Journal of Theoretieal Probability, Vol. 4, No. 1, 1991
Recurrent Networks and a Theorem of
Nash-Williams
Scan M c G u i n n e s s 1
Received June 5, 1989; revised August 3, 1990
In this paper, we generalize a result of Nash-Williams concerning recurrence of
locally finite networks, by extending his result to networks with possibly vertices
of infinite degree.
KEY WORDS:
Random walks; recurrent networks; reversible Markov chains.
In Ref. 8, Nash-Williams gives necessary and sufficient conditions for an
infinite, locally finite network to be recurrent. He shows that a locally
finite network N is recurrent if we can partition the vertices of N into
finite sets, say, P0, P1, P 2 .... which have the properties that there are no
edges between Pi and Pj for all i and j such that li-jl ~>2, and
Z•=o c([Pi, Pj]) 1= oo where c([Pi, Pj]) denotes the sum of the conductances of the edges going between Pi and Pj. Furthermore, he proves a sort
of converse to this by showing that if N is recurrent, then we can subdivide
the edges of N and assign conductances to the edges of the subdivision in
such a way that we obtain a network which has a partition of its vertex set
as above. Among other things, this result translates into necessary and sufficient conditions for certain reversible Markov chains to be recurrent and
thus is of some importance to probabilists. For example, in Ref. 5 Griffeath
and Liggett use Nash-Williams' result to examine the relationship between
the critical values of the finite and infinite versions of Spitzer's reversible
nearest-particle systems. The main purpose of this paper is to give our
generalization of Nash-Williams' result by showing his result essentially
holds even when the assumption of local finiteness is dropped. First, we
extend the sufficiency part of Nash-Williams' result, Which improves upon
Work completed at the University of Waterloo and at the University of Milan. Present
address: Odense Universitet, Campusvej 55, Odense M, Denmark, D K 5230.
87
0894-9840/91/0100-0087506.50/0 9 1991 PlenumPublishingCorporation
88
McGuinness
similar results in Refs. 5 and 6. We show that a network N, locally finite
or not, is recurrent if we can partition its vertices into sets Po, P1, P2 .... as
before only with the condition that the Pi's be finite be replaced by the condition that for each Pi a random walk starting at a vertex of Po must eventually visit Pi (with probability 1). We also give a sort of converse to this
as Nash-Williams did in the locally finite case. The most important step in
the proof is finding a nonnegative, unbounded function on the vertices
which is subharmonic at all vertices but one where the function has value
0. Specifically, if N is recurrent and has transition matrix P, where N's vertices are enumerated as 0, 1..... we give a simple combinatorial construction
of an infinite vector v = (v~) where Vo = 0, v~-~ ~ , and v~~ (Pv)~ for all i r 0.
Nash-Williams did the same for locally finite networks by using a result of
Ref. 4, but this result does not seem to apply in general to networks which
are not locally finite.
It is well known that for finite networks the effective conductance
between two vertices is proportional to their so-called transmission probability (see Ref. 2 or 7). We prove this for certain infinite networks having
possibly infinite vertex degrees. In doing so, we extend Nash-Williams'
definition of a system and reprove some of his results in a more general
setting. We also discuss in the last section the work of Lyons (6) and
Griffeath and Liggett r pertaining to the sufficient conditions of NashWilliams' result.
1. T E R M I N O L O G Y A N D N O T A T I O N
A network is a pair N = [G, c] where G is a connected graph and c is
a positive, real-valued function defined on E(G). Here E(G) and V(G)
denote the set of edges and vertices of G, respectively. We refer to c(e)
as the conductance of e and for A c E ( G ) we let c(A)=~eEAc(e). We
denote the set of edges incident with a vertex v in G by iG(v), and we let
c(v) - e(iG(v)). We refer to the cardinality of iG(v) as the degree of v.
Throughout we shall assume N is the network [G, c]. We shall also
assume (unless stated otherwise) that c(v) is finite for all v. We allow the
degree of a vertex to be infinite, but we require that the set of vertices be
countable.
If e is an edge incident with a vertex v, then we let e\v denote the other
endvertex of e. In general, if X is any set of vertices and e is an edge having
exactly one end in X, then we let e \ X denote the other end of e which is
not in X. For sets of vertices X and Y, we let IX, Y] denote the set of edges
having one endvertex in X and the other in Y. When X = {u} and Y = {v},
we denote IX, Y] by [u, v], in which case e ~ [u, v] means u and v are the
endvertices of e.
Recurrent Networks and a Theorem of Nash-Williams
89
If V is a subset of vertices and we contract V into a single vertex v,
then we say that we identify V with a vertex v.
