Exercise 1 in Random Walks. Worth at most 30 points.
(reminder- there will be about 4-5 exe, and you should hand in part of each)
Due the week after Passover.
By graph we always mean undirected connected and finite degree unless otherwise stated. Usually we also mean uniformly bounded degrees.
1.
(9) Let G=(V,E) be a finite d-regular graph. Let {X i
} be a
Simple random walk on G, and recall P x
denotes the distribution of the random walk started from X
0
=x.
Let x,y

V , and let

be the first (positive) hitting time of
{x,y} (i.e. the first time the walk hits either x or y) a.
Show that P x
(X  = y) = P y
(X  = x) b.
Use this to prove that E x
(#visits to y before returning to x) = 1 for every y

V c.
What happens in non-regular bounded degree graphs?
2.
(9) a. Show the above result true also for infinite recurrent graphs? b. What about transient graphs? c. Use (a) to show that the expected return time to a vertex in any infinite bounded degree graph is infinite.
3.
(4) Prove that the probability of a random walk staying for n steps in a fixed finite subset of a connected graph decreases exponentially in n.
4.
(12) * Find a recurrent graph where 2 independent lazy random meet only finitely many times (with probability 1) .
This is quite hard, and I recommend looking at the paper http://arxiv.org/abs/math/0406487 and summarizing the main steps of the proof.

5.
(5) Prove by counting paths that SRW on Z^3 is transient. (similar to the proof we gave for Z)
6.
(8) Find an example of a recurrent graph not satisfying Nash-
Williams. Hint: - try to construct a graph where nested cut-sets must grow fast , but is still recurrent.
7.
(5) A. Show that the uniform measure is stationary for SRW on any finite regular graph
B. Show that the counting measure is stationary for any regular graph
C. Show that the measure pi(v)~deg(v) is stationary for SRW on any graph with finite degrees. (Actually, showing C is enough of course)
8. (6) Show that the radius of convergence of Greens function G(x,y) does not depend on the choice of x and y.
Further hints, if necessary, will be given on demand 