Wei Su
In this paper we study the metastability of the contact process on a random regular graph. We show that the extinction time of the contact process, when initialized so that all vertices are infected at time 0, grows exponentially with the vertex number. Moreover, we show that the extinction time divided by its mean converges to a unit exponential distribution in law.
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We consider the contact process on the preferential attachment graph. The work of Berger, Borgs, Chayes and Saberi [BBCS1] confirmed physicists predictions that the contact process starting from a typical vertex becomes endemic for an arbitrarily small infection rate $λ$ with positive probability. More precisely, they showed that with probability $λ^{Θ(1)}$, it survives for a time exponential in the largest degree. Here we obtain sharp bounds for the density of infected sites at a time close to exponential in the number of vertices (up to some logarithmic factor).
We study the contact process on the configuration model with a power law degree distribution, when the exponent is smaller than or equal to two. We prove that the extinction time grows exponentially fast with the size of the graph and prove two metastability results. First the extinction time divided by its mean converges in distribution toward an exponential random variable with mean one, when the size of the graph tends to infinity. Moreover, the density of infected sites taken at exponential times converges in probability to a constant. This extends previous results in the case of an expone…
We consider a symmetric finite-range contact process on $\mathbb{Z}$ with two types of particles (or infections), which propagate according to the same supercritical rate and die (or heal) at rate $1$. Particles of type 1 can occupy any site in $(-\infty, 0]$ that is empty or occupied by a particle of type 2 and, analogously, particles of type 2 can occupy any site in $[1,+\infty)$ that is empty or occupied by a particle of type 1. We consider the model restricted to a finite interval $[-N + 1,N] \cap \mathbb{Z}$. If the initial configuration is $\mathbf{1}_ {(-N,0]}+2\mathbf{1}_{[1,N)}$, we p…
Metastability of the contact process on slowly evolving scale-free networks
本文研究慢速演化无标度网络上接触过程的亚稳态，确定了依赖参数$\eta$和幂律指数$\tau$的亚稳态指数，揭示了多种效应导致的丰富一阶相变，并开发了结合局部与全局分析的鞅证明技术。
We introduce a method to prove metastability of the contact process on Erdős-Rényi graphs and on configuration model graphs. The method relies on uniformly bounding the total infection rate from below, over all sets with a fixed number of nodes. Once this bound is established, a simple comparison with a well chosen birth-and-death process will show the exponential growth of the extinction time. Our paper complements recent results on the metastability of the contact process: under a certain minimal edge density condition, we give explicit lower bounds on the infection rate needed to get metast…
We consider the contact process on the model of hyperbolic random graph, in the regime when the degree distribution obeys a power law with exponent $χ\in(1,2)$ (so that the degree distribution has finite mean and infinite second moment). We show that the probability of non-extinction as the rate of infection goes to zero decays as a power law with an exponent that only depends on $χ$ and which is the same as in the configuration model, suggesting some universality of this critical exponent. We also consider finite versions of the hyperbolic graph and prove metastability results, as the size of…
Phase transitions for contact processes on sparse random graphs via metastability and local limits
通过亚稳态密度和局部收敛技术，研究稀疏有限图上接触过程的快慢灭绝渐近区域，给出临界阈值与局部极限中存活/灭绝阈值一致的条件，并证明稀疏图上快灭绝可能发生在拉伸指数时间尺度。
We study the contact process in the regime of small infection rates on finite scale-free networks with stationary dynamics based on simultaneous updating of all connections of a vertex. We allow the update rates of individual vertices to increase with the strength of a vertex, leading to a fast evolution of the network. We first develop an approach for inhomogeneous networks with general kernel and then focus on two canonical cases, the factor kernel and the preferential attachment kernel. For these specific networks we identify and analyse four possible strategies how the infection can surviv…
We study the metastability of the ferromagnetic Ising model on a random $r$-regular graph in the zero temperature limit. We prove that in the presence of a small positive external field the time that it takes to go from the all minus state to the all plus state behaves like ${\exp(β(r/2+ \mathcal{O}(\sqrt{r}))n)}$ when the inverse temperature $β\rightarrow\infty$ and the number of vertices $n$ is large enough but fixed. The proof is based on the so-called pathwise approach and bounds on the isoperimetric number of random regular graphs.
