Uniform Spanning Trees of Planar Graphs

Let G be a finite connected graph. A spanning tree T of G is a connected subgraph of G that contains no cycles and such that every vertex of G is incident to at least one edge of T. The set of spanning trees of a given finite connected graph is obviously finite and hence we may draw one uniformly at random. This random tree is called the uniform spanning tree (UST) of G. This model was first studied by Kirchhoff who gave a formula for the number of spanning trees of a given graph and provided a beautiful connection with the theory of electric networks. In particular, he showed that the probability that a given edge {x, y} of G is contained in the UST equals \(\mathcal {R}_{\mathrm {eff}}(x \leftrightarrow y; G)\); we prove this fundamental formula in Sect. 7.2 (see Theorem 7.2).

formula mentioned earlier and the spatial Markov property for the UST. In Sect. 7.3 we prove Pemantle's [68] result (7.1) showing that FUSF G exists. We will also define there the wired uniform spanning forest which is obtained by taking a limit of the UST probability measures over exhaustions with wired boundary. We will also need some fairly basic notions of electric networks on infinite graphs that we have not discussed in Sect. 2.5. Next, in Sect. 7.4 we will restrict to the setting of planar graph and employ planar duality to obtain an extremely useful connection between the free and wired spanning forests which will be useful later. Using these tools we have collected we will prove Theorem 7.1 in Sect. 7.5.

Kirchhoff's Effective Resistance Formula
Theorem 7.2 (Kirchoff [49]) Let G be a finite connected graph and denote by T a uniformly drawn spanning tree of G. Then for any edge e = (x, y) we have P(e ∈ T ) = R eff (x ↔ y) .
Proof Let a = z be two distinct vertices of G (later we will take a = x and z = y) and note that any spanning tree of G contains precisely one path connecting a and z. Thus, a uniformly drawn spanning tree induces a random path from a to z. By Claim 2.46 we obtain a unit flow θ from a to z. To be concrete, for each edge e we have that θ( e) is the probability that the random path from a to z traverses e minus the probability that it traverses e. We will now show that θ satisfies the cycle law (see Claim 2.14), so it is in fact the unit current flow (see Definition 2.19).
Let e 1 , . . . , e m be a directed cycle in G. Our goal is to show that where f + i (t) equals 1 if the unique path from a to z in t traverses e i and 0 otherwise, and similarly, f − j (t) equals 1 if this path traverses e j and 0 otherwise. For 1 ≤ i ≤ m we denote by T + i the set of pairs (t, i) for which f + i (t) = 1. Similarly define T − j as the set of pairs (t, j ) for which f − j (t) = 1. To prove (7.2) it suffices to show that Let (t, i) ∈ T + i . The graph t \ {e i } has two connected components. Let e j be the first edge after e i , in the order of the cycle e 1 , . . . , e m , that is incident to both connected components and consider the spanning tree t = t ∪ {e j } \ {e i }. Note that the unique path in t from a to z traverses e j , so (t , j) ∈ T − j . This procedure defines a bijection from i T + i to j T − j . Indeed, given (t , j) from before, we can erase e j and go on the cycle in the opposite order until we reach e i which has to be the first edge incident to the two connected components of t \{e j }. This shows (7.2) and concludes the proof.

Spatial Markov Property of the UST
We would like to study the UST probability measure conditioned on the event that some edges are present in the UST and others not. It turns out that sampling from this conditional distribution amounts to drawing a UST on a modified graph.
Let G = (V , E) be a finite connected graph and let A and B be two disjoint subsets of edges. We write (G − B)/A for the graph obtained from G by erasing the edges of B and contracting the edges of A. We identify the edges of (G − B)/A with the edges E \ B. Denote by T G and T (G−B)/A a UST on G and (G − B)/A, respectively, and assume that This assumption is equivalent to G − B being connected and that A contains no cycles.
Then, conditioned on the event that T G contains the edges A and does not contain any edge of B the distribution of T G is equal to the union of A with T (G−B)/A . In other words, for a set A of spanning trees of G we have that The proof of (7.3) follows immediately from the observation that the set of spanning trees of G not containing any edge of B is simply the set of spanning trees of G−B.
Similarly, the set of spanning trees of G containing all the edges of A is simply the union of A to each spanning tree of G/A, and (7.3) follows.

