ALEA, Lat. Am. J. Probab. Math. Stat.

18, 289–308 (2021)

DOI: 10.30757/ALEA.v18-13

Critical parameters for loop

and Bernoulli percolation

Peter Mühlbacher
University of Warwick
CV4 7AL,
Coventry, England, United Kingdom.
E-mail address: peter@muehlbacher.me
URL: http://peter.muehlbacher.me

Abstract. We consider a class of random loop models (including the random in-
terchange process) that are parametrised by a time parameter β ≥ 0. Intuitively,
larger β means more randomness. In particular, at β = 0 we start with loops of
length 1 and as β crosses a critical value βc , infinite loops start to occur almost
surely. Our random loop models admit a natural comparison to bond percolation
with p = 1−e−β on the same graph to obtain a lower bound on βc . For those graphs
of diverging vertex degree where βc and the critical parameter for percolation have
been calculated explicitly, that inequality has been found to be an equality. In
contrast, we show in this paper that for graphs of bounded degree the inequality is
strict, i.e. we show existence of an interval of values of β where there are no infinite
loops, but infinite percolation clusters almost surely.

1. Introduction

The loop models considered here are percolation type probabilistic models with
intimate connections to the correlation functions of certain quantum spin systems.
These connections were first discovered in Tóth (1993); Aizenman and Nachtergaele
(1994). Let G = (V, E) be a graph and β > 0, u ∈ [0, 1] be two parameters. To
each edge e ∈ E is assigned a time interval [0, β], and an independent Poisson
point process Xe with two kinds of outcomes: “crosses" occur with intensity u and
“double bars" occur with intensity 1 − u.
Given a realisation (Xe )e∈E , we consider the loop passing through a point (x, t) ∈
V ×[0, β] that is defined as follows (see Fig. 1.1). The loop is a closed trajectory with
support on V × [0, β]per where [0, β]per is the interval [0, β] with periodic boundary
conditions, i.e. the torus of length β. Starting at (x, t), move “up" until meeting

Received by the editors 2020-06-28; accepted 2020-10-03.

2010 Mathematics Subject Classification. 60K35, 82B26.
Key words and phrases. Random interchange, percolation, critical parameter.
289
290 P. Mühlbacher

the first cross or double bar with endpoint x; then jump onto the other endpoint,
and continue in the same direction if a cross, in the opposite direction if a double
bar; repeat until the trajectory returns to (x, t). A loop is called infinite if it visits
infinitely many different vertices at time 0, and finite otherwise. This property
is more well-known in the context of random stirring models, i.e. u = 1, first
introduced by Harris (1972) as finite/infinite permutation cycles. Here, however,
we want to extend it to u 6= 1 and avoid technical difficulties encountered when
defining permutations of infinite sets. We note furthermore that this unorthodox
naming may be justified by the following observation (pointed out already, e.g.,
in Hammond and Hegde, 2019): On connected graphs of bounded degree having
loops visiting infinitely (or finitely) many vertices at any time is characterised by
the almost sure presence (or absence), of an “infinite loop" as defined here.
For graphs of sufficiently high vertex degree one expects a phase transition in
the sense that there is a critical (loop) parameter βc > 0 such that (a) for β < βc
there are only finite loops and (b) for β > βc there are infinite loops almost surely.
Resolving (b) is subject of ongoing research: Results have been obtained for the
complete graph (Schramm, 2005; Berestycki, 2011 for u = 1, Björnberg et al., 2019
for u ∈ [0, 1]), the hypercube (Kotecký et al., 2016 for u = 1), trees (Angel, 2003;
Hammond, 2013, 2015 for u = 1, Björnberg and Ueltschi, 2018a; Hammond and
Hegde, 2019 for u ∈ [0, 1]) and the Hamming graph (Miłoś and Şengül, 2019 for
u = 1) and it remains an open problem for G = Zd with d ≥ 2. We can, however,
easily show (a) as follows: Loop models possess a natural percolation structure
when viewing any edge e with Xe not empty as opened; this occurs independently
for all e ∈ E with probability 1 − e−β . We call this the percolation model with
the corresponding parameter. Since the set of vertices visited by any loop (at any
time) must be contained in a single percolation cluster, only finite loops occur when
percolation clusters are finite. Choosing the critical loop parameter for percolation
per
βcper such that 1 − e−βc = pc (G, bond), the critical parameter for bond percolation
on G, we conclude that for β < βcper all loops visit only finitely many vertices almost
surely and hence βc ≥ βcper .
For a number of models with vertices of diverging degree it has been shown
that that the above bound is in fact sharp in the sense that βc = βcper (Schramm,
2005; Miłoś and Şengül, 2019). This is conjectured more generally for any graphs of
diverging vertex degree, most notably the hypercube. For d-regular trees it has been
shown in Angel (2003); Björnberg and Ueltschi (2018a) (for u = 1 and u ∈ [0, 1])
that βc and βcper agree to first order in d−1 as d → ∞. Ultimately we are interested
in proving statements like (a) and (b) for “small dimensions", e.g. G = Z3 , and
in this case, as suggested by the asymptotic expansion around d = ∞ in Angel
(2003); Björnberg and Ueltschi (2018a) and numerics Barp et al., 2015, we do not
expect βc = βcper . In fact this follows for d-regular trees and all u ∈ [0, 1] from Betz
et al. (2018b, Remark 2.13). The contribution of this paper is to give a rigorous
and robust proof of this statement for connected, countably infinite graphs G of
uniformly bounded degree and u ∈ (0, 1], thus extending one particular implication
of Betz et al. (2018b) to more general graphs. This leaves out the case u = 0.
We discuss in Subsection 4.4 why this is hard and why new results on dependent
percolation might be needed.
In Section 2 we introduce some more notation and state our main result, Theo-
rem 2.1, rigorously. In Section 3 we prove our main result. In Section 4 we discuss
Critical parameters for loop and Bernoulli percolation 291

