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• W1609190689 abstract "We use elementary methods to derive a sharp local central limit theorem for 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 starting point. More specifically, we give explicit asymptotic expressions in terms of $n$ and $x$ in dimensions one and two for $P(S(n)=x)$, the probability that simple random walk $S$ is at some point $x$ at time $n$, that are valid for all $|x|leq n/log^2(n)$. We also show that the behavior of planar simple random walk differs radically from that of planar standard Brownian motion outside of the disk of radius $n^{3/4}$, where simple random walk ceases to be approximately rotationally symmetric. Indeed, if $n^{3/4}=o(|S(n)|)$, S(n) is more likely to be found along the coordinate axes. This loss of rotational symmetry is not surprising, since if $|S(n)|=n$, there are only four possible locations for S(n). In this paper, we show how the transition from approximate rotational symmetry to complete concentration of $S$ along the coordinate axes occurs." @default.
• W1609190689 created "2016-06-24" @default.
• W1609190689 creator A5085497354 @default.
• W1609190689 date "2013-02-13" @default.
• W1609190689 modified "2023-09-27" @default.
• W1609190689 title "A Local Central Limit Theorem and Loss of Rotational Symmetry of Planar Simple Random Walk" @default.
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