Let $D_n$ be the dihedral group of order $2n$. Consider a continuous-time random walk on $D_n$ driven by arbitrary symmetric rates whose support generates $D_n$. For $p\in[1,\infty]$, we say the pair $(D_n,p)$ is rate-monotonic if for each fixed time $t$, the $\ell^p$-distance between the random walk's distribution at time $t$ and the uniform distribution is monotonically decreasing as a function of the rates. Lyons and White proved that $(D_n,2)$ and $(D_n,\infty)$ are rate-monotonic. Somewhat counterintuitively, they found several pairs $(D_n,p)$ with ${p\in[1,1.997]\cup[2.001,3.999]\cup[4.001,5.995]}$ that are not rate-monotonic, and they asked whether any such pairs exist with $p=4$ or $p=6$. We resolve their question, proving that $(D_n,2m)$ is rate-monotonic for all positive integers $m$ and $n$. In fact, we prove a generalization of this result to a broader family of groups that includes generalized dihedral groups, dicyclic groups, and generalized quaternion groups. In the other direction, we prove that for every real $p\geq 1$ that is not an even integer, there exists a positive integer $n$ such that $(D_n,p)$ is not rate-monotonic. The results of this paper were formally verified in Lean by AxiomProver assuming standard literature.
We prove that, for every constant $\varepsilon>0$ and every function $f:\Sigma^n\to\{0, 1\}$, there is a coalition of $O(n/\sqrt{\log n})$ coordinates and a target output $b\in\{0, 1\}$ such that, after the remaining coordinates are sampled uniformly and independently, the coalition can choose its values to make the ou...
Let $\mathbb{P}(x\prec y)$ be the probability that $x$ precedes $y$ in a uniformly random linear extension of an $n$-element poset $P$, and define the balancing coefficient to be $\delta(x,y)=\min(\mathbb{P}(x\prec y),\mathbb{P}(y\prec x))$ with $\delta(P)=\max_{x,y}\delta(x,y)$. We prove (Theorem 1) that sufficiently...
Let $X_H$ denote the number of copies of a fixed graph $H$ in $G_{n, p}$. Gilmer and Kopparty conjectured that $X_H$ satisfies a local central limit theorem (LCLT) provided that $H$ is connected, $p \gg n^{-1/m(H)}$, and $n^2 (1-p) \gg 1$, where $m(H)$ is the maximum density. Following the work of Berkowitz, Sah and Sa...
Asaf Cohen Antonir, Ilay Hoshen, M. Zhukovskii· 1 citation
Let $P$ be a finite nonempty poset with $n$ elements, let $f:P\to\{1,\ldots,n\}$ be a uniformly random order-preserving bijection, and put $h_P(x)=\mathbb{E}[f(x)]$. Aires and Kahn (2025) introduced $\operatorname{gap}(P)$ as the largest difference between consecutive values in the ordered list consisting of $0$, $n+1$...
Let $d \geq 5$, $0<\gamma<d - 2$, and $\Omega_N$ be the binomial random subset of $Q_N = [-N,N]^d \cap \mathbb Z^d$ with retention probability $p_N = N^{-\gamma}$. We prove that, with failure probability of optimal exponential order, every subset $B \subseteq \Omega_N$ of fixed positive relative density realizes, at ea...
Let $G= C_{n_1}\oplus\cdots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\cdots\mid n_r$, and let $\rr(G)=r$ denote its rank. The Davenport constant $\DD(G)$ is the least integer $\ell$ such that every sequence of $\ell$ elements of $G$ contains a nonempty zero-sum subsequence, and $\DD^*(G)=1+\sum_{i=1}^r(n...
Guo-Qing Wang· 2 citations· ⚡1
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