We let the set of points of 7/d also denote the network whose vertices
correspond to points in 77~, with the incidence relation that two vertices in
7/~ are joined by an edge with conductance 1 if their corresponding points
differ in exactly one coordinate by + 1. We refer to 7/d as the d-dimensional
grid.
2. R A N D O M
WALKS
In this section, we introduce the notion of a random walk on a
network and the property of recurrence. For a network N = [G, c] we
define a reversible M a r k o v chain whose state space is V(G) by letting the
transition probability Pu~ of stepping from u to v equal e([u, v])/c(u). Since
G is connected, this M a r k o v chain is irreducible. It should also be noted
that any irreducible, reversible M a r k o v chain with countable state space
can be realized as a M a r k o v chain defined on the vertices of a network.
To each finite walk Voeovlel".v,_le=_lv= on N we associate a
probability I-[7-~[c(ei)/e(v~)] representing the chance of taking such a
walk given that we start at v0. We define a random walk on N to be an
infinite walk, and we say that a vertex v of N is recurrent if any random
walk starting at v must return to v with probability one. Since the M a r k o v
chain associated with N is irreducible, it follows by a well-known result
that v is recurrent if and only if all the vertices of N are recurrent (see
Ref. 3, w
or Ref. 1, w
Thus we say that a network is recurrent if a
random walk must eventually return to its origin with probability one. A
network which is not recurrent is transient. For example, if N is finite, then
it follows that N is recurrent from the well-known fact that every finite,
irreducible M a r k o v chain is recurrent (see Ref. 1, w
Due to the close relationship that networks have with electrical
networks, we can translate methods for determining the electrical conductivity of an electric network into methods for determining the type of a
network; that is, whether a network is recurrent or transient. This was
done by Nash-Williams, (8) whose methods we discuss.
3. N A S H - W I L L I A M S '
THEOREM
A system is a quadruple S---[G, c,a, U] where a E V(G), and
U c V ( G ) - a. We shall always assume our systems are such that a random
walk starting at v must eventually arrive at a or U (with probability 1).
Associated with S is a probability function Ps defined on the vertices of S,
where ps(V) is the probability a random walk starting at v reaches a before
90
McGuinness
U. We have that ps(a)= 1 and ps(V)= 0 for all v E U. The transm&sion
probability of a system S is given by
1
z(S)=c(a)
~
(1-ps(e\a))c(e)
e ~ iG(a)
which is the probability a random walk starting at a will reach U before
a, after leaving a. We will show later that for certain subsets V~ = V2
V3... of vertices of V(G)-a, the network N is recurrent if and only if
limi~ co r ( S i ) = 0, where Si = [G, c, a, Vii, i = 1, 2, 3.....
Let N = {P~}i~ o be a partition of V(G) such that [ P ~ , P j ] = ~
for li--ill>2. We call N an N-constriction and denote the sum
ZF=oc([p~,p~+l]) 1 by ~bu(~ ). Here we take c([Pi, Pi+l]) 1 = 0 if
c([Pi, P~+~])=oo. We note that [Pe, P i + l ] r
since G is connected.
For example, let N=77 2 and P~={(x,y);x= +_i}, i = 0 , 1 , 2 , . . . . Then
~ = {p ~}i=o is an N-constriction for which c([Pi, P~+I]) = 0o for all i and
thus ~bz2(~ ) = 0.
Suppose N = [G, c] is locally finite (all vertices of finite degree) and
let S = [G, c, a, U] where O is finite. Let Cs = c(a) z(S). In terms of electric
networks, Cs represents the effective conductance between a and U
(identified as a single vertex) in the network where each edge e represents
a conductor with conductance c(e) (see Ref. 2). We thus refer to Cs as
the effective conductance of S. Now suppose that ~ = {P~}F~o is an
N-constriction where a ~ Po, and P~ is finite for all i. For i = 1, 2, 3 .... let
Si= [G,c,a, ~)j~=IP~]. We will show later that N is recurrent
if and only if lim~o~ Cs,=O. Nash-Williams (8) showed that Cs~<~
i 1 c([P~, PI+ i]) -1) 1, a result which immediately follows by applying
(Y~t=0
the shorting law for effective conductance (see Ref. 2). Thus we have
limi~ ~ C S i ~ N ( ~ )
- 1 . T h u s if ~ N ( ~ ) = (30, then l i m ~ ~ Cs,=O and N is
recurrent.
Proposition 1. (Nash-Williams. (8)) Let N = [G, c] be a locally finite
network. If there exists an N-constriction ~' whose sets are finite and for
which ~bN(~)= ~ , then N is recurrent.
