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Statistical properties of sites visited by independent random walks

Journal Article · · Journal of Statistical Mechanics
 [1];  [2]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. Boston Univ., MA (United States); Santa Fe Inst. (SFI), Santa Fe, NM (United States)
+ Show Author Affiliations
The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. Here, we consider two independent random walks on hyper-cubic lattices and study ordering probabilities associated with these characteristics. The first is the probability that during the time interval (0, t), the number of sites visited by a walker never exceeds that of another walker. The second is the probability that the sites visited by a walker remain a subset of the sites visited by another walker. Using numerical simulations, we investigate the leading asymptotic behaviors of the ordering probabilities in spatial dimensions d = 1, 2, 3, 4. We also study the time evolution of the number of ties between the number of visited sites. We show analytically that the average number of ties increases as a1 ln t with a1 = 0.970 508 in one dimension and as (ln t)2 in two dimensions.
Research Organization:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Sponsoring Organization:
USDOE Laboratory Directed Research and Development (LDRD) Program
Grant/Contract Number:
89233218CNA000001
OSTI ID:
1902097
Report Number(s):
LA-UR-22-26901
Journal Information:
Journal of Statistical Mechanics, Journal Name: Journal of Statistical Mechanics Journal Issue: 10 Vol. 2022; ISSN 1742-5468
Publisher:
IOP PublishingCopyright Statement
Country of Publication:
United States
Language:
English

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Figures / Tables (13)

FIG. 1(p. 2)figure FIG. 1
FIG. 2(p. 3)figure FIG. 2
FIG. 3(p. 4)figure FIG. 3
FIG. 4(p. 4)figure FIG. 4
FIG. 5(p. 5)figure FIG. 5
FIG. 6(p. 5)figure FIG. 6
FIG. 7(p. 6)figure FIG. 7
FIG. 8(p. 6)figure FIG. 8
FIG. 9(p. 6)figure FIG. 9
FIG. 10(p. 7)figure FIG. 10
FIG. 11(p. 7)figure FIG. 11
FIG. 12(p. 8)figure FIG. 12
TABLE I(p. 9)table TABLE I

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Related Subjects

97 MATHEMATICS AND COMPUTING
Brownian motion
Mathematics
Persistence