Abstract
We completely characterize when the free effective resistance of an infinite graph whose vertices have finite degrees can be expressed in terms of simple hitting probabilities of the random walk on the graph.
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1 Introduction
We consider undirected, connected graphs with no multiple edges and no self-loops. Each edge (x, y) is given a positive weight c(x, y). A possible interpretation is that (x, y) is a resistor with resistance 1/c(x, y). The graph then becomes an electrical network.
More precisely, a (weighted) graph \(G = (V,c)\) consists of an at most countable set of vertices V and a weight function \(c : V \times V \rightarrow {\mathbb {R}}_{\ge 0}\) such that c is symmetric and for all \(x \in V\), we have \(c(x,x) = 0\) and
We think of two vertices \(x,y \in V\) as being adjacent if \(c(x,y) > 0\). For \(x \in V\), let \(N(x) := \left\{ y \in V ~ | ~ c(x,y) > 0 \right\} \) be the set of neighbors of x. Throughout this work, we assume that every vertex has finite degree in G, i.e., \(|N(x)| < \infty \) for every \(x \in V\).
For \(x \in V\), let \(\mathbb {P}_x\) be the random walk on G starting at x. It is the Markov chain defined by the transition matrix
and initial distribution \(\delta _x\). We will think of \(\mathbb {P}_x\) as a probability measure on \(\Omega = V^{{\mathbb {N}}_0}\) equipped with the \(\sigma \)-algebra \((2^V)^{\otimes {\mathbb {N}}_0}\). If not explicitly stated otherwise, we will from now on assume that every occurring graph is connected. In that case, \(\mathbb {P}_x\) is irreducible.
For a set of vertices \(A \subseteq V\) and \(\omega = (\omega _k)_{k \in {\mathbb {N}}_0} \in \Omega \), let
be hitting times of A. For \(x \in V\), we use the shorthand notation \(\tau _{\left\{ x \right\} } =: \tau _x\).
Suppose that G is finite. Ohm’s Law states that the effective resistance R(x, y) between two vertices x, y is the voltage drop needed to induce an electrical current of exactly 1 ampere from x to y.
The relationship between electrical currents and the random walk of G has been studied intensively [3, 5,6,7]. For finite graphs, \(x \ne y\), one has the following probabilistic representations
Note that \((c_z)_{z \in V}\) is an invariant measure of p. A proof of the first equality in the unweighted case can be found in [7] and can be extended to fit our more general context. To see that (1.1) equals (1.2), realize that \(\sum _{k=0}^{\tau _y-1} \mathbbm {1}_x(\omega _k)\) is geometrically distributed with parameter \(\mathbb {P}_x[\tau _x^+ < \tau _y]\). For the last equality, use that any finite graph is recurrent and thus \(\mathbb {P}_x[\tau _x^+ =\tau _y = \infty ] = 0\).
The subject of effective resistances gets much more complicated on infinite graphs since those may admit multiple different notions of effective resistances. Recurrent graphs, however, have a property which is often referred to as unique currents [6] and consequently also have one unique effective resistance. In this case, the above representation holds [1, 8]. Indeed, [1, Lemma 2.61] states the more general inequalities
for the free effective resistance \(R^F\) (see Sect. 2) of any infinite graph whose vertices have only finitely many neighbors. For the convenience of the reader, we include a proof of the result, see Lemma 2.4.
In [5, Corollaries 3.13 and 3.15], it is suggested that one seems to have
on all transient graphs. However, this is false as our example in Sect. 3 shows.
The main result of this work (Corollary 6.3) states that the free effective resistance of a transient graph \(G = (V,c)\) admits the representation (1.5) for all \(x,y \in V\) if and only if G is a subgraph of an infinite line. Corollary 6.5 states that the lower bound in (1.4) is attained if and only if G is recurrent.
2 Free Effective Resistance
Let \(G = (V,c)\) be an infinite connected graph. For any \(W \subseteq V\), let \(G \mathord \restriction _W := (W, c \mathord \restriction _{W \times W})\) be the subgraph of G induced by W. We say a sequence \((V_n)_{n \in {\mathbb {N}}}\) of subsets of V is a finite exhaustion of V if \(|V_n|< \infty \), \(V_n \subseteq V_{n+1}\) and \(V = \cup _{n \in {\mathbb {N}}} V_n\). Define \(G_n = (V_n, c_n):= G\mathord \restriction _{V_n}\).
