Indistinguishability of Trees in Uniform Spanning Forests
Tom Hutchcroft
Joint work with Asaf Nachmias
BIRS, June 2015
Pictures due to subsets of Benjamini, Lyons, Peres and Schramm
Uniform Spanning Forests
The uniform spanning forests of an infinite graph G are defined as
infinite-volume limits of uniformly chosen random spanning trees on large
finite induced subgraphs of G .
Uniform Spanning Forests
The uniform spanning forests of an infinite graph G are defined as
infinite-volume limits of uniformly chosen random spanning trees on large
finite induced subgraphs of G .
We can take these limits with respect to two extremal boundary conditions,
obtaining the free uniform spanning forest (FUSF) and wired uniform
spanning forest (WUSF). Both limits shown to exist by Pemantle (’91).
Uniform Spanning Forests
The uniform spanning forests of an infinite graph G are defined as
infinite-volume limits of uniformly chosen random spanning trees on large
finite induced subgraphs of G .
We can take these limits with respect to two extremal boundary conditions,
obtaining the free uniform spanning forest (FUSF) and wired uniform
spanning forest (WUSF). Both limits shown to exist by Pemantle (’91).
Free and wired boundary conditions
Let G be an infinite graph and let Gn be an exhaustion
S of G , i.e. an
increasing sequence of finite induced subgraphs with Gn = G .
Free and wired boundary conditions
Let G be an infinite graph and let Gn be an exhaustion
S of G , i.e. an
increasing sequence of finite induced subgraphs with Gn = G .
The free uniform spanning forest is defined to be the weak limit of the
uniform measure on the spanning trees of Gn : for every finite S ⊆ E ,
FUSFG (S ⊆ F) = lim USTGn (S ⊆ T ).
n→∞
Free and wired boundary conditions
Let G be an infinite graph and let Gn be an exhaustion
S of G , i.e. an
increasing sequence of finite induced subgraphs with Gn = G .
The free uniform spanning forest is defined to be the weak limit of the
uniform measure on the spanning trees of Gn : for every finite S ⊆ E ,
FUSFG (S ⊆ F) = lim USTGn (S ⊆ T ).
n→∞
Alternatively, for each n we can form a graph Gn∗ from G by identifying
every vertex in G \ Gn into a single vertex.
Free and wired boundary conditions
Let G be an infinite graph and let Gn be an exhaustion
S of G , i.e. an
increasing sequence of finite induced subgraphs with Gn = G .
The free uniform spanning forest is defined to be the weak limit of the
uniform measure on the spanning trees of Gn : for every finite S ⊆ E ,
FUSFG (S ⊆ F) = lim USTGn (S ⊆ T ).
n→∞
Alternatively, for each n we can form a graph Gn∗ from G by identifying
every vertex in G \ Gn into a single vertex. The wired uniform spanning
forest is defined to be the weak limit
WUSFG (S ⊆ F) = lim USTGn∗ (S ⊆ T ).
n→∞
Free and wired boundary conditions
Let G be an infinite graph and let Gn be an exhaustion
S of G , i.e. an
increasing sequence of finite induced subgraphs with Gn = G .
The free uniform spanning forest is defined to be the weak limit of the
uniform measure on the spanning trees of Gn : for every finite S ⊆ E ,
FUSFG (S ⊆ F) = lim USTGn (S ⊆ T ).
n→∞
Alternatively, for each n we can form a graph Gn∗ from G by identifying
every vertex in G \ Gn into a single vertex. The wired uniform spanning
forest is defined to be the weak limit
WUSFG (S ⊆ F) = lim USTGn∗ (S ⊆ T ).
n→∞
The FUSF stochastically dominates the WUSF, i.e. the two measures can
be coupled so that the FUSF contains the WUSF.
Connections
The uniform spanning forests are closely related to many other interesting
models
1
Loop-erased random walk.
2
Conformally invariant scaling limits.
3
The Abelian sandpile model.
4
Domino tiling.
5
Random cluster model.