Limits over Exhaustions: The Free and Wired USF
Let G be an infinite connected graph and let {G n } be a finite exhaustion of it. In this section we will show that (7.1) holds and that the UST measures with wired boundary conditions also converge. Let us first explain the latter. Denote by G * n the graph obtained from G by identifying the infinite set of vertices G \ G n to a single vertex z n and erasing the loops at z n formed by this identification. We say that {G * n } is a wired finite exhaustion of G. We postpone the proof for a little longer and first discuss some of its implications. As mentioned earlier, Theorem 7.3 together with Kolmogorov's extension theorem [24,Theorem A.3.1] implies that there exists two probability measures μ F and μ W on infinite subsets of the edges of E arising as the unique limits of the laws T n and T * n . That is, the samples F f and F w of μ F and μ W satisfy We call μ F and μ W the free uniform spanning forest and the wired uniform spanning forest and denote them by FUSF G and WUSF G respectively. We have seen earlier (one paragraph below (7.1)) that both F f and F w are almost surely spanning forests, that is, spanning graphs of G with no cycles and that every connected component of them is infinite. Thus μ F and μ W are supported on what are known as essential spanning forests of G, that is, spanning forests of G in which every component is infinite.
Are the probability measures FUSF G and WUSF G equal? Not necessarily. It is easy to see that on the infinite path Z the WUSF Z and the FUSF Z are equal and are the entire graph Z with probability 1. Conversely, it is not very difficult to see that they are different on a 3-regular tree, see exercise 1 of this chapter. Pemantle [68] has shown that FUSF Z d = WUSF Z d for any d ≥ 1 and a very useful criterion for determining whether there is equality was developed in [12]. We refer the reader to [61, Chapter 10] for further reading.
Before presenting the proof of Theorem 7.3 let us make a few short observations regarding the effective resistance between two vertices in an infinite graph, extending what we proved in Sect. 2.5.

Effective Resistance in Infinite Networks
Let G be an infinite connected graph. We have seen in Sect. 2.5 that for any vertex v the electric resistance R eff (v ↔ ∞) from v to ∞ is well defined as the limit of R eff (a ↔ z n ; G * n ) where {G * n } is a wired finite exhaustion and z n is the vertex resulting in the identification of the vertices G \ G n .
To define the electric resistance between two vertices v, u of an infinite graph, one has to take exhaustions and specify boundary conditions since the limits may differ depending on them.
Claim 7.4 Let G be an infinite connected graph, {G n } a finite exhaustion and {G * n } a wired finite exhaustion. Then for any two vertices u, v of G we have that the limits exist and do not depend on the exhaustion {G n }.
Proof For the first limit we note that by Rayleigh's monotonicity (Corollary 2.29), the sequence R eff (u ↔ v; G n ) is non-increasing and non-negative since G n ⊂ G n+1 , hence it converges. A sandwiching argument as in the proof of Claim 7.4 shows that the limit does not depend on the exhaustion {G n }.
For the second limit, since G n can be obtained by gluing vertices of G n+1 we deduce by Corollary 2.30 that the sequence R eff (u ↔ v; G * n ) is non-decreasing and bounded (by the graph distance in G between u and v for instance), hence it converges. The limit does not depend on the exhaustion by an identical sandwiching argument.
the free effective resistance and wired effective resistance between u and v respectively.

Proof of Theorem 7.3
We will prove the assertion regarding the first limit; the second is almost identical. Write A = {e 1 , . . . , e k } and e i = (x i , y i ) for each 1 ≤ i ≤ k. Assume without loss of generality that G n contains A for all n. As before, denote by T n a UST of G n . By (7.3) and Theorem 7.2 we have that Note that G n /{e 1 , . . . , e i−1 } is a finite exhaustion of the infinite graph G/{e 1 , . . . , e i−1 } and so by Claim 7.4 we obtain that the limit exists and does not depend on the exhaustion.
Since we know this limit exists for all finite edge sets A, it follows by the inclusion-exclusion formula that P(A ⊂ T n , B ∩ T n = ∅) converges for any finite sets A, B, concluding our proof.
It is now quite pleasant to see that the symbiotic relationship between electric network and UST theories continues to flourish in the infinite setting. Indeed, by combining Theorems 7.3 and Claim 7.4 we obtain the extension of Kirchhoff's formula for infinite connected graphs.
Theorem 7.5 Let G be an infinite connected graph and denote by F F and F W a sample from FUSF G and WUSF G respectively. Then for any edge e = (x, y) of G we have that

