The Range of a Random Walk on a Comb
János Pach∗
Gábor Tardos†
EPFL, Lausanne, Switzerland
and Rényi Institute, Budapest, Hungary
Rényi Institute, Budapest, Hungary
tardos@renyi.hu
pach@cims.nyu.edu
Submitted: July 10, 2013; Accepted: Sep 25, 2013; Published: Oct 7, 2013
Mathematics Subject Classification: 05C81
Abstract
The graph obtained from the integer grid Z × Z by the removal of all horizontal
edges that do not belong to the x-axis is called a comb. In a random walk on a
graph, whenever a walker is at a vertex v, in the next step it will visit one of the
neighbors of v, each with probability 1/d(v), where d(v) denotes the degree of v.
We answer a question of Csáki, Csörgő, Földes, Révész, and Tusnády by showing
that theexpected number
√ of vertices visited by a random walk on the comb after n
1
steps is 2√2π + o(1)
n log n. This contradicts a claim of Weiss and Havlin.
1
Introduction
The theory of finite Markov chains or, in graph-theoretic language, the theory of random
walks on graphs is a classical topic in probability theory. It has many applications from
flows in networks, through statistical physics to complexity theory in computer science
(see Lovász [Lo96] and Woess [Wo00]).
To obtain a random walk in a locally finite graph G with vertex set V (G), we start at
any vertex v ∈ V (G) and in the next step we move to one of its neighbors, independently
of all previous events, with probability 1/d(v). Here, d(v) denotes the number of edges in
G incident to v. Every neighbor of v is equally likely to come next. Perhaps the simplest
example is a random walk on the d-dimensional integer grid Zd , studied by Pólya [Po21].
He proved that for d = 1 and 2, with probability 1, a random walk will return to its
starting point infinitely often, while for d > 3 only a finite number of times.
∗
Supported by NSF Grant CCF-08-30272, by OTKA under EUROGIGA projects GraDR and ComPoSe 10-EuroGIGA-OP-003, and by Swiss National Science Foundation Grants 200021-137574 and
200020-144531.
†
Supported by an NSERC grant, and the OTKA grant NN-102029
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1
The 2-dimensional comb C2 is a spanning tree of the integer grid Z2 obtained by removing all of its “horizontal” edges (that is, edges parallel to the x-axis) that do not
belong to the x-axis. In this graph, all vertices (or “sites”) (x, y) have degree 2, except
for the vertices of the form (x, 0), which have degree 4. For any nonnegative integer n, let
Wn = (Xn , Yn ) be the random variable denoting the position of the walker after n steps.
We assume the walk starts at the origin, so W0 = (0, 0). The study of random walks on
the comb was initiated by Weiss and Havlin [WeH86], as a model of “anomalous diffusion
on fractal structures.” These investigations were later extended to combs of higher dimensions by Gerl [Ge86] and Cassi and Regina [CaR92]. Krishnapur and Peres [KrP04]
proved that, on the 2-dimensional comb, with probability 1, two independent walkers
meet only a finite number of times. This is a rather surprising phenomenon, in view of
the fact that the random walk is recurrent, that is, a single random walker visits each
site an infinite number of times with probability 1. Some insight was provided by Bertacchi and Zucca [BeZ03] and by Bertacchi [Be06], whose asymptotic estimates suggested
that a walker spends most of her time moving vertically along a “tooth” of the comb.
Several strong approximation and limit theorems for random walks on a comb have been
established by Csáki, Csörgő, Földes, and Révész [CsCs09, CsCs11].
Let Vn denote the number of vertices (sites) visited during the first n steps of the
random walk Wn = (Xn , Yn ) on the 2-dimensional comb. According to the main result in
[WeH86], the expected value of Vn is asymptotically proportional to n3/4 , for large n. It
is not hard to see that almost surely the deviation of the horizontal projection Xn of the
walk is roughly n1/4 , while the expected length of the vertical projection is of order n1/2 .
