Record statistics for random walks and Lévy flights with resetting
Satya N. Majumdar, Philippe Mounaix, Sanjib Sabhapandit, Grégory Schehr
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>r</mml:mi> <mml:mo>∈</mml:mo> <mml:mfenced close="" open="["/> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mfenced close="" open="["/> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">⟨</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">⟩</mml:mo> </mml:mrow> <mml:mo>≈</mml:mo> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msqrt> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>ln</mml:mi> <mml:mspace width="0.17em"/> <mml:mi>N</mml:mi> </mml:math> 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.