Definition 2.1
Let \((V_n)_{n \in {\mathbb {N}}}\) be any finite exhaustion of V such that \(G_n\) is connected. For \(x,y \in V\), the free effective resistance \(R^F(x,y)\) of G is defined by
Remark 2.2
The fact that \(R_{G_n}(x,y)\) converges with the limit independent of a choice of \((V_n)_{n \in {\mathbb {N}}}\) is due to Rayleigh’s monotonicity principle (see, e.g., [2, 4]).
We denote by \(\mathbb {P}^n_x\) the random walk on \(G_n\) starting at x with transition matrix \(p_n\) associated with \(c_n\). Since we can extend it to a function on V by defining \(p_n(x,y) = 0\) whenever \(x \notin V_n\) or \(y \notin V_n\), \(\mathbb {P}^n_x\) is a probability measure on \(\Omega = V^{{\mathbb {N}}_0}\) for each \(x \in V_n\) and we have
for all \(x,y \in V_n\).
Remark 2.3
Note that for any \(x,y \in V_n\),
Lemma 2.4
([1, Lemma 2.61]). Let \(G = (V,c)\) be an infinite, connected graph such that \(|N(x)| < \infty \) for all \(x \in V\) and let \(R^F\) be the free effective resistance of G. Then,
holds for all \(x,y \in V\) with \(x \ne y\).
Proof
For any \(v \in V\), we have \((c_n)_v \rightarrow c_v\) as \(n \rightarrow \infty \) and thus \(p_n(v,w) \rightarrow p(v,w)\) for all \(v,w \in V\). By the definition of \(R^F\) and (1.3), we know that
In particular, \(\lim _{n \rightarrow \infty } \mathbb {P}^n_x[\tau _y < \tau _x^+]\) exists. Hence, the claim is equivalent to
Consider the discrete topology on V and its product topology on \(\Omega = V^{{\mathbb {N}}_0}\). Since \(p_n \rightarrow p\) point-wise and \(\left\{ y \in V_n ~ | ~ c_n(x,y) > 0 \right\} \subset N(x)\) and \(|N(x)|<\infty \) for all \(n \in {\mathbb {N}}\) and all \(x \in V_n\), it follows that \((\mathbb {P}^n_x)_{n \in {\mathbb {N}}}\) converges weakly to \(\mathbb {P}_x\). In the product topology, the sets \(\left\{ \omega \in \Omega ~ | ~ \tau _y(\omega ) < \tau _x^+(\omega ) \right\} \) and \(\left\{ \omega \in \Omega ~ | ~ \tau _y(\omega ) \le \tau _x^+(\omega ) \right\} \) are open and closed, respectively. By the Portmanteau theorem, it follows that
and
\(\square \)
In view of (2.1), equation (1.5) holds if and only if
Analogously, the lower bound of (1.4) is attained if and only if
3 The Transient \(\mathcal {T}\)
We will now show that (1.5) does not hold in general. Consider the graph \(\mathcal {T}\) shown in Fig. 1. It is transient and we have \(R^F(B,T) = 2\). However,
Due to the symmetry of \(\mathcal {T}\) we have \( \mathbb {P}_0[\tau _B< \tau _T] = \mathbb {P}_0[\tau _T < \tau _B]\). Together with the transience of \(\mathcal {T}\), this implies
and
More precisely, one can compute
Hence,
and
4 Probability of Paths
To check whether (2.2) holds, it is useful to write both sides as sums of probabilities of paths.
A sequence \(\gamma = (\gamma _0, \ldots , \gamma _n) \in V^{n+1}\) is called a path (of length n) in G if \(c(\gamma _k, \gamma _{k+1}) > 0\) for all \(k = 0, \ldots , n-1\). We denote by \(L(\gamma )\) the length of \(\gamma \) and by \(\Gamma _G\) the set of all paths in G. A path \(\gamma \) is called simple if it does not contain any vertex twice. The probability of \(\gamma \) with respect to \(\mathbb {P}_x\) is defined by
We say \(\gamma \) is \(x \rightarrow y\) if \(\gamma _0 = x, \gamma _{L(\gamma )} = y\) and \(\gamma _k \notin \{x,y\}\) for all \(k = 1,\ldots , L(\gamma ) - 1\). We denote by \(\Gamma _G(x,y)\) the set of all paths \(x \rightarrow y\) in G.
