4
SOME DISCRETE PROCESSES
If t 0 and we have observed 7(s),0 s t, then the conditional
distribution of 7(5), t s 00, is the same (up to time change) as the
original distribution mapped conformally onto H \ 7[0, i\. In other words,
if g is a conformal transformation of HI \ 7[0,t] onto H with g(j(t)) =
0,#(oo) = 00, then the distribution of
VW(s):=g(i(t
+ s)), 0 s o o ,
is the same as the distribution of 7(5), 0 s 00 (modulo time change).
Searching for processes with the above property led Oded Schramm to define the
processes that are now called chordal Schramm-Loewner evolutions (chordal SLEK).
As we will see, this is a one-parameter family of processes parametrized by K 0.
For loop-erased random walk in Z+, it has been proved [57] that the scaling limit
is chordal SLE2. The determination n = 2 comes from particular analysis of the
loop-erased walk, i.e., it does not follow only from the assumptions above.
FIGURE
0.2. Loop-erased walks obtained from walks in Figure 0.1.
One can also define the loop-erased random walk in all of Z2. There is a
technical difficult in that the random walk is recurrent and hence one cannot just
take an infinite walk and erase the loops. However, any of the reasonable limiting
operations one might take wil give the same answer, e.g.:
Consider simple random walks of n steps and erase the loops. This gives
a measure on self-avoiding walks (of a random length k n). Take the
limit as n 00 of this measure.
Do the same thing, except taking simple random walks until they first
visit a point of absolute value at least N.
Use the transition probabilities as in (0.1) replacing p(z,w) with p =
1/4 and qyn with qyn, which is the unique function from
Z2
into [0, 00)
satisfying
qVn(z) = Q, zeVn; qVn(z) ~ log \z\, z - 00.
Here A denotes the usual discrete Laplacian,
^vn (z) = -
Yl
\QV"
M ~
SV n (*)]

\z-w\ = l
Consider a candidate for the scaling limit of the loop-erased random walk in Z2.
We expect that this is a random, simple 7 : [0, 00) C with the following property.
Suppose 7[0,t] is given. Let g = gt be a conformal transformation of C \ 7[0,£]
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