Pierre Youssef

Let $d$ be a fixed large integer. For any $n$ larger than $d$, let $A_n$ be the adjacency matrix of the random directed $d$-regular graph on $n$ vertices, with the uniform distribution. We show that $A_n$ has rank at least $n-1$ with probability going to one as $n$ goes to infinity. The proof combines the method of simple switchings and a recent result of the authors on delocalization of eigenvectors of $A_n$.

We derive a lower bound on the smallest singular value of a random $d$-regular matrix, that is, the adjacency matrix of a random $d$-regular directed graph. More precisely, let $C_1<d< c_1 n/\log^2 n$ and let $\mathcal{M}_{n,d}$ be the set of all $0/1$-valued square $n\times n$ matrices such that each row and each column of a matrix $M\in \mathcal{M}_{n,d}$ has exactly $d$ ones. Let $M$ be uniformly distributed on $\mathcal{M}_{n,d}$. Then the smallest singular value $s_{n} (M)$ of $M$ is greater than $c_2 n^{-6}$ with probability at least $1-C_2\log^2 d/\sqrt{d}$, where $c_1$, $c_2$, $C_1$, and $C_2$ are absolute positive constants independent of any other parameters.

Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices. We show that, as long as $d\to\infty$ with $n$, the empirical spectral distribution of appropriately rescaled matrix $A_n$ converges weakly in probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical distribution in directed $d$-regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of $A_n$ based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between the matrix entries.

We show that any $$n\times m$$ matrix $$A$$ can be approximated in operator norm by a submatrix with a number of columns of order the stable rank of $$A$$. This improves on existing results by removing an extra logarithmic factor in the size of the extracted matrix. Our proof uses the recent solution of the Kadison–Singer problem. We also develop a sort of tensorization technique to deal with constraint approximation problems. As an application, we provide a sparsification result with equal weights and an optimal approximate John’s decomposition for non-symmetric convex bodies. This enables us to show that any convex body in $$\mathbb{R}^n$$ is arbitrary close to another one having $$O(n)$$ contact points and fills the gap left in the literature after the results of Rudelson and Srivastava by completely answering the problem. As a consequence, we also show that the method developed by Guédon, Gordon, and Meyer to establish the isomorphic Dvoretzky theorem yields to the best known result once we inject our improvements.

We show that any $n\times m$ matrix $A$ can be approximated in operator norm by a submatrix with a number of columns of order the stable rank of $A$. This improves on existing results by removing an extra logarithmic factor in the size of the extracted matrix. Our proof uses the recent solution of the Kadison-Singer problem. We also develop a sort of tensorization technique to deal with constraint approximation problems. As an application, we provide a sparsification result with equal weights and an optimal approximate John's decomposition for non-symmetric convex bodies. This enables us to show that any convex body in $\mathbb{R}^n$ is arbitrary close to another one having $O(n)$ contact points and fills the gap left in the literature after the results of Rudelson and Srivastava by completely answering the problem. As a consequence, we also show that the method developed by Gu\'edon, Gordon and Meyer to establish the isomorphic Dvoretzky theorem yields to the best known result once we inject our improvements.

Suppose that $m,n\in \mathbb{N}$ and that $A:\mathbb{R}^m\to \mathbb{R}^n$ is a linear operator. It is shown here that if $k,r\in \mathbb{N}$ satisfy $k<r\le \mathrm{\bf rank(A)}$ then there exists a subset $\sigma\subseteq \{1,\ldots,m\}$ with $|\sigma|=k$ such that the restriction of $A$ to $\mathbb{R}^{\sigma}\subseteq \mathbb{R}^m$ is invertible, and moreover the operator norm of the inverse $A^{-1}:A(\mathbb{R}^{\sigma})\to \mathbb{R}^m$ is at most a constant multiple of the quantity $\sqrt{mr/((r-k)\sum_{i=r}^m \mathsf{s}_i(A)^2)}$, where $\mathsf{s}_1(A)\geqslant\ldots\geqslant \mathsf{s}_m(A)$ are the singular values of $A$. This improves over a series of works, starting from the seminal Bourgain--Tzafriri Restricted Invertibility Principle, through the works of Vershynin, Spielman--Srivastava and Marcus--Spielman--Srivastava. In particular, this directly implies an improved restricted invertibility principle in terms of Schatten--von Neumann norms.

For any $\alpha\in (0,1)$ and any $n^{\alpha}\leq d\leq n/2$, we show that $\lambda(G)\leq C_\alpha \sqrt{d}$ with probability at least $1-\frac{1}{n}$, where $G$ is the uniform random $d$-regular graph on $n$ vertices, $\lambda(G)$ denotes its second largest eigenvalue (in absolute value) and $C_\alpha$ is a constant depending only on $\alpha$. Combined with earlier results in this direction covering the case of sparse random graphs, this completely settles the problem of estimating the magnitude of $\lambda(G)$, up to a multiplicative constant, for all values of $n$ and $d$, confirming a conjecture of Vu. The result is obtained as a consequence of an estimate for the second largest singular value of adjacency matrices of random {\it directed} graphs with predefined degree sequences. As the main technical tool, we prove a concentration inequality for arbitrary linear forms on the space of matrices, where the probability measure is induced by the adjacency matrix of a random directed graph with prescribed degree sequences. The proof is a non-trivial application of the Freedman inequality for martingales, combined with boots-trapping and tensorization arguments. Our method bears considerable differences compared to the approach used by Broder, Frieze, Suen and Upfal (1999) who established the upper bound for $\lambda(G)$ for $d=o(\sqrt{n})$, and to the argument of Cook, Goldstein and Johnson (2015) who derived a concentration inequality for linear forms and estimated $\lambda(G)$ in the range $d= O(n^{2/3})$ using size-biased couplings.

In this note, we show that the norm of an $n\times n$ random jointly exchangeable matrix with zero diagonal can be estimated in terms of the norm of its $n/2\times n/2$ submatrix located in the top right corner. As a consequence, we prove a relation between the second largest singular values of a random matrix with constant row and column sums and its top right $n/2\times n/2$ submatrix. The result has an application to estimating the spectral gap of random undirected $d$-regular graphs in terms of the second singular value of {\it directed} random graphs with predefined degree sequences.

Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability at least $1-C\ln^{3} d/\sqrt{d}$ for $C\leq d\leq cn/\ln^2 n$, where $c, C$ are positive absolute constants. To this end, we establish a few properties of $d$-regular directed graphs. One of them, a Littlewood-Offord type anti-concentration property, is of independent interest. Let $J$ be a subset of vertices of $G$ with $|J|\approx n/d$. Let $\delta_i$ be the indicator of the event that the vertex $i$ is connected to $J$ and define $\delta = (\delta_1, \delta_2, ..., \delta_n)\in \{0, 1\}^n$. Then for every $v\in\{0,1\}^n$ the probability that $\delta=v$ is exponentially small. This property holds even if a part of the graph is "frozen".