The area enclosed by Pólya's walk

Extract

1. Pólya's walk in the Euclidean plane is the motion of a particle, which takes successive independent random steps of unit length parallel to the coordinate axes, the probabilities of each of the four possible directions at each step being ¼. We imagine the particle starts at the origin, and that, in taking the ith step (i = 1,2,…) from (x_i-1, y_i-1) to (x_i, y_i), it leaves a straight track of unit length between these two points. Let T_n denote the track thus marked out in the first n steps; so that T_n is a connected chain of n straight segments each of unit length. Define the interior of T_n to be the set of points, which neither lie on T_n nor can be connected to the point at infinity without crossing T_n at least once; and let A_n denote the area of the interior of T_n. Associated with each walk ω, there is a sequence of random numbers {A_n(ω)}; what can be said about the behaviour of this sequence, especially for large n? We shall proveTheorem 1. For all ω,0 ≤ A_n(ω) ≤ 1/16 n².