The two-type Richardson model with unbounded initial configurations
The two-type Richardson model with unbounded initial configurations
复制标题
具有无界初始构型的二型理查森模型
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
O. Haggstrom
中科院分区：
文献类型：
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作者：
Maria Deijfen;O. Haggstrom
两型Richardson模型描述了Z^d上两个相互竞争的传染病的增长，主要问题是这两种传染病是否可以同时增长到占据Z^d的无限部分.对于有界的初始构形，这一点已经得到了深入的研究.本文考虑由超平面H={x in Z^d：x_1=0}中的点x=(x_1，…，x_d)组成的无界初始构形。结果表明，从H{0}中的所有点都是1型感染，而原点0是2型感染的构型出发，当且仅当它的强度严格大于1型感染时，2型感染有一个无界增长的正概率.相反，如果初始的1型感染被限制在负的x1轴上，那么当感染类型相同时，在起始处的2型感染也可以无限增长。
The two-type Richardson model describes the growth of two competing infections on Z^d and the main question is whether both infection types can simultaneously grow to occupy infinite parts of Z^d. For bounded initial configurations, this has been thoroughly studied. In this paper, an unbounded initial configuration consisting of points x=(x_1,...,x_d) in the hyperplane H={x in Z^d: x_1=0} is considered. It is shown that, starting from a configuration where all points in H{0} are type 1 infected and the origin 0 is type 2 infected, there is a positive probability for the type 2 infection to grow unboundedly if and only if it has a strictly larger intensity than the type 1 infection. If instead the initial type 1 infection is restricted to the negative x_1-axis, it is shown that the type 2 infection at the origin can grow unboundedly also when the infection types have the same intensity.