Some discrete processes

This book focuses on continuous time conformally invariant processes taking

values in C. One of the main motivations for studying such processes is that they

arise as limits of simple discrete processes taking values in a planar lattice. While

we will not discuss convergence of discrete processes to continuous processes in this

book, it will be helpful to the reader to know the discrete processes that motivate

some of the processes in this book.

0.1. Simple random walk

Let Z2 = Z-fiZ denote the integer lattice embedded in C. Simple random walk

starting at z G Z2 is the process Sn given by So — z and

Sn = z + Xi -\ + Xn,

where Xi,... , Xn are independent random variables each uniformly distributed on

{l,—l,i, —%}. We define St for positive, noninteger t by linear interpolation. For

every N oo, we can define

B

W

=

N-l/2 ^

Note that for positive integers k,

? W \ 2) l

-T?\J^fu(N)\2] k

E [ R e ( 5 ^ 1 = E [ I m ( ^ n = ^ .

As N — oo, the processes B±

)

converge to a complex Brownian motion Bt — B\ +

il?2,

where B\,B\ are independent one-dimensional standard Brownian motions.

Let Z+ = Z -f iZ

+

= {j + ik G

Z2

: k 0} be the discrete upper half plane. If

k is an integer, let u^ denote the stopping time

(j/- = min{n 1 : Im[5n] = k}.

A well-known result for one-dimensional walks, sometimes called the gambler's ruin

estimate, states that if 0 k m,

k

P{^m ^o | Im[S0] =*:} = —,

m

which implies for 0 &;, k' m,

P{o-m CTQ 1 Im[5Q] = k}

=

k_

P{am a0 | Im[S0] =

*;}

~ *''

Let Sn denote the Markov chain that corresponds to simple random walk stopped

at time am and conditioned so that crm ao- Prom the last equation we can see

the transition probabilities for this chain are given (for 0 k m) by

P(J + ikj - 1 + ifc) = p(j + ik, j + 1 -h ik) = - ,

l