We consider collisions of simple random walks on comb graphs $\mathrm{Comb}(\mathbb{Z},H)$, which are obtained by attaching vertical segments of the form $[0,H_x] \cap \mathbb{Z}$ to any point $x$ of the integer axis. For $\mathrm{Comb}(\mathbb{Z},H)$ with profile $H_x(x) = |x| \log^\gamma(|x| \vee 1)$, we show that two independent simple random walks starting from the same site collide infinitely often almost surely if $\gamma \leq 2$. If the tooth profile is taken as a typical realization of i.i.d. heavy-tailed random variables with $\textbf{P}(H_x>z) \sim Cz^{-\gamma}$ (with some $C>0$) as $z$ tends to infinity, we show that infinitely many collisions occur almost surely for two independent random walks if $\gamma>1/3$, whereas finitely many collisions occur almost surely if $\gamma \in (0,1/3)$, and for any $\gamma \in (0,1]$, three independent random walks only collide finitely many times, almost surely.
We study the classical $XY$-model on random geometric graphs $\mathcal{G}_{n, \varepsilon}$, which are obtained by sampling $n \in \mathbb{N}$ independent points in a finite domain $\Omega \subset \mathbb{R}^d$, $d \geq 2$, and connecting two points by and edge if their distance is of order $\varepsilon>0$. We refer to...
We study the maximal multiplicative persistence of $k$-cycles in random \v{C}ech complexes. Let $f:\mathbb{R}^d\to\mathbb{R}$ be a probability density function, let ${P}_n$ be a Poisson process with intensity $nf$, and let $\Pi_{k,n}$ denote the largest death-to-birth ratio among all non-essential $k$-cycles ($1\le k \...
Let $\mathrm{Sym(n,k)}$ denote the set of permutations on $\{1,2,\ldots,n\}$ with exactly $k$ cycles. A family $\mathcal{F}\subset\mathrm{Sym}(n,k)$ is said to be intersecting if $\sigma^{-1}\tau$ has a fixed point for all $\sigma,\tau\in\mathcal{F}$. In this paper, we investigate the size and structure of maximum-size...
Let $w \colon (0,1) \to \mathbb{R}_{+}$ be a weight. We prove that for an arbitrary Schauder basis $\{e^{i \lambda_n t}\}_{n \in \mathbb{Z}}$ in $L^2(0,1)$ and an arbitrary lacunary defect set $A \subset \mathbb{Z}$ the system $\{w(t) r_n(t)\}_{n \in \mathbb{Z} \setminus A}$ is always complete and never minimal for any...
In this paper, we study favorite sites of one-dimensional transient random walk on $\mathbb{Z}$ beyond the simple case. Denote by $\mathcal{K}(n)$ the set of favorite sites at time $n$. The goal of this paper is twofold. First, we study $\# \mathcal{K}(n)$, the cardinality of $\mathcal{K}(n)$, and show that, under some...
Ze-Chun Hu, Ren-Ming Song, Guang-Shuo Zhou et al.· 0 citations
We determine the limiting distribution of the cover time of simple random walk on the two-dimensional discrete torus. For the continuous-time walk with total jump rate one on $(\mathbb Z/N\mathbb Z)^2$, let $T_N$ denote its cover time. We prove that $\frac{T_N}{(2/\pi)N^2\log N}-2\log N+\log\log N \Longrightarrow G+\lo...
Ye Zhou· 0 citations
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