Planar Duality
When G is planar there is a very useful relationship between FUSF G and WUSF G . Recall that given a planar map G, the dual graph of G is the graph G † whose vertex set is the set of faces of G and two faces are adjacent in G † if they share an edge in G. Thus, G † is locally-finite if and only if every face of G has finitely many edges.
To each edge e ∈ E(G) corresponds a dual edge e † ∈ E(G † ) which is the pair of faces of G incident to e; this is clearly a one-to-one correspondence. When G is a finite planar graph, this correspondence induces a one-to-one correspondence between the set of spanning trees of G and the set of spanning trees of G † . Given a spanning tree of t of G we slightly abuse the notation and write t † for the set of edges {e † : e ∈ G \ t}, that is If t † has a cycle, then t is disconnected. Furthermore, if there is a vertex G † not incident to any edge of t † , then all the edges of the corresponding face in G are present in t hence t contains a cycle. We deduce that if t is a spanning tree of G, then t † is a spanning tree of G † . The converse also holds since (t † ) † = t and (G † ) † = G. Now assume that G is an infinite planar maps such that G † is locally finite. Given an essential spanning forest F of G we similarly define F † as the set of edges {e † : e ∈ G \ F}. A similar argument shows that F † is an essential spanning forest of G † . This raises the natural question: when F is a sample of FUSF G , what is the law of F † ? The answer in general is an object known as the transboundary uniform spanning forest [44, Proposition 5.1]. However, when G is additionally assumed to be one-ended (in particular, in the setting of Theorem 7.1) it turns out that F † is distributed as WUSF G † : Proposition 7.6 Let G be an infinite, one-ended planar map with a locally finite dual G † and let F be a sample of FUSF G . Then the law of F † is WUSF G † .
Proof Let G n be a finite exhaustion of G. Let F n be a finite exhaustion G † defined by letting f ∈ F n if and only if every vertex of f in G belongs to G n . Then G † n is obtained from G † by contracting G † \ F n into a single vertex which corresponds to the outer face of G n . Thus, G † n is a wired exhaustion of G † and the statement follows.
We use to obtain an important criterion of connectivity of FUSF G in the planar case.
Proposition 7.7 Let G be an infinite, one-ended planar map with a locally finite dual G † . Then a sample of FUSF G is connected almost surely if and only if each component of a sample of WUSF G is one-ended almost surely.
Proof By Proposition 7.6 it suffices to show that if F is an essential spanning forest of G, then F is connected if and only if every component of F † is one-ended. Indeed, if F is disconnected, then the boundary of a connected component of F induces an bi-infinite path in F † . Conversely, if F † contains a bi-infinite path, then by the Jordan curve theorem F is disconnected.

Last Note on Infinite Networks
We make two more useful and natural definitions. Given two disjoint finite sets A and B in an infinite connected graph G we define the free and wired effective resistance between them R W eff (A ↔ B; G) and R F eff (A ↔ B; G) as the free and wired effective resistance between a and b in the graph obtained from G by identifying A and B to the vertices a and b. Lastly, given a graph G, a wired finite exhaustion {G * n } of G and two disjoint finite sets A and B we define where the last limit exists since the sequence is non-increasing from n that is large enough so that G n contains A and B. In the proof of Theorem 7.1 we will require the following estimate.

Lemma 7.8 Let A and B be two finite sets of vertices in an infinite connected graph G. Then
Proof For any three distinct vertices u, v, w in a finite network we have by the union bound that P u (τ {v,w} < τ + u ) ≤ P u (τ v < τ + u ) + P u (τ w < τ + u ). Hence by Claim 2.22 we get that Let {G * n } be a wired finite exhaustion of G and assume without loss of generality that A and B are contained in G * n for all n. Then by the previous estimate Denote by M the maximum in the statement of the lemma and take n → ∞ in the last inequality. We obtain that Rearranging gives that By symmetry, the same inequality holds when we replace the roles of A and B. We put this together with the triangle inequality for effective resistances (2.9) and get that which by rearranging gives the desired inequality.

Method of Random Sets
We present the following weakening of the method of random paths as in Sect. 2.6. Let μ be the law of a random subset W of vertices of G. Define the energy of μ as

Lemma 7.9 (Method of Random Sets) Let A, B be two disjoint finite sets of vertices in an infinite graph G. Let W be a random subset of vertices of G and denote by μ its law. Assume that the subgraph of G induced by W almost surely contains a simple path starting at A that is either infinite or finite and ends at B. Then
Proof Given W let γ be a simple path, contained in W , connecting A to B or an infinite path starting at A. We choose γ according to some prescribed lexicographical ordering. Then, letting ν be the law of γ , where by e ∈ γ we mean that the directed edge e is traversed (in its direction) by γ , and by E(ν) we mean the energy of the flow induced by γ , as in Claim 2.46. Let γ be an independent random path having the same law as γ . Then the sum above is precisely the expected number of directed edges traversed both by γ and γ . Since these are simple paths, they each contain at most one directed edge emanating from each vertex v ∈ W . Thus, the expected number of directed edges used by both paths is at most the number of vertices used by both paths. Hence, and the proof is concluded by Thomson's principle (Theorem 2.28).

