Fourier-Walsh coefficients for a coalescing flow (discrete time)

TL;DR

This paper analyzes the Fourier-Walsh coefficients of a coalescing flow generated by independent random signs, revealing the nonlinear and noise-sensitive nature of the walk's position after N steps.

Contribution

It introduces a detailed analysis of the Fourier-Walsh expansion for coalescing random walks, highlighting the complexity and noise sensitivity of the function.

Findings

Fourier-Walsh coefficients involve products of approximately sqrt(N) signs

The walk's position is a highly nonlinear, noise-sensitive function of the signs

The analysis provides insights into the structure of coalescing flows

Abstract

A two-dimensional array of independent random signs produces coalescing random walks. The position of the walk, starting at the origin, after N steps is a highly nonlinear, noise sensitive function of the signs. A typical term of its Fourier-Walsh expansion involves the product of about square roof of N signs.