Intersections of Random Walks pp 139-161 | Cite as

# Two and Three Dimensions

Chapter

## Abstract

In this chapter we study where
so we would expect that
for some ζ; = ζ

$$ f\left( n \right) = P\{ {S^1}\left( {0,n} \right) \cap {S^2}(0,n] = \phi \} $$

*S*^{1},*S*^{2}are independent simple random walks in*Z*^{2}or*Z*^{3}. By (3.29),$$ {c_1}{n^{\left( {d - 4} \right)/2}} \le f\left( n \right) \le {c_2}{n^{\left( {d - 4} \right)/4}} $$

(5.1)

$$ f\left( n \right) \approx {n^{ - \zeta }} $$

_{ 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,$${P^{x,y}}\left\{ {{B^1}\left[ {0,\infty } \right) \cap {B^2}\left[ {0,\infty } \right) \ne \phi } \right\} = 1$$

## Keywords

Brownian Motion Random Walk Variational Formulation Conformal Invariance Harmonic Measure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1991