Bounds for the expected return probability of the delayed random walk on finite clusters of an invariant percolation on transitive unimodular graphs are derived. They are particularly suited for the case of critical Bernoulli percolation and the associated heavy-tailed cluster size distributions. The upper bound relies on the fact that cartesian products of finite graphs with cycles of a certain minimal size are Hamiltonian. For critical Bernoulli bond?percolation on the homogeneous tree this bound is sharp. The asymptotic type of the expected return probability for large times $t$ in this case is of order $t^{-3/4}$.
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机译：推导了传递单模图上不变渗流的有限簇上的延迟随机游动的预期返回概率的界。它们特别适用于临界伯努利渗滤和相关的重尾簇大小分布的情况。上限取决于这样一个事实，即具有一定最小尺寸的循环的有限图的笛卡尔积是哈密顿量。对于齐次树上的临界伯努利键渗流，该边界是尖锐的。在这种情况下，大笔$ t $的预期收益概率的渐近类型为$ t ^ {-3/4} $阶。
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