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 $E_n$ denote the number of alternating permutations of $\{1,\dots,n\}$, equivalently characterized by $\sum_{n\ge0}E_nz^n/n!=\sec z+\tan z$. For every $q\ge1$, the sequence $(E_n\bmod q)_{n\ge0}$ is eventually periodic; let $d(q)$ and $s(q)$ denote its minimal eventual period and preperiod. For every odd prime $p$,...
Let $A=(A_n)_{n\ge2}$ be a triangular array of random matrices, where $A_n=(a_{ij})_{1\le i,j\le n}$ is an $n\times n$ random matrix with independent real entries satisfying $\mathbb E a_{ij}=0$ and $\mathbb Ea_{ij}^2=1$, and put $\mathcal L_n=\log|\det A_n|$ and \[ W_n^{\mathrm d}(A_n):=\frac{\mathcal L_n - \frac12\lo...
Let $p$ be an odd prime, let $\varnothing\neq A\subseteq\mathbb F_p$ have cardinality $N$, and let $f\in\mathbb F_p[x,y]$ be a non-degenerate quadratic polynomial. Writing $S=|A+A|$ and $M=|f(A,A)|$, we prove the full-range trade-off $S^8M^6\gtrsim N^{17}(1+N^3/p^2)^{-3}$. Consequently, $\max\{|A+A|,|f(A,A)|\}\gtrsim \...
We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_n\in \operatorname{Mat}_n(\mathbb{F}_p)$ has independent and $\alpha_n$-balanced entries. We prove that if $$ \liminf_{n\to\infty}\frac{n\alpha_n}{\log n}>1, $$ then, for every fixed collection of distinct monic irreducible p...
Criteria for the permutation and complete permutation properties of the mapping F(X)=T(X+B^tf(AX)) over $\mathbb{F}_{q}^n$ are established and it is shown that any arbitrary function from $\mathbb{F}_q^m$ to $\mathbb{F}_q^{\,n-m}$ can be extended to a bijection over $\mathbb{F}_{q}^n
Sartaj Ul Hasan, Ramandeep Kaur, Hridesh Kumar et al.· 0 citations
A subspace of $\mathbb{F}_q^n$ is called cyclically covering if the union of $\sigma^i(U)$ can cover the whole space $\mathbb{F}_q^n$, where $\sigma$ is the cyclic shift, $0 \leqslant i \leqslant n-1$. Let $h_q(n)$ be the largest possible co-dimension of a cyclically covering subspace of $\mathbb{F}_q^n$. We show that...
Xuan Wang, Minjia Shi· arXiv.org· 0 citations
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