Using Proposition 1, the networks 7/1 and Z 2 can be easily shown to
be recurrent. For 7/1 let P i = {i, - i } , i = 1, 2 .... and let ~ - - {Pi}i~o. Then
~b~l(~)=Z=o~ 89
For 772 let Pi={(x,y);lxl+[yl=i}, i = 0 , 1 , 2 .....
Then ~bz2(~)= ~2p=o[1/(8i + 4)] = or. One might ask, if N is recurrent does
there exist an N-constriction ~ such that ~bN(~)= m? In general, the
answer is no. For example, suppose we view 773 as an infinite stack of
copies of g 2, the copies in the stack being indexed as .... - 2 , - 1, 0, 1, 2,....
Recurrent Networks and a Theorem of Nash-Williams
91
It is easy to see that we can subdivide the edges between the 7/2's in such
a way as to obtain a recurrent network. If we have a constriction N =
{Pi}F_o of such a network, and we assume (without loss of generality) that
Po contains a point in the zeroth 2 2 of the stack, then it is easy to see that
U~=oPj must contain at least O(i 2) vertices of the zeroth Z 2, and thus
I[Pi, P i + I ] [ is at least O(i 2) and Z~=o [[Pi, P i + 1 1 1 - 1 < oQ. However,
Nash-Williams (8~ showed that if N is locally finite, then by subdividing the
edges of N in a certain way and assigning new conductances, one obtains
a network M = [-H, d] which is of the same type as N and which has an
M-constriction N such that ~b~(N) = Go. More specifically, M = [H, d] is a
refinement of N if H is a subdivision of G and d is such that if e s E(G) is
subdivided by ej ' e 2 ..... e, e E ( H ) , then 52"i = 1 d(ei) 1= c(e)-l.
It is well known that for an electric network consisting of conductors
connected in series, that the effective resistance (reciprocal of effective conductance) between its endpoints is obtained by simply adding the resistance
(reciprocal of conductance) of each conductor. Thus the condition that
Y~'/~ 1 d(ei) -1= c(e)-1 ensures that the effective resistance of the subdivided
edge remains the same. It will become evident later on when we establish
the connexion between recurrence and effective conductance for networks,
locally finite or not, that this is enough to establish that a network and any
refinement of it are of the same type. Denoting the set of refinements of N
by N(N), we give Nash-Williams' main result of Ref. 8.
T h e o r e m 1. (Nash-Williams. (8)) A locally finite network N is
recurrent if and only if there exists M ~ ( N )
such that M has an
M-constriction ~ whose sets are finite and for which ~bM(~) -- oQ.
In Theorem 2 below, we prove a more general version of Theorem 1
where N may or may not be locally finite. Let a be a vertex of G let
U~ V(G)-a. We define zN(u) to be the probability that a random walk
in N starting at a visits a vertex in U.
T h e o r e m 2. Let N = [G, c] and let a ~ V(G). Then N is recurrent if
and only if there exists M e N ( N ) having an M-constriction N = {Pi}F=o
such that a e Po, ~ff(Pi) -- 1 for all i, and ~bM(N) = oo.
The assumption that zff(P~) = 1 for all i is indispensable in Theorem 2,
for suppose N is the network constructed by attaching a binary tree to each
vertex of Z 1. Here a binary tree is an infinite tree rooted at a vertex of
degree two such that each nonroot vertex has degree 3 and the conductance of each edge equals 1. Such a tree is easily seen to be transient (see
Ref. 2 or 7), and thus it is clear that N is also transient. If for i = 0, 1, 2 ....
we let Pi be the set of vertices of the binary trees attached to vertices i and
McGuinness
92
- i in Z 1, then N = {Pi}~0 is an N-constriction for which ~ N ( ~ 0 ) --- 0 0 .
However, this does not contradict T h e o r e m 2 since zN(P~)<I for
i = 1 , 2 .....
