Frequently visited sets for random walks

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corollary_1_3: For a symmetric transient walk on Z^d with finite second moments and any fixed K > 0, the largest occupation time up to time n of a pair of sites at distance at most K, divided by log n, has almost-sure limit strictly below -2/log(1 - gamma_d), twice the one-site constant.

equation_4_1: For a symmetric transient walk on Z^d started at the origin and y not 0, the total number of visits to {0, y}, time zero included, exceeds u with probability (1 - gamma_d/(1 + t_y))^u, where gamma_d is the escape probability and t_y the probability of ever hitting y.

lemma_2_1: For a symmetric transient walk on Z^d and a finite set A containing the origin, the tail of the total occupation time of A is a finite sum of powers ((lambda_j - 1)/lambda_j)^u over the eigenvalues lambda_j of the Green matrix of A, with weights from its eigenvectors.

lemma_2_2: For a symmetric transient walk on Z^d with finite second moments and a finite set A and all large u, the probabilities that A is occupied at least u times by time n, for n at least u^6, and ever, both lie within constant factors of exp(-theta* u), where theta* = log(Lambda_A/(Lambda_A - 1)).

theorem_1_1: For a symmetric transient walk on Z^d with finite second moments and a finite set A, the maximal occupation time up to time n of a translate of A, divided by log n, tends almost surely to -1/log(1 - 1/Lambda_A), where Lambda_A is the largest eigenvalue of the Green matrix of A.

theorem_1_2: Evaluates the limit of Theorem 1.1 for a two-point set {0, y} as the constant -1/log(1 - gamma_d/(1 + t_y)), and for simple random walk on the unit sphere and the unit ball of Z^d in terms of the return probability.

theorem_1_4: For a walk whose step has d - 1 moments and identity covariance, the largest Green-matrix eigenvalue of the rescaled lattice discretization of a compact set K, times epsilon squared, tends to the norm of the Brownian Newtonian-potential operator on K.

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Endre Csáki, Antónia Földes, Pál Révész, Jay Rosen and Zhan Shi, Frequently visited sets for random walks, Stochastic Processes and their Applications 115 (2005), 1503–1517. DOI.

Canonical source. The 15-page PDF comes from Rosen's publication collection. It has the journal's pagination and running heads. The first page records acceptance on 7 April 2005 and online availability on 4 May 2005. The earlier arXiv version is a 19-page preprint; no full version-equivalence comparison is claimed. The file, the publisher's typeset article as posted on the author's site, prints "© 2005 Elsevier B.V. All rights reserved." on p. 1503, every other right reserved. Labels and pages below are those of this print.

Digest. The paper works with a symmetric transient random walk $X_n$ in $\mathbb Z^d$, $d\ge3$, not supported on a proper subgroup, and its occupation measure $\mu_n^X(A)$, counting time zero (p. 1504). For a finite set $A$, $\Lambda_A$ is the largest eigenvalue of the Green matrix $G_A(x,y)=G(x-y)$, $x,y\in A$. Theorem 1.1 (p. 1504) shows, under finite second moments, that the largest occupation time of a translate $x+A$ up to time $n$, and of a translate $X_m+A$ by a point of the path, is almost surely $(-1/\log(1-1/\Lambda_A)+o(1))\log n$; for $A=\{0\}$ this is the Erdős–Taylor constant $-1/\log(1-\gamma_d)$, with $\gamma_d$ the probability of no return. Theorem 1.2 (pp. 1504–1505) evaluates the constant for two-point sets $\{0,y\}$ and, for simple random walk, for the unit sphere and ball. Corollary 1.3 (p. 1505) deduces that two sites at bounded distance cannot both be nearly maximally visited: the pair's joint occupation has limit strictly below twice the one-site constant. Theorem 1.4 (p. 1506) is an invariance principle linking $\Lambda_A$ for rescaled discretizations of a compact set $K$ to the norm of the Brownian potential operator on $K$; the Brownian analogue (1.9)–(1.10) of Theorem 1.1 is stated on p. 1505 with its proof omitted.

The engine is Lemma 2.1 (p. 1507), an exact spectral formula for the tail of the total occupation of a finite set containing the origin, and the localization Lemma 2.2 (p. 1509), which brackets the occupation tails by constant multiples of $e^{-\theta^*u}$, $\theta^*=\log(\Lambda_A/(\Lambda_A-1))$. For a two-point set the spectral formula collapses to the single geometric law (4.1) (p. 1513).

Connection. Hao, Li, Okada and Zheng quote (4.1) for simple random walk in the proof of their Lemma 3.2, which separates visits to thick sites in time in their proof of the favorite-site law in dimensions d ≥ 3.

Read status. Claims checked: the setting, Theorems 1.1, 1.2 and 1.4, Corollary 1.3, Lemmas 2.1 and 2.2, Remark 2.3, equation (4.1) and the statements (1.9)–(1.10) were read clause by clause on the page images of the print. The derivation of (4.1) and the proof of Corollary 1.3 were followed; the other proofs were read for structure only, and (1.9)–(1.10) are stated in the paper without proof. Nothing here is independently reviewed.

Bears on. Problem 1165, indirectly: the problem concerns planar simple random walk, which the paper does not treat, and the paper proves nothing about it. Equation (4.1) is an input, through Hao–Li–Okada–Zheng's Lemma 3.2, to their favorite-count law for $d\ge3$, the transient companion of their planar result on the problem's question.

Results.

• Theorem 1.1 (p. 1504): the maximal occupation of a translate of a finite set $A$ is almost surely $(-1/\log(1-1/\Lambda_A)+o(1))\log n$.
• Theorem 1.2 (pp. 1504–1505): the constants for $\{0,y\}$, and for simple random walk for the unit sphere and the unit ball.
• Corollary 1.3 (p. 1505): pairs at distance at most $K$ have joint occupation constant below $-2/\log(1-\gamma_d)$.
• Theorem 1.4 (p. 1506): $\varepsilon^2\Lambda_{\varepsilon^{-1}\mathcal L_\varepsilon(K)} \to\Lambda_K^0$.
• Lemma 2.1 (p. 1507): spectral formula for the tail of $\mu_\infty^X(A)$.
• Lemma 2.2 (p. 1509): the localization lemma.
• Equation (4.1) (p. 1513): the geometric law of the total occupation of $\{0,y\}$.

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