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Statement

Setting (p. 137). The walk is the symmetric nearest-neighbor walk on $\mathbb Z^2$ started at the origin.

Theorem 7 (p. 151). Let $f$ be a monotonic function increasing to $+\infty$, and let $E_n$ be the event that the planar walk does not return to the origin between times $n$ and $n^{f(n)}$. Then $\mathbf P\{E_n\text{ i.o.}\}$ is $0$ or $1$ according as

$$\sum_{k=1}^{\infty}\frac1{f\bigl(2^{2^k}\bigr)}$$

converges or diverges. The series runs over the doubly exponential sequence $2^{2^k}$, the sequence $n_k$ of the proof. The print writes the growth hypothesis as "increases to $+\infty$ as $x\to\infty$" [sic], with the variable $x$ for $n$.

The paper motivates the theorem (p. 151) as the question of how long the gaps between returns can be: for which monotone $g$ the interval $(n,n+g(n))$ contains a return for all but finitely many $n$.

Theorem 7A (p. 153). The paper states without proof the line analogue, saying it can be proved by similar methods: for the walk on $\mathbb Z$, with $f$ as above and $E_n$ the event of no return to the origin between $n$ and $n\{f(n)\}^2$, $\mathbf P(E_n\text{ i.o.})$ is $0$ or $1$ according as $\sum 1/f(2^k)$ converges or diverges.

Source. P. Erdős and S. J. Taylor, Some problems concerning the structure of random walk paths, Acta Math. Acad. Sci. Hungar. 11 (1960), 137--162: the walk on p. 137, (2.16) on p. 141, Theorem 7 and (4.8)--(4.9) on p. 151, the proof on pp. 151--153, Theorem 7A on p. 153. The edition read is identified on the source card.

Read depth. Claims checked: Theorems 7 and 7A were read clause by clause on the printed pages. The proof on pp. 151--153 was read for the pointer below and not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pages 151--153. The paper's estimate (2.16) (p. 141) for the probability that a planar walk started at distance $\varrho$ avoids the origin for $n$ steps gives, from the position at time $n$, two-sided bounds $\frac13\cdot\frac1{f(n)}<\mathbf P(E_n)<\frac3{f(n)}$ for large $n$ ((4.8)--(4.9), p. 151). Convergence of the series gives the zero case by Borel--Cantelli along $n_k=2^{2^k}$, with $f$ replaced by $f/2$ to cover the intermediate $n$. For divergence the paper bounds the overlap $\mathbf P(E_{n_{5k}}\cap E_{n_{5r}})$ in two cases ((4.14)--(4.15), p. 153), so that disjointified events carry at least half the probability, and concludes with the zero-one law.

Dependencies

The planar avoidance estimate (2.16) of the same paper (p. 141), which rests on (2.5) and the local estimates (2.9)--(2.10).

Bears on

No problem page of this corpus.