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This is sometimes known as the \"critical case\" because large clusters of zero-weight edges force passage times to grow at most logarithmically, giving zero time constant. Denote $T(\\mathbf{0}, \\partial B(n))$ as the passage time from the origin to the boundary of the box $[-n,n] \\times [-n,n]$. We characterize the limit behavior of $T(\\mathbf{0}, \\partial B(n))$ by conditions on the distribution function $F$. 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