Two and Three Dimensions

Abstract

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),

$$ {c_1}{n^{\left( {d - 4} \right)/2}} \le f\left( n \right) \le {c_2}{n^{\left( {d - 4} \right)/4}} $$

(5.1)

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,

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