A local limit theorem and loss of rotational symmetry of planar symmetric simple random walk

• Published in: Transactions of the American Mathematical Society 2019-04, Vol.371 (4), p.2553-2573
• Main Author: Christian Beneš
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/7399

We derive a local limit theorem for normal, moderate, and large deviations for symmetric simple random walk on the square lattice in dimensions one and two that is an improvement of existing results for points that are particularly distant from the walk's starting point. More specifically, we give explicit asymptotic expressions in terms of n and x, where x is thought of as dependent on n, in dimensions one and two, for P(S_n=x), the probability that symmetric simple random walk S started at the origin is at some point x at time n, that are valid for all x. We also show that the behavior of planar symmetric simple random walk differs radically from that of planar standard Brownian motion outside the disk of radius n^{3/4}, where the random walk ceases to be approximately rotationally symmetric. Indeed, if n^{3/4}=o(\vert S_n\vert), S_n is more likely to be found along the coordinate axes. In this paper, we give a description of how the transition from approximate rotational symmetry to complete concentration of S along the coordinate axes occurs.