β β

Figure 1.1. Graphs and realizations of Poisson point processes,

and their loops. In both cases, there are exactly two loops, one in
red and one in blue.

briefly some natural generalisations and what is expected to happen when certain
assumptions of Theorem 2.1 are dropped. This includes a short proof of an anal-
ogous result for some expander graphs, as well as discussions how straightforward
generalisations to other physically relevant models and parameter regimes are not
covered by our methods or expected to fail.

1.1. Notation. To avoid confusion we highlight briefly some notation:

• For two edges e, ẽ we write ẽ ∼ e as a shorthand for “e is adjacent to ẽ",
i.e. they share a vertex.
• We write cycles for permutation cycles (subsets of V ) and loops for closed
trajectories in V × [0, β].
• A local configuration is a configuration of links (crosses or double bars) on
finitely many edges, e.g. “two adjacent edges with at least one link each".

2. Setting and Result

Consider any countably infinite, connected, undirected graph G = (V, E) of uni-

formly bounded degree, i.e. deg(v) ≤ ∆ for some ∆ ∈ N and all v ∈ V . From now
on all graphs will be considered to be undirected, at most countably infinite and of
uniformly bounded degree. In our notation we will sometimes suppress dependency
on ∆, which we think of as fixed from now on. We consider bond percolation on G.
More precisely we take Ω := {0, 1}E and equip it with the σ-algebra generated by
finite cylinder sets, as well as the (Bernoulli) product measure with parameter p:
πp := ⊗e∈E νe , where νe (1) = p = 1 − νe (0), to obtain a probability space. Define
its critical parameter pc and its critical loop parameter for percolation βcper by
pc = pc (G, bond) := inf{p : πp (|C(v)| = ∞) > 0} and βcper := − ln(1 − pc ),
(2.1)
292 P. Mühlbacher

where C(v) ⊆ V is the set of vertices that are connected to v via a path of open
edges. Since G is connected, a standard argument shows that the definition is
indeed independent of the choice of v.
Denote the joint law of the Poisson point processes (Xe )e∈E =: X (henceforth
also referred to as configuration) as given in the introduction by Pβ,u and let the
random variable L(v) be the set of vertices that the loop starting at (v, 0) ∈ V ×[0, β]
visits at time 0, i.e.
L(v) = L(v)(X) := {w ∈ V : (v, 0) ↔ (w, 0)}. (2.2)
Here {(v, t) ↔ (w, t )} is the event that the two space-time points (v, t) and (w, t0 )
0

are traversed by the same loop. Similarly to the percolation case one can easily
show, see e.g. Angel (2003, Proposition 5), that
βc = βc (u) := inf{β : Pβ,u (|L(v)| = ∞) > 0} (2.3)
is well-defined, i.e. it does not depend on the choice of v.
We can now state the main theorem:
Theorem 2.1. For all countably infinite, connected graphs G of uniformly bounded
degree with pc (G, bond) < 1 and all u ∈ (0, 1] we have βc (u) > βcper .
The condition pc (G, bond) < 1 is a mere technicality, needed to exclude boring
graphs like G = Z. Note that pc = 1 implies βcper = ∞ and by the observation in
the introduction (βc (u) ≥ βcper ) we get the rather uninteresting equality “∞ = ∞"
in these cases.
The result can easily seen to hold for graphs of unbounded degree as long as
their critical percolation parameter coincides with that of an induced subgraph of
uniformly bounded degree.
To put it in the language of percolation theory, the main ingredient of our proof,
Proposition 3.2, implies that there is an interval of β for which infinite percolation
clusters for (the naturally coupled) bond percolation, but no infinite loops occur
almost surely. More precisely, we obtain (as a special case) the following:
Corollary 2.2. Consider G = Zd , d ≥ 2, u ∈ (0, 1]. There exist 0 < β1 < β2 < ∞
(depending only on d, u) such that for all β ∈ (β1 , β2 ) we have constants a, b > 0
depending only on d such that:
• Pβ,u (|L(0)| = k) ≤ ae−bk for all k ∈ N (small loops),
• πp (|C(0)| = ∞) > 0 for p = 1 − e−β (infinite percolation cluster).
We stress once more that this result is not expected to be true for graphs of
diverging degree; see Conjecture 4.2 for a rigorous statement. For u = 1 it has
already been ruled out for the complete graph (Schramm, 2005), the hypercube
(Kotecký et al., 2016), and the Hamming graph (Miłoś and Şengül, 2019).

