Generalised directed last passage percolation: invariant laws on the cylinders

The directed last passage percolation (LPP) on the quarter-plane is a growing model. To come into the growing set, a cell needs that the cells on its bottom and on its left to be in the growing set, and then to wait a random time. We present here a generalisation of directed last passage percolation (GLPP). In GLPP, the waiting time of a cell depends on the difference of the coming times of its bottom and left cells. We explain in this article the physical meaning of this generalisation. In this first work on GLPP, we study them as a growing model on the cylinders rather than on the quarter-plane, the eighth-plane or the half-plane. We focus, mainly, on the law of the front line. In particular, we prove, in some integrable cases, that this law could be given explicitly as a function of the parameters of the model. These new results are obtained by the use of probabilistic cellular automata (PCA) to study LPP and GLPP.