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Xian-Yuan Wu

Publications and source records attributed to Xian-Yuan Wu.

16 recordsLinked to original sources
arXiv2026

Infinite collisions of simple random walks on random recursive trees generated by Bernoulli sequences

In this paper, we study random recursive trees generated by Bernoulli sequences. Starting from a graph with two vertices and one edge, each new vertex is connected to the last vertex with probability $ p $, or to the second-last vertex with probability $ q = 1-p $, this recursive construction yields a random infinite recursive tree $T$. We prove that $T$ almost surely has exactly one topological end. Furthermore, we establish that $T$ has the infinite collision property: two independent simple random walks on $T$ collide infinitely often almost surely.

math.PR↗
arXiv2025

An Explicit Description of Extreme Points of the Set of Couplings with Given Marginals: with Application to Minimum-Entropy Coupling Problems

Given probability distributions ${\bf p}=(p_1,p_2,\ldots,p_m)$ and ${\bf q}=(q_1,q_2,\ldots, q_n)$ with $m,n\geq 2$, denote by ${\cal C}(\bf p,q)$ the set of all couplings of $\bf p,q$, a convex subset of $\R^{mn}$. Denote by ${\cal C}_e({\bf p},{\bf q})$ the finite set of all extreme points of ${\cal C}(\bf p,q)$. It is well known that, as a strictly concave function, the Shannan entropy $H$ on ${\cal C}(\bf p,q)$ takes its minimal value in ${\cal C}_e({\bf p},{\bf q})$. In this paper, first, the detailed structure of ${\cal C}_e({\bf p},{\bf q})$ is well specified and all extreme points are enumerated by a special algorithm. As an application, the exact solution of the minimum-entropy coupling problem is obtained. Second, it is proved that for any strict Schur-concave function $Ψ$ on ${\cal C}(\bf p,q)$, $Ψ$ also takes its minimal value on ${\cal C}_e({\bf p},{\bf q})$. As an application, the exact solution of the minimum-entropy coupling problem is obtained for $(Φ,\hbar)$-entropy, a large class of entropy including Shannon entropy, Rényi entropy and Tsallis entropy etc. Finally, all the above are generalized to multi-marginal case.

math.PR↗
arXiv2022

Minimal Joint Entropy and Order-Preserving Couplings

This paper focuses on the extreme-value problem for Shannon entropy of the joint distribution with given marginals. It is proved that the minimum-entropy coupling must be of order-preserving, while the maximum-entropy coupling coincides with the independent one. Note that in this sense, we interpret entropy as a measure of system disorder.

cs.IT↗
arXiv2019

A note on the asymptotic behavior of the height for a birth-and-death process

This paper focuses on the asymptotic behaviors of the {\it height} for a birth-and-death process which related to a mean-field model \cite{FFS}(or the Anick-Mitra-Sondhi model \cite{DDM}). Recently, the asymptotic mean value of the height for the model is given in \cite{LAV}. In this paper, first, the asymptotic variance of the height is given, and as a consequence, a weak Law of Large Number for the height is obtained. Second, the centered and normalized height is proved to converge in distribution to a degenerate distribution, this indicates that the desired Central Limit Theorem fails.

math.PR↗
arXiv2017

On The Time Constant for Last Passage Percolation on Complete Graph

This paper focuses on the time constant for last passage percolation on complete graph. Let $G_n=([n],E_n)$ be the complete graph on vertex set $[n]=\{1,2,\ldots,n\}$, and i.i.d. sequence $\{X_e:e\in E_n\}$ be the passage times of edges. Denote by $W_n$ the largest passage time among all self-avoiding paths from 1 to $n$. First, it is proved that $W_n/n$ converges to constant $μ$, where $μ$ is called the time constant and coincides with the essential supremum of $X_e$. Second, when $μ<\infty$, it is proved that the deviation probability $P(W_n/n\leq μ-x)$ decays as fast as $e^{-Θ(n^2)}$, and as a corollary, an upper bound for the variance of $W_n$ is obtained. Finally, when $μ=\infty$, lower and upper bounds for $W_n/n$ are given.

math.PR↗
arXiv2017

On The Modified Newman-Watts Small World and Its Random Walk

It is well known that adding "long edges (shortcuts)" to a regularly constructed graph will make the resulted model a small world. Recently, \cite{W} indicated that, among all long edges, those edges with length proportional to the diameter of the regularly constructed graph may play the key role. In this paper, we modify the original Newman-Watts small world by adding only long special edges to the $d$-dimensional lattice torus (with size $n^d$) according to \cite{W}, and show that the diameter of the modified model and the mixing time of random walk on it grow polynomially fast in $\ln n$.

math.PR↗
arXiv2017

On The Waiting Time for A M/M/1 Queue with Impatience

This paper focuses on the problem of modeling the correspondence pattern for ordinary people. Suppose that letters arrive at a rate $λ$ and are answered at a rate $μ$. Furthermore, we assume that, for a constant $T$, a letter is disregarded when its waiting time exceeds $T$, and the remains are answered in {\it last in first out} order. Let $W_n$ be the waiting time of the $n$-th {\it answered} letter. It is proved that $W_n$ converges weekly to $W_T$, a non-negative random variable which possesses a density with {\it power-law} tail when $λ=μ$ and with exponential tail otherwise. Note that this may provide a reasonable explanation to the phenomenons reported by Oliveira and Barabási in \cite{OB}.

