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Statement

Setting (p. 1504). $X_n$, $n=0,1,\ldots$, is a symmetric transient random walk in $\mathbb Z^d$, $d\ge3$, started at the origin and not supported on any subgroup strictly smaller than $\mathbb Z^d$; these assumptions stand throughout the paper. Its occupation measure is $\mu_n^X(A)=\sum_{j=0}^n\mathbf 1_A(X_j)$, so time zero is counted. With $q_n(x)=\mathbf P(X_n=x)$, the Green function is $G(x)=\sum_{k\ge0}q_k(x)$ (1.1). For a finite $A\subseteq\mathbb Z^d$, $\Lambda_A$ is the largest eigenvalue of the $|A|\times|A|$ matrix $G_A(x,y)=G(x-y)$, $x,y\in A$ (1.2).

Theorem 1.1 (p. 1504). If $X$ has finite second moments, then for every finite $A\subseteq\mathbb Z^d$, almost surely,

$$\lim_{n\to\infty}\sup_{x\in\mathbb Z^d}\frac{\mu_n^X(x+A)}{\log n} =-\frac1{\log(1-1/\Lambda_A)} \tag{1.3}$$

and

$$\lim_{n\to\infty}\sup_{0\le m\le n}\frac{\mu_n^X(X_m+A)}{\log n} =-\frac1{\log(1-1/\Lambda_A)}. \tag{1.4}$$

For $A=\{0\}$ the paper notes (p. 1504) that $\Lambda_A=G(0)=1/\gamma_d$, where $\gamma_d$ is the probability of no return to the origin, so the limit is $-1/\log(1-\gamma_d)$; for simple random walk this recovers Theorem 13 of Erdős and Taylor (the paper's reference [3]), recorded at Erdős–Taylor, Theorem 13. Remark 2.3 (p. 1511) notes that $G_{z+A}=G_A$, so $\Lambda_{z+A}=\Lambda_A$.

Proof pointer

Section 3, pp. 1511–1512, following the method of Erdős and Taylor (Section 7 of their paper). Write $\theta^*=\log(\Lambda_A/(\Lambda_A-1))$, the exponent in Lemma 2.2. For the lower bound of (1.4) the paper cuts $[0,n]$ into blocks of length $(\log n)^8$, applies the lower half of Lemma 2.2 to the independent block occupations and uses Borel–Cantelli, giving (3.1). For the upper bound it splits $\mu_n^X(X_m+A)$ at time $m$ into a backward and a forward piece, which by symmetry of $X_1$ are independent copies (3.2), bounds the tail of their sum by (2.28) to get (3.3), and interpolates along $n_k=n^k$. The lower bound of (1.3) follows from (3.1); the upper bound reduces to the case of (1.4) through (3.4) and Remark 2.3.

Read depth

Claims checked: the setting, Theorem 1.1 and Remark 2.3 were read clause by clause on the page images of the print; the proof in Section 3 was read for its structure. Nothing here is independently reviewed.

Dependencies

Lemma 2.2 and its estimate (2.28); the Erdős–Taylor block method.

Source. E. Csáki, A. Földes, P. Révész, J. Rosen and Z. Shi, Frequently visited sets for random walks, Stochastic Process. Appl. 115 (2005), 1503–1517, doi:10.1016/j.spa.2005.04.003; the edition read is named on the source card.

Bears on

No Erdős problem directly. The paper treats only transient walks in dimension $d\ge3$.