Phase Transition for the Large-Dimensional Contact Process with Random Recovery Rates on Open Clusters

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• 作者：Xiaofeng Xue
• 关键词：Contact process ; Random recovery rates ; Percolation phase ; Transition
• 刊名：Journal of Statistical Physics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：165
• 期：5
• 页码：845-865
• 全文大小：589 KB
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Statistical Physics
  Mathematical and Computational Physics
  Physical Chemistry
  Quantum Physics
  
• 出版者：Springer Netherlands
• ISSN：1572-9613
• 卷排序：165

文摘

In this paper we are concerned with the contact process with random recovery rates on open clusters of bond percolation on \(\mathbb {Z}^d\). Let \(\xi \) be a random variable such that \(P(\xi \ge 1)=1\), which ensures \(\mathrm{E}\frac{1}{\xi }<+\infty \), then we assign i. i. d. copies of \(\xi \) on the vertices as the random recovery rates. Assuming that each edge is open with probability p and the infection can only spread through the open edges, then we obtain that $$\begin{aligned} \limsup _{d\rightarrow +\infty }\lambda _d\le \lambda _c=\frac{1}{p\mathrm{E}\frac{1}{\xi }}, \end{aligned}$$where \(\lambda _d\) is the critical value of the process on \(\mathbb {Z}^d\), i.e., the maximum of the infection rates with which the infection dies out with probability one when only the origin is infected at \(t=0\). To prove the above main result, we show that the following phase transition occurs. Assuming that \(\lceil \log d\rceil \) vertices are infected at \(t=0\), where these vertices can be located anywhere, then when the infection rate \(\lambda >\lambda _c\), the process survives with high probability as \(d\rightarrow +\infty \) while when \(\lambda <\lambda _c\), the process dies out at time \(O(\log d)\) with high probability.KeywordsContact processRandom recovery ratesPercolation phaseTransitionReferences1.Bertacchi, D., Lanchier, N., Zucca, F.: Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions. Ann. Appl. Probab. 21, 1215–1252 (2011)MathSciNetCrossRefMATHGoogle Scholar2.Bramson, M., Durrett, R., Schonmann, R.H.: The contact process in a random environment. Ann. Probab. 19, 960–983 (1991)MathSciNetCrossRefMATHGoogle Scholar3.Chen, X.X., Yao, Q.: The complete convergence theorem holds for contact processes on open clusters of \(\mathbb{Z}^d\times \mathbb{Z}^+\). J. Stat. 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Appl. 462, 793–806 (2016)MathSciNetCrossRefGoogle ScholarCopyright information© Springer Science+Business Media New York 2016Authors and AffiliationsXiaofeng Xue1Email author1.School of ScienceBeijing Jiaotong UniversityBeijingChina About this article CrossMark Print ISSN 0022-4715 Online ISSN 1572-9613 Publisher Name Springer US About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s10955-016-1660-3_Phase Transition for the Large-Dim", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s10955-016-1660-3_Phase Transition for the Large-Dim", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. 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