Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.15294

Sharp Exponential Asymptotics for Normal Sign Matrices

Caden Young
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $M_n$ be an $n\times n$ random matrix whose entries are independent Rademacher random variables, and put $N=\binom n2$. We prove \[ Pr(M_nM_n^T=M_n^TM_n)=2^{-N+O(n)}. \] This gives the sharp exponential order for the probability that a random sign matrix is normal. The lower bound is supplied by symmetric sign matrices. We also record the immediate consequence that random $0$-$1$ matrices have the same sharp exponential normality probability. The proof of the matching upper bound is combinatorial: after conditioning on the symmetric/skew-symmetric type pattern of the off-diagonal entries, a mod-$4$ reduction gives a system of linear equations over $F_2$; a rank-duality argument converts the sum over type patterns into a count of commuting symmetric pairs over $F_2$; and this count is bounded by summing over rational canonical types, using balanced symmetric bilinear forms and the standard finite-field centralizer formula.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Caden Young. 2026-07-01. Sharp Exponential Asymptotics for Normal Sign Matrices. https://arxiv.org/abs/2607.15294

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rooted Spider Embeddings and the Erd\H os-Sós Conjecture

Under a local density condition, we prove that every $k$-edge spider embeds at any prescribed center of degree at least $k$, unless all legs are even and the host graph has one of two specified structures. These structures contain complete bipartite subgraphs with prescribed neighborhoods. The proof uses path rerouting and three exchange lemmas that describe equality in neighborhood estimates. As a consequence, we recover the Erd\H os-Sós bound for all spiders.

math.CO↗
arXiv2026

Random algebraic constructions for extremal and Ramsey problems

Building on Bukh's random algebraic method, we develop a framework for extremal and Ramsey problems involving apex hypergraphs. If $\mathcal{H}$ is a $(d-1)$-partite $(d-1)$-uniform hypergraph with $S$ edges and $\mathcal{H}(t)$ is obtained by adjoining $t$ vertices with common link $\mathcal{H}$, we prove that $\operatorname{ex}(n,\mathcal{H}(t))=Ω_{\mathcal{H}}(n^{d-1/S})$ for $t>9^{S+o_d(S)}$, which is best possible when $\mathcal{H}$ is Sidorenko. Our framework also yields sharper sided Zarankiewicz bounds, quantitative generalized Tur'an bounds, and diagonal multicolor Ramsey constructions. For each fixed $s\geq 2$ and $K\geq 3$, we further prove $\operatorname{r}_K(\mathcal K_{s,t};\mathcal K_n) =Θ_{s,t,K}((n/\log n)^s)$ for $t>9^{s+o(s)}$, extending a theorem of Alon and Rödl from factorial to exponential $t$. The main ingredients are interpolation on $m$-independent varieties, control of the dependencies imposed by symmetry, and linear spaces of forms whose nonzero members remain regular after a common algebraic slice. Limited edge independence then gives the spectral and local-density estimates needed for the Ramsey application.

math.CO↗
arXiv2026

A note on vertex-critical induced subgraphs of shift graphs

Shift graphs, introduced by Erdős and Hajnal in 1964, form one of the simplest known non-recursive constructions of triangle-free graphs with arbitrarily large chromatic number. In this note, we identify a surprising property: for each integer $k \geq 1$, the smallest $k$-chromatic shift graph contains a \emph{unique} induced $k$-vertex-critical subgraph. We give an explicit description of this subgraph and prove its uniqueness. This provides a new family of vertex-critical triangle-free graphs of arbitrarily large chromatic number.

math.CO↗