Delay-capacity tradeoffs for mobile networks have been analyzed through a number of research works. However, Levy mobility known to closely capture human movement patterns has not been adopted in such work. Understanding the delay-capacity tradeoff for a network with Levy mobility can provide important insights into understanding the performance of real mobile networks governed by human mobility. This paper analytically derives an important point in the delay-capacity tradeoff for Levy mobility, known as the critical delay. The critical delay is the minimum delay required to achieve greater throughput than what conventional static networks can possibly achieve (i.e., O(1/root n) per node in a network with nodes). The Levy mobility includes Levy flight and Levy walk whose step-size distributions parametrized by alpha is an element of (0, 2] are both heavy-tailed while their times taken for the same step size are different. Our proposed technique involves: 1) analyzing the joint spatio-temporal probability density function of a time-varying location of a node for Levy flight, and 2) characterizing an embedded Markov process in Levy walk, which is a semi-Markov process. The results indicate that in Levy walk, there is a phase transition such that for alpha is an element of (0, 1), the critical delay is always Theta(n(1/2)), and for alpha is an element of [1, 2] it is Theta(n(alpha/2)). In contrast, Levy flight has the critical delay Theta(n(alpha/2)) for alpha is an element of (0, 2].