Favorite sites for simple random walk in two and more dimensions

TL;DR

This work resolves the long-standing question of how many favorite sites a simple random walk on ${\mathbb Z}^d$ can realize over time. It proves that in dimension $2$ three favorite sites occur infinitely often while four never do, and for $d\ge3$ it establishes sharp asymptotics for the growth of the number of favorite sites via a precise limsup involving the non-return probability $\gamma_d$. The authors develop a time-change approach using the maximal local time $m$, decompose the 2D local time into external and lazy components, and employ urn-model analogies and domino tilings to control near-maxima; in higher dimensions they exploit approximate independence of thick points and a Csáki-type bound to derive exact asymptotics. These methods settle a classical problem of Erdős and Révész (1987) and connect local-time maxima to the geometry of thick points and Gaussian-free-field related structures, with implications for occupation measures and extreme-point behavior in random walks.

Abstract

On the trace of a discrete-time simple random walk on $\mathbb{Z}^d$ for $d\geq 2$, we consider the evolution of favorite sites, i.e., sites that achieve the maximal local time at a certain time. For $d=2$, we show that almost surely three favorite sites occur simultaneously infinitely often and eventually there is no simultaneous occurrence of four favorite sites. For $d\geq 3$, we derive sharp asymptotics of the number of favorite sites. This answers an open question of Erdős and Révész (1987), which was brought up again in Dembo (2005).

Figures (3)

• Figure 1: Excursions $\{e^{(i)}:i\ge 1\}$. The two concentric disks are $D(x,R_n)$ and $D(x,r^0_n)$. The blue curve (both solid and dashed parts) is the sample path of $S$, the three solid sections of which are $e^{(1)}$, $e^{(2)}$, and $e^{(3)}$ respectively.
• Figure 2: Excursions $\{v_{i,j}:j=1,\ldots,Z^i\}$. The disk $D(x,r_n)$ is added in this figure. The two red dashed sections are $v_{1,1}$ and $v_{1,2}$ respectively and $Z^1=2$ in this case.
• Figure 3: $D(0,K_n)$, $D(x,r_{n,k})$, and $U_n$.

Theorems & Definitions (64)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Remark 1.4
• Lemma 2.1: $d=2$
• Lemma 2.2: $d\ge3$
• proof
• Lemma 2.3: Local central limit theorem
• Lemma 2.4: Moderate deviation
• Lemma 2.5: $d=2$