Speed of Random Walks and Geometry on Graphs
Balint Virag
UC Berkeley
Berkeley, CA 94720 USA
On rapidly growing graphs, random walk does not obey the usual diusive law which says
that the distance of the walker from its starting point is on the order of the square root of the
time. In graphs with exponential growth such as regular trees, the distance tends to be on the
order of the time spent. It is therefore interesting to dene the speed of the walk at a given
epoch, that is the distance from the starting point divided by the time spent. The most natural
questions about this speed process concern the connection between its limiting behavior and
the growth properties of the graph. The lower and upper speed of the walk are dened as the
respective limits of the speed process.
For a b + 1-regular tree, the walk usually has b directions which increase its distance from
the starting point and 1 direction which decreases this distance. The law of large numbers then
implies that the lower and upper speed are both equal (b , 1)=(b + 1). A heuristic conclusion
is that the faster the graph grows, the faster the walk can be. If we consider a tree which does
not branch for k , 1 generations and then branches into bk ospring and continues similarly,
one can easily see that the speed of the walk on this tree will get slower as k increases. In
particular, there are graphs which grow as fast as the regular trees but the walk has slower
speed on them. But are the ones where the walk is faster?
It is clear that increasing k introduces long \pipes", that is subgraphs which do not
\branch" rapidly, and this is what slows down the walk. This concept of \branching rapidly"
can be made rigorous through isoperimetric inequalities. Dene surface of a vertex set as the
number of edges between the subgraph and its complement, and dene the volume as the sum
of the degrees. We say that a graph satises an isoperimetric inequality with constant i if
the surface of each vertex set is at least i times its volume. If we require that the graph has
bounded degrees, then such an isoperimetric inequality implies a bound on the kernel of the
walk, which in turns implies a lower bound on the lower speed.
The above strong isoperimetric inequality is a very fragile property, while the speed of a
walk is more robust. Indeed, if we add a long pipe to a graph satisfying such an inequality, the
constant inequality will be readily destroyed, even though the speed does not change. If we
add a sequence of pipes of increasing length to a scarce set of vertices to a binary tree, then it
will not satisfy such inequality with any constant, but the walk still can have positive speed.
Supercritical Galton-Watson trees, percolation with high retention parameter on graphs with
positive isoperimetric constant, and random geometric edge-stretchings of such graph all have
the property that the random walk on them has positive speed, while they clearly do not satisfy
a strong isoperimetric inequality.
It was therefore suggested by Benjamini, Lyons and Schramm that strong isoperimetric
inequalities be replaced by weak ones in many probability questions. Our formulation of a weak
isoperimetric inequality with constant i requires that each vertex set has surface at least i times
it volume, with possibly some exceptions, provided that each vertex of the graph is contained
in only nitely many exceptional vertex sets (of course, there need not be a uniform bound on
the number of such exceptions). The above authors conjectured that such a weak inequality
together with a bound on the degree, imply positive speed. Chen and Peres have shown that the
above examples of supercritical Galton-Watson trees, percolation with high retention parameter
on graphs with isoperimetric constant, and random geometric edge-stretchings all satisfy such a
weak isoperimetric inequality. This, together with conjecture of Benjamini, Lyons and Schramm
proved by this author (Virag 1999), gives a geometric explanation for positive speed in these
graphs.
For an inequality in the other direction, dene the upper and lower growth of a graph as
the corresponding limit of the n-th root of the number of vertices at distance n from a xed
vertex. These might depend on the choice of the xed vertex. The signicance of the branching
number for trees was discovered by Lyons. For b 1, add thickness to each edge given by 1=b
to the power of the distance of the edge from a xed vertex. The branching number br is the
inmum of the b-s for which the total thickness of each cutset separating the xed vertex from
innity is bounded away from 0. Lyons showed that 1=br is the critical probability for Bernoulli
percolation. The branching number is never greater than the lower growth; recurrent graphs
have branching number 1, while there are recurrent graphs with lower growth greater than 1.
b + 1 regular trees have growth and branching number b.
Benjamini and Peres conjectured that for trees, the lower speed of the walk is never grater
than (br , 1)=(br +1) (see Peres 1997). This is true for general graphs, as proved by this author
(Virag 1998).
Another conjecture by the same authors claimed that the lim sup speed in trees is always
bounded above by (gr , 1)=(gr + 1). It is interesting to note that the lower growth gr here
cannot be replaced by lower growth. Indeed, it is possible to attach binary trees of rapidly
increasing depth to a scarce set of vertices of an innite pipe to get a graph with lim sup speed
1=3 but lower growth 1.
In an upcoming paper, we will prove this conjecture for general graphs. In what follows,
we illustrate the proof of the lim sup seed result in a much simpler setting of trees that are
spherically symmetric. These are trees in which vertices at a given distance from the root have
the same number of neighbors. The random walk on these trees can be thought of as a weighted
nearest neighbor random walk on the nonnegative integers, with weight wi on the edge between
i and i + 1 is given by the number of vertices at distance i from the root in the tree. Each step,
the walk moves up or down with odds given by the edge weights.