See, e.g., Bertacchi [Be06] (cp. Panny and Prodinger [PaP85]). This suggests that the
expected number of sites visited by the random walk on C2 is around n1/4 · n1/2 = n3/4 , as
was stated by Weiss and Havlin [WeH86]. The aim of this note is to show that the truth
is closer n1/2 , than to n3/4 .
All logarithms used in this paper are natural logarithms.
Theorem 1 The expected value of Vn , the number of vertices visited during the first n
steps of a random walk on the 2-dimensional comb, satisfies
√
1
√ + o(1)
E[Vn ] =
n log n.
2 2π
2
Elementary properties of a random walk on Z
We collect some well-known and easy facts about 1-dimensional random walks on Z; all
of them can be found, e.g., in [Fe68] or [Ré07]. For any pair of integers n, i > 0, let pn,i
denote the probability that starting at 0, after n steps we end up at the vertex (integer)
i. We have
n 1
pn,i = n+i n ,
2
2
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where the value of the above binomial coefficient is considered 0, whenever
integer. It follows that
!
r
n
1
2
1
pn,i 6
=
+ o(1) √ ,
n
⌊n/2⌋ 2
π
n
as n tends to infinity. Moreover, we have
r
pn,i =
2
+ o(1)
π
!
1
√ ,
n
n+i
2
is not an
(1)
√
whenever i/ n → 0 and n + i is even.
Let An stand for the number of times the random walk visits the origin during the
first n steps and Bn for the number of sites visited during the the first n steps. We have
!
r
n
X
√
2
E[An ] =
pm,0 =
+ o(1)
n,
(2)
π
m=0
E[Bn ] =
2
r
!
√
2
n.
+ o(1)
π
(3)
See, e.g., [Ré07], p. 253. Finally, for any j > 0, let rj denote the probability that starting
at position 0, the infinite random walk on Z reaches j before it would return to 0. We
have
1
rj = .
(4)
2j
3
Proof of Theorem 1
The vertices and edges of the comb C2 that belong to the x-axis form the backbone. The
connected components of the graph obtained from C2 after the removal of the backbone
are called teeth.
We consider the projections (Xi ) and (Yi ) of the two-dimensional walk (Xi , Yi ), separately. First, we reduce these one-dimensional walks by getting rid of the steps when
the value does not change. In this way, horizontal steps contribute only to the reduced
projection (Xi′ ) and vertical steps contribute only to the reduced projection (Yi′ ). Note
that with probability 1 both reduced walks are infinite and they are distributed as the
standard random walk on the line. Let us consider the random walk on the comb up to
(and including) the nth vertical move. We call this walk W . Its reduced projections are
(Xi′ )ai=0 and (Yi′ )ni=0 , where the random variable a is the number of horizontal moves in W .
Let c stand for the number of sites on the backbone reached by W . It is easy to describe
the asymptotic behavior of E[a] and E[c].
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Lemma 2
r
E[a] =
!
2
+ o(1) n1/2 ;
π
E[c] = Θ(n1/4 ).
Proof: Let d denote the number of times W moves from a tooth to a position on the
backbone, including the starting position at the origin, but not including the final position
after the last move, even if it satisfies this condition. Clearly, d is distributed as An−1 . W
has d chances to make horizontal moves. At each chance, it makes precisely i consecutive
horizontal moves with probability 2−i−1 . Thus, for the expected value of a, the number
of horizontal moves in W , we have
!
r
∞
X
√
2
+ o(1)
n.
E[a] = E[d
i2−i−1 ] = E[d] = E[An−1 ] =
π
i=0
Here the last equality follows by (2).
Obviously, the conditional distribution of c, given a, is the same as the distribution of
Ba , so that by (3) we obtain
!
r
√
2
E[c|a] = 2
+ o(1)
a.
π
Hence, we have
r
E[c] =
2
!
√
2
+ o(1) E[ a] 6
π
2
r
!
7/4
p
2
2
+ o(1) n1/4 .