3. Proof of Theorem 2.1

3.1. Intuition. Intuitively Theorem 2.1 tells us that for β = βcper and all v ∈ V we
have that L(v) is significantly smaller than the cluster C(v) of the naturally coupled
percolation process, i.e. bond percolation with p = 1 − e−β . This is due to cancel-
lations. Consider, for example, the empty configuration. Adding one cross at some
edge e = {v, w} will result in {(v, 0) ↔ (w, 0)} since the time interval is periodic
and there are no other links, but after adding a second cross on the same edge that
Critical parameters for loop and Bernoulli percolation 293

connectivity is lost again. The main idea of this proof is to identify and remove
a local configuration that increases the likelihood of Bernoulli percolation without
making the loop larger and show that it occurs sufficiently often in a sufficiently in-
dependent manner. The main challenge is the “sufficiently independent" part, since
it is not enough to show that such a local configuration occurs on a positive fraction
of edges in the graph: Consider bond percolation on Z2 , barely above criticality,
and let us close a positive fraction of open edges. Can we conclude that there is no
infinite cluster of open edges anymore? We want to draw attention to Aizenman
and Grimmett (1991) (and corrections in Balister et al., 2014) to emphasise that
this is not a trivial question. Their results do not imply ours, however. One obvi-
ous limitation is that their results apply to Zd only. Another one is that they only
prove the difference of critical parameters which does not imply Corollary 2.2. We
also want to stress the main conceptual difference: While Aizenman and Grimmett
(1991) proves that infinite percolation clusters can be split up into finite ones by
removing certain edges, we show the significantly stronger statement that removing
certain dependent enhancements results in a process that is bounded by a strictly
smaller Bernoulli percolation process whose entries are again independent. This
constitutes a more robust approach that also generalises to several other settings
some of which are mentioned in Section 4.
So unless the edges to be closed are sampled from another (independent)
Bernoulli percolation measure, the answer is far from obvious. If they are, however,
then it is easy to see that we just obtain another percolation process with strictly
smaller parameter p0 . This will be our goal.

3.2. Preliminary reductions. To make that intuition precise we need to introduce

some more notation. For all e ∈ E and a < b let Ne (a, b] be the number of links
(a link is a cross or a double bar) of the configuration Xe in the interval of heights
(a, b]. For all β > 0 let ne := Ne (0, β] be the total number of links on e and for all
edges e with ne = 2 let 0 ≤ ae < be ≤ β denote the heights of its two links.
Now we couple (Xe )e∈E to a Bernoulli bond percolation process S = (Se )e∈E by
colouring each edge e ∈ E either red, blue, or uncoloured:
(1) Red (Re = 1) if e has exactly two links (ne = 2), both of which are crosses
and such that Nẽ (ae , be ] = 0 for all ẽ ∼ e.
(2) Blue (Be = 1) if e has at least one link (ne ≥ 1) and e is not red.
(3) Uncoloured /closed (Se = 0) if e is neither red, nor blue and
coloured /open (Se = 1) if e is red or blue (or, equivalently, if ne ≥ 1).
Introduce the shorthands R = (Re )e∈E , B = (Be )e∈E , S = (Se )e∈E and note
that S = B + R. Thus S is a Bernoulli bond percolation process with parameter
1 − e−β . It is easy to see that vertices connected by loops at time 0 must be subsets
of percolation clusters, i.e. for any v ∈ V we have L(v) ⊆ C(v), where C(v) is the
vertex set of the connected component containing v of the subgraph G0 obtained
by removing all edges e with Se = 0. This percolation bound is too generous. Note
that such vertices must in fact be subsets of blue clusters CB (v), the set of vertices
connected by blue edges to v:

Lemma 3.1. For all v ∈ V we have

L(v) ⊆ CB (v) ⊆ C(v). (3.1)

294 P. Mühlbacher

Proof : We have CB (v) ⊆ C(v) from the definition: Blue clusters are subsets of
coloured clusters.
Now fix any u, v ∈ V and assume u ∈ L(v). We prove that u must be in CB (v):
By definition it suffices to prove that there exists a path of adjacent, blue edges
(e1 , . . . , em ) with v ∈ e1 , u ∈ em . In particular each ei has at least one link. Now
assume there was no such sequence, but u ∈ L(v): This is either because there is
no such path from v to u of open (potentially also red) edges at all or there are
paths of open edges from v to u, but all of them have at least one red edge in them.
The former case readily yields a contradiction, since the loop emanating from (v, 0)
cannot possibly visit u at any time if there exists a cutset of edges without any
links on them. In the latter case, we get a contradiction as well since red edges
are defined so that removing them leaves L(v) invariant and then the previous case
applies.
It remains to show that L(v) is invariant under removing red edges for all v ∈ V :
Consider any configuration X with at least one uncoloured edge and make some
uncoloured edge a red edge. We can obtain every configuration this way, i.e. by
putting two crosses (without any links on adjacent edges in between) on an edge
that didn’t have any links on it before. So it suffices to prove that adding one red
edge cannot change L(v) for any v. Fix any loop that is affected by adding this
red edge: It is easy to see that its position at times t = kβ, k ∈ Z is not changed
since its position is only different from before at times (kβ + a, kβ + b) for some
0 < a < b < β. Hence L(v) stays invariant. 

