This document provides an analytic-combinatoric proof of Pólya's recurrence theorem for simple random walks on lattices Zd. It introduces generating functions to represent combinatorial classes of random walks. The generating function for simple random walks yields a formula for the probability of being at a given position at a given time. Laplace's method is then used to estimate these probabilities asymptotically, showing the random walk is recurrent if d = 1, 2 and transient if d ≥ 3, proving Pólya's theorem.

1. An analytic-combinatoric proof of P´olya’s recurrence
theorem
Brian Burns
November 20, 2016
Abstract
We provide a direct proof of P´olya’s famous recurrence theorem based on asymptotic
techniques and the Cauchy integral formula. We also provide an introduction to the
basic notions from the theory of analytic combinatorics to show how one might “guess”
a proof of P´olya’s famous theorem.
1 Introduction.
“A drunk man returns to the bar, but a drunk bird may never return home”. The origin
of this confusing phrase is in the behavior of a simple random walk (SRW) on the lattice
Zd
. The SRW on Zd
is the path a particle traces on Zd
if it is placed at the origin at time
0 and if, at each second, it randomly chooses to move one unit length in the 2d available
directions. More formally, a SRW is a sequence of random variables {Yn}∞
n=1, with
Yn =
n
i=1
Xi
where the Xi are independently, identically distributed in {−1, 1}d
.
A SRW on Zd
is said to be recurrent if it eventually returns to the origin with probability
1, and transient if not. An incredible theorem due to P´olya [1] says that once d hits 3, space
becomes too “big” for the SRW on Zd
to be recurrent:
Theorem 1.1 (P´olya’s recurrence theorem). The SRW on Zd
is recurrent if d = 1, 2
and is transient for d ≥ 3.
Remarkably, this theorem can be proven with essentially no combinatorics; complex analysis
and asymptotic estimates of integrals stand in its stead. The purpose of this note is to
explain how, and to explain a correspondence between “counting” problems and “calculus”
problems which greatly generalizes and illuminates the techniques used in the proof.
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2. 2 Analytic combinatorics.
A remarkable theory of analytic combinatorics due to computer scientists Philippe Flajolet
and Robert Sedgewick provides a systematic framework for the often ad-hoc use of generating
functions in combinatorial problems [2]. To a countable class of combinatorial objects C and
associated size function |·| 1
, one assigns an ordinary generating function
C(z) =
c∈C
z|c|
=
∞
n=0
(#{c ∈ C|n = |c|})zn
.
This method of baking a combinatorial problem into a function is well-known, with its origins
in the work of Euler. The insight of Sedgewick and Flajolet, however, is twofold:
• Combinatorial constructions “play nice” with this mapping: combinatorial sentences
are mapped to algebraic or differential equations, which may then be solved.
• The behavior of the function C(z) near its singularities can determine asymptotic
properties of the combinatorial objects in C as their size grows large.2
We acquiant the reader with the first principle via a few examples.
2.1 Examples of combinatorial sentences.
Given combinatorial classes A and B with size functions |·|A and |·|B, we may create the
combinatorial class A OR B whose set of objects is A ∪ B and whose size function is given
by
|c| =
|c|A c ∈ A
|c|B c ∈ B.
Similarly we may create the combinatorial class A AND B whose set of objects is the
cartesian product A × B and whose size function is given by (a, b) = |a|A + |b|B. If A and
B have associated generating functions A(z) and B(z), it is easy to see that A OR B and
A AND B have associated generating functions A(z) + B(z) and A(z)B(z) respectively.
We can iterate these basic constructions to make more complicated combinatorial classes.
For example, consider SEQ A, the set of all (possibly empty) sequences of elements from A,
whose size function is (a1, . . . an) = |a1|A + · · · + |an|A. Note that a sequence of length n is
just an element of the n-fold cartestian product A × · · · × A, and therefore letting denote
the empty sequence (with size 0), we have
SEQ A = OR A OR (A AND A) OR (A AND A AND A) OR . . .
and therefore SEQ A has associated generating function
1 + A(z) + A(z)2
+ · · · =
1
1 − A(z)
.
1
think of C, for example, as the set of all binary trees with |T| = number of nodes in the tree T.
2
This idea actually goes back to Hardy and Ramanujan in [3].
2

