2

SOME DISCRETE PROCESSES

k + 1 k — 1

K7+ifc,.7+i(fc + l)) - - ^ - , p(j + i/c,j + i(/c-l) ) = —^-.

Note that the transition probabilities do not depend on m. Hence, we can let m —

oo and get a process defined on Z2^ called random walk half-plane excursion. We

can also extend the transition probability to Z = {j + ik : k = 0} by p(j, j + i) = 1.

If we scale these processes in the same way that we scaled the simple random walk,

we get the H-excursion discussed in §5.3.

oS3

R£ s

2*

FIGURE

0.1. A simple random walk and a random walk half-plane excursion.

We call a subset A of

Z2

simply connected if A and 1? \ A are nonempty

connected subsets of

Z2.

Let

dA = {z G Z2 \ A : dist(*, A) = 1}.

We call a finite path

LU

=

[LUQ,LUI,...

,o;n] an excursion in A (of length n) if it

is a nearest neighbor path (i.e.,

\LUJ

— &j-i\ = 1 for all j) in Z2 with

LUO,LUU

G

dA, (Ji,... ,o;n_i G A. We can consider an excursion of length n as a curve

a; : [0, n] — » C by linear interpolation. Let £4 denote the set of excursions in A.

The random walk excursion measure (see [43]) is the measure on SA that assigns

measure

4~n

to each excursion in EA of length n. In other words, the excursion

measure of LU = [LUQ, ... ,ujn] is the probability that the first n steps of a simple

random walk starting at UJQ are the same as UJ. Suppose D is a bounded simply

connected domain in C containing the origin. For each N 00, let Djsr denote the

connected component containing the origin of the set of z = x + iy G

Z2

such that

{x' H- iy' : \x — x'\ 1/2, \y — y'\ 1/2} is contained in ND. For each N, we get a

measure on paths by considering the random walk excursion measure on DJV and

scaling the excursions by Brownian scaling, uj^N\t) = N~1^2cj(2tN). As N — 00,

these measures approach an infinite measure on paths called excursion measure on

D, which is discussed in §5.2.

We will see that Brownian motion in C is invariant under conformal transfor-

mation up to a change of time. The excursion measure is also conformally invariant.

If 2, w are distinct points in 3D, then conditioning the excursion measure to have

endpoints 2, w gives a probability distribution on excursions from z to w in D. This

is the same (up to a time change) as the conformal image of M-excursions under a

conformal transformation of C onto D sending 0 to z and 00 to w.

A random walk loop (of length 2n) is a nearest neighbor path LU = [uo,... , u)2n)

with CJQ = cj2n- The rooted random walk loop measure (see [60]) is the measure that

gives each random walk loop of length 2n measure (2n) _ 1 4 _ 2 n . We can think of this

measure as a measure on unrooted loops that gives measure 4 _ 2 n to every (unrooted)

loop of length 2n and then chooses a root uniformly among {CJI, ... , cc^n}- By linear

interpolation, we can consider loops of length 2n as curves u : [0,2n] — C. For