Pَlya number and first return of bursty random walk: Rigorous solutions

The recurrence properties of random walks can be characterized by Pَlya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we investigate Pَlya number and first return for bursty random walk on a line, in which the walk has different step size and moving probabilities. Using the concept of the Catalan number, we obtain exact results for first return probability, the average first return time and Pَlya number for the first time. We show that Pَlya number displays two different functional behavior when the walk deviates from the recurrent point. By utilizing the Lagrange inversion formula, we interpret our findings by transferring Pَlya number to the closed-form solutions of an inverse function. We also calculate Pَlya number using another approach, which corroborates our results and conclusions. Finally, we consider the recurrence properties and Pَlya number of two variations of the bursty random walk model.