First gap statistics of long random walks with bounded jumps

We obtain exact analytical results for the joint statistics of the gap and time interval between the first two maxima of long, one-dimensional, random walks (RWs) with bounded jumps. Both discrete and continuous time settings are considered. For discrete time RWs, we find that the joint distribution exhibits a concentration effect in the sense that a gap close to its maximum possible value is much more likely to be achieved by a single jump (i.e. by realizations with adjacent first two maxima) rather than by a long walk between the first two maxima. We show that a similar, albeit slightly different, concentration phenomenon also occurs for continuous time random walks (CTRWs). Our numerical simulations confirm this concentration effect.