Abstract
Abstract
We compute exactly the mean number of records ⟨R
N
⟩ for a time-series of size N whose entries represent the positions of a discrete time random walker on the line with resetting. At each time step, the walker jumps by a length η drawn independently from a symmetric and continuous distribution f(η) with probability 1 − r (with 0 ⩽ r < 1) and with the complementary probability r it resets to its starting point x = 0. This is an exactly solvable example of a weakly correlated time-series that interpolates between a strongly correlated random walk series (for r = 0) and an uncorrelated time-series (for (1 − r) ≪ 1). Remarkably, we found that for every fixed
r
∈
0
,
1
and any N, the mean number of records ⟨R
N
⟩ is completely universal, i.e. independent of the jump distribution f(η). In particular, for large N, we show that ⟨R
N
⟩ grows very slowly with increasing N as
⟨
R
N
⟩
≈
(
1
/
r
)
ln
N
for 0 < r < 1. We also computed the exact universal crossover scaling functions for ⟨R
N
⟩ in the two limits r → 0 and r → 1. Our analytical predictions are in excellent agreement with numerical simulations.
Subject
General Physics and Astronomy,Mathematical Physics,Modeling and Simulation,Statistics and Probability,Statistical and Nonlinear Physics
Cited by
21 articles.
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