6
`2 -Betti numbers and the fixed price problem: the two USFs of a
Cayley graph differ if and only if the first `2 -Betti number of the
group is non-zero.
7
Potential theory: the two USFs of a graph differ if and only if the
graph admits non-constant Harmonic functions of finite energy.
Connectivity
Although they are defined as limits of trees, the USFs need not be
connected.
Connectivity
Although they are defined as limits of trees, the USFs need not be
connected.
Pemantle (’91) proved that the FUSF and WUSF of Zd coincide for all d,
and that they are almost surely connected (i.e. a single tree) if and only if
d ≤ 4.
Connectivity
Although they are defined as limits of trees, the USFs need not be
connected.
Pemantle (’91) proved that the FUSF and WUSF of Zd coincide for all d,
and that they are almost surely connected (i.e. a single tree) if and only if
d ≤ 4.
Benjamini, Lyons, Peres and Schramm (BLPS) (’01) gave a complete
characterisation of the connectivity of the WUSF, showing that the WUSF
of a graph is almost surely connected if and only if two independent simple
random walks on the graph intersect infinitely often almost surely.
Connectivity
Although they are defined as limits of trees, the USFs need not be
connected.
Pemantle (’91) proved that the FUSF and WUSF of Zd coincide for all d,
and that they are almost surely connected (i.e. a single tree) if and only if
d ≤ 4.
Benjamini, Lyons, Peres and Schramm (BLPS) (’01) gave a complete
characterisation of the connectivity of the WUSF, showing that the WUSF
of a graph is almost surely connected if and only if two independent simple
random walks on the graph intersect infinitely often almost surely.
BLPS also showed that the number of components of the WUSF in any
graph is a constant, and in a transitive graph the WUSF is either
connected or has infinitely many components a.s.
Recall that an infinite tree is said to be one-ended if and only if it does
not contain a bi-infinite self-avoiding path, or equivalently if any two
half-infinite paths intersect.
Recall that an infinite tree is said to be one-ended if and only if it does
not contain a bi-infinite self-avoiding path, or equivalently if any two
half-infinite paths intersect.
One-ended trees are ‘small’ and ‘close to being finite’, e.g. are recurrent
for simple random walk and have pc = 1.
It is known that every component of the WUSF is one-ended almost surely
for several large classes of graphs, e.g. every transient transitive graph.
(Pemantle ‘91, BLPS ‘01, Lyons, Morris and Schramm ‘07, Aldous and
Lyons ‘07, H. ‘15, H. and Nachmias ‘15+).
Recall that an infinite tree is said to be one-ended if and only if it does
not contain a bi-infinite self-avoiding path, or equivalently if any two
half-infinite paths intersect.
One-ended trees are ‘small’ and ‘close to being finite’, e.g. are recurrent
for simple random walk and have pc = 1.
It is known that every component of the WUSF is one-ended almost surely
for several large classes of graphs, e.g. every transient transitive graph.
(Pemantle ‘91, BLPS ‘01, Lyons, Morris and Schramm ‘07, Aldous and
Lyons ‘07, H. ‘15, H. and Nachmias ‘15+).
It’s also known (Morris ‘03) that every component of the WUSF is
recurrent almost surely on any graph.
What about the FUSF?
The FUSF is far less well understood.
What about the FUSF?
The FUSF is far less well understood.
No characterisation of its connectivity is known, and it is not even known
whether the number of components of the FUSF of any graph is constant.
What about the FUSF?
The FUSF is far less well understood.
No characterisation of its connectivity is known, and it is not even known
whether the number of components of the FUSF of any graph is constant.
Question (BLPS ‘01)
Let G be a Cayley graph and let F be a sample of the FUSF of G . Is the
number of components of F one or ∞ almost surely?
What about the FUSF?
The FUSF is far less well understood.
No characterisation of its connectivity is known, and it is not even known
whether the number of components of the FUSF of any graph is constant.