Theorem 2 is important in the following sense. A random walk on a
network corresponds in a one-to-one fashion with an irreducible, reversible
Markov chain having a countable state space. Nash-Williams' result
(Theorem 1) applies only to locally finite networks and thus is only
applicable to the corresponding restricted class of reversible Markov
chains. However, Theorem 2 applies to all networks and hence to all
irreducible reversible Markov chains, and can be thought of as giving an
equivalent condition for recurrence for such Markov chains. There may be
many instances where a network is not locally finite and where Theorem 2
becomes useful: for example, random walks on groups with infinitely many
generators. As another example, consider the network which is Z 1 with
additional edges (n, 2n), (n, 2n + 1), (n, 2n + 2),... for each n > 0, such that
(n, 2n + j ) has conductance (n + j ) - 2 - ~ , where 6 > 0. Then such a network
can be shown to be recurrent using Theorem 2. As mentioned at the start
of this paper, Griffeath and Liggett <5) use the sufficiency of Nash-Williams'
criterion to examine the relationships between the critical values of the
finite and infinite versions of Spitzer's reversible particle systems. In fact,
they first provide a reworking of the sufficiency of Nash-Williams' result,
generalizing it in a similar way as is done in Theorem 2. Then they use the
more general version. Thus it would seem that Nash-Williams' result could
have many applications in probability theory and stochastic processes, and
that the development of a more general theorem is needed in this regard.
One aspect of the proof of T h e o r e m 2 is that it involves a combinatorial construction of an infinite vector v = (v~), where Vo= 0, v~ --* oe
and (Pv)i>~v~ for i C0, P being the transition matrix of a recurrent,
reversible Markov chain. Our construction is attractive because of its
simplicity and thus may have further applications in probability theory or
possibly in potential theory.
4. T H E E F F E C T I V E C O N D U C T A N C E O F A SYSTEM
Let S = [G, c, a, U] be a system. A finite partition ~ =
{pi}n_ 0 of
V(G) is called an S-constriction if aePo, Uc-Pn, and [Pi, P i ] = ~ for
li-jl >/2. We denote the sum ~."-lc([Pi,
Pi+l])-I by ~bs(~) which
i~0
represents the effective resistance between the endpoints of a parallel-series
network corresponding to the network N where P0, P1 ..... Pn are identified
with single nodes. We mentioned earlier that Nash-Williams showed for
locally finite S that Cs <~O~s(~)-1. In this section, we extend the notion of
Recurrent Networks and a Theorem of Nash-Williams
93
effective conductance to infinite systems and show that the above inequality
still holds.
For an infinite system S = [G, c, a, U] we define the effective conductance of S to be Cs= supr Cr, where the supremum is taken over all
systems T = [H, c, a, U'], U' being a finite nonempty subset of U, and H
being a finite, connected subgraph of G containing a and U'. This
corresponds to the definition given in Ref. 10. It follows easily from this
definition of effective conductance of S and the above result of NashWilliams for locally finite S, that if ~ is an S-constriction, then C s <~
~bs(~) -a. As in the locally finite case, we also have Cs= c(a)z(S). To see
this we reason as follows. Let G1 - G2 - G 3 ~- . . . be an infinite sequence of
connected graphs each containing a, and also having the properties that
E(G)=U~IE(G~), and U~=V(Ge)c~U#ffJ for all i. For i=1,2,.., let
Se= [Gi, % a, U~] where c~(e)= c(e) for all e 9
Then we have C s =
q(a) ~(S~), and lim~_. ~o Cs, = Cs. It remains to show that limi_. ~ z(Si) =
v(S). For this we use the fact that limi_ ~ ps~(V) = ps(v) for all v (see the
appendix) and applying Fatou's Lemma (Ref. 9, p. 226), we obtain that
limi_. ~ r(Si) = ~(S).
Proposition 2. Let S = [G, c,a, U] be a system and let ~ be an
S-constriction. Then c(a) ~(S) = Cs <~Os(~)
5. A G E N E R A L I Z A T I O N
OF NASH-WILLIAMS' THEOREM
Our main objective in this section is to prove Theorem 2. Let a 9 V(G)
p oo
and suppose N = { /}~=o is an N-constriction such that aePo and
r N ( P i ) = I for all i. For k = 1 , 2 , 3 , . . , let Sk = [G,c,a, Uj=k
~ e j]. If G is
locally finite and P,. is finite for all i, then it easily follows that N is
recurrent if and only if lim~_ ~ z(Si)= 0. In Lemma 1 below, we prove a
more general result for all networks.
For a system S = [G, c, a, U] let zj(S) be the probability that a random walk starting at a visits U for the first time in j steps, without visiting
a in between. Then z ( S ) = Zj~> lzj(S). For F~_ V ( G ) - a, let r(j, F) denote
the probability a random walk starting at a visits F in fewer than j steps.
L e m m a 1. Let N = [G, c] and let U o _ U1 --- U 2 ~ - - . be a sequence
of sets of vertices such that a E U o , z N ( u i ) = I for all i and
limi_oo r(i, U i ) = 0 . For i = 0 , 1,2,... let S i = I-G,c,a, Ue]. Then N is
recurrent if and only if limi~ ~ z(S~) = O.