For \(A \subseteq V\), let
be the set of all paths \(x \rightarrow y\) in G that use only vertices in A.
Using this notion and \(\Gamma _{G_n}(x,y) = \Gamma _G(x,y;V_n)\), we see that (2.2) becomes
Since \(\Gamma _G(x,y;V_n)\) increases to \(\Gamma _G(x,y)\), this might look like an easy application of either the Monotone Convergence Theorem or the Dominated Convergence Theorem. However, neither is applicable since \(\mathbb {P}^n_x(\gamma )\) may be strictly greater than \(\mathbb {P}_x(\gamma )\).
To investigate when exactly (4.1) holds, we will introduce another random walk on V which can be considered an intermediary between \(\mathbb {P}^n_x\) and \(\mathbb {P}_x\).
5 Extended Finite Random Walk
The difference in the behavior of \(\mathbb {P}_x\) and \(\mathbb {P}^n_x\) occurs only when \(\mathbb {P}_x\) leaves \(V_n\). Instead, \(\mathbb {P}^n_x\) is basically reflected back to a vertex in \(V_n\). We will now construct an intermediary random walk which still has a finite state space, models the behavior of stepping out of \(V_n\) and has the same transition probabilities as \(\mathbb {P}_x\) in \(V_n\). This is done by adding boundary vertices to \(G_n\) wherever there is an edge from \(V_n\) to \(V \setminus V_n\).
For any set \(A \subseteq V\), let
be the inner boundary and \(\partial _o A := \partial _i (V \setminus A)\) be the outer boundary of A in G.
For any \(v \in \partial _i A\), let \(\overline{v}\) be a copy of v. Define \(\overline{G_n} = (\overline{V_n}, \overline{c_n})\) where
and \(\overline{c_n}\) is defined as follows. For \(x,y \in \overline{V_n}\), let
In particular, we have \((\overline{c_n})_x = c_x\) for all \(x \in V_n\). We denote by \(\overline{\mathbb {P}^n_x}\) the random walk on \(\overline{G_n}\) starting at x with transition matrix \(\overline{p_n}\) given by
Furthermore, let \(V_n^* := \overline{V_n} \setminus V_n\).
Example 5.1
Let G be the lattice \({\mathbb {Z}}^2\) with unit weights, see Fig. 2. Furthermore, let \(V_n := \left\{ -n\ldots ,0,\ldots ,n \right\} ^2\). \(G_1\) and \(\overline{G_1}\) are illustrated in Fig. 3. Note that \(c((1,1), \overline{(1,1)}) = 2\) since (1, 1) has two edges leaving \(V_1\) in G.
Lemma 5.2
(Relation of \(p_n, \overline{p_n}\) and p). For \(x,y \in V_n\) we have
For \(x,y \in V\) and \(m \in {\mathbb {N}}\) such that \(x,y \in V_m\), we have
Note that for \(n \in {\mathbb {N}}\) and \(x,y \in V_n\), we have
By Lemma 5.2, the following holds for all \(x,y \in V_n\).
The connection between \(\mathbb {P}_x^n(\gamma )\) and \(\overline{\mathbb {P}^n_x}(\gamma )\) is a bit more intricate. In order to investigate this connection, first consider what kind of paths exist in \(\overline{G_n}\). Let \(x,y \in V_n\), \(x \ne y\) and \(\overline{\gamma } \in \Gamma _{\overline{G_n}}(x,y)\) with \(L(\overline{\gamma }) \ge 2\). Then, by the definition of \(\overline{G_n}\), there exist \(l\in {\mathbb {N}}\) with \(l \ge 2\), \(v_1, \ldots , v_{l-1} \in V_n \setminus \left\{ x,y \right\} \) and \(k_1, \ldots , k_{l-1} \in {\mathbb {N}}_0\) such that \(k_j = 0\) for any \(j \in \left\{ 1, \ldots , l-1 \right\} \) with \(v_j \notin \partial _iV_n\) and
where \((v)_k := (v, \underbrace{\overline{v}, v, \ldots , \overline{v}, v}_{k \text { times}})\) for \((v,k) \in ((\partial _iV_n)\times {\mathbb {N}}_0) \cup ((V_n \setminus \partial _iV_n) \times \left\{ 0 \right\} )\). Note that the representation (5.1) is unique for \(\overline{\gamma }\) since \(G_n\) does not contain any self-loops.
