Exact record and order statistics of random walks via first-passage ideas

TL;DR

This paper reviews recent exact results on record and order statistics of one-dimensional random walks, highlighting the impact of strong correlations using first-passage techniques, and compares them to classical i.i.d. variable results.

Contribution

It provides a comprehensive review of recent exact findings on records and order statistics in correlated random walks, employing first-passage methods.

Findings

Exact results for record statistics of random walks

Comparison with i.i.d. variable results

Application of first-passage techniques

Abstract

While records and order statistics of independent and identically distributed (i.i.d.) random variables X_1, ..., X_N are fully understood, much less is known for strongly correlated random variables, which is often the situation encountered in statistical physics. Recently, it was shown, in a series of works, that one-dimensional random walk (RW) is an interesting laboratory where the influence of strong correlations on records and order statistics can be studied in detail. We review here recent exact results which have been obtained for these questions about RW, using techniques borrowed from the study of first-passage problems. We also present a brief review of the well known (and not so well known) results for records and order statistics of i.i.d. variables.