arXiv subjects

Xiaofeng Xue

Publications and source records attributed to Xiaofeng Xue.

51 records · Page 3Linked to original sources

arXiv2017

The contact process with semi-infected state on the complete graph

In this paper we are concerned with the contact process with semi-infected state on the complete graph $C_n$ with $n$ vertices. In our model, each vertex is in one of three states that `healthy', `semi-infected' or `wholly-infected'. Only wholly-infected vertices can infect others. A healthy vertex becomes semi-infected when being infected while a semi-infected vertex becomes wholly-infected when being further infected. Each (semi- and wholly-) infected vertex becomes healthy at constant rate. Our main result shows the phase transition for the time wholly-infected vertices wait for to die out. Conditioned on all the vertices are wholly-infected when $t=0$, we show that wholly-infected vertices survive for $\exp\{O(n)\}$ units of time when the infection rate $λ>4$ while die out in $O(\log n)$ units of time when $λ<4$.

math.PR↗

arXiv2017

The contact process on the regular tree with random vertex weights

This paper is concerned with contact process with random vertex weights on regular trees, and study the asymptotic behavior of the critical infection rate as the degree of the trees increasing to infinity. In this model, the infection propagates through the edge connecting vertices $x$ and $y$ at rate $λρ(x)ρ(y)$ for some $λ>0$, where $\{ρ(x),{x\in T^d}\}$ are $i.i.d.$ vertex weights. We show that when $d$ is large enough there is a phase transition at $λ_c(d)\in(0,\infty)$ such that for $λ<λ_c(d)$ the contact process dies out, and for $λ>λ_c(d)$ the contact process survives with a positive probability. Moreover, we also show that there is another phase transition at $λ_e(d)$ such that for $λ<λ_e(d)$ the contact process dies out at an exponential rate. Finally, we show that these two critical values have the same asymptotic behavior as $d$ increases.

math.PR↗

arXiv2017

Mean field limit for survival probability of the high-dimensional contact process

In this paper we are concerned with the contact process on the squared lattice. The contact process intuitively describes the spread of the infectious disease on a graph, where an infectious vertex becomes healthy at a constant rate while a healthy vertex is infected at rate proportional to the number of infectious neighbors. As the dimension of the lattice grows to infinity, we give a mean field limit for the survival probability of the process conditioned on the event that only the origin of the lattice is infected at t=0. The binary contact path process is a main auxiliary tool for our proof.

math.PR↗

arXiv2016

Asymptotic for critical value of the large-dimensional SIR epidemic on clusters

In this paper we are concerned with the SIR (Susceptible-Infective-Removed) epidemic on open clusters of bond percolation on the squared lattice. For the SIR model, a susceptible vertex is infected at rate proportional to the number of infective neighbors while an infective vertex becomes removed at a constant rate. A removed vertex will never be infected again. We assume that there is only one infective vertex at $t=0$ and define the critical value of the model as the maximum of the infection rates with which infective vertices die out with probability one, then we show that the critical value is $\big(1+o(1)\big)/(2dp)$ as $d\rightarrow+\infty$, where $d$ is the dimension of the lattice and $p$ is the probability that a given edge is open. Our result is a counterpart of the main theorem in \cite{Hol1981} for the contact process.

math.PR↗

arXiv2016

Law of large numbers for the SIR model with random vertex weights on Erdős-Rényi graph

In this paper we are concerned with the SIR model with random vertex weights on Erdős-Rényi graph $G(n,p)$. The Erdős-Rényi graph $G(n,p)$ is generated from the complete graph $C_n$ with $n$ vertices through independently deleting each edge with probability $(1-p)$. We assign i. i. d. copies of a positive r. v. $ρ$ on each vertex as the vertex weights. For the SIR model, each vertex is in one of the three states `susceptible', `infective' and `removed'. An infective vertex infects a given susceptible neighbor at rate proportional to the production of the weights of these two vertices. An infective vertex becomes removed at a constant rate. A removed vertex will never be infected again. We assume that at $t=0$ there is no removed vertex and the number of infective vertices follows a Bernoulli distribution $B(n,θ)$. Our main result is a law of large numbers of the model. We give two deterministic functions $H_S(ψ_t), H_V(ψ_t)$ for $t\geq 0$ and show that for any $t\geq 0$, $H_S(ψ_t)$ is the limit proportion of susceptible vertices and $H_V(ψ_t)$ is the limit of the mean capability of an infective vertex to infect a given susceptible neighbor at moment $t$ as $n$ grows to infinity.

math.PR↗

arXiv2016

Contact processes with random recovery rates and edge weights on complete graphs

In this paper we are concerned with the contact process with random recovery rates and edge weights on complete graph with $n$ vertices. We show that the model has a critical value which is inversely proportional to the product of the mean of the edge weight and the mean of the inverse of the recovery rate. In the subcritical case, the process dies out before a moment with order $O(\log n)$ with high probability as $n\rightarrow+\infty$. In the supercritical case, the process survives at a moment with order $\exp\{O(n)\}$ with high probability as $n\rightarrow+\infty$. Our proof for the subcritical case is inspired by the graphical method introduced in \cite{Har1978}. Our proof for the supercritical case is inspired by approach introduced in \cite{Pet2011}, which deal with the case where the contact process is with random vertex weights.

math.PR↗

arXiv2016

Phase transition for the SIR model with random transition rates on complete graphs

