zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Unpredictable paths and percolation. (English) Zbl 0937.60070
Summary: We construct a nearest-neighbor process {S n } on 𝐙 that is less predictable than simple random walk, in the sense that given the process until time n, the conditional probability that S n+k =x is uniformly bounded by Ck -α for some α>1/2. From this process, we obtain a probability measure μ on oriented paths in 𝐙 3 such that the number of intersections of two paths, chosen independently according to μ, has an exponential tail. (For d4, the uniform measure on oriented paths from the origin in 𝐙 d has this property.) We show that on any graph where such a measure on paths exists, oriented percolation clusters are transient if the retention parameter p is close enough to 1. This yields an extension of a theorem of G. R. Grimmet, H. Kesten and Y. Zhang [Probab. Theory Relat. Fields 96, No. 1, 33-44 (1993; Zbl 0791.60095)], who proved that supercritical percolation clusters in 𝐙 d are transient for all d3.

60J45Probabilistic potential theory
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
60J65Brownian motion
60K35Interacting random processes; statistical mechanics type models; percolation theory
60G50Sums of independent random variables; random walks