Question (BLPS ‘01)
Let G be a Cayley graph and let F be a sample of the FUSF of G . Is the
number of components of F one or ∞ almost surely?
BLPS proved that if G is a Cayley graph and FUSF 6= WUSF, then the
FUSF of G contains a multiply-ended, transient component almost surely.
What about the FUSF?
The FUSF is far less well understood.
No characterisation of its connectivity is known, and it is not even known
whether the number of components of the FUSF of any graph is constant.
Question (BLPS ‘01)
Let G be a Cayley graph and let F be a sample of the FUSF of G . Is the
number of components of F one or ∞ almost surely?
BLPS proved that if G is a Cayley graph and FUSF 6= WUSF, then the
FUSF of G contains a multiply-ended, transient component almost surely.
Question (BLPS ‘01)
For a Cayley graph with FUSF 6= WUSF, is every component of the FUSF
infinitely-ended and transient almost surely?
What about the FUSF?
The FUSF is far less well understood.
No characterisation of its connectivity is known, and it is not even known
whether the number of components of the FUSF of any graph is constant.
Theorem (H. and Nachmias ‘15, Timar ‘15)
Let G be a Cayley graph and let F be a sample of the FUSF of G . Then F
is either connected or has infinitely many connected components almost
surely.
BLPS proved that if G is a Cayley graph with FUSF 6= WUSF, then the
FUSF of G contains a multiply-ended, transient component almost surely.
Theorem (H. and Nachmias ‘15, Timar ‘15)
For a Cayley graph with FUSF 6= WUSF, every component of the FUSF is
infinitely-ended and transient almost surely.
Indistinguishability
Let’s think about the question
Question (BLPS ‘01)
For a Cayley graph with FUSF 6= WUSF, is every component of the FUSF
infinitely-ended and transient almost surely?
Indistinguishability
Let’s think about the question
Question (BLPS ‘01)
For a Cayley graph with FUSF 6= WUSF, is every component of the FUSF
infinitely-ended and transient almost surely?
We might wonder more generally: When the FUSF or WUSF of a graph
is disconnected, how different can the components be from each other?
Indistinguishability
Let’s think about the question
Question (BLPS ‘01)
For a Cayley graph with FUSF 6= WUSF, is every component of the FUSF
infinitely-ended and transient almost surely?
We might wonder more generally: When the FUSF or WUSF of a graph
is disconnected, how different can the components be from each other?
Conjecture (BLPS ‘01)
Let G be a Cayley graph and let F be a sample of either the FUSF or the
WUSF of G . Then the components of F are indistinguishable from each
other in the following sense.
If A is an automorphism-invariant set of subgraphs of G , then either every
component of F is in A or no components of F are in A almost surely.
Indistinguishability
When the FUSF or WUSF of a graph is disconnected, how different can
the components be from each other?
Conjecture (BLPS ‘01)
Let G be a Cayley graph and let F be a sample of either the FUSF or the
WUSF of G . Then the components of F are indistinguishable from each
other in the following sense.
If A is an automorphism-invariant set of subgraphs of G , then either every
component of F is in A or no components of F are in A almost surely.
For example, A could be the set of one-ended subgraphs of G , the set of
transient subgraphs of G , the set of subgraphs of G with some specified
growth rate, etc.
Indistinguishability
When the FUSF or WUSF of a graph is disconnected, how different can
the components be from each other?
Conjecture (BLPS ‘01)
Let G be a Cayley graph and let F be a sample of either the FUSF or the
WUSF of G . Then the components of F are indistinguishable from each
other in the following sense.
If A is an automorphism-invariant set of subgraphs of G , then either every
component of F is in A or no components of F are in A almost surely.
For example, A could be the set of one-ended subgraphs of G , the set of
transient subgraphs of G , the set of subgraphs of G with some specified
growth rate, etc.
Indistinguishability
When the FUSF or WUSF of a graph is disconnected, how different can
the components be from each other?
