Zeros of a two-parameter random walk

TL;DR

This paper establishes that the number of zeros of a two-parameter simple random walk grows almost linearly with N, contrasting with the higher power growth of zero crossings, and also characterizes zeros on the diagonal.

Contribution

It provides a rigorous proof of the asymptotic behavior of zeros in a two-parameter random walk, including growth rates and diagonal zeros, extending previous work.

Findings

Number of zeros in the first N-by-N steps is N^{1+o(1)} almost surely.

Number of zero crossings is N^{3/2+o(1)} as previously shown.

Zeros on the diagonal grow logarithmically as (c+o(1)) log N, with c=2π.

Abstract

We prove that the number gamma(N) of the zeros of a two-parameter simple random walk in its first N-by-N time steps is almost surely equal to N to the power 1+o(1) as N goes to infinity. This is in contrast with our earlier joint effort with Z. Shi [4]; that work shows that the number of zero crossings in the first N-by-N time steps is N to the power (3/2)+o(1) as N goes to infinity. We prove also that the number of zeros on the diagonal in the first N time steps is (c+o(1)) log N as N goes to infinity, where c is 2\pi.