We investigate the scaling limit of the range (the set of visited vertices) for a general class of critical lattice models, starting from a single initial particle at the origin. Conditions are given on the random sets and an associated “ancestral relation” under which, conditional on longterm survival, the rescaled ranges converge weakly to the range of super-Brownian motion as random sets. These hypotheses also give precise asymptotics for the limiting behaviour of the probability of exiting a large ball, that is for the extrinsic one-arm probability . We show that these conditions are satisfied by the voter model in dimensions $$d\ge 2$$ d ≥ 2 , sufficiently spread out critical oriented percolation and critical contact processes in dimensions $$d>4$$ d > 4 , and sufficiently spread out critical lattice trees in dimensions $$d>8$$ d > 8 . The latter result proves Conjecture 1.6 of van der Hofstad et al. (Ann Probab 45:278–376, 2017 ) and also has important consequences for the behaviour of random walks on lattice trees in high dimensions.

中文翻译：

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论高维格模型的范围

我们从原点的单个初始粒子开始研究一般类别的临界晶格模型的范围（访问顶点集）的缩放限制。对随机集和相关的“祖先关系”给出了条件，在这种关系下，以长期生存为条件，重新调整的范围作为随机集弱收敛到超布朗运动的范围。这些假设还为退出大球的概率的限制行为给出了精确的渐近线，即外在单臂概率。我们表明，选民模型在 $$d\ge 2$$ d ≥ 2 维度上满足这些条件，在 $$d>4$$ d > 4 维度上充分展开了临界导向渗透和临界接触过程，并且充分在维度 $$d>8$$ d > 中展开临界格树 8. 后一个结果证明了 van der Hofstad 等人的猜想 1.6。(Ann Probab 45:278–376, 2017) 并且对随机游走在高维格树上的行为也有重要影响。