Heavy-tailed random variables play an important role in applied probability and feature prominently in insurance mathematics. The joint distribution of n independent identically integer-valued distributed random variables, conditional on the value of the sum, also appears as the stationary distribution for the zero-range process, a continuous-time Markov process for interacting particles distributed on n lattice sites. In this context the transition in large deviations from Cramer-type collective behavior ("small steps") to big-jump is interpreted as a transition from gas to condensed phase, and large deviations for stretched exponential variables act as a toy model for phase transitions, including a regime of supersaturated gas (a big jump that does not swallow all of the excess). The talk presents local large deviations theorems for sums of i.i.d. \(\mathbb N_0\)-valued, heavy-tailed random variables. The proofs are inspired by singularity analysis and random combinatorial structures. Based on joint work with N. M. Ercolani and D. Ueltschi (J. Theoret. Probab., 2018).