3. To show this is not an exercise in mindless generalization, take the following two examples.
A binary tree is either empty, or is a node with two trees. In addition, a simple combinatorial
argument shows that the nth
Fibonacci number is given by the number of ways to make a
2 × n tile out of 2 × 1 and 1 × 2 tiles, see [2]. The size of a tree is the number of nodes, and
the size of a tile is its horizontal length. Therefore we have two combinatorial classes T and
F satisfying
T = OR (N AND T AND T ), F = SEQ G
where N is one-element combinatorial class containing a single node of size one, and G is
the two-element combinatorial class containing a vertical tile of size 1 and a horizontal tile
of size 2. Therefore we get equations in generating functions
T(z) = 1 + zT(z)2
, F(z) =
1
1 − (z + z2)
.
Solving the quadratic3
in T and applying partial fraction decomposition to F give the for-
mulae
T(z) =
1 −
√
1 − 4z
2z
, F(z) =
1
√
5
1
1 − (1+
√
5
2
)z
−
1
1 − (1−
√
5
2
)
Applying Newton’s binomial formula to T and expanding F in geometric series give the well-
known formulae for the number of binary trees and Fibonacci numbers of size n respectively:
Tn =
1
n + 1
2n
n
, Fn =
1
√
5

 1 +
√
5
2
n
−
1 −
√
5
2
n

 .
3 The generating function Ω(z, t).
3.1 Caveats.
In the proof that follows, we will deal with combinatorial classes whose elements have many
sizes rather than just one, and we allow some of the sizes to be negative. This is to keep
track of the motion of the random walk, whose position in Zd
has d integer coordinates, not
all of which of course must be positive.
Therefore instead of dealing with a single-variable power series in z, on a d-dimensional
lattice our generating function will be a many-variable Laurent series in zα := zα1
1 . . . zαd
d .
Moreover since we are dealing with a random variable, the coefficients in our generating
functions will be real-valued probabilities rather than integer-valued counts. Despite these
modifications, analogues of the statements in section 2 carry over essentially verbatim.
3.2 SRW ordinary generating function.
With this in mind, we try to bake the behavior of a SRW into a generating function. To do
this, first note a SRW in Zd
is a sequence of simple moves: stepping once forward in time
3
We throw out one solution as a result of monotonicity considerations.
3

4. and randomly stepping backwards or forwards in one of the d directions available. This
simple move therefore has probabilistic specification
M = T AND (S1 OR S2 OR . . . Sd), where
• T is the one-element combinatorial class consisting of a single step forward in time
with generating function T(t, z1, . . . , zd) = t, and
• Si is the two-element combinatorial class containing a backwards move and a forwards
move in the ith
direction, each with probability 1
2d
and therefore generating function
Si(t, z1, . . . zd) = zi + z−1
i .
By the discussion in section one, our simple move thus is associated to the generating function
M(t, z1, . . . , zd) =
t
2d
d
i=1
zi + z−1
i .
Since an SRW is a sequence of such simple moves, we have probabilistic specification SRW =
SEQ M, so therefore the simple random walk on Zd
has associated generating function
Ω(z1, . . . , zd, t) =
1
1 − t
2d
d
i=1 zi + z−1
i
.
The coefficient of zα
tn
in the Laurent series expansion of Ω about the origin4
is the probability
that the simple random walk is at position (α1, . . . , αd) at time t = n. Note we may also
take coefficients in time and space alone rather than together. For example with d = 3,
[z1
1z−3
2 z5
3]Ω(z, t)
is a function of t alone, whose power series expansion n antn
has the following property:
an is the chance that at time n, the random walk is at position (1, −3, 5).
4 Pure walks.
With this in mind, consider a special type of finite random walk: one which starts at the
origin and returns at time n, but never returns beforehand. Call such a walk a “pure” walk,
and let pn be the probability that a random walk of size n is pure. Recurrence is equivalent
to the assertion ∞
n=0 pn = 1, so we take this line of attack.
Associate to the combinatorial class of pure walks the probability generating function
P(t) = ∞
n=0 pntn
with the convention that p0 = 0. Note that a random walk with position
(0, . . . , 0) is a sequence of pure walks. So as before, we have
1
1 − P(z)
= [z0
]Ω(z, t) =⇒ P(z) = 1 −
1
[z0]Ω(z, t)
.
4
Which from hereon out we will write as [zα
tn
]Ω(z, t).
4

