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Source. Erdős–Taylor (1960), Theorem 13, printed pp. 161–162 (PDF pp. 25–26). This proof expands the source's union bound and independent-piece argument, with integer cutoffs and an explicit all-future consequence used in Hao–Li–Okada–Zheng, Lemma 3.2.

For symmetric nearest-neighbor simple random walk on $\mathbb Z^d$, $d\ge3$, write $\xi(x,n)=\sum_{j=0}^n1_{\{S_j=x\}}$ and $\xi^*(n)=\max_x\xi(x,n)$. Let $\gamma\in(0,1)$ be the probability of never returning to zero after time zero, $q=1-\gamma$, and $\alpha=-1/\log q$. Then, almost surely,

$$\frac{\xi^*(n)}{\log n}\longrightarrow\alpha. \tag{1}$$

More quantitatively, for every $0<\epsilon<1$, define

$$D_n^\epsilon= \{(1-\epsilon/2)\alpha\log n\le\xi^*(n) \le(1+\epsilon/2)\alpha\log n\}.$$

There are $C=C(d,\epsilon)$ and $N_0=N_0(d,\epsilon)$ such that

$$\mathbb P(\exists n\ge N:(D_n^\epsilon)^c) \le C N^{-\epsilon/4}\qquad(N\ge N_0). \tag{2}$$

Classical input. The only analytic estimate needed below is $\sup_x\mathbb P(S_j=x)\le C_dj^{-d/2}$ for $j\ge1$. This is the standard transient heat-kernel estimate. The summation proof in Hao–Li–Okada–Zheng, Lemma 2.2 gives $\mathbb P(\exists j>h:S_j=0)\le C_dh^{1-d/2}$. No conclusion of Hao's Theorem 1.2 is used.

Upper tail. Starting at a site, the number of visits to that site, including the starting visit, has tail $q^{r-1}$ at integer threshold $r\ge1$, by the strong Markov property at successive returns. If a site has local time at least $r$ by time $n$, its first visit occurs at some $i\in\{0,\ldots,n\}$ and the walk from $i$ makes at least $r$ visits to $S_i$. Discarding the first-visit restriction only enlarges the event. A union bound and the independent increments after each fixed $i$ give

$$\mathbb P(\xi^*(n)\ge r)\le(n+1)q^{r-1}. \tag{3}$$

In particular, for each fixed $\lambda>\alpha$ and large $n$,

$$\mathbb P(\xi^*(n)>\lambda\log n) \le C_{d,\lambda}n^{1-\lambda/\alpha}. \tag{4}$$

Lower tail. Fix $0<\lambda<\alpha$. Set $h=\lceil\log n\rceil$, $r=\lceil\lambda\log n\rceil$ and $b=rh$. Write $q_h$ for the probability of a first return to zero within $h$ steps. The heat-kernel consequence above gives $q-C_dh^{1-d/2}\le q_h\le q$, hence $q_h\to q>0$.

Within a block of $b$ steps, the event that each of the first $r$ successive return times to the block's starting site is at most $h$ has probability $p_n=q_h^r$. Strong Markov gives the product; on this event all $r$ returns fit in the block and produce $r+1>\lambda\log n$ visits. Moreover,

$$\frac{\log p_n}{\log n} =\frac r{\log n}\log q_h\longrightarrow \lambda\log q=-\lambda/\alpha.$$

There are $v=\lfloor n/b\rfloor$ disjoint blocks. The success event for each is determined by its own increments, so these events are independent even if their spatial ranges intersect. A success implies $\xi^*(n)>\lambda\log n$. Therefore

$$\mathbb P(\xi^*(n)\le\lambda\log n) \le(1-p_n)^v\le\exp(-vp_n). \tag{5}$$

Since $b=O((\log n)^2)$ and $vp_n=n^{1-\lambda/\alpha+o(1)}$, for every fixed $0<\kappa<1-\lambda/\alpha$ the last bound is at most $\exp(-n^\kappa)$ for all sufficiently large $n$. This is an inequality: success in a selected block is a sufficient way to obtain a thick site, not a necessary one.

All late times. Apply (5) with $\lambda=(1-\epsilon/2)\alpha$ and $\kappa=\epsilon/4$. The sum of $\exp(-n^{\epsilon/4})$ over integers $n\ge N$ is at most $C N^{-\epsilon/4}$ for large $N$; for example, its terms are eventually at most $n^{-2-\epsilon/4}$.

For the upper side, if $2^k\le n<2^{k+1}$ violates the upper bound in $D_n^\epsilon$, monotonicity gives

$$\xi^*(2^{k+1})>(1+\epsilon/2)\alpha\log(2^k) \ge(1+\epsilon/4)\alpha\log(2^{k+1})$$

for all sufficiently large $k$. By (4) the probability is at most $C2^{-k\epsilon/4}$. Summing over $k\ge\lfloor\log_2N\rfloor$ proves the upper half of (2), and hence (2) itself. Here $\log_2N$ in the floor means logarithm to base two; all other logarithms are natural.

Letting $N\to\infty$ in (2) shows that $D_n^\epsilon$ holds eventually almost surely. Taking a countable sequence of positive $\epsilon$ tending to zero proves (1). Counting only times $1,\ldots,n$, as in the original paper, changes the maximum by at most one. $\square$

Used in. Hao–Li–Okada–Zheng, Lemma 3.2 and its transient favorite-site theorem. This logarithmic law concerns $d\ge3$; the separate planar upper bound has a squared logarithm.

Bears on. Problem 1165 only by comparison: the theorem feeds Hao, Li, Okada and Zheng's Theorem 1.2 on favorite sites in dimension $d\ge3$, which the problem page cites as a contrast to the planar answer. It is not an input to that answer.