Related papers: Phase transition for the bottom singular vector of…

We study the problem of symmetric matrix completion, where the goal is to reconstruct a positive semidefinite matrix $\rm{X}^\star \in \mathbb{R}^{d\times d}$ of rank-$r$, parameterized by $\rm{U}\rm{U}^{\top}$, from only a subset of its…

In this first paper, we demonstrate a theorem that establishes a first step toward proving a necessary topological condition for the occurrence of first or second order phase transitions: we prove that the topology of certain submanifolds…

We continue the study of the time synchronization model from arXiv:1201.2141 . There are two types $i=1,2$ of particles on the line $R$, with $N_{i}$ particles of type $i$. Each particle of type $i$ moves with constant velocity $v_{i}$.…

We consider a one-dimensional network in which the nodes at Euclidean distance $l$ can have long range connections with a probabilty $P(l) \sim l^{-\delta}$ in addition to nearest neighbour connections. This system has been shown to exhibit…

Matrix completion algorithms recover a low rank matrix from a small fraction of the entries, each entry contaminated with additive errors. In practice, the singular vectors and singular values of the low rank matrix play a pivotal role for…

We show a simple local norm regularization algorithm that works with high probability. Namely, we prove that if the entries of a $n \times n$ matrix $A$ are i.i.d. symmetrically distributed and have finite second moment, it is enough to…

Computing the top eigenvectors of a matrix is a problem of fundamental interest to various fields. While the majority of the literature has focused on analyzing the reconstruction error of low-rank matrices associated with the retrieved…

Let $A$ be a $n \times n$ symmetric matrix with $(A_{i,j})_{i\leq j} $, independent and identically distributed according to a subgaussian distribution. We show that $$\mathbb{P}(\sigma_{\min}(A) \leq \varepsilon/\sqrt{n}) \leq C…

We numerically investigate the localization transition in a one-dimensional system subjected to a composite potential consisting of periodic and quasi-periodic components. For the rational wave vector $\alpha=1/2$, the periodic component…

We study the hierarchical analogue of power-law random band matrices, a symmetric ensemble of random matrices with independent entries whose variances decay exponentially in the metric induced by the tree topology on $\mathbb{N}$. We map…

We consider a generalization of the classical pinning problem for integer-valued random walks conditioned to stay non-negative. More specifically, we take pinning potentials of the form $\sum_{j\geq 0}\epsilon_j N_j$, where $N_j$ is the…

We establish a localization phase diagram for light in a random three-dimensional (3D) ensemble of motionless two-level atoms with a three-fold degenerate upper level, in a strong static magnetic field. Localized modes appear in a narrow…

We use a localization procedure to weaken the growth assumptions of Royer [8], Miclo [4] and Zitt [9] concerning the continuous-time simulated annealing in $\mathbf{R}^d$. We show that a transition occurs for potentials growing like $a \log…

We consider a family of random locations, called intrinsic location functionals, of periodic stationary processes. This family includes but is not limited to the location of the path supremum and first/last hitting times. We first show that…

Inspired by the idea of Bernoulli decomposition, we give a simple proof for a generalization of Hal\'asz anti--concentration result about random sum of vectores in $\mathbb{R}^d$. From our results, we can give one upper bound for the…

We study n by n symmetric random matrices H, possibly discrete, with iid above-diagonal entries. We show that H is singular with probability at most exp(-n^c), and the spectral norm of the inverse of H is O(sqrt{n}). Furthermore, the…

Random defects do not constitute the unique source of electron localization in two dimensions. Lattice quasidisorder generated from two inplane superimposed rotated, main and secondary, square lattices, namely monolayers where moir\'e…

Low-rank modeling has many important applications in computer vision and machine learning. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better empirical…

There has been a recent revival in understanding the spontaneous scalarization phenomenon in scalar-tensor gravity as a phase transition. Using the tools of the Landau theory, we now know that first-order transitions where scalarization…

We investigate eigenvectors of rank-one deformations of random matrices $\boldsymbol B = \boldsymbol A + \theta \boldsymbol {uu}^*$ in which $\boldsymbol A \in \mathbb R^{N \times N}$ is a Wigner real symmetric random matrix, $\theta \in…