Let $\Lambda \subset \mathbb{R}^n$ be a full rank lattice with the property that $\det \Gamma \ge 1$ for all sublattices $\Gamma \subseteq \Lambda^*$. We prove that there exists a symmetric random vector $X \in \mathbb{R}^n$ with a smooth, strictly positive density $f : \mathbb{R}^n \to (0, \infty)$ satisfying that $f$...
Brayden Letwin, Victor Reis, T. Rothvoss· 0 citations
We confirm the Kannan--Lov\'asz--Simonovits conjecture for quadratic forms: if $X \sim \mu$ is an isotropic log-concave random vector in $\mathbb{R}^n$, then for any symmetric matrix $M$ one has $$ \operatorname{Var}_{X \sim \mu}(\langle MX,X\rangle) \leq 2\,\mathbb{E}_{X \sim \mu}|\nabla\langle MX,X\rangle|^2. $$ As a...
Let $A=(a_{ij})$ be an $n\times n$ real-valued random matrix with independent, mean-zero, variance-one entries whose fourth moments are uniformly at most $K$. Suppose that there exists $\kappa \in (0, 1)$ such that the entries of $A$ satisfy $$ \max_{i,j}\sup_{u \in \mathbb{R}} \mathbb{P}(\lvert a_{ij} - u\rvert<1) \le...
B. Letwin, Achintya Raya Polavarapu· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
