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
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
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