In this paper we are concerned with the Susceptible-Infective-Removed model with random transition rates on complete graphs $C_n$ with $n$ vertices. We assign i. i. d. copies of a positive random variable $ξ$ on each vertex as the recovery rates and i. i. d copies of a positive random variable $ρ$ on each edge as the edge infection weights. We assume that a susceptible vertex is infected by an infective one at rate proportional to the edge weight on the edge connecting these two vertices while an infective vertex becomes removed with rate equals the recovery rate on it, then we show that the model performs the following phase transition when at $t=0$ one vertex is infective and others are susceptible. When $λ<λ_c$, the proportion of vertices which have ever been infective converges to $0$ weakly as $n\rightarrow+\infty$ while when $λ>λ_c$, there exist $c(λ)>0$ and $b(λ)>0$ such that for each $n\geq 1$ with probability at least $b(λ)$ the proportion of vertices which have ever been infective is at least $c(λ)$. Furthermore, we prove that $λ_c$ is the inverse of the production of the mean of $ρ$ and the mean of the inverse of $ξ$.

math.PR↗

arXiv2016

Phase transition for large dimensional contact process with random recovery rates on open clusters

In this paper we are concerned with contact process with random recovery rates on open clusters of bond percolation on $\mathbb{Z}^d$. Let $ξ$ be a positive random variable, then we assigned i. i. d. copies of $ξ$ on the vertices as the random recovery rates. Assuming that each edge is open with probability $p$ and $\log d$ vertices are occupied at $t=0$, we prove that the following phase transition occurs. When the infection rate $λ<λ_c=1/({p{\rm E}\frac{1}ξ})$, then the process dies out at time $O(\log d)$ with high probability as $d\rightarrow+\infty$, while when $λ>λ_c$, the process survives with high probability.

math.PR↗

arXiv2015

Mean field limit for bias voter model on regular trees

In this paper we are concerned with bias voter models on trees and lattices, where the vertex in state 0 reconsiders its opinion at a larger rate than that of the vertex in state 1. For the process on tree with product measure as initial distribution, we obtain a mean field limit at each moment of the probability that a given vertex is in state 1 as the degree of the tree grows to infinity. Furthermore, for our model on trees and lattices, we show that the process converges weakly to the configuration where all the vertices are in state 1 when the rate at which a vertex in state 0 reconsiders its opinion is sufficiently large. The approach of graphical representation and the complete convergence theorem of contact process are main tools for the proofs of our results.

math.PR↗

arXiv2015

Critical value for the contact process with random edge weights on regular tree

In this paper we are concerned with contact processes with random edge weights on rooted regular trees. We assign i.i.d weights on each edge on the tree and assume that an infected vertex infects its healthy neighbor at rate proportional to the weight on the edge connecting them. Under the annealed measure, we define the critical value λ_c as the maximum of the infection rate with which the process will die out and define λ_e as the maximum of the infection rate with which the process dies out at exponential rate. We show that these two critical values satisfy an identical limit theorem and give an precise lower bound of λ_e. We also study the critical value under the quenched measure. We show that this critical value equals that under the annealed measure or infinity according to a dichotomy criterion. The contact process on a Galton-Watson tree with binomial offspring distribution is a special case of our model.

math.PR↗

arXiv2015

Convergence rates for subcritical threshold-one contact processes on lattices

In this paper we are concerned with threshold-one contact processes on lattices. We show that the probability that the origin is infected converges to 0 at an exponential rate I in the subcritical case. Furthermore, we give a limit theorem for I as the degree of the lattice grows to infinity. Our results also hold for classic contact processes on lattices.

math.PR↗

arXiv2014

Contact processes with random vertex weights on oriented lattices

In this paper we are concerned with contact processes with random vertex weights on oriented lattices. In our model, we assume that each vertex x of Z^d takes i. i. d. positive random value ρ(x). Vertex y infects vertex x at rate proportional to ρ(x)ρ(y) when and only when there is an oriented edge from y to x. We give the definition of the critical value λ_c of infection rate under the annealed measure and show that λ_c=[1+o(1)]/(dEρ^2) as d grows to infinity. Classic contact processes on oriented lattices and contact processes on clusters of oriented site percolation are two special cases of our model.

math.PR↗

arXiv2014

Critical infection rates for contact processes on open clusters of oriented percolation in $Z^d$

In this paper we are concerned with contact processes on open clusters of oriented percolation in $Z^d$, where the disease spreads along the direction of open edges. We show that the two critical infection rates in the quenched and annealed cases are equal with probability one and are asymptotically equal to $(dp)^{-1}$ as the dimension $d$ grows to infinity, where $p$ is the probability of edge `open'.

math.PR↗

arXiv2014

Fluid limit of threshold voter models on tori

In this paper, we are concerned with threshold voter models on tori. Assuming that the initial distribution of the process is product measure with density p, we obtain a fluid limit of the proportion of vertices in state 1 as the dimension of the torus grows to infinity. The fluid limit performs a phase transition phenomenon from p < 1/2 to p > 1/2.

math.PR↗

arXiv2013

Asymptotic Behavior of Critical Infection Rates for Threshold-one Contact Processes on Lattices and Regular Trees

In this paper we study threshold-one contact processes on lattices and regular trees. The asymptotic behavior of the critical infection rates as the degrees of the graphs growing to infinity are obtained. Defining λ_c as the supremum of infection rates which causes extinction of the process at equilibrium, we prove that nλ_c^{T^n}\rightarrow1 and 2dλ_c^{Z^d}\rightarrow1 as n,d\rightarrow+\infty. Our result is a development of the conclusion that λ_c^{Z^d}\leq\frac{2.18}{d} shown in \cite{Dur1991}. To prove our main result, a crucial lemma about the probability of a simple random walk on a lattice returning to zero is obtained. In details, the lemma is that \lim_{d\rightarrow+\infty}2dP\big(\exists n\geq1, S_n^{(d)}=0\big)=1, where S_n^{(d)} is a simple random walk on Z^d with S_0^{(d)}=0.

math.PR↗