Record statistics for random walk bridges

TL;DR

This paper analyzes the statistics of records in a random walk bridge, revealing how the bridge constraint affects universality in record statistics and deriving general analytical results for various jump distributions.

Contribution

It introduces the first comprehensive analysis of record statistics in random walk bridges, showing the impact of the bridge constraint on universality and deriving results for arbitrary jump distributions.

Findings

Record statistics depend on the Lévy index for large n.

Bridge constraints break strong universality present in free walks.

Analytical results are verified by numerical simulations.

Abstract

We investigate the statistics of records in a random sequence $\{x_B(0)=0,x_B(1),\cdots, x_B(n)=x_B(0)=0\}$ of $n$ time steps. The sequence $x_B(k)$'s represents the position at step $k$ of a random walk `bridge' of $n$ steps that starts and ends at the origin. At each step, the increment of the position is a random jump drawn from a specified symmetric distribution. We study the statistics of records and record ages for such a bridge sequence, for different jump distributions. In absence of the bridge condition, i.e., for a free random walk sequence, the statistics of the number and ages of records exhibits a `strong' universality for all $n$, i.e., they are completely independent of the jump distribution as long as the distribution is continuous. We show that the presence of the bridge constraint destroys this strong `all $n$' universality. Nevertheless a `weaker' universality still remains for large $n$, where we show that the record statistics depends on the jump distributions only through a single parameter $0<\mu\le 2$, known as the L\'evy index of the walk, but are insensitive to the other details of the jump distribution. We derive the most general results (for arbitrary jump distributions) wherever possible and also present two exactly solvable cases. We present numerical simulations that verify our analytical results.