Theorem (H., Nachmias ‘15)
Let G be a Cayley graph and let F be a sample of either the FUSF or the
WUSF of G . Then the components of F are indistinguishable from each
other in the following sense.
If A is an automorphism-invariant set of subgraphs of G , then either every
component of F is in A or no components of F are in A almost surely.
Partial results were obtained independently by Timar (‘15), who confirmed
the conjecture for the FUSF in the case that FUSF 6= WUSF.
The Lyons-Schramm Argument
Lyons and Schramm (‘99) proved that for any automorphism-invariant,
insertion-tolerant percolation process on a Cayley graph, the infinite
components are indistinguishable from each other in this sense.
The Lyons-Schramm Argument
Lyons and Schramm (‘99) proved that for any automorphism-invariant,
insertion-tolerant percolation process on a Cayley graph, the infinite
components are indistinguishable from each other in this sense.
Here, a percolation (i.e. random subgraph) ω is insertion tolerant if the
law of ω ∪ {e} is absolutely continuous with respect to the law of ω for
every edge e of G .
The Lyons-Schramm Argument
Lyons and Schramm (‘99) proved that for any automorphism-invariant,
insertion-tolerant percolation process on a Cayley graph, the infinite
components are indistinguishable from each other in this sense.
Here, a percolation (i.e. random subgraph) ω is insertion tolerant if the
law of ω ∪ {e} is absolutely continuous with respect to the law of ω for
every edge e of G .
They also applied indistinguishability to prove theorems not of the form
covered by the theorem (connectivity decay, monotonicity of uniqueness in
Bernoulli percolation).
The Lyons-Schramm Argument
Lyons and Schramm (‘99) proved that for any automorphism-invariant,
insertion-tolerant percolation process on a Cayley graph, the infinite
components are indistinguishable from each other in this sense.
Here, a percolation (i.e. random subgraph) ω is insertion tolerant if the
law of ω ∪ {e} is absolutely continuous with respect to the law of ω for
every edge e of G .
They also applied indistinguishability to prove theorems not of the form
covered by the theorem (connectivity decay, monotonicity of uniqueness in
Bernoulli percolation).
Since adding an edge to a forest can create a cycle, the USFs are clearly
not insertion tolerant in general.
The Lyons-Schramm Argument
We say that
a component C of ω has type A if C ∈ A and type ¬A otherwise.
The Lyons-Schramm Argument
We say that
a component C of ω has type A if C ∈ A and type ¬A otherwise.
a vertex v has type A if its component in ω has type A , and type
¬A otherwise.
The Lyons-Schramm Argument
We say that
a component C of ω has type A if C ∈ A and type ¬A otherwise.
a vertex v has type A if its component in ω has type A , and type
¬A otherwise.
We call an edge e pivotal if there is an infinite component C of ω such
that the connected component C 0 of ω ∪ {e} containing C has a different
type than C .
The Lyons-Schramm Argument
We say that
a component C of ω has type A if C ∈ A and type ¬A otherwise.
a vertex v has type A if its component in ω has type A , and type
¬A otherwise.
We call an edge e pivotal if there is an infinite component C of ω such
that the connected component C 0 of ω ∪ {e} containing C has a different
type than C .
Assume for contradiction that infinite components of types A and ¬A
both exist. Our goal is to contradict the measurability of A .
The Lyons-Schramm Argument
We say that
a component C of ω has type A if C ∈ A and type ¬A otherwise.
a vertex v has type A if its component in ω has type A , and type
¬A otherwise.
We call an edge e pivotal if there is an infinite component C of ω such
that the connected component C 0 of ω ∪ {e} containing C has a different
type than C .
Assume for contradiction that infinite components of types A and ¬A
both exist. Our goal is to contradict the measurability of A .
Some preliminary steps:
1 If both types of infinite components coexist, then pivotal edges
exist.
The Lyons-Schramm Argument
We say that
a component C of ω has type A if C ∈ A and type ¬A otherwise.
a vertex v has type A if its component in ω has type A , and type
¬A otherwise.