Definition 5.3
For \(x,y \in V_n\), \(x \ne y\), let \(\pi : \Gamma _{\overline{G_n}}(x,y) \rightarrow \Gamma _{G_n}(x,y)\) be the projection of \(\Gamma _{\overline{G_n}}(x,y)\) onto \(\Gamma _{G_n}(x,y)\) which replaces all occurrences of \((v, \overline{v}, v)\) for any \(v \in \partial _i V_n\) by (v).
More precisely, let \(\overline{\gamma } \in \Gamma _{\overline{G_n}}(x,y)\). If \(L(\overline{\gamma }) = 1\), then \(\overline{\gamma } = (x,y)\) and we define \(\pi (\overline{\gamma }) := (x,y)\). If \(L(\overline{\gamma }) \ge 2\), it is of the form (5.1) and we define
Lemma 5.4
For all \(x,y \in V_n\), \(x \ne y\) and \(\gamma \in \Gamma _{G_n}(x,y)\), we have
Proof
For any \(v,w \in V_n\), we have
and, if \(v \in \partial _i V_n\),
For \( \gamma \in \Gamma _{G_n}(x,y)\) with \(L(\gamma ) = 1\), we have \(\gamma = (x,y)\). Since any \(\overline{\gamma } \in \pi ^{-1}(\gamma )\) visits x and y only once, \(\pi ^{-1}(\gamma ) = \left\{ \gamma \right\} \) holds and
Now let \(\gamma = (x,v_1, \ldots , v_{l-1}, y) \in \Gamma _{G_n}(x,y)\) with \(l=L(\gamma ) \ge 2\). We define
It follows that
and we compute
\(\square \)
Proposition 5.5
For \(x,y \in V_n\), \(x \ne y\), we have
Proof
Using
we compute
\(\square \)
Since we now have clarified the relation between \(\mathbb {P}^n_x\), \(\overline{\mathbb {P}^n_x}\) and \(\mathbb {P}_x\), we can return our attention to (2.2).
Proposition 5.6
For \(x,y \in V\), \(x \ne y\), we have
if and only if
Proof
We have
and
Since \(\Gamma _G(x,y;V_n) = \Gamma _{\overline{G_n}}(x,y;V_n)\) and \((c_n)_x \rightarrow c_x\), it follows that \(\mathbb {P}^n_x[\tau _y< \tau _x^+] \rightarrow \mathbb {P}_x[\tau _y < \tau _x^+]\) holds if and only if
This is the same as
\(\square \)
Using the same approach, we can also characterize when (2.3) holds.
Proposition 5.7
For \(x,y \in V\), \(x \ne y\), we have
if and only if
which in turn is equivalent to
Proof
Using (5.6) and (5.7) from the proof of Proposition 5.6, we have
and
provided either one of these two limits exists.
Hence, we have convergence as desired if and only if
On the other hand, we have
which implies the second claim. \(\square \)
Remark 5.8
An equivalent approach would be to consider a lazy random walk on \(G_n\) which has the same transition probabilities p(v, w) as \(\mathbb {P}_x\) for \(v \ne w\) but stays at v with probability
In that case the notion of “stepping out of \(V_n\)” would be modeled by staying at any vertex \(v \in V_n\).
6 Embedding \(\mathcal {T}\) into Transient Graphs
We will show that whenever a graph G is transient and not part of an infinite line, one can find a subgraph of G which is similar to \(\mathcal {T}\) from Sect. 3. We will also show that this is sufficient for (5.5) not to hold.