5. From Abel’s theorem on the convergence of power series or the monotone convergence the-
orem applied to the counting measure on N, we may take t ↑ 1 to get
∞
n=0
pn = lim
t↑1
P(t) = 1 −
1
[z0]Ω(z, 1)
.
Since recurrence is equivalent to n pn = 1, recurrence occurs if and only if
∞ = [z0
]Ω(z, 1) = [z0
]
1
1 − 1
2d
d
i=1(zi + z−1
i )
=
∞
n=0
[z0
]


d
i=0
zi + z−1
i
2d


n
.
With this in mind let
wn = [z0
]


d
i=0
zi + z−1
i
2d


n
.
We seek to put an estimate on these numbers to see if the sum converges or diverges.
5 Probabilities as Cauchy integrals.
If γ is a positively-oriented circle around the origin, the fundamental calculation from com-
plex analysis
1
2πi γ
zn
dz =
1 z = −1
0 otherwise
applied to the multinomial expansion of ( d
i=1 zi + z−1
i )n
gives
wn =
1
(2πi)d
γ
. . .
γ
1
z1 . . . zd


d
i=0
zi + z−1
i
2d


n
dz1 . . . dzd
where we do d iterated contour integrals around the origin. Parametrizing the contour by
γ(t) = eit
and using cos(t) = 1
2
(eit
+ e−it
), we get
wn =
1
(2π)d
[0,2π]d

1
d
d
i=1
cos(ti)


n
dt1 . . . dtd.
6 Laplace’s Method.
It’s helpful to now stop and examine the situation we find ourselves in. The function f(t) :=
1
d
d
i=1 cos(ti) is bounded by 1, so as n → ∞ the only meaningful contributions to the above
integral will come from portions of space where f is very near 1 - other contributions will
decay exponentially in n. There are two of these portions of space, one localized about the
point (0, . . . , 0) and one localized about the point (π, . . . , π); we’ll consider the first of them
and multiply by two at the end to get the right answer.
5

6. Since the above integral is essentially determined by the local behavior about t = 0, we
blithely expand in a power series and notice the coincidence
1
d
d
i=1
cos(ti) ≈ 1 −
1
2d
(t2
1 + · · · + t2
d) ≈ e−1
2
(t2
1+...t2
d)
.
Therefore to approximate the integral formula for wn, we should be able to replace f(t) by
a Gaussian and call it a day. This is precisely the idea behind Laplace’s method, which is
used to estimate extremely localized or extremely oscillatory integrals which crop up in both
real and complex analysis. For a more detailed discussion, see, for example [4]: the proof
is simply Fourier inversion. With this in mind we invoke Laplace’s method to obtain, as
n → ∞ the asymptotic estimate
wn ∼
2
(2π)2
Rd
e−n
2
(t2
1+···+t2
d)
dt1 . . . dtd =
dd/2
2d−1πd/2
n−d/2
= Cdn−d/2
.
Then by the comparison test for series convergence and our earlier discussion, we get
SRW Recurrent ⇐⇒
∞
n=1
n−d/2
diverges
which holds if and only if the dimension d is 1 or 2.
References
[1] P´olya, G. “ ¨Uber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im
Strassennetz.” Math. Ann., 84:149-160, 1921.
[2] Philippe F., Sedgewick, R. Analytic Combinatorics. Cambridge University Press, 2009.
[3] Hardy, G. H. and Ramanujan S., “Asymptotic formulae in combinatory analysis”, Proc.
London Math. Soc., 17(2) (1918) 75?115
[4] Bruijn, N. G. De. Asymptotic Methods in Analysis. Amsterdam. North-Holland Publish-
ing Co., 1958.
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