We call an edge e pivotal if there is an infinite component C of ω such
that the connected component C 0 of ω ∪ {e} containing C has a different
type than C .
Assume for contradiction that infinite components of types A and ¬A
both exist. Our goal is to contradict the measurability of A .
Some preliminary steps:
1 If both types of infinite components coexist, then pivotal edges
exist. In particular, a pivotal edge for the component at the root
exists with positive probability, and we may assume w.l.o.g. that with
positive probability the root has type A and a pivotal edge for the
component of the root exists.
The Lyons-Schramm Argument
Assume for contradiction that infinite components of types A and ¬A
both exist. Our goal is to contradict the measurability of A .
Some preliminary steps:
1
If both types of infinite components coexist, then pivotal edges
exist. In particular, a pivotal edge for the component at the root
exists with positive probability, and we may assume w.l.o.g. that with
positive probability the root has type A and a pivotal edge for the
component of the root exists.
2
Stationarity of the delayed on the component of the root.The
delayed simple random walk hXn in≥0 is the random walk on G that
rejects steps along edges not in ω. The invariance of ω implies that
the sequence
hω, hXn+k in∈Z ik∈Z
is stationary.
3
Every infinite component is transient almost surely.
The Lyons-Schramm Argument: Main Part
Pick r large enough so that there exists a pivotal edge for the cluster
at the origin at distance exactly r from the origin.
The Lyons-Schramm Argument: Main Part
Pick r large enough so that there exists a pivotal edge for the cluster
at the origin at distance exactly r from the origin.
For each n ∈ Z, let en be an edge chosen uniformly from the set of
edges at distance r from Xn .
The Lyons-Schramm Argument: Main Part
Pick r large enough so that there exists a pivotal edge for the cluster
at the origin at distance exactly r from the origin.
For each n ∈ Z, let en be an edge chosen uniformly from the set of
edges at distance r from Xn .
Using transience, we can show that there is a positive probability
that
en is a pivotal edge for the component of the origin and
The Lyons-Schramm Argument: Main Part
Pick r large enough so that there exists a pivotal edge for the cluster
at the origin at distance exactly r from the origin.
For each n ∈ Z, let en be an edge chosen uniformly from the set of
edges at distance r from Xn .
Using transience, we can show that there is a positive probability
that
en is a pivotal edge for the component of the origin and
neither endpoint of en is visited by hXn in∈Z
On this event, let ω 0 = ω ∪ {e}.
Observation: The probability that hXn in∈Z visits an endpoint of e
given ω doesn’t depend on whether e is present in ω or not.
The Lyons-Schramm Argument: Main Part
Pick r large enough so that there exists a pivotal edge for the cluster
at the origin at distance exactly r from the origin.
For each n ∈ Z, let en be an edge chosen uniformly from the set of
edges at distance r from Xn .
Using transience, we can show that there is a positive probability
that
en is a pivotal edge for the component of the origin and
neither endpoint of en is visited by hXn in∈Z
On this event, let ω 0 = ω ∪ {e}.
Observation: The probability that hXn in∈Z visits an endpoint of e
given ω doesn’t depend on whether e is present in ω or not.
Implication: On this event, we have that ω 0 is ‘at least as likely’ as ω
up to a controllable factor by insertion tolerance.
The Lyons-Schramm Argument: Main Part
Pick r large enough so that there exists a pivotal edge for the cluster
at the origin at distance exactly r from the origin.
For each n ∈ Z, let en be an edge chosen uniformly from the set of
edges at distance r from Xn .
Using transience, we can show that there is a positive probability
that
en is a pivotal edge for the component of the origin and
neither endpoint of en is visited by hXn in∈Z
On this event, let ω 0 = ω ∪ {e}.
Observation: The probability that hXn in∈Z visits an endpoint of e
given ω doesn’t depend on whether e is present in ω or not.
Implication: On this event, we have that ω 0 is ‘at least as likely’ as ω
up to a controllable factor by insertion tolerance.