Proof Let Pn be the probability that a random walk starting at a
does not return to a in n or fewer steps. Then the probability that a
94
McGuinness
r a n d o m walk starting at a never returns to a equals l i m , + co P~. We also
have ~j~>, zj(S,) ~< p , , and therefore
z(S,,) = ~ rj(S~) <~ ~ rj(S,) + p, <~r(n, U,) + p,
j>/l
j<n
Thus if N is recurrent, then lim,~o~ r ( S ~ ) = 0 . Suppose instead that
l i m , ~ co r ( S , ) = 0. Since r X ( u i ) = 1 for all i, it follows that a r a n d o m walk
starting at a which does not return to a must visit U~ (with probability 1)
for all i. Thus lira, ~ o~ P, <~r(S~) for all i, and hence lim~ ~ co p~ = 0 and N
is recurrent.
[]
F o r Si in L e m m a 1 it now follows that N is recurrent if and only if
l i m , ~ oo Cs~ = 0. We will now show that for any network there exists a
sequence of sets { U~} ~ o as in L e m m a 1, with the additional p r o p e r t y that
U~ is finite for all i.
L e m m a 2. Let N = [ G , c ] and let a~ V(G). Then there exists an
infinite sequence of sets of vertices U1 c U2 c ... such that a e O1, 0 i is
finite for all i, and limi+ o~ r(i, Ui)= O.
Proof We assume that for all v E V(G) the set ia(v) is enumerated in
some order. F o r each v E V(G) and 0 < fl < 1, if ia(v) is enumerated as
el, e2,..., we let fie(v)= {ej\v;j= 1, 2 ..... nB} where
nt~ = min Y.7=1 c(ei)
.~z+
c(v)
>l~
F o r Y c V(G) let 6B(Y)= 0~+ y~5~(v).
We construct the sets U~, i = 1, 2 ..... as follows. Let {s i}i=
+
be a
sequence of positive terms converging to zero such that 1 > s~ > s2 > ....
F o r i = 1 , 2 , . . . , let 1 - s ~ / i < f l i < l . F o r i=1,2,..., let ft V j~+
i J j = o' be sets of
vertices defined by V ~
and V{=6~'(V~ 1), j = l , 2 ..... N o w let
Ui.= U /i=- 1o Vi,
j i = 1, 2,..., and let p{ be the probability a r a n d o m walk starting at a first reaches U~ in exactly j steps (i-- 1, 2 , . . . ; j = 0, 1, 2 ..... i - 1). By
the construction of Ui, p~ < 1 -fl+. Thus r( i, U~) = Z / ~-1
= o P: < i(1 -- fli) < si,
and therefore limz ~ o~ r(i, U~) = 0. Since 6:(v) is finite for all 0 < fl < 1 and
v ~ V(G), it follows that 0~ is finite for all i. We note that the sTs can be
chosen so that U~ c U2 c ....
We are now ready to prove T h e o r e m 2.
Proof of Theorem 2. Sufficiency is easy. Suppose there exists
p
ov
M~(N)
having M-constriction ~ = { i}~=o such that a~Po, rY(P~) l
for all i, and ~bM(~) = oo. By the discussion in Section 4 and the c o m m e n t s
following L e m m a 1, it follows that M is recurrent and thus N is recurrent.
Recurrent Networks and a Theorem of Nash-Williams
95
Proving the necessity of T h e o r e m 2 is m o r e difficult. In Ref. 8, N a s h Williams used a result of Foster (4) to show that if N is locally finite and
recurrent, then there exists x: V(G) ~ N + u {0} such that
i. x(a)=O.
~ , i.e., {v;x(v)<e} is finite for all e > 0 .
ii.
x~
iii.
•e ~iG(x)(X(v)-- x(e\v)) c(e) >~0 for all v # a.
We shall also find such an x, only we avoid Foster's complicated result
which m a y be difficult to use when the assumption of local finiteness is
dropped. F o r example, if N is such that there exists vertices vl, v2,.., such
that limi~ ~ Pv, a = 1, then (8) of Ref. 4 will not be satisfied.
Suppose N is recurrent. By L e m m a 2 we can find a sequence of sets of
vertices UI ~ U2 ~ --- for which a ~ U1, O~ is finite for all i, and lim i~ co
r(i, U i ) = 0 . Since U~ is finite for all i, z N ( u ~ ) = I for all i and v. F o r
i = 1, 2 .... let S~= [G, c, a, Ug]. Since N is recurrent, it follows from
L e m m a 1 that limi~ ~ z(S~)= 0. Thus we can find a sequence {ik}k~=0 such
that z(Sek ) ~<k 3 for k = 1, 2 ..... Let Q0 = Ui0, and for j = 1, 2,... let Q j =
Oij-U~j_~. F o r v~ V(G), let h(v) be the expected highest k for which a
r a n d o m walk starting at v would visit Qk before visiting a. Since N is
recurrent, all r a n d o m walks must eventually visit a (with probability 1).
