Related papers: The large $N$ factorization does not hold for arbi…

We establish, under a moment matching hypothesis, the local universality of the correlation functions associated with products of $M$ independent iid random matrices, as $M$ is fixed, and the sizes of the matrices tend to infinity. This…

Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. The recent surge of transport maps offers a mathematical foundation and new insights for tackling this challenge by…

We prove that if C is a tensor C*-category in a certain class, then there exists an uncountable family of pairwise non stably isomorphic II_1 factors (M_i) such that the bimodule category of M_i is equivalent to C for all i. In particular,…

We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with…

We survey the current phenomenological status of semi-inclusive deep inelastic scattering at moderate hard scales and in the limit of very large transverse momentum. As the transverse momentum becomes comparable to or larger than the…

Tensor factorization arises in many machine learning applications, such knowledge base modeling and parameter estimation in latent variable models. However, numerical methods for tensor factorization have not reached the level of maturity…

Tensors are widely used to represent multiway arrays of data. The recovery of missing entries in a tensor has been extensively studied, generally under the assumption that entries are missing completely at random (MCAR). However, in most…

Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four…

Let $\M$ be a type ${\rm II_1}$ factor and let $\tau$ be the faithful normal tracial state on $\M$. In this paper, we prove that given finite elements $X_1,\cdots X_n \in \M$, there is a finite decomposition of the identity into $N \in…

A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin-Kasteleyn random…

A singular $k$-tuple of a tensor $T$ of format $(n_1,\ldots,n_k)$ is essentially a complex critical point of the distance function from $T$ constrained to the cone of tensors of format $(n_1,\ldots,n_k)$ of rank at most one. A generic…

We propose a method to construct a tensor network representation of partition functions without singular value decompositions nor series expansions. The approach is demonstrated for one- and two-dimensional Ising models and we study the…

In this article, we obtain a super-exponential rate of convergence in total variation between the traces of the first $m$ powers of an $n\times n$ random unitary matrices and a $2m$-dimensional Gaussian random variable. This generalizes…

We study the number $M_n(T)$ be the number of integer $n\times n$ matrices $A$ with entries bounded in absolute value by $T$ such that the Galois group of characteristic polynomial of $A$ is not the full symmetric group $S_n$. One knows…

We study in this article multiplicities of eigenvalues of tensors. There are two natural multiplicities associated to an eigenvalue $\lambda$ of a tensor: algebraic multiplicity $\operatorname{am}(\lambda)$ and geometric multiplicity…

Let $\boldsymbol{X}_1,\boldsymbol{X}_2,\dots$ be independent copies of a random vector $\boldsymbol{X}$ with values in $\mathbb{R}^d$ and with a continuous distribution function. The random vector $\boldsymbol{X}_n$ is a complete record, if…

Anisotropic invariants play an important role in continuum mechanics. Knowing the number of independent invariants is crucial in modelling and in a rigorous construction of a constitutive equation for a particular material, where it is…

We obtain exact analytic expressions of real tensor eigenvalue/vector distributions of real symmetric order-three tensors with Gaussian distributions for $N\leq 8$. This is achieved by explicitly computing the partition function of a…

We prove rigorously that the symmetric traceless and the antisymmetric tensor models in rank three with tetrahedral interaction admit a $1/N$ expansion, and that at leading order they are dominated by melon diagrams. This proves the recent…

Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(k + 1)-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this…