On sequences of records generated by planar random walks

TL;DR

This paper analyzes the statistical properties of three types of records in planar random walks, revealing universal power-law growth and providing detailed asymptotic distributions for these records.

Contribution

It extends the analysis of record statistics to two-dimensional random walks, deriving new asymptotic results and full distributions for diagonal, simultaneous, and radial records.

Findings

Mean number of records grows as power laws with exponents 1/4, 1/3, 1/2.

Full asymptotic distributions for diagonal and simultaneous records.

Radial records grow with a super-universal square-root law.

Abstract

We investigate the statistics of three kinds of records associated with planar random walks, namely diagonal, simultaneous and radial records. The mean numbers of these records grow as universal power laws of time, with respective exponents 1/4, 1/3 and 1/2. The study of diagonal and simultaneous records relies on the underlying renewal structure of the successive hitting times and locations of translated copies of a fixed target. In this sense, this work represents a two-dimensional extension of the analysis made by Feller of ladder points, i.e., records for one-dimensional random walks. This approach yields a variety of analytical asymptotic results, including the full statistics of the numbers of diagonal and simultaneous records, the joint law of the epoch and location of the current diagonal record and the angular distribution of the current simultaneous record. The sequence of radial records cannot be constructed in terms of a renewal process. In spite of this, their mean number is shown to grow with a super-universal square-root law for isotropic random walks in any spatial dimension. Their full distribution is also obtained. Higher-dimensional diagonal and simultaneous records are also briefly discussed.