Now note that Theorem 2.1, i.e. βcper < βc , is equivalent to the existence of a β
such that
(i) there is an infinite percolation (i.e. coloured) cluster with positive proba-
bility, but
(ii) there is no infinite cycle a.s.
By Lemma 3.1, (ii) follows from
(ii’) there is no infinite blue cluster a.s.
To this end we want many edges to be red so that if we choose β barely above
criticality (for percolation) and remove all red edges, this would split up all infinite
percolation clusters into finite (blue) ones.
Note that Be = Se (1 − Re ). So if S and R were independent Bernoulli bond
percolation processes with parameters p, pR ∈ (0, 1), respectively, then B would
be a Bernoulli bond percolation process with parameter p(1 − pR ) < p. Hence
choosing p = pc (G) + ε with ε sufficiently small would give us a corresponding
loop parameter β = − ln(1 − p) for which there is an infinite coloured cluster with
positive probability, but no infinite blue cluster a.s.
Two problems arise: Re is not independent of Re0 , i.e. the law of R is not a
product measure. But it clearly suffices to show that it still dominates a non-
degenerate product measure. This is the hard part and will be done in Proposi-
tion 3.2. The other problem is that R and S are not independent. This can easily
be overcome by noting that it suffices to consider R restricted to coloured edges
E 0 = {e ∈ E : Se = 1} instead and showing that R|E 0 dominates a non-degenerate
product measure on E 0 .
Note furthermore that without loss of generality we might assume u = 1, since
considering u ∈ (0, 1) merely decreases the intensity of red edges by a factor of
Critical parameters for loop and Bernoulli percolation 295

u2 > 0; a factor that can be absorbed by δ in Proposition 3.2. It will not matter if
the other links are crosses or double bars, so henceforth we will suppress u in the
notation and talk about crosses, not links.
We conclude that to prove Theorem 2.1 it suffices to prove the following:

be 3

be 1
ae3
ae1

e1 e2 e3
G

Figure 3.2. An example of a configuration on three edges

e1 , e2 , e3 . All edges are coloured (Se1 = Se2 = Se3 = 1) because
ne1 = ne2 = ne3 = 2 > 0. e1 is red (Re1 = 1) since e1 has two
crosses and no neighbours with a link between ae1 and be1 . e2 is
blue (Be2 = 1) because ne2 > 0, but not both links are crosses. e3
is blue too because it has a neighbour, e2 , with a link between its
two crosses, i.e. Ne2 (ae3 , be3 ] > 0. Note that, given that e1 is red,
the leftmost vertex cannot possibly be in the same loop as the
others.

Proposition 3.2. Fix any β > 0, u = 1 and some G as in Theorem 2.1. Let
G0 = (V 0 , E 0 ) be a (not necessarily connected) subgraph of G. Then there exists
δ > 0 (depending only on β, ∆, but independent of G0 and other properties of G)
such that the Bernoulli product measure πδ is stochastically dominated by the law
of (Re )e∈E 0 conditioned on ne > 0 for all e ∈ E 0 .

Proof that Proposition 3.2 implies Theorem 2.1: As in the above discussion it suf-
fices to prove that there exists a β such that there is an infinite coloured cluster
with positive probability, but no infinite blue cluster a.s. The process S of coloured
edges is just a Bernoulli bond percolation process with parameter p = 1 − e−β . So
it suffices to prove that B is dominated by some Bernoulli bond percolation process
with parameter p0 < p, i.e. there exists a ε > 0 such that for all e0 ∈ E we have

sup Pβ,u (Be0 = 1 | (Be )e∈E\{e0 } ) ≤ p − ε. (3.2)

(Be )e∈E\{e0 }
296 P. Mühlbacher

Now note that

l.h.s. of (3.2)
= sup Pβ,u (Se0 (1 − Re0 ) = 1 | (Se )e∈E\{e0 } , (Re )e:Se =1 )
(Se ,Re )e∈E\{e0 }

= sup Pβ,u (Re0 = 0 | Se0 = 1, (Se )e∈E\{e0 } , (Re )e6=e0 :Se =1 ) · p

(Se ,Re )e∈E\{e0 }

= sup Pβ,u (Re0 = 0 | Se0 = 1, (Se )e∈E 0 = 1, (Se )e6=e0 ∈E

/ 0 = 0, (Re )e∈E 0 ) · p
E 0 ⊆E\{e0 }
(Re )
e∈E 0

≤ (1 − u2 inf Pβ,u=1 (Re0 = 1 | (Se )e∈E 0 ∪{e0 } = 1, (Se )e6=e0 ∈E

/ 0 = 0, (Re )e∈E 0 )) · p
E 0 ⊆E\{e0 }
(Re )
e∈E 0

≤ (1 − u2 δ)p,
where the last inequality is due to Proposition 3.2. 
0
Henceforth we only consider the (now fixed) subgraph G , so we introduce the
shorthand
µ := Pβ,u ( · |ne > 0 ∀e ∈ E 0 (G0 )), (3.3)
where Pβ,u is as in the introduction, but restricted to configurations on edges of G0 .
By Liggett et al. (1997, Lemma 1.1), it suffices to show that
inf µ (Re0 = 1|Re1 = ε1 , . . . , Rem = εm ) ≥ δ, (3.4)
e0 ∈E 0

