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,£]