Threshold θ ≥ 2 contact processes on homogeneous trees

• 成果类型: Article
• 署名作者: Fontes, Luiz Renato; Schonmann, Roberto H.
• 署名单位: University of California System; University of California Los Angeles
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0092-z
• 发表日期: 2008
• 页码: 513-541
• 关键词: population-growth model; bootstrap percolation; sexual reproduction; STABILITY

摘要

We study the threshold theta >= 2 contact process on a homogeneous tree T-b of degree k = b + 1, with infection parameter lambda >= 0 and started from a product measure with density p. The corresponding mean-field model displays a discontinuous transition at a critical point lambda(MF)(c) (k, theta) and for lambda >= lambda(MF)(c) (k, theta) it survives iff p >= p(c)(MF)(k, theta, lambda), where this critical density satisfies 0 < p(c)(MF)(k,theta,lambda) < 1, lim(lambda ->infinity) p(cMF)(k,theta,lambda) = 0. For large b, we show that the process on T-b has a qualitatively similar behavior when lambda is small, including the behavior at and close to the critical point lambda(c)(T-b,theta). In contrast, for large lambda the behavior of the process on T-b is qualitatively distinct from that of the mean-field model in that the critical density has p(c)(Tb,theta,infinity) := lim(lambda ->infinity) p(c)( Tb,theta,lambda) > 0. We also show that limb ->infinity b lambda(c)(Tb,theta) = Phi(theta), where 1 < Phi(2) < Phi (3) <..., lim(theta ->infinity) Phi(theta)= infinity, and 0 < lim inf(b ->infinity)b(theta/(theta-1)) p(c)(T-b,theta,infinity) <= lim sup(b ->infinity) b(theta/theta -1)) p(c)( T-b,theta,infinity) < infinity.