Proposition 6.1
Let G be a transient, connected graph which is not a subgraph of a line. Then, there exist \(x,y,z \in V\) such that \(x \ne y\), (x, z, y) is a path in G and
Proof
Since G is transient, it is infinite. If G is not a subgraph of a line, then there exists some \(z \in V\) with at least three adjacent vertices. Let F be a set of exactly three neighbors of z. Since G is transient and F is finite, there exists \(v \in \partial _o F\) such that
If \(v = z\), we can choose \(x,y \in F\), \(x \ne y\), and get
If \(v \ne z\), then there exists \(v' \in F\) such that \((z,v',v)\) is a path in G. Let \(x,y \in V\) be such that \(F = \left\{ x,y,v' \right\} \), see Fig. 4. It follows that
\(\square \)
Theorem 6.2
Let G be a transient, connected graph. Then,
holds if and only if G is a subgraph of an infinite line.
Proof
First, assume that G is a subgraph of an infinite line and let \(x,y \in V\), \(x \ne y\). Then, for any \(n \in {\mathbb {N}}\) sufficiently big, we have
i.e., there exists no path \(x \rightarrow y\) which leaves \(V_n\) before reaching y. Hence,
To prove the converse direction, suppose that G is not a subgraph of a line. By Proposition 6.1, we know that there exist distinct vertices \(x,y,z \in V\) such that (x, z, y) is a path in G and \(\mathbb {P}_z[\tau _x = \tau _y = \infty ] > 0\). Hence,
Without loss of generality assume that \(\limsup _{n \rightarrow \infty } \overline{\mathbb {P}_z^n}[\tau _{V_n^*}< \tau _y < \tau _x] > 0\). It follows that \(\limsup _{n \rightarrow \infty }\overline{\mathbb {P}_x^n}[\tau _{V_n^*}< \tau _y < \tau _x^+] > 0\) because for all \(n \in {\mathbb {N}}\) with \(\left\{ x,y,z \right\} \subseteq V_n\), we have
\(\square \)
Corollary 6.3
Let G be a transient, connected graph with \(|N(x)| < \infty \) for any \(x \in V\). Then,
holds for all \(x,y \in V\) with \(x \ne y\) if and only if G is a subgraph of an infinite line.
Proof
As seen in (2.2), the desired probabilistic representation (1.5) holds if and only if
By Proposition 5.6, this is equivalent to
and the claim follows by Theorem 6.2. \(\square \)
Theorem 6.4
Let G be an infinite, connected graph. If
holds, then G is recurrent.
Proof
By Proposition 5.7, we have
Suppose that G is transient and not a subgraph of a line. Using the same arguments as in the proof of Theorem 6.2, we see that there exist distinct vertices \(x,y,z \in V\) such that \((x,z,y) \in \Gamma _G(x,y)\) and
Since the same argument as in (6.1) yields
for all \(n \in {\mathbb {N}}\) with \(\left\{ x,y,z \right\} \subseteq V_n\), it follows from (6.2) that
which implies
However, we also have
for all \(n \in {\mathbb {N}}\) with \(\left\{ x,y,z \right\} \subseteq V_n\) by the same argument as in (6.1), and it follows that
which is a contradiction to (6.2).
Hence, if G is transient, then it must be a subgraph of a line. In this case,
follows for all \(x,y \in V\) with \(x\ne y\) by Theorem 6.2. Together with (6.2) and (5.10), this implies
for all \(x,y \in V\) with \(x\ne y\). However, this is a contradiction to the transience of G. \(\square \)
Corollary 6.5
Let G be an infinite, connected graph with \(|N(x)| < \infty \) for any \(x \in V\). Then,
holds for all \(x,y \in V\) with \(x \ne y\) if and only if G is recurrent.
Proof
If G is recurrent, we have \(\mathbb {P}_x[\tau _x^+ = \tau _y = \infty ] = 0\) for all \(x,y \in V\). Hence,
and (1.4) implies the claim.
If (6.3) holds for all \(x,y \in V\) with \(x \ne y\), then by (2.3) we have
for all \(x,y \in V\) with \(x \ne y\), and Proposition 5.7 and Theorem 6.4 imply the recurrence of G. \(\square \)
This shows that the lower bound in (1.4) is actually a strict inequality for some \(x,y \in V\) with \(x \ne y\) for any transient graph \(G = (V,c)\).
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Weihrauch, T., Bachmann, S. On the Probabilistic Representation of the Free Effective Resistance of Infinite Graphs. J Theor Probab 36, 1956–1971 (2023). https://doi.org/10.1007/s10959-022-01218-5
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DOI: https://doi.org/10.1007/s10959-022-01218-5