uniformly over all choices of ei ∈ E 0 \ {e0 }, εi ∈ {0, 1} for i = 1, . . . , m and m ∈ N,

such that µ(Re1 = ε1 , . . . , Rem = εm ) > 0. In what follows we consider e0 ∈ E 0 to
be an arbitrary fixed edge unless explicitly stated otherwise.
Remark 3.3 (Notation). In what follows we will repeatedly abuse notation to ab-
breviate statements like (3.4) and its following paragraph to the shorter “µ(Re0 =
1|{Re1 , . . . , Rem }) ≥ δ uniformly for all {Re1 , . . . , Rem }". We will consider more
general events than Re0 = 1 and more general conditions than {Re1 , . . . , Rem }.
Conditions {C1 = c1 , C2 = c2 , . . . , Cm = cm } will be called admissible if there
exists a configuration X such that C1 (X) = c1 , C2 (X) = c2 , . . . . One might worry
about conditioning on such admissible conditions since they may be probability
zero events, but we do this only for the sake of an accessible presentation. Instead
one might also discretise time so that there is no need to condition on probability
0 events and take limits in the end.
For example Re1 = 1 and Re2 = 0 with e1 = e2 is not admissible. A more
interesting example of conditions that are not admissible are Xe1 having two points
at heights a < b, Re1 = 1 (so far it is admissible) and Ne2 (a, b] > 0 for some e2 ∼ e1 .

3.3. A spatial Markov property. In order to avoid having to deal with complicated
long-range correlations, we refine the σ-algebra by allowing to condition not only
on finite collections {Re }e∈IbE 0 , but also on some finite collection {Xe }e∈I 0 bE 0 to
obtain a spatial Markov property. For functions fe : X 7→ fe (X) that depend only
on Xe and its nearest neighbour configurations {Xẽ }ẽ∼e we will abuse notation
and write fe (Xe , Xẽ∼e ) instead of fe (X) to emphasise its dependency on only these
variables. Endow the set of edges E 0 with its natural metric d = dE 0 , given by the
graph distance of edges and define for each A ⊆ E 0 its mollification
(k)
A := {e0 : ∃e ∈ A s.t. d(e, e0 ) ≤ k}.
Critical parameters for loop and Bernoulli percolation 297

The following lemma is essentially a spatial Markov property (given that we have
something like a “security layer" that we have full control over):

Lemma 3.4. For any A ⊆ E 0 and any collection of functions {fe }e∈E 0 , with fe
depending only on the configuration on e and its nearest neighbours (Xe0 )e0 :e0 ∼e , we
have:

 
µ (fe )e∈A |(Xe )e∈E 0 \A , (fe )e∈E 0 \A = µ (fe )e∈A |(Xe )e∈A(2) \A , (fe )e∈A(1) \A .
(3.5)

Proof : Note that none of the functions (fe )e∈E 0 \A(1) depend on the values of
(Xe )e∈A . Similarly, none of the functions that depend on Xe for any e ∈ A,
i.e. (fe )e∈A(1) , depend on (Xe )e∈E 0 \A(2) . Since (Xe )e∈E 0 \A(2) and (Xe )e∈A(2) are
independent under µ, the result follows. 

We will use Lemma 3.4 for fe = Re for e ∈ / A. If for e ∈ A we set fe = id,

we obtain that, conditioning as in (3.5), Xe is independent of the outside, i.e.
(Xe )e∈E\A(2) , (Re )e∈E\A(1) or in other words: It only depends on the “boundary"
(2)
A \ A. The same holds for fe = Re and fe = ne := Ne (0, β], i.e. the occurrence
of double crosses and the number of crosses within A, respectively.

Corollary 3.5. For arbitrary A ⊆ E 0 such that e0 ∈ A, all choices of Ẽ ⊆ E 0 with

|Ẽ| < ∞, e0 ∈
/ Ẽ and all choices of (Re )e∈Ẽ we have
   
µ Re0 = 1 (Re )e∈Ẽ ≥ inf µ Re0 = 1 (Re )e∈Ẽ , (Xe )e∈A(2) \A
(Xe )e∈A(2) \A
 
(3.5)
= inf µ Re0 = 1 (Re )e∈Ẽ∩A(1) , (Xe )e∈A(2) \A ,
(Xe )e∈A(2) \A

(3.6)

where the infimum goes over all admissible configurations (Xe )e∈A(2) \A .

