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 an application, we apply the above to $M=\mathbb{E}_{X \sim \mu}(\langle X,\theta\rangle X\otimes X)$ for $\theta\in S^{n-1}$ and show that the Kannan--Lov\'asz--Simonovits constant $\psi_n$ satisfies $$ \psi_n\leq C\log^{1/4}n $$ for some absolute constant $C>0$.
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 $$ \max_{i,j}\sup_{u \in \mathbb{R}} \mathbb{P}(\lvert a_{ij} - u\rvert<1) \le \kappa. $$ We prove that there are constants $c,C>0$, depending only on $K$ and $\kappa$, such that for every fixed invertible $n\times n$ matrix $M$ and every $\varepsilon\ge0$, $$ \mathbb{P}!\left(s_{\min}(MA) \le \frac{\varepsilon}{\lVert M^{-1}\rVert_{\mathrm{HS}}}\right) \le C\varepsilon + e^{-cn}. $$ In the Gaussian case, we also show that the above estimate is sharp in the sense that $\mathbb{E}[s_{\min}(MA)]\asymp \lVert M^{-1}\rVert_{\mathrm{HS}}^{-1}.$
B. Letwin, Achintya Raya Polavarapu· 0 citations
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