Two and Three Dimensions

Part of the Probability and Its Applications book series (PA)


In this chapter we study
$$ f\left( n \right) = P\{ {S^1}\left( {0,n} \right) \cap {S^2}(0,n] = \phi \} $$
where S 1, S 2 are independent simple random walks in Z 2 or Z 3. By (3.29),
$$ {c_1}{n^{\left( {d - 4} \right)/2}} \le f\left( n \right) \le {c_2}{n^{\left( {d - 4} \right)/4}} $$
so we would expect that
$$ f\left( n \right) \approx {n^{ - \zeta }} $$
for some ζ; = ζ d . We show that this is the case and that the exponent is the same as an exponent for intersections of Brownian motions. Let B 1, B 2 be independent Brownian motions in R d starting at distinct points x, y. It was first proved in [19] that if d < 4,
$${P^{x,y}}\left\{ {{B^1}\left[ {0,\infty } \right) \cap {B^2}\left[ {0,\infty } \right) \ne \phi } \right\} = 1$$


Brownian Motion Random Walk Variational Formulation Conformal Invariance Harmonic Measure 
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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  1. 1.Department of MathematicsDuck UniversityDurhamUSA

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