math.PR↗
arXiv2017

Mixing Time of Random Walk on Poisson Geometry Small World

This paper focuses on the problem of modeling for small world effect on complex networks. Let's consider the supercritical Poisson continuous percolation on $d$-dimensional torus $T^d_n$ with volume $n^d$. By adding "long edges (short cuts)" randomly to the largest percolation cluster, we obtain a random graph $\mathscr G_n$. In the present paper, we first prove that the diameter of $\mathscr G_n$ grows at most polynomially fast in $\ln n$ and we call it the Poisson Geometry Small World. Secondly, we prove that the random walk on $\mathscr G_n$ possesses the rapid mixing property, namely, the random walk mixes in time at most polynomially large in $\ln n$.

math.PR↗
arXiv2009

Phase Transition on The Degree Sequence of a Mixed Random Graph Process

This paper focuses on the problem of the degree sequence for a mixed random graph process which continuously combines the {\it classical} model and the BA model. Note that the number of step added edges for the mixed model is random and non-uniformly bounded. By developing a comparing argument, phase transition on the degree distributions of the mixed model is revealed: while the {\it pure} classical model possesses a {\it exponential} degree sequence, the {\it pure} BA model and the mixed model possess {\it power law} degree sequences. As an application of the methodology, phase transition on the degree sequence of {\it another} mixed model with {\it hard copying} is also studied, especially, in the power law region, the inverse power can take any value greater than 1.

math.PR↗
arXiv2008

The Degree Sequence of a Scale-Free Random Graph Process with Hard Copying

In this paper we consider a simple model of random graph process with {\it hard} copying as follows: At each time step $t$, with probability $0<α\leq 1$ a new vertex $v_t$ is added and $m$ edges incident with $v_t$ are added in the manner of {\it preferential attachment}; or with probability $1-α$ an existing vertex is copied uniformly at random. In this way, while a vertex with large degree is copied, the number of added edges is its degree and thus the number of added edges is not upper bounded. We prove that, in the case of $α$ being large enough, the model possesses a mean degree sequence as $ d_{k}\sim Ck^{-(1+2α)}$, where $d_k$ is the limit mean proportion of vertices of degree $k$.

math.PR↗
arXiv2008

On a Lower Bound for the Time Constant of First-Passage Percolation

We consider the Bernoulli first-passage percolation on $\mathbb Z^d (d\ge 2)$. That is, the edge passage time is taken independently to be 1 with probability $1-p$ and 0 otherwise. Let ${μ(p)}$ be the time constant. We prove in this paper that \[ μ(p_1)-μ({p_2})\ge \frac{μ(p_2)}{1-p_2}(p_2-p_1)\] for all $ 0\leq p_1<p_2< 1$ by using Russo's formula.

math.PR↗
arXiv2008

On the Degree Sequence and its Critical Phenomenon of an Evolving Random Graph Process

In this paper we focus on the problem of the degree sequence for the following random graph process. At any time-step $t$, one of the following three substeps is executed: with probability $α_1$, a new vertex $x_t$ and $m$ edges incident with $x_t$ are added; or, with probability $α-α_1$, $m$ edges are added; or finally, with probability $1-\a$, $m$ random edges are deleted. Note that in any case edges are added in the manner of preferential attachment. we prove that there exists a critical point $α_c$ satisfying: 1) if $α_1<α_c$, then the model has power law degree sequence; 2) if $α_1>α_c$, then the model has exponential degree sequence; and 3) if $α_1=α_c$, then the model has a degree sequence lying between the above two cases.

math.PR↗
arXiv2006

A Geometrical Structure for an Infinite Oriented Cluster and its Uniqueness

We consider the supercritical oriented percolation model. Let ${\fK}$ be all the percolation points. For each $u\in {\fK}$, we write $γ_u$ as its right-most path. Let $G=\cup_u γ_u$. In this paper, we show that $G$ is a single tree with only one topological end. We also present a relationship between ${\fK}$ and $G$ and construct a bijection between ${\fK}$ and $\Z$ using the preorder traversal algorithm. Through applications of this fundamental graph property, we show the uniqueness of an infinite oriented cluster by ignoring finite vertices.

math.PR↗
arXiv2003

Two-dimensional Poisson Trees converge to the Brownian web

The Brownian web can be roughly described as a family of coalescing one-dimensional Brownian motions starting at all times in $\R$ and at all points of $\R$. It was introduced by Arratia; a variant was then studied by Toth and Werner; another variant was analyzed recently by Fontes, Isopi, Newman and Ravishankar. The two-dimensional \emph{Poisson tree} is a family of continuous time one-dimensional random walks with uniform jumps in a bounded interval. The walks start at the space-time points of a homogeneous Poisson process in $\R^2$ and are in fact constructed as a function of the point process. This tree was introduced by Ferrari, Landim and Thorisson. By verifying criteria derived by Fontes, Isopi, Newman and Ravishankar, we show that, when properly rescaled, and under the topology introduced by those authors, Poisson trees converge weakly to the Brownian web.

math.PR↗