Thus to prove Proposition 3.2 it suffices to prove that there is a δ > 0 such that
the r.h.s. of (3.6) is bounded from below by δ for all choices of (Re )e∈A(1) \{e } for
0
some A as in the corollary.
We want to choose A as small as possible. It may be tempting to choose A =
{e0 }, but this does not work since one could prescribe an arbitrarily high density
of crosses on one of the neighbours of e0 making the probability of satisfying the
conditions to colour an edge red arbitrarily small. Instead we choose A = {e0 }∪{ẽ ∈
E 0 : ẽ ∼ e0 } and prove that the probability of colouring e0 red cannot be made
arbitrarily small in this case.

3.4. Pivotal edges. From now on we consider e0 and A = {e0 } ∪ {ẽ ∈ E 0 : ẽ ∼ e0 }

fixed. Motivated by Lemma 3.4 we will refer to conditioning on some admissible
event B = {(Re )e∈A(1) \{e } , (Xe )e∈A(2) \A } as boundary conditions B. We formalise
0
the intuition that we may restrict our attention to finitely many classes of config-
urations that “influence" their adjacent edges’ configurations. For these classes the
desired properties can then be checked directly.
298 P. Mühlbacher

Definition 3.6. Given some configuration X, an edge e0 ∈ E 0 , and an event

(written as a function f : X → {0, 1}), we say e0 is pivotal for f , if f is not
invariant under changes to Xe0 when keeping (Xe )e6=e0 fixed.
For all edges ẽ we denote by Pẽ the random event that e0 is pivotal for Rẽ .
See Figure 3.3 for an illustration. Clearly no e0 outside of the support of f is
pivotal for f . Here we define the support of local functions f as in Friedli and
Velenik (2018, Definition 3.11) for local functions: The smallest subset Ẽ ⊆ E 0
such that if X agrees with X 0 on Ẽ, then f (X) = f (X 0 ). The following lemma will
provide more interesting examples:
Lemma 3.7. Fix any ẽ ∼ e0 and let B be any boundary condition that does not
prescribe Rẽ . Then the event Pẽ (e0 is pivotal for Rẽ ) is true if and only if nẽ = 2
and there is no edge ẽ˜ 6= e0 adjacent to ẽ with Nẽ˜(aẽ , bẽ ] > 0.
Proof : Recall the condition for colouring e red. The condition that nẽ = 2 is clear,
since if that was not the case Rẽ = 0 irrespectively of the configurations of any of
its neighbours. Note furthermore that once there is a cross between the two crosses
on ẽ, Rẽ = 0 irrespectively of all the other configurations. On the other hand, if Pẽ
is satisfied, one can always choose two different configurations Xe0 such that ẽ is
not red with one configuration (place a cross inside (aẽ , bẽ ]), but red with the other
(place no cross inside (aẽ , bẽ ]). 

Remark 3.8. The reason we are interested in non-pivotal edges is that conditioning
on f such that e is not pivotal for f leaves the distribution of Xe invariant. More
precisely we have that the (conditional) law of Xe0 can be written as



L Xe0 |{Xe , Re }e∈E 0 \{e0 } = F {Xe , Re }e:1Pe =1 , (3.7)
with F some function not depending on any other parameters (other than the fixed
∆, β) such that F (∅) is µ restricted to e0 , i.e. the law of a Poisson point process of
intensity 1 on [0, β], conditioned to have at least one cross.
Note that in our setting (Xe )e∼e0 will be random and hence the right hand side
of (3.7) (Pe , to be precise) will in general still depend on configurations of these
non-pivotal edges.
Definition 3.9. Given any boundary conditions B, we call nearest neighbours
e ∼ e0 with 1Pe = 1 and Re = 0 (or Re = 1) neighbours of type 0 (or type 1).
In the next lemma we will show that, at the price of a constant that is uniform
over all admissible boundary conditions, we may assume that e0 has no type 0
neighbours.
Lemma 3.10. Fix arbitrary boundary conditions B. Then
!
\
µ Pec B ≥ 2(∆−1) =: η, (3.8)
e:Re =0

where  ∈ (0, 1] is bounded away from 0 uniformly over all admissible boundary
conditions B, for any fixed β > 0, ∆ ∈ N.
Proof : Note that it suffices to prove that
µ Pẽc Oẽ ≥ ,


(3.9)
Critical parameters for loop and Bernoulli percolation 299

e0 ẽ ẽ˜ G
β

e0 ẽ ẽ˜ G

Figure 3.3. Examples for e0 being pivotal for f = Rẽ , i.e. Pẽ
(left) and Pẽc (right). The shaded region indicates that Xe0 is not
conditioned on, i.e. is still random. Note that conditioning on Rẽ
being equal to 0 (or 1) will make ẽ a type 0 (or a type 1) neighbour
of e0 in the left picture. In the right picture conditioning on Rẽ = 1
is not admissible.

for  > 0 as in the lemma, and some outer configuration Oẽ := {(Xe , Re )e∈E 0 \{e0 ,ẽ} ,
Rẽ = 0} arbitrary with any ẽ such that ẽ ∼ e0 . This is sufficient because the
family of outer configurations Oẽ is strictly larger than that of arbitrary boundary
conditions B and hence can only decrease the lower bound. Denote by K the event
(defined on {nẽ = 2}) that there is an edge ẽ˜ 6= e0 adjacent to ẽ which has a cross
between the two crosses on ẽ; by Lemma 3.7 we have Pẽc = {nẽ 6= 2}∪(K∩{nẽ = 2}).
Clearly
µ Pẽc Oẽ = µ K ∩ {nẽ = 2} Oẽ + µ nẽ 6= 2 Oẽ .