Proof of Theorem 7.1
In Theorem 7.1 we assume that G = (V , E) is a bounded-degree, one-ended triangulation. Hence G † is a bounded degree (in fact, 3-regular), one-ended and transient planar map with faces of uniformly bounded size. We leave this verification as an exercise for the reader. To avoid carrying the † symbol around, and with a slight abuse of notation, let G = (V , E) be a graph satisfying these assumptions on G † , that is, we assume that G is a one-ended, transient, infinite planar map with bounded degrees and face sizes. We will prove under these assumptions that every component of WUSF G is one ended almost surely which implies Theorem 7.1 by Proposition 7.7.
Let T be the bounded-degree one-ended triangulation obtained from G by adding a vertex inside each face of G and connecting it by edges to the vertices of that face according to their cyclic ordering. By Theorem 4.4 there exists a circle packing of T in the unit disc U. We identify the vertices of T as the vertices V (G) and faces F (G) of G, and denote this circle packing as P Given z ∈ U and r ≥ r > 0 denote by A z (r, r ) the annulus {w ∈ C : r ≤ |w − z| ≤ r }.
Definition 7.10 Write V z (r, r ) for the set of vertices v of G such that either We emphasize that V z (r, r ) contains only vertices of G; no vertices of T that correspond to faces of G belong to it.