We have h(a) = 0 and h(v) >>.k for all v ~ Qk. Also, by the choice of the U;s,
Qk is finite and n o n e m p t y for all k and thus h ~ oo. It can also be seen that
for k = 0 , 1 , 2 , . . , if vEQk and v~a, then h(v)=max{k,~e~c(~)h(e\v )
[c(e)/c(v)] }. Thus for all v # a, ~ic(~)(h(v)- h(e\v)) c(e) >~O. We will
now show that h(v)< ~ for all v. Let v ~ V ( G ) - a and let qk(v) be the
probability a r a n d o m walk starting at v has Qk as the highest set visited
before reaching a. Let p be the probability a r a n d o m walk starting at a
visits v before returning to a. Since G is connected, p > 0. We also have
~(S~) >>,pqk(v), and thus for k = 1, 2 ..... qk(v) <~z(Si~)/p < 1/pk 3. Therefore,
h(v)= ~
k=O
kqk(v)<~ p-1 ~ k - 2 < o(3
k=l
Using h we can n o w construct our desired refinement of N, the construction of which follows that of Nash-Williams. (s) Since h ~ ~ , we can
assume the values realized by h are 0 = ho < hi < h2 < .--. F o r all e ~ E(G),
do the following. If e E [-u, v] where h(u)= hi, h(v)= hi, and j - i = k > 1,
then subdivide e by a path of length k, say Uoeou~e~...uk_ ~ek_ ~u~, where
Uo = u and uk = v. F o r l = 0, 1, 2 ..... k - 1 let
h j - hi
t
d(et)=\hi+-~+ i---hi+i/ c(e)
96
McGuinness
Extend h to the new vertices /,/1, u2
u k ~ by letting h(u~)=he+~. If
j - i ~ < 1, do not subdivide e but let d ( e ) = e(e). The resulting network is a
refinement of N, and by the way in which we extended h to V(H), we have
.....
(h(v) - h(e\v)) c(e) =
e e ia(v )
~,
( h ( v ) - h(e\v)) d(e)
for all v ~ V(G)
e e ill(v)
}-~'ee i11(v)(h(v)--h(e\v))d(e)=
0 for all v e V ( H ) - V(G).
We will now construct the desired M-constriction. Let ~ = {p i}i=o
where Pi = {v E V(H); h(v)= hi}. The collection N is easily seen to be an
M-constriction. Recall that the sets Ui, i-- 1, 2 .... were such that zN(ui)= 1
for all i. By the comments at the end of Section 3, we have zM(Ui)= 1 for
all i. If we choose j large enough so that Pi c V(H) - Uj, then any random
walk visiting Uj must first visit Pi. It follows from this that vY(Pi) = 1 for
all i. We observe that the vertices u0, ul,..., uk for the subdivided edge
above are such that
and
(h(u,) - h(u,+ ,)) d(e,) = ( h i - hi) c(e) = (h(u) - h(v)) c(e)
Using this observation we see that
(h(e\P i + ~) -- h(e\Pi) ) d(e)
E
ee [Pi,Pi+l]
e~E(H)
=
E
(h(e\ Yi) - h(e\ Ye)) c(e) <
e~E(G)
where IT,.= (U j=0 Pj)c~ V(G).