+ o(1)
E[a] =
π
π 3/4
(5)
√
√
On the other hand, the limiting distribution of d/ n (i.e., that of An−1 / n) is equal to the
distribution of the absolute value of a random variable
√ with standard normal distribution
(see, e.g., [Ré07], p. 506). Therefore, we have P (d/ n > 1/2) > 1/2 + o(1). Using the
fact that, given d, the q
inequality a > d holds with probability exactly 1/2, we obtain that
√
√
E[ a] > (1/4 + o(1)) 12 n = Θ(n1/4 ). In view of (5), this implies the second part of
the lemma.
✷
We prove Theorem 1 by estimating the expected number of sites Vn′ reached by W . As
W makes n + a moves, it reaches the Vn sites visited in the first n steps and potentially
at most a further sites. According to the first part of Lemma 2, this potential increase is
so small that it does not change the asymptotic behavior of the expected value.
First, we establish the upper bound. We classify a site (i, j) as close if |j| < n1/4 ,
far if |j| > 2n1/2 log1/2 n, and intermediate if n1/4 6 |j| 6 2n1/2 log1/2 n, and estimate the
expected number of sites reached in each class, separately.
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The number of close sites reached is less than 2n1/4 + 1 times the number c of sites
reached
√ on the backbone. Thus, by Lemma 2, the expected number of close sites reached
is O( n). The number of far sites reached can easily be bounded by the number of
steps 0 6 i 6 n with |Yi′ | > 2n1/2 log1/2 n. For each i 6 n, the Chernoff bound gives
P (|Yi′ | > 2n1/2 log1/2 n) < 1/n. Therefore, the expected number of far sites reached is at
most 1.
Let us call the tooth of the comb containing the site where W ends the final tooth. In
case Yn′ = 0, there is no final tooth. The total number of intermediate sites on the final
tooth is less than n1/2 log1/2 n, so this also bounds the expected number of intermediate
sites reached on the final tooth.
Finally, we consider the number w of intermediate sites reached outside the final tooth.
When such a site at distance j is reached, it must be reached through a vertical move.
If the i-th vertical move reaches it for the last time in W , then we have |Yi′ | = j and
the part of the walk after the i-th vertical move (starting from this site) must reach the
backbone before it comes back to the same site. The probability for |Yi′ | = j is 2pi,j .
Assuming that this happens, according to (4), the probability that the infinite random
walk reaches the backbone before it returns to the same site is rj = 1/(2j). Hence, the
total probability is at most pi,j /j. We bound E[w] by summing pi,j /j over all 1 6 i 6 n
1/2
and n1/4
log1/2 n. Using the fact that pi,j = 0 if i + j is odd and, by (1),
p 6 j 6 2n √
pi,j 6 ( 2/π + o(1))/ i, it follows that
!
!
r
r
XX
X
2
2
1√
1
√ 6
E[w] 6
pi,j /j 6
+ o(1)
+ o(1)
n log n. (6)
π
π
4
j i
i,j
j
i
i+j
even
Summing over all the sites reached by W , we obtain the upper bound in the theorem:
√
1
′
√ + o(1)
E[Vn ] 6 E[Vn ] 6
n log n.
2 2π
Before turning to the lower bound, we introduce the symbol uj to denote the probability that W reaches the site (0, j). We need the following lemma.
Lemma 3
uj = O
n1/4 log n
|j| + 1
.
Proof:PLet ti stand for the probability that W reaches the site (i, 0). Clearly, we have
′
E[c] = +∞
i=−∞ ti . Let W denote the random walk on the comb starting at the origin and
′
of the sites W ′ reached, we have
ending with the 2n-th vertical move. For the number V2n
√
′
(7)
E[V2n
] = O( n log n),
by the upper bound (6) we have just proved. For every i and every j, the probability
that W ′ reaches the site (i, j) is at least ti uj . Indeed, with probability ti the walk reaches
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5
(i, 0) before the n-th vertical move, and after that it reaches (i, j) within n further vertical
steps with probability uj . Note that if (i, j) is reached, then at least |j| + 1 sites of the
form (i, j ′ ) are reached (along the same tooth). Therefore, we obtain
X
′
]>
(|j| + 1)ti uj = (|j| + 1)uj E[c].