Random walks on $\mathbb{Z}$ with metastable Gaussian distribution caused by linear drift with application to the contact process on the complete graph
本文研究带局部线性向零漂移的整数随机游走，证明其在快速增长时间窗口内呈亚稳高斯分布，并将该理论应用于完全图上的超临界接触过程。
We study the performance of Markov chains for the $q$-state ferromagnetic Potts model on random regular graphs. It is conjectured that their performance is dictated by metastability phenomena, i.e., the presence of "phases" (clusters) in the sample space where Markov chains with local update rules, such as the Glauber dynamics, are bound to take exponential time to escape. The phases that are believed to drive these metastability phenomena in the case of the Potts model emerge as local, rather than global, maxima of the so-called Bethe functional, and previous approaches of analysing these pha…
Consider Glauber dynamics for the Ising model on the hypercubic lattice with a positive magnetic field. Starting from the minus configuration, the system initially settles into a metastable state with negative magnetization. Slowly the system relaxes to a stable state with positive magnetization. Schonmann and Shlosman showed that in the two dimensional case the relaxation time is a simple function of the energy required to create a critical Wulff droplet. The dilute Ising model is obtained from the regular Ising model by deleting a fraction of the edges of the underlying graph. In this paper …
The Eyring-Kramers Law for the Extinction Time of the Contact Process on Stars
本文推导了星图上接触过程灭绝时间的埃林-克拉默斯定律，通过特殊函数理论和势论方法，确定了灭绝时间的指数主因子和亚指数预因子。
We study the discrete-time threshold-$θ\geq 2$ contact process on random graphs of general degrees. For random graphs with a given degree distribution $μ$, we show that if $μ$ is lower bounded by $θ+2$ and has finite $k$th moments for all $k>0$, then the discrete-time threshold-$θ$ contact process on the random graph exhibits a discontinuous phase transition in the emergence of metastability, thus answering a question of Chatterjee and Durrett \cite{cd13}. To be specific, we establish that (i) for any large enough infection probability $p>p_1$, the process started from the all-infected state w…
Consider random $d$-regular graphs, i.e., random graphs such that there are exactly $d$ edges from each vertex for some $d\ge 3$. We study both the configuration model version of this graph, which has occasional multi-edges and self-loops, as well as the simple version of it, which is a $d$-regular graph chosen uniformly at random from the collection of all $d$-regular graphs. In this paper, we discuss mixing times of Glauber dynamics for the Ising model with an external magnetic field on a random $d$-regular graph, both in the quenched as well as the annealed settings. Let $β$ be the inverse …
The Second Phase Transition of the Contact Process on a Random Regular Graph
本文研究大型随机正则图上的接触过程，证明在第二临界值处存在区分线性再感染与长健康期后再感染的相变。
We show that the contact process on a random $d$-regular graph initiated by a single infected vertex obeys the "cutoff phenomenon" in its supercritical phase. In particular, we prove that when the infection rate is larger than the critical value of the contact process on the infinite $d$-regular tree there are positive constants $C, p$ depending on the infection rate such that for sufficiently small $ε> 0$, when the number $n$ of vertices is large then (a) at times $t< (C - ε) \log n$ the fraction of infected vertices is vanishingly small, but (b) at time $(C + ε) \log n$ the fraction of infec…
We consider the contact process with infection rate $λ$ on a random $(d+1)$-regular graph with $n$ vertices, $G_n$. We study the extinction time $τ_{G_n}$ (that is, the random amount of time until the infection disappears) as $n$ is taken to infinity. We establish a phase transition depending on whether $λ$ is smaller or larger than $λ_1(\mathbb{T}^d)$, the lower critical value for the contact process on the infinite, $(d+1)$-regular tree: if $λ< λ_1(\mathbb{T}^d)$, $τ_{G_n}$ grows logarithmically with $n$, while if $λ> λ_1(\mathbb{T}^d)$, it grows exponentially with $n$. This result differs f…
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 pr…
Zero-range processes with decreasing jump rates are known to exhibit condensation, where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value. We study such a process on a one-dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed, super-critical particle density. We show that the process exhibits metastability with respect to the condensate location, i.e. the suitably accelerated process of the rescaled location converges to a limiting Markov process on the unit torus. This process has sta…
We consider the contact process on a dynamic graph defined as a random $d$-regular graph with a stationary edge-switching dynamics. In this graph dynamics, independently of the contact process state, each pair $\{e_1,e_2\}$ of edges of the graph is replaced by new edges $\{e_1',e_2'\}$ in a crossing fashion: each of $e_1',e_2'$ contains one vertex of $e_1$ and one vertex of $e_2$. As the number of vertices of the graph is taken to infinity, we scale the rate of switching in a way that any fixed edge is involved in a switching with a rate that approaches a limiting value $\mathsf{v}$, so that l…
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 gi…
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 latt…
The contact process on dynamic regular graphs: monotonicity and subcritical phase
本文研究动态随机 $d$-正则图上的接触过程及 herds 过程，证明临界感染率随边动态参数严格递减、亚临界灭绝时间呈指数尾，并推出全感染初始态下灭绝时间对顶点数为对数阶。
It is known that the limiting behavior of the contact process strongly depends upon the geometry of the graph on which particles evolve: while the contact process on the regular lattice exhibits only two phases, the process on homogeneous trees exhibits an intermediate phase of weak survival. Similarly, we prove that the geometry of the graph can drastically affect the limiting behavior of multitype versions of the contact process. Namely, while it is strongly believed (and partly proved) that the coexistence region of the multitype contact process on the regular lattice reduces to a subset of…
Percolation of the contact process on the regular tree
本文研究正则树上接触过程的渗流相变，证明在 $d\geq 3$ 时 $\lambda_1<\lambda_p$，并在 $d\geq 7$ 时给出完整的相变顺序，解决了长期未决问题。
Competitive displacement in the grass-bushes-trees process on a scale-free network
该研究在幂律随机图（擦除配置模型）上分析草-灌木-树过程，探究图参数对共存或竞争取代（灌木联合亚稳态或快速灭绝）的影响。
Estimate of the exit time for the Long Range Ising model on random regular graphs
本文针对随机正则图上的长程 Ising 模型，利用考虑相互作用衰减与图结构的路径方法估计能量势垒及亚稳态退出时间，推广了短程情形结论。
We consider the discrete-time threshold-$θ\ge 2$ contact process on a random r-regular graph on n vertices. In this process, a vertex with at least θoccupied neighbors at time t will be occupied at time t+1 with probability p, and vacant otherwise. We show that if $θ\ge 2$ and $r \ge θ+2$, $ε_1$ is small and p is at least $p_1(ε_1)$, then starting from all vertices occupied the fraction of occupied vertices stays above $1-2ε_1$ up to time $\exp(γ_1(r)n)$ with probability at least $1 - \exp(-γ_1(r)n)$. In the other direction, we show that for $p_2 < 1$ there is an $ε_2(p_2)>0$ so that if $p \le…
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