We are interested in exploring the relationship between the time T = Tn it takes for the
walk to get from the origin to n + 1 to the weight wn which measures the size of \level n" of
the tree. Toward this end, we introduce a new heaven state to the chain and send the walk to
heaven before every step with probability 1 , , and stop it at time T when it reaches n + 1.
The probability that the chain survives is then given by
X1 P(T = k)
k=0
k
(1)
and if we set e, = , then (1), as a function of , becomes the Laplace transform of T . Thus
the distribution of T can be recovered from the survival probabilities.
Now let fi , bi be the expected number of steps the walk moves forward or backward along
the edge (i; i + 1). Let also f,1 = 1, b,1 = 0. Note that on the vertices 1:::n the expected
number of outgoing steps equals times the expected number of incoming steps. In fact, we
have for i = 0:::n:
fi,1 + bi = (fi + bi,1 ); or equivalently: fi,1 , bi,1 = fi , bi :
(2)
Note that fn gives the survival probability, but it is a very complicated function of the wi. To
solve this problem, we introduce the quantity
i =
fi , bi
:
fi , bi
Note rst that ,1 = n = , and so
2 0 n,1 = ,1 n =
f,1 , b,1
f,1 , b,1
!
(3)
f0 , b0
f0 , b0
!
fn , bn
fn , bn
!
The product telescopes by (2) and we are left with
fn , bn
f,1 , b,1
= fn:
Now note that ratio of expected number of steps forward and backward from a vertex equals
the ratio of the corresponding edge weights: fi=bi,1 = wi=wi,1. Thus, by another telescoping
argument we can write
wn =
w1
wn
w0
wn,1
= fn
f0
Y fi
n,1
i=0 bi
= f0,1
Y 1 , i
n,1
i=0
i
, i
for the last equality we substituted the expression for fn and wrote fi=bi in terms of i by (3).
To summarize, we have
Ee,T = fn =
wn
=
0 :::n,1
,1 1 , nY
i
i
f0 i=0 , i
Note that this is almost symmetric in , and a boundary eect appears only in the f0 of
the second expression. This asymmetry causes some diculties, which we choose to avoid by
noting that f0 = is bounded by the expected number of returns to the origin of the random
walk without killing. If the walk is transient, as we will assume in this paper, then this quantity
is bounded by some constant not depending on n, while wn is unbounded.
We will proceed to bound Ee,T in terms of wnf0 = . If this quantity is xed, then the
standard convexity argument implies that Ee,T is maximized if all the i are equal. If we set
gn to be the nth root of wn times the expected number of hits to 0, then we get the bound
q
0
1n
2 , 4e,2 g
g
+
1
,
(
g
+
1)
n
n
n
A
Ee,T e, @
2e,
The expression on the left hand side is the Laplace transform of a random variable T. Consider
the standard biased random walk on the integers, with probability of moving right given by
gn=(gn + 1). Then T is 1 plus the time it takes for this walk to move n steps to the right. We
have thus proved a stochastic bound T >> T.
To get the result for the lim sup speed, assume that the tree is transient, and let g0 >
lim sup(wn)1=n . Then for some " > 0 and all large n, we have gn < g0 , ". Let Tn be the hitting
time of n + 1, then we have
(4)
lim sup jXk j=k lim sup n=Tn = lim sup n2=Tn2
(the last equality holds for any nondecreasing sequence of real numbers Tn). Consider the
events An given by n2 =Tn2 > (g0 , 1)=(g0 + 1). Note that
ETn = 1 + (gn + 1)=(gn , 1) n
Var Tn = O(n)
and it follows from Chebyshev's inequality that PAn = O(n,2). Hence only nitely many of
the An happen by the Borel-Cantelli lemma. Thus by (4) have shown
Theorem 1 Let fXk g be a random walk on a transient spherically symmetric tree with upper
growth gr = lim sup wn1=n. Then the lim sup speed satises
lim sup jXk j=k (gr , 1)=(gr + 1):
A similar theorem holds for general graphs, as we will prove in a subsequent paper. For
general graphs, we rst decompose the graph into weighted paths, and then prove the inequality
on those paths using the methods of Theorem 1.
References
Benjamini, I., Lyons, R. and Schramm, O. (1998) Percolation perturbations in potential
theory and random walks, To appear.
Chen, D. and Peres, Y. (1998) Anchored expansion, percolation and speed, Preprint.
Lyons, R. (1990) Random walks and percolation on trees, Ann. Probab. 18, 931{958.
Peres, Y. (1997) Probability on Trees: An Introductory Climb, Lecture notes from St.
Flour summer school.
Virag, B. (1998) On the speed of random walks on graphs, Preprint.
Virag, B. (1999) On the anchored expansion property, Preprint.
Resume
Nous etudions la relation entre la geometrie et la vitesse des marches aleatoires sur des
graphs.