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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; $\mu_n^X(A)=\sum_{j=0}^n\mathbf 1_A(X_j)$, and $\Lambda_A$ is the largest eigenvalue of the Green matrix $G_A$ of a finite set $A$.

Lemma 2.2 (p. 1509, the localization lemma). Let $\{X_n\}$ be a symmetric transient random walk in $\mathbb Z^d$ with finite second moments, and let $A$ be a finite set in $\mathbb Z^d$. Set $\theta^*=\log(\Lambda_A/(\Lambda_A-1))$. The paper's quantifiers read "for some $1<c_1<\infty$, $n\geqslant u^6$, and all $u>0$ sufficiently large"; under them

$$c_1^{-1}e^{-\theta^*u}\le\mathbf P(\mu_n^X(A)\ge u) \le\mathbf P(\mu_\infty^X(A)\ge u)\le c_1e^{-\theta^*u}. \tag{2.16}$$

The paper calls this the crucial lemma (p. 1506); $A$ need not contain the origin.

Proof pointer

Pp. 1509–1511. When $A$ contains the origin, the dominant term of Lemma 2.1 is the one for $\lambda_1=\Lambda_A$, with $h_1>0$, so $\mathbf P(\mu_\infty^X(A)>u)e^{\theta^*u}\to h_1\in(0,\infty)$ (2.18), which gives the upper bound. For the lower bound the paper stops the walk at the exit time from a ball of radius $z$, which exceeds $n$ only with probability at most $c_1\exp(-c_2nz^{-2})$ (2.19), bounds the occupation after that exit by an independent copy times the probability of returning to $A$ from distance $z$, which is $O(z^{-1})$ by the Green-function bound $G(x)\le c/|x|$ (2.25)–(2.27), controls the tail of the sum of two independent copies by $C(1+u)e^{-u\theta^*}$ (2.28), and takes $z=u^2$. A general $A$ is handled by decomposing at the first hitting time of $A$ (2.30).

Read depth

Claims checked: the statement was read clause by clause on the page image of the print; the proof was read for its structure. Nothing here is independently reviewed.

Dependencies

Lemma 2.1; external: the bound $G(x)\le c/|x|$ from Lawler's notes, as cited by the paper.

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.