(3.10)
Now assume for contradiction that for every ε > 0 there is an outer configuration
Oẽ (ε) such that both terms on the right hand side of (3.10) are smaller than ε/2.
In particular (negating the first term) this implies that
ε
µ (K c ∩ {nẽ = 2}) ∪ {nẽ 6= 2} Oẽ (ε) ≥ 1 − .


(3.11)
2
300 P. Mühlbacher

By assumption we have µ(nẽ 6= 2|Oẽ (ε)) < ε/2 and thus

µ K c ∩ {nẽ = 2} Oẽ (ε) ≥ 1 − ε.

(3.12)

It remains to show that for ε > 0 sufficiently small there cannot be an outer configu-
ration Oẽ (ε) such that both µ(nẽ 6= 2|Oẽ (ε)) < ε/2 and (3.12) hold simultaneously.
Note that to satisfy the former constraint for ε sufficiently small we surely have to
have at least one condition that changes the law of Xẽ . By Lemma 3.7 and Re-
mark 3.8 (with ẽ taking the role of e0 ) this can only be done by having neighbours
of type 0 or 1. We cannot condition on neighbours being of type 0, however, since
K would be true in this case and hence we could not possibly satisfy (3.12). On the
other hand it is easy to see that neighbours of type 1 cannot possibly increase the
number of crosses in probability – note that the result of conditioning only on type
1 neighbours is again a Poisson point process of the same intensity (and conditioned
to have at least one cross) on a strict subset of [0, β]. Noting that under the law
of a Poisson point process of intensity 1 on [0, β] conditioned to have at least one
cross there is a strictly positive probability of having less than 2 crosses gives the
desired contradiction and thus finishes the proof. 

We showed that with uniformly positive probability e0 is not pivotal for ẽ ∼ e0

with Rẽ = 0 and hence the law of Xe0 is not influenced. So we still need to
show that for these edges nẽ < N with uniformly positive probability for some
sufficiently large, but finite N — otherwise the probability of {Re0 = 1} could be
made arbitrarily small.

Lemma 3.11. Fix arbitrary boundary conditions B. Then

 
 X \  1
µ ne < N Pec , B ≥ , (3.13)
2
e:Re =0 e:Re =0
e∼e0

for sufficiently large N ∈ N that depends only on β, ∆.

Proof : We show a stronger statement. For any fixed edge e0 with arbitrary condi-
tioning “on the outside" Oe0 = {(Xe )e∈E\{e0 } , (Re )e∈E } we have
! !
\ 3.10 \
µ ne0 > Ñ Pec , Oe0 η ≤ µ ne0 > Ñ , Pec |Oe0
e:Re =0 e:Re =0
Eµ (ne0 |Oe0 )
≤
Ñ
1 + |{e : e ∼ e0 }| + β 1 + 2(∆ − 1) + β
≤ ≤ .
Ñ Ñ
(3.14)

The third inequality follows from Lemma 3.7 and Remark 3.8 (with e0 taking the
role of e0 ). Note again that type 0 neighbours only decrease the number of points
Critical parameters for loop and Bernoulli percolation 301

and every type 1 neighbour can force at most one additional cross on e0 . Hence
 \ \ 
µ {ne ≤ Ñ } Pec , B
e:e∼e0
Re =0
e:Re =0
 
≥ inf µ̃ 1{ne0 ≤ Ñ } 1{ne ≤ Ñ : ∀e ∈ F } Oe0 ,F
Oe0 ,F | {z }| {z }
=:f =:g
Z Z
= inf dµ̃((Xe )e∈F |Oe0 ,F )g((Xe )e∈F ) dµ̃(Xe0 |Oe0 ,F , (Xe )e∈F )f (Xe0 )
Oe0 ,F
 
(3.14) 1 + 2(∆ − 1) + β −1
Z
≥ inf dµ̃((Xe )e∈F |O )g((Xe )e∈F ) × 1 −
e0 ,F η
Oe0 ,F Ñ
 2(∆−1)
1 + 2(∆ − 1) + β −1
≥ ··· ≥ 1 − η , (3.15)
Ñ
where e0 ∼ e0 , µ̃ = µ(· | e:Re =0 Pec ), F = {e ∈ E : e 6= e0 , e ∼ e0 } and Oe0 ,F =
T

{(Xe )e∈E\{e:e∼e0 } , (Re )e∈E }. For Ñ sufficiently large, implies the result. 
Now that we can be sure there is not going to be an “unlimited number" of crosses
on edges adjacent to e0 (Lemma 3.11) and that we may condition on the case that
e0 is only pivotal for Re that are conditioned to be 1 (Lemma
S 3.10), the last thing to
prove is that there are no boundary conditions that make e:e∼e0 (ae , be ] arbitrarily
Re =1
close to [0, β]. (Recall that 0 ≤ ae < be ≤ β are the heights of the first and the
second cross on the edge e respectively.)
Lemma 3.12. Fix arbitrary boundary conditions B. Then
 
[ β \ X
µ  [0, β] \ (ae , be ] ≥ Pec , ne < N, B ≥ 0 , (3.16)
e:e∼e0 2 e:e∼e0
Re =1
e:Re =0 Re =0

for some sufficiently small 0 > 0 that depends only on β, ∆.

