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Statement

Setting (p. 1504). $X_n$ is a symmetric transient random walk in $\mathbb Z^d$, $d\ge3$, started at the origin and not supported on a proper subgroup; $\mu_n^X(A)=\sum_{j=0}^n\mathbf 1_A(X_j)$; $\gamma_d$ is the probability of no return to the origin. For $y\in\mathbb Z^d$, $t_y=\mathbf P(T_y<\infty)$ with $T_y=\inf\{s>0:X_s=y\}$. $S(0,1)=\{e_1,\ldots,e_d,-e_1,\ldots,-e_d\}$ and $B(0,1)=\{0\}\cup S(0,1)$ are the Euclidean sphere and ball of radius 1 about the origin.

Theorem 1.2 (pp. 1504–1505). If $X$ has finite second moments, then for any $0\ne y\in\mathbb Z^d$, almost surely,

$$\lim_{n\to\infty}\sup_{x\in\mathbb Z^d} \frac{\mu_n^X(x+\{0,y\})}{\log n} =-\frac1{\log\bigl(1-\gamma_d/(1+t_y)\bigr)}. \tag{1.5}$$

For the simple random walk, almost surely,

$$\lim_{n\to\infty}\sup_{x\in\mathbb Z^d} \frac{\mu_n^X(x+S(0,1))}{\log n} =-\frac1{\log\bigl(1-\gamma_d/(2d(1-\gamma_d))\bigr)} \tag{1.6}$$

and

$$\lim_{n\to\infty}\sup_{x\in\mathbb Z^d} \frac{\mu_n^X(x+B(0,1))}{\log n} =-\frac1{\log\Bigl(\frac{p+\sqrt{p^2+2/d}}2\Bigr)}, \qquad p=1-\frac1{2d(1-\gamma_d)}. \tag{1.7}$$

The print writes the fractions in (1.6) and in $p$ inline as "$\gamma_d/2d(1-\gamma_d)$" and "$1-1/2d(1-\gamma_d)$"; the readings above are the ones the proof on pp. 1513–1515 computes, with $\Lambda_{S(0,1)}=2d(1-\gamma_d)/\gamma_d$ and $p=1-1/\Lambda$ for $\Lambda=\Lambda_{S(0,1)}/G(0)=2d(1-\gamma_d)$.

Proof pointer

Section 4, pp. 1513–1515: each case evaluates $\Lambda_A$ in Theorem 1.1. For $A=\{0,y\}$ the Green matrix has eigenvalues $G(0)\pm G(y)$, and $G(y)=t_yG(0)$ gives $\Lambda_A=(1+t_y)/\gamma_d$; the same computation yields the exact law (4.1). For $S(0,1)$, Perron–Frobenius and the symmetry relation $2d\gamma_d=\mathbf P(T_{2e_1}=\infty)+(2d-2)\mathbf P(T_{e_1-e_2}=\infty)$ give $\Lambda_{S(0,1)}=2d(1-\gamma_d)/\gamma_d$, together with the geometric law (4.2) for $\mu_\infty^X(S(0,1))$. For $B(0,1)$ the largest eigenvalue comes from a $2\times2$ reduction, (4.3)–(4.7), and the paper records in (4.8) that the law of $\mu_\infty^X(B(0,1))$ is a combination of two geometric terms which is not a mixture of geometric laws.

Read depth

Claims checked: the statement and the computations of Section 4 for (1.5) and (1.6) were read clause by clause on the page images of the print; the ball computation (4.3)–(4.8) was read for structure. Nothing here is independently reviewed.

Dependencies

Theorem 1.1 and Lemma 2.1.

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 $d\ge3$.