Lattice paths with catastrophes

In queuing theory, it is usual to have some models with a "reset" of thequeue. In terms of lattice paths, it is like having the possibility of jumpingfrom any altitude to zero. These objects have the interesting feature that theydo not have the same intuitive probabilistic behaviour as classical Dyck paths(the typical properties of which are strongly related to Brownian motiontheory), and this article quantifies some relations between these two types ofpaths. We give a bijection with some other lattice paths and a link with acontinued fraction expansion. Furthermore, we prove several formulae forrelated combinatorial structures conjectured in the On-Line Encyclopedia ofInteger Sequences. Thanks to the kernel method and via analytic combinatorics,we provide the enumeration and limit laws of these "lattice paths withcatastrophes" for any finite set of jumps. We end with an algorithm to generatesuch lattice paths uniformly at random.Comment: 32 page