Universal first-passage properties of discrete-time random walks and Lévy flights on a line: Statistics of the global maximum and records

Abstract

In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence { x 0 = 0 , x 1 , x 2 , … , x n } up to n steps where x i represents the position at step i of a random walker hopping on a continuous line by drawing independently, at each time step, a random jump length from an arbitrary symmetric and continuous distribution (it includes, e.g., the Lévy flights). I will focus on the statistics of two extreme observables associated with the sequence: (i) its global maximum and the time step at which the maximum occurs and (ii) the number of records in the sequence and their ages. I will demonstrate how the universal statistics of these observables emerge as a consequence of Pollaczek–Spitzer formula and the associated Sparre Andersen theorem.