fractional geometry and more general factorizable measures. In spacetimes where coordinates and momenta span the whole real line, Heisenberg's principle is proven and the…
In this paper, following the recent work of Fathi (2018) in the classical case, we provide by two different methods a sharp symmetrized free Talagrand inequality for the semicircular law, which improves the free TCI of Biane and Voiculescu…