The latter sum is finite since Yi is finite, and hence both sums
are absolutely convergent since h(e\Pi+ 1 ) - h(e\Pi) = h l - hi+ 1 < O, and
h(e\ Y t ) - h(e\ Yi)< O. Thus by rearranging terms in the latter sum we have
( h ( e \ Y i ) - h ( e \ Y i ) ) c(e)= ~
e ~ [ Y i , Yi]
e~E(G)
~
( h ( v ) - h ( e \ Y i ) ) c(e)
VE Yi e E [ v , Pi]
= ~
~,, ( h ( v ) - h ( e \ v ) ) c(e)
v E Yi e ~ i G ( v )
Since Ze~ia(v)(h(v) - h(e\v)) c(e) >~0 for all v ~ a, it follows that
(k(e\Yi)-h(e\Yi))c(e)>~
e ~ [ Yi, Yi]
e ~ E(G)
~
(h(a)-h(e\a))c(e)
e E iG(a)
= -
~
e ~ iG(a)
h(e\a) c(e)
Recurrent Networks and a Theorem of Nash-Williams
97
Let 6 = ~-'~-eceG(a)h(e\a) c(e). Then
(hi- hi+ ~) d([Pi, Pi+ l) ] =
(h(e\Pi+ l) - h(e\Pi)) d(e) >~ - 6
e~ [Pi,Pi+l]
e ~ E(H)
and d([P i, P i + l ] ) ~ -6/(hi+l-hi) for i = 0 , 1, 2,.... Therefore
~bM(~)= ~ d([Pi, Pi+l])-l>~ ~ hi+~--hi-6-1 lim hi = ~
i=0
i=0
[~
i~oo
6. OTHER RESULTS
Lyons (6) and Griffeath and Liggett (5) investigated, among other things,
sufficient conditions for a reversible Markov chain with countable state
space to be recurrent. Their results, translated in terms of networks, extend
Proposition 1 by allowing infinite vertex degrees. We will rephrase this
work in terms of networks and show that the sufficiency of Theorem 2 is
a stronger result.
In Ref. 5, Griffeath and Liggett show that a well-known result from
analysis called Dirichlet's principle has an interesting analog for reversible
Markov chains. We sidetrack here and give an equivalent formulation of
Dirichlet's principle for networks.
A function 0:~2 __. ~ is harmonic in a region R if it satisfies Laplace's
equation
~2 0
A20 = ~
(~20
+ dy----~= 0
at every point inside R. Suppose we have the following boundary value
problem. Given a closed region R of R 2 with boundary dR, and a continuous, real-valued function g defined on ~R, does there exist a harmonic
function defined on R which equals g on dR? This is known as Dirichlet's
problem for the region R. For certain restrictions to R, a function 0 which
is harmonic in R and takes on certain prescribed values on the boundary
of R, is unique. Furthermore, if R is a bounded domain, then among those
continuously differentiable functions taking on prescribed values on the
boundary of R, the integral
2 ~h2
~x+-~ydxdy
ffROh
is minimized by a harmonic function on R. This is known as Dirichlet's
principle (see Ref. 5). Dirichlet's principle for reversible Markov chains with
860/4/1-7
98
McGuinness
countable state space is given by Griffeath and Liggett, (5) and we give its
equivalent formulation in terms of systems.
For a network N = [G, c] and 0: V(G) ~ R, let qsN(0 ) denote the sum
Z .... v~a) Y ~ E~,~l(O(u) -- O(v)) 2 c(e). For S = [G, c, a, U] and v ~ V(G) let
f pss ( v ) = ~,~ie~)(ps(v) -- ps(e\v)) c(e).
Proposition 3. (Dirichlet's principle.) Let S = I-G, c, a, U] and let
F = {0: V ( G ) ~ E; 0 ( a ) = 1 and 0 ( u ) = 0 for all u~ U}. Then
f~s(a) = ~s(Ps)
Proof
and
qSs(Ps ) = min ~s(O)
O~F
See Ref. 5, Theorem 2.1.
[]
Proposition 3 is an extension to infinite systems of Thomson's principle for finite systems (see Ref. 2 or Ref. 10, Proposition 2.1). Griffeath and
Liggett ~5) used Proposition 3 to derive a sufficient condition for a reversible
Markov chain with countable state space to be recurrent. In terms of
networks, their result extends Proposition 1 by allowing infinite vertex
degrees. We state their result in terms of networks.
oO
Theorem 3. Let N = [G,c], a6 V(G), and let ~ = {p i}/=0
be an
N-constriction such that a ~ Po, rU(p~)= 1 for all i, and ~ p ~ c(v)< ~ for
all i. For i = 1 , 2 .... let S~=[G,c,a, Oj~=~Pj]. Then we have Cs~<~
~- ' c(I- P j, Pj+I"])-I] -1, and N is recurrent if ~ U ( ~ ) = (30.
[SZj:0
Proof
See Ref. 5, Theorem 2.10.
[]
We note that the sufficiency of Theorem 2 does not require that
Z ~ e , c(v) be finite for all i, and thus is a stronger result than Theorem 3.
In fact, we can show that if ~ e , c(v) is finite for all i, then r~'(Pi)= 1 for
all i. For this we use the following lemma, which is an extension of the fact
that every finite network is recurrent.
Lemma 3.
Let N = [G, c]. If c(E(G)) is finite, then N is recurrent.