Lemma 7.11
There exists a constant C < ∞ depending only on the maximal degree such that for any z ∈ U and any positive integer n satisfying |z| ≥ 1 − C −n the sets Proof By the Ring Lemma (Lemma 4.2) there exists a constant B < ∞ such that for any C > 1, any z satisfying z ≥ 1 − C −n and any 1 ≤ i ≤ n, if a circle of P intersects A z (C −i , 2C −i ) or is tangent to a circle that intersects A z (C −i , 2C −i ), then its radius is at most BC −i . Hence, this set of circles is contained in the disc of radius (2 + 4B)C −i around z. Furthermore, since |z| ≥ 1 − C −n , by the Ring Lemma again there exists b > 0 such that any such circle must be of distance at least bC −i from z. Hence, any fixed C > 4+4B b satisfies the assertion of the lemma.
Lemma 7.12 Let z ∈ U and r > 0. Let U be a uniform random variable in [1,2] and denote by μ r the law of the random set V z (U r, U r) (as defined in Definition 7.10). Then there exists a constant C < ∞ depending only on the maximal degree such that Proof For each vertex v, the event v ∈ V z (U r, U r) implies that the circle {w ∈ C : |w − z| = Ur} intersects the circle P (v) or intersects P (f ) for some face f incident to v. The union of P (v) and P (f ) over all such faces f is contained in the Euclidean ball around the center of P (v) of radius r(v) + 2 max f :v∈f r(f ). Since T has finite maximal degree we have that r(f ) ≤ Cr(v) for all f with v ∈ f where C < ∞ depends only on the maximal degree by the Ring Lemma (Lemma 4.2). Hence, (7.6) We claim that v∈V z (r,2r) min{r(v), r} 2 ≤ 16r 2 . (7.7) Indeed, consider a vertex v ∈ V z (r, 2r) for which the corresponding circle P (v) has radius larger than r. By Definition 7.10 this circle must intersect {w ∈ C : |w − z| ≤ 2r}. We replace each such P (v) with a circle of radius r that is contained in the original circle and intersects {w ∈ C : |w − z| ≤ 2r}. The circles in this new set still have disjoint interiors and are contained in {w ∈ C : |w − z| ≤ 4r}. Therefore their area is at most π16r 2 and (7.7) follows. The proof of lemma is now concluded by combining (7.6) and (7.7).
Proof of Theorem 7.1 Let F be a sample of WUSF G and given an edge e = (x, y) we define A e to be the event that x and y are in two distinct infinite connected components of F \ {e}. It is clear that every component of F is one-ended almost surely if and only if P(e ∈ F , A e ) = 0 (7.8) for every edge e of G. Consider the triangulation T described above Definition 7.10 and its circle packing P in U. By choosing the proper Möbius transformation we may assume that the tangency point between P (x) and P (y) is the origin, and that the centers of P (x) and P (y) lie on the negative and positive real axis, respectively. Fix now an arbitrary ε > 0 and let V ε be all the vertices of G such that the center z(v) of P (v) satisfies |z(v)| ≤ 1 − ε. Denote by B e ε the event that every connected On the event A e ε , the paths η x and η y split V ε into two pieces, L and R. Right: We define a random set containing a path (solid blue) from η x to η y ∪ {∞} in G \ K c using a random circle (dashed blue). Here we see two examples, one in which the path ends at η y , and the other in which the path ends at the boundary (i.e., at infinity) component of F\{e} intersects V \V ε . Note that A e ⊂ ∩ ε>0 B e ε but this containment is strict since it is possible that e ∈ F and x is connected to y in F inside V ε .
Assume that B e ε holds. Let η x be the rightmost path in F \ {e} from x to V \ V ε when looking at x from y, and let η y be the leftmost path in F \ {e} from y to V \ V ε when looking at y from x. As mentioned above, the paths η x and η y are not necessarily disjoint. Nonetheless, concatenating the reversal of η x with e and η y separates V ε into two sets of vertices, L and R, which are to the left and right of e (when viewed from x to y) respectively. See Fig. 7.1 for an illustration of the case when η x and η y are disjoint (when they are not, R is a "bubble" separated from V \ V ε ).
On the event B e ε , let K be the set of edges that are either incident to a vertex in L or belong to the path η x ∪ η y , and set K = E off of this event. Note that the edges of K do not touch the vertices of R. The condition that η x and η y are the rightmost and leftmost paths to V \ V ε from x and y is equivalent to the condition that K does not contain any open path from x to V \ V ε other than η x , and does not contain any open path from y to V \ V ε other than η y . We note that K can be explored algorithmically, without querying the status of any edge in E \ K, by performing a right-directed depth-first search of x's component in F and a left-directed depth-first search of y's component in F, stopping each search when it first leaves V ε .
Denote by A e ε the event that η x and η y are disjoint, or equivalently, that K does not contain an open path from x to y (and in particular, no path starting at η x and ending at η y ). The event A e ε is measurable with respect to the random set K and A e = ∩ ε>0 A e ε . Hence P(e ∈ F , A e ) ≤ P(e ∈ F | A e ε ) = E[P(e ∈ F | A e ε , K)] . (7.9) Denote by K o the open edge of K (that is, the edge of K in F) and by K c the closed edges of K (that is, the edges of K not belonging to F). In particular, η x and η y are contained in K o . Then by the UST Markov property (7.3), conditioned on K and the event A e ε , the law of F is equal to the union of K o with a sample of the WUSF on (G − K c )/K o . In particular, by Kirchhoff's formula Theorem 7.5 we have that where in the last inequality we used the fact that gluing cannot increase the resistance (Corollary 2.30). We will show that the last quantity tends to 0 as ε → 0 which gives (7.8). To that aim, let v x be the endpoint of the path η x and let z 0 be the center of the P (v x ). On the event A e ε , for each 1 − |z 0 | ≤ r ≤ 1/4, we claim that the set V z 0 (r, r), as defined in Definition 7.10, contains a path in G from η x to η y that is contained in R ∪ η x ∪ η y or an infinite simple path starting at η x that is contained in R ∪ η x . Either of these paths are therefore a path in G − K c .
To see this, consider the arc A (z 0 , r) = {z ∈ U : |z − z 0 | = r} viewed in the clockwise direction and let A(z 0 , r) be the subarc beginning at the last intersection of A (z 0 , r) with a circle corresponding to a vertex in the trace of η x , and ending at the first intersection after this time of A (z 0 , r) with either ∂U or a circle corresponding to a vertex in the trace of η y (see Fig. 7.1). Hence, if A e ε holds, then the set of vertices of T whose circles in P intersect A(z 0 , r) contains a path in T starting at η x and ending η y or does not end at all, for every 1 − |z 0 | ≤ r ≤ 1/4. To obtain a path in G rather than T we divert the path counterclockwise around each face of G. That is, whenever the path passes from a vertex u of G to a face f of G and then to a vertex v of G, we replace this section of the path with the list of vertices of G incident to f that are between u and v in the counterclockwise order. By Definition 7.10 this diverted path is in V z 0 (r, r) and so this construction shows that the subgraph of G − K c induced by the set V z 0 (r, r) contains a path from η x to η y or an infinite path from η x , as claimed.
Let r i = C −i for i = 1, . . . , N where C < ∞ the constant from Lemma 7.11 and N = log C (ε) . Assume without loss of generality that C ≥ 4 so that ε ≤ r i ≤ 1/4 for all i = 1, . . . , N. By Lemma 7.11 the measures μ r i defined in Lemma 7.12 are supported on sets that are contained in the disjoint sets V z (r i , 2r i ). Thus, by Lemma 7.9 and Lemma 7.12 we have where B < ∞ is a constant depending only on the maximum degree. By symmetry we also have .