Let $M_n$ be an $n\times n$ matrix with independent uniform sign entries. We prove that there exist absolute constants $C,c>0$ such that, for all sufficiently large $n$, \[ \mathbb{P}\!\left( \left|\operatorname{Per}(M_n)\right| \ge e^{-Cn}\sqrt{n!} \right) \ge 1-n^{-c}. \] Our proof tracks the total squared permanent of minors under successive row exposure. Up to $k=\lfloor n/2\rfloor$, the total squared permanent grows deterministically via the Boolean lattice up-operator; for larger $k$, the row exposure increments are governed by positive semidefinite Rademacher quadratic forms. Therefore, we confirms the exponential scale lower bound suggested by Tao and Vu.
We prove an improved restricted isometry bound for Gaussian partial circulant matrices with arbitrary prescribed sampling sets. There is a universal constant $C>0$ such that the following holds. Let $1\leq K\leq m\leq N$ be positive integers, let $\Omega\subset\mathbb Z_N$ be any fixed set with $|\Omega|=m$, and let $g\sim\mathcal N(0,I_N)$. For every $\delta,\eta\in(0,1)$, the normalized partial circulant matrix generated by $g$ has the RIP of order $K$ with constant at most $\delta$, with probability at least $1-\eta$ over the draw of $g$, provided \[ m\geq C\delta^{-2}K \max\{\log^2(eK)\log(2N)\log(em),\log(2/\eta)\}. \] The proof refines the Maurey entropy step in the chaos-process argument by combining a noncommutative Khintchine inequality with a Schatten moment estimate controlled by $m$, replacing one factor $\log(2N)$ in the Krahmer--Mendelson--Rauhut bound by $\log(em)$.
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 $q$ 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.
Consider $n$ independent, non-negative, mean at most one random variables, $X_1,X_2,\ldots$. We show the following bound on the probability of their sum exceeding a threshold $t$: \[ \mathbb{P}\left[\sum_{i=1}^n X_i\ge t\right] \leq 1-\left(1-\frac{1}{t}\right)^n \text{ for all } t\ge 2n+1 \,. \] To prove this, we consider a relaxed optimization problem over a set of sequences of ordered, but non-independent random variables. This allows us to reformulate it recursively as dynamic programming problem. The bound becomes an equality for the binary i.i.d.~random variables satisfying $\mathbb{P}\left[X_i=0\right]= 1-\frac{1}{t}$ and $\mathbb{P}\left[X_i=t\right]=\frac{1}{t}$, which remains the maximizer in the relaxed problem.
For the $n\times n$ lower-triangular all-ones matrix $Q$, we prove a near-optimal lower bound \[ \gamma_{2,1}(Q) := \inf_{Q=AB} \|A\|_{2\to\infty}\|B\|_{1\to1} = \Omega\!\left( \frac{\log^{3/2}n}{(\log\log n)^{3/2}} \right), \] where the infimum ranges over real factorizations of arbitrary finite inner dimension. This cost is a central parameter in space bounds for factorization-based rank and quantile estimation in turnstile streams and in error bounds for matrix mechanisms for continual counting under pure differential privacy. The proof combines right-sided Haar projections with a scale-dependent numerical-sparsity decomposition of the rows of $B$. At each scale, a rank--Frobenius argument shows that the numerically sparse rows cannot account for all of the required Schatten $2/3$ mass, while a Haar projection estimate bounds the contribution of the remaining rows. Summing these bounds over the dyadic scales yields the result. The proof was obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors verified the proof and made minor revisions.
Honghao Lin, V. Mirrokni, David P. Woodruff· 0 citations
We settle the real half of a conjecture of Bandeira, Kennedy and Singer on the dimension dependence of the Gaussian constant governing the little Grothendieck problem over the orthogonal group. For an $N\times N$ standard real Gaussian matrix $G_N$, the average singular value $\alpha_{\mathbb R}(N)=N^{-3/2}\,\mathbb E\|G_N\|_*$ satisfies the quantitative estimate \[ \alpha_{\mathbb R}(N+1)-\alpha_{\mathbb R}(N)>\frac{1}{1000N^2}, \qquad N\ge1. \] Thus, the real constants increase strictly to the Marchenko--Pastur limit $8/(3\pi)$. The proof is finite-dimensional and exposes a mechanism not visible in the limiting spectral law. We decompose the Laguerre-orthogonal mean into its Laguerre-unitary counterpart and an explicit correction, then complete the resulting finite Laguerre sums to infinite diagonal tails. A bivariate generating function yields a positive diagonal kernel with a dimension-monotone remainder. This puts consecutive orthogonal corrections in common positive coordinates, where the nearest diagonal alone supplies an $N^{-2}$ reserve that dominates the unitary one-step term. The required unitary estimate is derived directly from Abreu's recurrence, and the first five dimensions are handled by exact closed forms.