E[V2n
i
From here, applying our upper bound (7) and Lemma 2, the result follows.
✷
Now we are ready to prove the lower bound in Theorem 1.
Let Zi stand for the site reached by the i-th vertical move of the random walk W . Let
qi,j stand for the probability that |Yi′ | = j and Zi is reached by the i-th vertical move for
the last time in W , 0 < i 6 n. Each site that does not belong to the backbone and is
reached by W , is reached for the last time by a single uniquely determined vertical move.
Thus, we have
X
E[Vn′ ] = E[c] +
qi,j .
0<j,0<i6n
Suppose that Zi is not on the backbone. If after the i-th vertical move the random
walk W returns to the backbone before it would revisit Zi , and after returning to the
backbone it still does not visit Zi during the following n vertical moves, then Zi was
reached in W for the last time by the i-th vertical move. Therefore, using the notation
in Section 2, we have
qi,j > 2pi,j rj (1 − uj ) for 0 < i 6 n, j > 0.
(8)
P
qi,j can be obtained by evaluating the terms qi,j . Let us
The lower bound for E[Vn′ ] >
consider only those terms qi,j for which n1/4 log2 n < j < n1/2 / log n, j 2 log√
n 6 i 6 n,
and i + j is even. For
these
values,
by
Lemma
3,
we
have
u
=
o(1).
Since
j/
i → 0, (1)
j
p
√
yields that pi,j = ( 2/π + o(1))/ i. Combining this with (4), inequality (8) gives
!
r
1
2
+ o(1) √ .
qi,j >
π
ij
Thus, for a fixed j, we get
X
j 2 log n6i6n, i≡j
qi,j >
r
2
+ o(1)
π
!√
n
.
j
Summing over all j, n1/4 log2 n < j < n1/2 / log n, we obtain
!
r
√ X1
√
2
1
′
√ + o(1)
n
n log n,
E[Vn ] >
+ o(1)
>
π
j
2
2π
j
as claimed. Note that our estimates for√the expectation of Vn′ carry over to the expectation
of Vn , as |Vn′ − Vn | 6 a and E[a] = O( n). This completes the proof of Theorem 1. ✷✷
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6
Acknowledgment
We are indebted to G. Tusnády for calling our attention to the problem addressed in this
note, and to E. Csáki, M. Csörgő, A. Földes, and P. Révész for their valuable remarks.
References
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D. Bertacchi, Asymptotic behaviour of the simple random walk on the 2dimensional comb, Electron. J. Probab. 11 (2006), no. 45, 1184–1203.
[BeZ03]
D. Bertacchi and F. Zucca, Uniform asymptotic estimates of transition probabilities on combs, J. Aust. Math. Soc. 75 no. 3, 325–353.
[CaR92]
D. Cassi and S. Regina, Random walks on d-dimensional comb lattices, Mod.
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[CsCs09] E. Csáki, M. Csörgő, A. Földes, and P. Révész, Strong limit theorems for a
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[CsCs11] E. Csáki, M. Csörgő, A. Földes, and P. Révész, On the local time of random
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7
The Range of a Random Walk on a Comb
The Electronic Journal of Combinatorics
The graph obtained from the integer grid $\mathbb{Z}\times\mathbb{Z}$ by the removal of all horizontal edges that do not belong to the $x$-axis is called a comb. In a random walk on a graph, whenever a walker is at a vertex $v$, in the next step it will visit one of the neighbors of $v$, each with probability $1/d(v)$, where $d(v)$ denotes the degree of $v$. We answer a question of Csáki, Csörgő, Földes, Révész, and Tusnády by showing that the expected number of vertices visited by a random walk on the comb after $n$ steps is $(\frac1{2\sqrt{2\pi}}+o(1))\sqrt{n}\log n.$ This contradicts a claim of Weiss and Havlin....Read more