The Lyons-Schramm Argument: Main Part
For large n, en is far away from the origin with high probability.
The Lyons-Schramm Argument: Main Part
For large n, en is far away from the origin with high probability.
If en is also pivotal (which occurs with probability bounded below),
the root has a different type in ω and ω 0 .
The Lyons-Schramm Argument: Main Part
For large n, en is far away from the origin with high probability.
If en is also pivotal (which occurs with probability bounded below),
the root has a different type in ω and ω 0 .
Since ω 0 and ω are the same in a large neighbourhood of the root,
this shows that we cannot predict the type of the origin with arbitrary
accuracy by examining ω in a large neighbourhood of the origin.
The Lyons-Schramm Argument: Main Part
For large n, en is far away from the origin with high probability.
If en is also pivotal (which occurs with probability bounded below),
the root has a different type in ω and ω 0 .
Since ω 0 and ω are the same in a large neighbourhood of the root,
this shows that we cannot predict the type of the origin with arbitrary
accuracy by examining ω in a large neighbourhood of the origin.
This contradicts the measurability of the property A .
Update Tolerance
We introduce a way of locally modifying the FUSF/WUSF, in which we
add an edge of our choice and, in return, are required to delete an edge
that depends on the forest. This replaces insertion-tolerance.
Update Tolerance
We introduce a way of locally modifying the FUSF/WUSF, in which we
add an edge of our choice and, in return, are required to delete an edge
that depends on the forest. This replaces insertion-tolerance.
The idea stems from the Markov chain-tree theorem
(Hill ‘66, Shubert ‘75, Leighton and Rivest ‘83, ‘86).
Update Tolerance
We introduce a way of locally modifying the FUSF/WUSF, in which we
add an edge of our choice and, in return, are required to delete an edge
that depends on the forest. This replaces insertion-tolerance.
The idea stems from the Markov chain-tree theorem
(Hill ‘66, Shubert ‘75, Leighton and Rivest ‘83, ‘86).
If G is a finite network, e is an oriented edge of G and T is a spanning
tree of G , we define the update U(T , e) as follows:
If e is already contained in T , or is a self-loop, let U(T , e) = T .
Update Tolerance
We introduce a way of locally modifying the FUSF/WUSF, in which we
add an edge of our choice and, in return, are required to delete an edge
that depends on the forest. This replaces insertion-tolerance.
The idea stems from the Markov chain-tree theorem
(Hill ‘66, Shubert ‘75, Leighton and Rivest ‘83, ‘86).
If G is a finite network, e is an oriented edge of G and T is a spanning
tree of G , we define the update U(T , e) as follows:
If e is already contained in T , or is a self-loop, let U(T , e) = T .
Otherwise, let d = d(T , e) be the first edge in the unique path in T
from e − to e + , and let
U(T , e) = T ∪ {e} \ {d}.
Update Tolerance
We introduce a way of locally modifying the FUSF/WUSF, in which we
add an edge of our choice and, in return, are required to delete an edge
that depends on the forest. This replaces insertion-tolerance.
The idea stems from the Markov chain-tree theorem
(Hill ‘66, Shubert ‘75, Leighton and Rivest ‘83, ‘86).
If G is a finite network, e is an oriented edge of G and T is a spanning
tree of G , we define the update U(T , e) as follows:
If e is already contained in T , or is a self-loop, let U(T , e) = T .
Otherwise, let d = d(T , e) be the first edge in the unique path in T
from e − to e + , and let
U(T , e) = T ∪ {e} \ {d}.
Easy fact: If v is a fixed vertex of G , E is a uniform edge with E − = v
d
and T is a UST, then U(T , E ) = T .
Update Tolerance
Easy fact: If v is a fixed vertex of G , E is a uniform edge with E − = v
d
and T is a UST, then U(T , E ) = T .
Corollary: update-tolerance. For any event A and any edge e,
P(T ∈ A ) ≥
1
P(U(T , e) ∈ A ).
deg(e − )
Update Tolerance
Easy fact: If v is a fixed vertex of G , E is a uniform edge with E − = v
d
and T is a UST, then U(T , E ) = T .