Proof Let a ~ V(G). By Lemma 2 there exists a sequence of cofinite
subsets of V(G) - - a, say U I = U 2 = ..., such that V ( G ) = 0 i =ool Ui and
lim;~o~ r(i, Ue)=0. For i = 1,2 .... let S i = [G,c,a, Ui], and let S * =
[G*, c*, a, U~] be the system obtained from S~ by identifying the vertices
of G ~ - a with a single vertex. Since 0~= ~ Ui-- V(G) and c(E(G)) is finite,
it follows that l i m ~ c([Ui, U i ] ) = 0 , and hence l i m i ~ c*(a)z(S*)=
l i m ~ o~ Cst = 0. We have from the monotonicity law for effective conductance (see Ref. 2 or 10) that Cs, <~Cs* for all i. Thus l i m ~ ~ Cs, = 0, and
therefore N i s recurrent (by the discussion after the proof of Lemma 1). []
Recurrent Networks and a Theorem of Nash-Williams
99
Let N and N be as in Theorem 3 except that we drop the condition
that zU(p~)= 1 for all i. Let Ni = [G~, c~] be the network obtained from N
by identifying 0 j = ; P ~ with a single vertex b~. Since Y'.j=oZ,~pjc(v) is
finite, it follows that ci(E(G~)) is finite, and thus N~ is recurrent by
Lemma 3. Therefore, a random walk in N~, starting at a, must eventually
reach b~ (with probability 1) and therefore vN(P~)= 1.
In Ref. 6, Lyons proved Theorem 3 (in terms of Markov chains) in the
case where the condition ZNa(p~)= 1 for all i is dropped. As we have shown
above, this result can be viewed as an application of the sufficiency of
Theorem 2.
APPENDIX
Let N = [ G , c ] and let S = [ G , c , a , U]. Recall that ps(v) is the
probability that a random walk starting at v reaches a before U, and vN(u)
is the probability that a random walk starting at v eventually reaches U.
Following the definition of a system, we assume that ~N(u) = 1 for all vertices v. Let G t _~ G 2 _~ . . . be connected subgraphs of G such that a E V(Gi)
for all i, Ui= V(Gi)c~ U r
and (J~=l E(G~)=E(G). For i = 1, 2,... let
N~= [G~, c~] and Se= [Gi, c~, a, U~] where G(e)=c(e) for all e~E(Gi).
Theorem.
limi~oo ps~(V)=ps(v ) for all v.
Proof For i = l, 2 .... and j = 0, 1, 2,... let pJsi(v) be the probability that
a random walk in Ni starting at v reaches a before U in exactly j steps. Let
pJs(V) be the corresponding probability for N. Then Ps( /) ) --- Z joo= o Ys(v) and
for i = 1, 2 ..... Psi(v) = Y'-j~o PJs,(v)9 The following is easily seen to be true:
(*)
Let W be any collection of finite walks of N whose lengths are
bounded above. Let Wf be those walks of W residing in N~. Let
p be the probability of taking a walk in W, and let p~ be the
corresponding probability for W~ in N~. Then
lim p; = p
i~OO
Thus by (*) it follows that for j = 1, 2.... and for all v
lim
i--+ GO
J
- p (v)
psi(v)-
Hence from Fatou's lemma (Ref. 9, p. 226), it follows that
p~(v)~<liminf ~. pJs,(v)
j=0
i
j=0
Ps(V) <<.lim.inf psi(V)
100
McGuinness
Let 0 < e < 1. Since r N ( u ) = 1, there exists l ~ N such t h a t the p r o b a b i l i t y a
r a n d o m walk in N s t a r t i n g at v reaches U or a in fewer t h a n l steps, is
g r e a t e r t h a n l - e / 2 . By ( , ) , we c a n choose I ~ N large e n o u g h such t h a t for
all i > I the p r o b a b i l i t y t h a t a r a n d o m walk in Ni s t a r t i n g at v reaches a o r
U in fewer t h a n l steps, is greater t h a n 1 - e/2 - ~/2 = 1 - ~. T h u s it follows
t h a t for all i > / , Z j = l
< e. T h u s
lim sup p s i ( v ) = l i m s u p
i
i
~ p~i(v)=limsup
j=O
l
~< lira s u p ~
i>I
i>l
1
~
pSsi(V)
j--O
t--1
Ps + g = Z Pss(v) + e
j=0
j=0
Since l t e n d s to infinity as e t e n d s to zero, we have
lira sup psi(V) <~ ~ pSs(v ) = ps(V)
i
j=0
T h u s we have
lim sup psi(V) <~ps(v) <<.lim inf psi(V)
i
i
It n o w follows t h a t
l i m i s u p psi(V) = l i m i n f psi(V) = ps(V)
a n d l i m i ~ ~ p s , ( V ) = ps(V).
[]
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