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We construct an efficient model for graphs of finitely generated subgroups of free groups. Using this we give a very short proof of Dicks's reformulation of the strengthened Hanna Neumann Conjecture as the Amalgamated Graph Conjecture. In…
Since the 1970's, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new…
Let $S$ be a punctured Riemann surface with Euler characteristic $\chi(S)<0$. For any unitary representation $\rho: \pi_1(S) \to U(N)$, we introduce its renormalized energy and its harmonic representatives, which are equivariant harmonic…
In this paper, we discuss the solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop…
We address the question of the asymptotic description of random tensors that are local-unitary invariant, that is, invariant by conjugation by tensor products of independent unitary matrices. We consider both the mixed case of a tensor with…
We study analogues of classical Hilbert transforms as fourier multipliers on free groups. We prove their complete boundedness on non commutative $L^p$ spaces associated with the free group von Neumann algebras for all $1<p<\infty$. This…
The main result of this paper is a generalization of Popa's free independence result for subalgebras of ultraproduct ${\rm II_1}$ factors [Po95] to the framework of ultraproduct von Neumann algebras $(M^\omega, \varphi^\omega)$ where $(M,…
Renyi entropies S_q are useful measures of quantum entanglement; they can be calculated from traces of the reduced density matrix raised to power q, with q>=0. For (d+1)-dimensional conformal field theories, the Renyi entropies across…
In this paper we describe the amalgamated free product of finite and semifinite hyperfinite von Neumann algebras over atomic type I subalgebras. To do this we extend the notions of free dimension and standard embeddings used in the related…
We show that the error probability of reconstructing kernel matrices from Random Fourier Features for the Gaussian kernel function is at most $\mathcal{O}(R^{2/3} \exp(-D))$, where $D$ is the number of random features and $R$ is the…
Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean…
The extension $k \mapsto \mu^{\boxplus k}$ of the concept of a free convolution power to the case of non-integer $k \geq 1$ was introduced by Bercovici-Voiculescu and Nica-Speicher, and related to the minor process in random matrix theory.…
In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In previous work, the first- and second-named authors provided conditions under which tensor products of unitary random…
Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.
We show that if one selects uniformly independently and identically distributed matrices $A_1, \ldots, A_s \in \mathrm{SL}_2(\mathbb{Z})$ from a ball of large radius $X$ then with probability at least $1 - X^{-1 + o(1)}$ the matrices $A_1,…
The study of generic properties of quantum states has led to an abundance of insightful results. A meaningful set of states that can be efficiently prepared in experiments are ground states of gapped local Hamiltonians, which are well…
In these notes we explain how the CFT description of random matrix models can be used to perform actual calculations. Our basic example is the hermitian matrix model, reformulated as a conformal invariant theory of free fermions. We give an…
Graph independence (also known as $\epsilon$-independence or $\lambda$-independence) is a mixture of classical independence and free independence corresponding to graph products or groups and operator algebras. Using conjugation by certain…
In the framework of quantum groups and additive R-matrices, the fusion procedure allows to construct higher-dimensional solutions of the Yang-Baxter equation. These solutions lead to integrable one-dimensional spin-chain Hamiltonians. Here…
We show how the combination of new "linearization" ideas in free probability theory with the powerful "realization" machinery -- developed over the last 50 years in fields including systems engineering and automata theory -- allows solving…