1.6. Exercises 47
the union of T1 and T2 and adding one more connection: we say that
o and ˜ o are adjacent.
Convince yourself that T is a connected tree, i.e., between
any two points of T there is a unique path in the tree that
does not go through any point more than once.
Let Sn denote simple random walk on the tree, i.e., the
process that at each step chooses one of the three nearest
neighbors at random, each with probability 1/3, with the
choice being independent of all the previous moves. Show
that Sn is transient, i.e., with probability one Sn visits the
origin only finitely often. (Hint: Exercise 1.18 could be
helpful.)
Show that with probability one the random walk does one
of the two following things: either the random walk visits
T1 only finitely often or it visits T2 only finitely often. Let
f(x) be the probability that the walk visits T1 finitely often.
Show that f is a nonconstant bounded harmonic function.
(A function f on T is harmonic if for every x T , f(x)
equals the average value of f on the nearest neighbors of x.)
Consider the space of bounded harmonic functions on T .
Show that this is an infinite dimensional vector space.
Exercise 1.27. Show that if A A1 are two subsets of
Zd,
then
λA λA1 . Show that if A1 is connected and A = A1, then λA λA1 .
Give an example with A1 disconnected and A a strict subset of A1
for which λA = λA1 .
Exercise 1.28. Consider β(j, N),j = 1,...,N 1 where β(j, N) is
the unique positive number satisfying
cosh
β(j, N)
N
+ cos

N
= 2.
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