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2020 On covering monotonic paths with simple random walk
Eviatar B. Procaccia, Yuan Zhang
Electron. J. Probab. 25: 1-39 (2020). DOI: 10.1214/20-EJP545


In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_{1}$ ball. We show that among all such paths, the one that maximizes the covering probability is the monotonic increasing one that stays within distance 1 from the diagonal. As a result, we can obtain an exponential upper bound on the decaying rate of covering probability of any such path when $d\geq 4$. The main tool is a general combinatorial inequality, that is interesting in its own right.


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Eviatar B. Procaccia. Yuan Zhang. "On covering monotonic paths with simple random walk." Electron. J. Probab. 25 1 - 39, 2020.


Received: 2 April 2020; Accepted: 5 November 2020; Published: 2020
First available in Project Euclid: 17 December 2020

Digital Object Identifier: 10.1214/20-EJP545

Primary: 60C05, 60G50


Vol.25 • 2020
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