arXiv subjects

Chenxu Feng

Publications and source records attributed to Chenxu Feng.

4 recordsLinked to original sources

arXiv2025

The Exact Limsup Constant for Once-Visited Sites of One-Dimensional Simple Random Walk

For a one-dimensional simple random walk, let $g_1(n)$ denote the number of sites visited exactly once at time $n$. Major (1988) proved that \begin{equation*} \limsup_{n\to\infty}\frac{g_1(n)}{\log^2 n}=C\qquad a.s. \end{equation*} where $C$ is a positive and finite constant. While this result settled the question of existence, the exact value of $C$ remained unknown. In this paper, we determine that $C=1/16$. The main novelty of our work lies in introducing a self-boosting iterative framework for analysis.

math.PR↗

arXiv2025

Structural Properties of the Geometric Preferential Attachment Model

This paper analyzes key properties of networks generated by geometric preferential attachment. We establish that the expected number of triangles is proportional to that of the standard preferential attachment model, with a proportionality constant equal to the ratio of the number of triangles between a random geometric graph and an Erd\H{o}s-R\'enyi graph. Furthermore, we prove that the maximum degree grows polynomially with the network size, sharing the same exponent as the standard model; however, the spatial constraint induces a slower growth rate in the network's early evolution. Finally, we extend prior results on connectivity and diameter to the case of networks with finite out-degrees.

math.PR↗

arXiv2025

Exact Limsup Growth of Rarely Visited Sites for One-Dimensional Simple Random Walk

We investigate the minimal local time $f(n)$ of a one-dimensional simple random walk up to time $n$, defined as the smallest number of visits to any site in the range. A conjecture formulated repeatedly by Erd\H{o}s and R\'{e}v\'{e}sz (1987, 1991) stated that $\limsup_{n\to\infty}f(n)=2$ almost surely, which was disproved by T\'{o}th (1996) who showed $\limsup_{n\to\infty}f(n)=\infty$. Subsequently, R\'{e}v\'{e}sz (2013) suggested studying the growth rate and established an upper bound of the order $\log n$. In this paper, we determine the precise asymptotic growth rate, proving that with probability one, $$ \limsup_{n\to\infty}\frac{f(n)}{\log\log n}=\frac{1}{\log 2}. $$ This result answers the open question posed in Section 13.2 of R\'{e}v\'{e}sz (2013).

math.PR↗

arXiv2025

Strong Detection Threshold for Correlated Erd\H{o}s-R\'enyi Graphs with Constant Average Degree

Consider a pair of correlated Erd\H{o}s-R\'enyi graphs $\mathcal G(n,\tfrac{\lambda}{n};s)$ that are subsampled from a common parent Erd\H{o}s-R\'enyi graph with average degree $\lambda$ and subsampling probability $s$. We establish a sharp information-theoretic threshold for the detection problem between this model and two independent Erd\H{o}s-R\'enyi graphs $\mathcal G(n,\tfrac{\lambda}{n})$, showing that strong detection is information-theoretically possible if and only if $s>\min\{ \tfrac{1}{\sqrt{\lambda}}, \sqrt{\alpha} \}$ where $\alpha\approx 0.338$ is the Otter's constant. Our result resolves a constant gap between arXiv:2203.14573 and arXiv:2008.10097.

math.PR↗