46 1. Random Walk and Discrete Heat Equation
Exercise 1.22. Suppose Sn is a simple random walk in Zd and A
Zd is finite with N points. Let TA be the smallest n such that Sn A.
Show that
P{TA kN} 1
1
2d
k
.
Exercise 1.23. Finish the proof of Theorem 1.9 by doing the follow-
ing:
Use connectedness of A to show that any nonzero eigen-
function φ with every component nonnegative must actually
have every component strictly positive.
Give an example of a disconnected A such that λ1 has mul-
tiplicity greater than one.
Given an example of a disconnected A such that λ1 has
multiplicity one. Does Theorem 1.9 hold in this case?
Exercise 1.24. Suppose A is a bounded subset of Zd. We call {x, y}
an edge of A if x, y A, |x y| = 1 and at least one of x, y is in A.
If F : A R is a function, we define its energy by
E(f) = [f(x)
f(y)]2,
where the sum is over the edges of A. For any F : ∂A R, define
E(F ) to be the infimum of E(f) where the infimum is over all f on A
that agree with F on ∂A. Show that if f agrees with F on ∂A, then
E(f) = E(F ) if and only if f is harmonic in A.
Exercise 1.25. Verify (1.31).
Exercise 1.26. We will construct a “tree” each of whose vertices has
three neighbors. We start by constructing T1 as follows: the vertices
of T1 are the “empty word”, denoted by o, and all finite sequences of
the letters a, b, i.e., “words” x1 · · · xn where x1,x2,...,xn {a, b}.
Both words of one letter are adjacent to o. We say that a word of
length n 1 and of length n are adjacent if they have the exact same
letters, in order, in the first n 1 positions. Note that each word of
positive length is adjacent to three words and the root is adjacent to
only two words. We construct another tree T2 similarly, calling the
root ˜ o and using the letters ˜ a,
˜.
b Finally, we make a tree T by taking
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