1.4. Expected time to escape 29
(−1)x1+···+xd . We call x even if par(x) = 1 and otherwise x is odd.
If n is a nonnegative integer, then
pn(x, y; A) = 0 if
(−1)npar(x
+ y) = −1.
If = λφ, then Q[parφ] = −λparφ.
Theorem 1.9. Suppose A is a finite connected subset of
Zd
with at
least two points. Then λ1 λ2, λN = −λ1 λN−1. The eigenfunc-
tion φ1 can be chosen so that φ1(x) 0 for all x A,
lim
n→∞
λ1
−n
pn(x, y; A) = [1 +
(−1)n
par(x + y)] φ1(x) φ1(y).
Example 1.10. One set in
Zd
for which we can compute the eigen-
functions and eigenvalues exactly is a d-dimensional rectangle,
A = {(x1,...,xd)
Zd
: 1 xj Nj 1}.
The eigenfunctions are indexed by
¯
k = (k1,...,kd) A,
φ¯(x1,...,xd)
k
= sin
k1πx1
N1
sin
k2πx2
N2
· · · sin
kdπxd
Nd
,
with eigenvalue
λ¯
k
=
1
d
cos
k1π
N1
+ · · · + cos
kdπ
Nd
.
1.4. Expected time to escape
1.4.1. One dimension. Let Sn denote a one-dimensional random
walk starting at x {0,...,N} and let T be the first time that the
walker reaches {0,N}. Here we study the expected time to reach 0
or N,
e(x) = E[T | S0 = x].
Clearly, e(0) = e(N) = 0. Now suppose x {1,...,N −1}. Then the
walker takes one step which goes to either x 1 or x + 1. Using this
we get the relation
e(x) = 1 +
1
2
[e(x + 1) + e(x 1)] .
Hence, e satisfies
(1.19) e(0) = e(N) = 0, Le(x) = −1, x = 1,...,N 1.
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