1.4 Acknowledgements
I would like to thank Wendelin Werner for suggesting this problem to me. This work was done while I was a graduate student at the University of Chicago and I am very grateful to my advisors Steve
Lalley and Greg Lawler for all their patient help and guidance.
2 Definitions and background
2.1 Irreducible bounded symmetric random walks
Throughout this paper, Λ will be a two-dimensional discrete lattice of R
2
. In other words, Λ is an additive subgroup of R
2
not generated by a single element such that there exists an open neighbor- hood of the origin whose intersection with Λ is just the origin. It can be shown see for example
[16, Proposition 1.3.1] that Λ is isomorphic as a group to Z
2
. Now suppose that V
⊂ Λ \ {0} is a finite generating set for Λ with the property that the first nonzero component of every x
∈ V is positive. Suppose that κ : V → 0, 1 is such that X
x ∈V
κx ≤ 1. Let px = p
−x = κx2 for x ∈ V and p0 = 1 − P
x ∈V
κx. Define the random walk S with distribution p to be
S
n
= X
1
+ X
2
+ · · · + X
n
. where the random variables X
k
are independent with distribution p. Then S is a symmetric, irre- ducible random walk with bounded increments. It is a Markov chain with transition probabilities
px, y = p y − x.
If X = X
1
, X
2
has distribution p, then Γ
i, j
= E
X
i
X
j
i, j = 1, 2
is the covariance matrix associated to S. There exists a unique symmetric positive definite matrix A such that Γ = A
2
. Therefore, if e S
j
= A
−1
S
j
, then e S is a random walk on the discrete lattice A
−1
Λ with covariance matrix the identity. Since a linear transformation of a circle is an ellipse, it is clear
that if we can show that the growth exponent α
2
is 5 4 for random walks whose covariance matrix
is the identity, then α
2
will be 5 4 for random walks with arbitrary covariance matrix. Therefore, to
simplify notation and proofs, throughout the paper S will denote a symmetric, irreducible random walk on a discrete lattice Λ with bounded increments and covariance matrix equal to the identity.
2.2 A note about constants
Kategori