Returns to the origin of the Pólya walk with stochastic resetting
Résumé
We consider the simple random walk (or Pólya walk) on the one-dimensional lattice subject to stochastic resetting to the origin with probability r at each time step. The focus is on the joint statistics of the numbers N × t of spontaneous returns of the walker to the origin and N • t of resetting events up to some observation time t. These numbers are extensive in time in a strong sense: all their joint cumulants grow linearly in t, with explicitly computable amplitudes, and their fluctuations are described by a smooth bivariate large deviation function. A nontrivial crossover phenomenon takes place in the regime of weak resetting and late times. Remarkably, the time intervals between spontaneous returns to the origin of the reset random walk form a renewal process described in terms of a single 'dressed' probability distribution. These time intervals are probabilistic copies of the first one, the 'dressed' first-passage time. The present work follows a broader study, covered in a companion paper, on general nested renewal processes.
Domaines
Physique [physics]
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