We construct graphs (trees of bounded degree) on which the contact process has critical rate (which will be the same for both global and local survival) equal to any prescribed value between zero and \(\lambda _c({\mathbb {Z}})\), the critical rate of the one-dimensional contact process. We exhibit both graphs in which the process at this target critical value survives (locally) and graphs where it dies out (globally).

中文翻译：

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具有指定临界速率的接触过程的图形构造

我们构建图（有界度树），在该图上接触过程的临界率（对于全局和局部生存率都是相同的）等于零和 \（\ lambda _c（{\ mathbb {Z} }）\），一维接触过程的临界速率。我们展示了两个图表，在该图表中，处于该目标临界值的过程得以幸存（局部），而在此图表中，该临界值消失（全局）。