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 $\mathbb{E}[s_{\min}(MA)]\asymp \lVert M^{-1}\rVert_{\mathrm{HS}}^{-1}.$
Let $A=(\xi_{ij})$ be an $n\times n$ random matrix with independent, not necessarily identically distributed, real entries satisfying \[ \mathbb E\xi_{ij}=0,\qquad \mathbb E\xi_{ij}^{2}=1,\qquad \sup_{z\in\mathbb R}\mathbb P(|\xi_{ij}-z|0$ and $b\in(0,1)$. We prove that, for every $\delta\in(0,1)$, there are constants $c,C>0$, depending only on $a,b,\delta$, such that \[ \mathbb P\left( s_{n+1-l}(A)>Ct\frac l{\sqrt n} \right) \le \exp\!\left(-c\min\{tl,n\}\right) \] for every $t\ge1$ and every $1\le l\le(1-\delta)n$. Thus, with no moment assumption beyond variance, all but a fixed proportion of the largest singular values satisfy the optimal upper bound of order $l/\sqrt n$ with an exponential upper-tail estimate. Combined with the lower bound of the rectangular least singular value bound, this gives $s_{n+1-l}(A)\asymp l/\sqrt n$ with failure probability exponentially small in $l$. The same argument gives the rectangular scale $\sqrt{N+1}-\sqrt{n-l+1}$ for $N\times n$ matrices whenever $N-n+l\le(1-\delta)N$.
Manuel Fernández, Achintya Raya Polavarapu· 0 citations
Let $q$ be an odd prime power, and let $A_n=(a_{ij})\in\mathbb F_q^{n\times n}$ be a random matrix whose entries are independent and uniformly distributed on $\mathbb F_q$. The permanent of $A_n$ is defined by $\operatorname{per}(A_n)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}$, where $S_n$ denotes the symmetric group on $[n]$. Ghasemi, Gross, and Kopparty conjectured the zero-mass asymptotic $\Pr[\operatorname{per}(A_n)=0]=1/q+o(1)$ for every fixed odd prime power $q$, and Hunter, Kwan, and Sauermann subsequently stated its equivalent full-distribution formulation: for every fixed $q$ and every $x\in\mathbb F_q$, \[ \lim_{n\to\infty}\Pr[\operatorname{per}(A_n)=x]=\frac1q. \] In this paper, we prove this conjecture. More precisely, we prove that there is an absolute constant $C>0$ such that \[\frac12\sum_{x\in\mathbb F_q}\left|\Pr[\operatorname{per}(A_n)=x]-\frac1q\right|\le C\frac{\log n}{n}\] for every odd prime power $q$ and every $n\ge 7$. The estimate is uniform in $q$, so the conclusion remains valid for every sequence $q=q(n)$ of odd prime powers.
Let $X=\{X_j\}_{j=1}^\infty$ be a sequence of independent random variables whose densities and moments of order $2d$ are uniformly bounded. For a random vector $f(X)=(f_1(X),f_2(X))$ whose components are polynomial functionals of degree at most $d$, we prove that \[ [[f]]_{\mu,\infty}^{\frac1{2d-1}}\mu(f\in A) \le C\bigl(\lambda_2(A)\bigr)^{\frac1{2d-1}} \] for every Borel set $A\subset\mathbb R^2$, where $C$ depends only on $d$ and the uniform density and moment bounds, and $\lambda_2$ denotes the Lebesgue measure on $\mathbb R^2$. Here $[[f]]_{\mu,\infty}$ measures the failure of proportionality of the highest-order orthogonal-chaos components of $f_1$ and $f_2$ with respect to the law $\mu$ of $X$. Consequently, whenever these components are not proportional, the law of $f$ admits a density in the weak Lorentz space $L^{\frac{2d-1}{2d-2},\infty}(\mathbb R^2)$. This recovers the dichotomy established by Nualart and Tudor for two-dimensional Wiener chaos vectors and extends it beyond the Gaussian setting. We also obtain the lower bound \[ \int_{\mathbb R^\infty}\Delta_f\,d\mu \ge C[[f]]_{\mu,\infty}^2, \] where $\Delta_f$ is the determinant of the Gram matrix of $\nabla f_1$ and $\nabla f_2$. In the special case of Gaussian measures, this gives a relaxed version of the estimate conjectured by Nourdin, Nualart, and Poly.
Let $(X_1,\ldots,X_n)$ be independent nonnegative random variables with $\mathbb{E} X_i\le1$, and write $S=\sum_iX_i$. For $\delta>0$, we prove that \[ \mathbb{P}\left(S<\mathbb{E} S+\delta\right)\ge b_{n,\delta}, \] where $b_{n,\delta}=\delta(n/(n+\delta))^n$ for $0<\delta<1$ and $b_{n,\delta}=(1-1/(n+\delta))^n$ for $\delta\ge1$. The bound is sharp for every $n$ and $\delta\ge 1$. In particular, since $b_{n,\delta} \ge e^{-1}$ for $\delta \ge 1$, our result proves Feige's conjecture [Feige, 2004] in the affirmative for $\delta\ge 1$. The proof is found by ChatGPT 5.6 Pro. It combines the exact Dirichlet calibration theorem of Vlassis and Thomas [Vlassis and Thomas, 2026], which resolves Gaffke's conjecture in statistics, with results in convex geometry including Gr\"unbaum's centroid theorem [Gr\"unbaum, 1960] and its generalization by Letwin and Yaskin [Letwin and Yaskin, 2024].
Weibo Fu, Yanjun Han, Guanyang Wang et al.· 7 citations· ⚡1
Let $W=n^{-1/2}\sum_{i=1}^n X_i$, where the $X_i$ are independent centered random vectors in ${\mathbb R}^p$ with $|X_{ij}|\le B$ almost surely. Suppose that $\text{Cov}(W)$ has unit diagonal and smallest eigenvalue at least $b^2>0$. We prove that the distance between $W$ and a Gaussian vector with the same covariance, uniformly over axis-aligned rectangles, is at most $C\min\{1,b^{-2}Bn^{-1/2}\log^{3/2}(ep)\}$. For fixed $b$, the dependence on summand size and dimension matches known lower bounds in growing-dimensional regimes. The proof combines a concentration estimate near rectangle boundaries with a carefully chosen Gaussian comparison.
This work develops an approach to the spectral norm of the matrix product $\Pi U_V$, based on entropy estimates for level sets of vectors $x\in V$, and shows that matching results hold for other random models with negatively associated entries.
K. Tikhomirov· arXiv.org· 0 citations
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