Corollary: update-tolerance. For any event A and any edge e,
X
1
1
P(T ∈ A ) =
P(U(T , ẽ) ∈ A ) ≥
P(U(T , e) ∈ A ).
−)
−)
deg(e
deg(e
−
−
ẽ =e
Update Tolerance
Easy fact: If v is a fixed vertex of G , E is a uniform edge with E − = v
d
and T is a UST, then U(T , E ) = T .
Corollary: update-tolerance. For any event A and any edge e,
Idea: extend to the USFs to get a property similar to insertion tolerance.
Lemma
Let hGn in≥1 be an exhaustion of an infinite graph G and let e be an edge
of G . Then there exists a pair (F, D(e)) such that, letting Tn be a UST of
Gn .
d
(Tn , d(T , e)) −−−→ (F, D(e)).
n→∞
Update Tolerance
Easy fact: If v is a fixed vertex of G , E is a uniform edge with E − = v
d
and T is a UST, then U(T , E ) = T .
Corollary: update-tolerance. For any event A and any edge e,
Idea: extend to the USFs to get a property similar to insertion tolerance.
Lemma
Let hGn in≥1 be an exhaustion of an infinite graph G and let e be an edge
of G . Then there exists a pair (F, D(e)) such that, letting Tn be a UST of
Gn .
d
(Tn , d(T , e)) −−−→ (F, D(e)).
n→∞
Proved using electrical interpretations. A corresponding statement also
holds for the WUSF.
Update Tolerance
Idea: extend to the USFs to get a property similar to insertion tolerance.
Lemma
Let hGn i be an exhaustion of an infinite graph G and let e be an edge of
G . Then there exists a pair (F, D) such that
d
(Tn , d(Tn , e)) −−−→ (F, D(e)).
n→∞
We can now define the update U(F, e) as follows:
If e is already contained in F, or is a self-loop, let U(F, e) = F.
Update Tolerance
Idea: extend to the USFs to get a property similar to insertion tolerance.
Lemma
Let hGn i be an exhaustion of an infinite graph G and let e be an edge of
G . Then there exists a pair (F, D) such that
d
(Tn , d(Tn , e)) −−−→ (F, D(e)).
n→∞
We can now define the update U(F, e) as follows:
If e is already contained in F, or is a self-loop, let U(F, e) = F.
Otherwise, sample D(e) from its conditional distribution given F and
let U(F, e) = F ∪ {e} \ {D(e)}.
Update Tolerance
Idea: extend to the USFs to get a property similar to insertion tolerance.
Lemma
Let hGn i be an exhaustion of an infinite graph G and let e be an edge of
G . Then there exists a pair (F, D) such that
d
(Tn , d(Tn , e)) −−−→ (F, D(e)).
n→∞
We can now define the update U(F, e) as follows:
If e is already contained in F, or is a self-loop, let U(F, e) = F.
Otherwise, sample D(e) from its conditional distribution given F and
let U(F, e) = F ∪ {e} \ {D(e)}.
Taking the limit of the finite update-tolerance, we get update-tolerance for
the FUSF.
FUSFG (F ∈ A ) ≥
1
FUSFG (U(F, e) ∈ A ).
deg(e − )
Indistinguishability for the FUSF when it differs from the WUSF.
Two further key ingredients are required in order to adapt the
Lyons-Schramm argument.
Every component is transient and infinitely-ended: if recurrent and
transient trees coexist, we can use updates to create a ‘transient tree
with a recurrent branch’.
Transient component,
no recurrent branches
Transient component
with a recurrent branch
e1 e2 e3
Recurrent
Component
The existence of such a component contradicts the Mass-Transport
Principle: Roughly speaking, transport mass from the recurrent
branch to where it meets the transient part of the tree.
Indistinguishability for the FUSF when it differs from the WUSF.
Two further key ingredients are required in order to adapt the
Lyons-Schramm argument.
