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Statement

Setting (pp. 1504–1506). $X_n$ is a symmetric transient random walk in $\mathbb Z^d$, $d\ge3$, not supported on a proper subgroup, with Green function $G$, and $\Lambda_A$ is the largest eigenvalue of $G_A(x,y)=G(x-y)$ on a finite $A$. $K\subseteq\mathbb R^d$ is a fixed compact neighborhood of the origin which is the closure of its interior. With $u^0(x)=c_d/|x|^{d-2}$ and $c_d=2^{-1}\pi^{-d/2}\Gamma(\frac d2-1)$, the 0-potential density of Brownian motion in $\mathbb R^d$ (1.8), $\Lambda_K^0$ is the norm of the operator $R_Kf(x)=\int_Ku^0(x-y)f(y)\,dy$ on $L^2(K,dx)$. For $x\in\mathbb R^d$ and $\varepsilon>0$ let $e_\varepsilon(x)=x+[0,\varepsilon]^d$, and put

$$\mathcal L_\varepsilon(K)=\{x\in\varepsilon\mathbb Z^d: e_\varepsilon(x)\subseteq K\},\qquad \mathcal C_\varepsilon(K)=\bigcup_{x\in\mathcal L_\varepsilon(K)} e_\varepsilon(x). \tag{1.11}$$

Assume that $\lim_{\varepsilon\to0}\lambda^d(\mathcal C_\varepsilon(K)) =\lambda^d(K)$, with $\lambda^d$ Lebesgue measure (1.12). Note $\varepsilon^{-1}\mathcal L_\varepsilon(K)\subseteq\mathbb Z^d$.

Theorem 1.4 (p. 1506). Assume that $X_1$ has $d-1$ moments and covariance matrix equal to the identity. Then

$$\lim_{\varepsilon\to0}\varepsilon^2 \Lambda_{\varepsilon^{-1}\mathcal L_\varepsilon(K)}=\Lambda_K^0 \tag{1.13}$$

and consequently

$$-\lim_{\varepsilon\to0}\varepsilon^2\big/ \log\bigl(1-1/\Lambda_{\varepsilon^{-1}\mathcal L_\varepsilon(K)}\bigr) =\Lambda_K^0. \tag{1.14}$$

The paper presents this (p. 1506) as linking the constant of Theorem 1.1 for the discretized, rescaled set with the Brownian constant. That Brownian analogue is stated on p. 1505 without proof: for convex $K$, with $K(x,r)=x+rK$ and $\nu_T^W$ the occupation measure of Brownian motion $W$, for any $S,T\in(0,\infty)$ almost surely $\lim_{\varepsilon\to0}\sup_{|x|\le S} \nu_T(K(x,\varepsilon))/(\varepsilon^2|\log\varepsilon|)=2\Lambda_K^0$ (1.9), and the same limit holds with the supremum of $\nu_T(K(W_t,\varepsilon))$ over $0\le t\le T$ (1.10). The paper says the proof is very similar to the case of balls treated by Dembo, Peres, Rosen and Zeitouni and omits it (p. 1506).

Proof pointer

Section 5, pp. 1515–1517. $R_K$ is a compact, strictly positive definite operator whose top eigenspace is one-dimensional. The paper approximates it by operators with kernels built from $u^0$ and from the rescaled Green function on the cubes of $\mathcal L_\varepsilon(K)$, using Uchiyama's asymptotic $G(x)=(1+\delta(x))u^0(x)$ with $\delta(x)\to0$ (5.4), and obtains convergence of the largest eigenvalues from Kato's perturbation theory (5.10). The eigenvalue equation (5.13) identifies $\Lambda_{\varepsilon^{-1}\mathcal L_\varepsilon(K)}$ with $\varepsilon^{-2}$ times the top eigenvalue of the approximating operator.

Read depth

Claims checked: the setting, Theorem 1.4 and the statements (1.9)–(1.10) were read clause by clause on the page images of the print; the proof in Section 5 was read for structure. Nothing here is independently reviewed.

Dependencies

External: Uchiyama's Green-function asymptotics, Kato's perturbation theorem, and the Perron–Frobenius theorem for positive operators, 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.