Every component is transient and infinitely-ended: if recurrent and
transient trees coexist, we can use updates to create a ‘transient tree
with a recurrent branch’.
Transient component,
no recurrent branches
Transient component
with a recurrent branch
e1 e2 e3
Recurrent
Component
The existence of such a component contradicts the Mass-Transport
Principle: Roughly speaking, transport mass from the recurrent
branch to where it meets the transient part of the tree.
If components of type A and ¬A coexist, then ‘good’ pivotal edges
exist: edges such that, when we update at that edge, infinitely many
vertices change their type.
Number of components.
Suppose for contradiction that there are 1 < k < ∞ components.
Let hYn in≥0 be a simple random walk on G . For every component T ,
N
1 X
1(Yn ∈ T ) exists and is non-random
n→∞ N
Freq(T ) := lim
n=1
(Burton-Keane, Lyons-Schramm).
Number of components.
Suppose for contradiction that there are 1 < k < ∞ components.
Let hYn in≥0 be a simple random walk on G . For every component T ,
N
1 X
1(Yn ∈ T ) exists and is non-random
n→∞ N
Freq(T ) := lim
n=1
(Burton-Keane, Lyons-Schramm).
P
Since there are only finitely many components, ki=1 Freq(Ti ) = 1
and indistinguishability implies Freq(T ) = 1/k for every
component T .
Number of components.
Suppose for contradiction that there are 1 < k < ∞ components.
Let hYn in≥0 be a simple random walk on G . For every component T ,
N
1 X
1(Yn ∈ T ) exists and is non-random
n→∞ N
Freq(T ) := lim
n=1
(Burton-Keane, Lyons-Schramm).
P
Since there are only finitely many components, ki=1 Freq(Ti ) = 1
and indistinguishability implies Freq(T ) = 1/k for every
component T .
Using updates, we can join an infinite piece of one component onto
another component.
Number of components.
Suppose for contradiction that there are 1 < k < ∞ components.
Let hYn in≥0 be a simple random walk on G . For every component T ,
N
1 X
1(Yn ∈ T ) exists and is non-random
n→∞ N
Freq(T ) := lim
n=1
(Burton-Keane, Lyons-Schramm).
P
Since there are only finitely many components, ki=1 Freq(Ti ) = 1
and indistinguishability implies Freq(T ) = 1/k for every
component T .
Using updates, we can join an infinite piece of one component onto
another component.
The infinite piece has to have a positive limsup frequency with
positive probability. (Similar to transient components not having
recurrent branches.)
Number of components.
Suppose for contradiction that there are 1 < k < ∞ components.
Let hYn in≥0 be a simple random walk on G . For every component T ,
N
1 X
1(Yn ∈ T ) exists and is non-random
n→∞ N
Freq(T ) := lim
n=1
(Burton-Keane, Lyons-Schramm).
P
Since there are only finitely many components, ki=1 Freq(Ti ) = 1
and indistinguishability implies Freq(T ) = 1/k for every
component T .
Using updates, we can join an infinite piece of one component onto
another component.
The infinite piece has to have a positive limsup frequency with
positive probability. (Similar to transient components not having
recurrent branches.)
This implies that the frequency of the component we added this piece
to must have increased, contradicting update tolerance.
Overview of our proof
(Indistinguishability for the WUSF.) This case is much less similar
to the Lyons-Schramm setting.
Every property is either a tail property, meaning that the type of a
component doesn’t change when we make finite modifications, or there
exist ‘good’ pivotal edges.
We analyse the two cases separately:
(non-tail properties.) We adapt the Lyons-Schramm argument. Since
the walk is recurrent, we need to find a different way of choosing edges
in such a way that the probability we chose a particular edge changes
in a controllable way when we update at that edge. We do this using
Bernoulli bond percolation.
(tail properties.) For this case we have a completely separate
argument, based on Wilson’s algorithm and the spatial Markov
property of the WUSF. Also works for general transitive graphs.