Proof : It is enough to show that
 
β 1
µ bẽ − aẽ < Oẽ ≥ 00 (3.17)
2 2(∆ − 1)
for any red ẽ adjacent to e0 and arbitrary admissible conditioning on its outside,
i.e. Oẽ := {(Xe )e∈E 0 \{ẽ} , (Re )e∈E 0 \{ẽ} , Rẽ = 1} and P some 00 > 0 independently of
ẽ and Oẽ . Note that the set of configurations where e:Re =0 ne < N is a subset of
e∼e0

the set of admissible Oẽ . The condition that e:Re =0 Pec is also included. To see
T
this, note that Pec either because (a) ne 6= 2 (in that case it is included in the Xe
conditioning), or (b) there is some ẽ˜ ∈ / {ẽ, e0 } s.t. Nẽ˜(ae , be ] > 0 – again, this is
included in the Xẽ˜ conditioning – or (c) Nẽ (ae , be ] > 0, but this is not admissible,
since ẽ is red by assumption.
By Lemma 3.7 and Remark 3.8 (with ẽ taking the role of e0 ) only type 0 and type
1 neighbours affect the distribution of Xẽ (and hence bẽ −aẽ which is a deterministic
function of Xẽ ). For all type 0 and type 1 neighbours we define their support to be
the interval between the heights of their two links. Recall that type 0 neighbours
e (of ẽ) “force a link" on ẽ at some height in their support. Having multiple type
0 neighbours such that the union of their supports has more than two connected
302 P. Mühlbacher

ẽ e0 ẽ0 G
Rẽ = 1 Rẽ0 = 1
β

ẽ e0 ẽ0 G

Figure 3.4. The left picture illustrates how the probability of

e0 being red is small if there are many crosses on its neighbours,
while the right picture illustrates what could go wrong with type 1
neighbours. The area shaded in red emphasises that, in order not
to violate Rẽ = 1 and Rẽ0 = 1, no crosses are allowed in that
region.

components is not admissible, since then we would require nẽ > 2. Multiple type
0 neighbours such that the union of their supports has exactly two connected com-
ponents is also not admissible since this would violate Rẽ = 1. Having type 0
neighbours such that the union of their supports (a, b] is connected, together with
Rẽ = 1, just forces both links on ẽ to be in (a, b], hence making bẽ − aẽ smaller in
probability.
For the same reason type 1 neighbours also only make bẽ − aẽ smaller in proba-
bility, as it results in Xẽ being a Poisson point process (conditioned to have at least
one cross) on a disconnected subset of [0, β] of the same intensity. Thus we get
   
β 1 β 1
µ bẽ − aẽ < Oẽ ≥ µ bẽ − aẽ < nẽ = 2 . (3.18)
2 2(∆ − 1) 2 2(∆ − 1)
Critical parameters for loop and Bernoulli percolation 303

Noting that a Poisson point process of intensity 1 on [0, β], conditioned to have two
crosses, has a strictly positive probability of placing both crosses within any given
(positive) distance finishes the proof. 

3.5. Reduction to a Poisson point process. With these lemmas we can now proceed
to prove Proposition 3.2 as follows:
!
\
c
µ (Re0 = 1|B) ≥ µ Re0 = 1, Pe B
e:Re =0
!
3.10 \
≥ µ R e0 = 1 Pec , B η
e:Re =0
 
3.11 \ X 1
≥ µ Re0 = 1 Pec , ne < N, B η
e:e∼e0 2
e:Re =0 Re =0
 
3.12 \ X β 1
≥ µ Re0 = 1 Pec , ne < N, [0, β] \ ∪ e:e∼e 0 (ae , be ] ≥ , B η0 .
e:e∼e0
Re =1 2 2
e:Re =0 Re =0

(3.19)
Now we are in a setting where e0 has only “small" type 1 neighbours and the
other neighbours have some uniformly bounded number of crosses. We proceed by
remarking once again that the law of Xe0 with only type 1 neighbours is that of
a Poisson S point process conditioned to have at least one cross with intensity 1 on
[0, β] \ ( e:Re =1 (ae , be ]) — the Lebesgue measure of which is ensured to be larger or
e∼e0

equal to β/2. Since the number of neighbours is bounded by 2(∆ − 1) it also has at
most 2(∆−1)+1T disconnected components. Moreover, since we do not condition on
Re0 and we have e:Re =0 Pec , it follows that, conditioning on the exact configuration
of type 1 neighbours, (Xe )e∼e0 :Re =0 and Xe0 are conditionally independent. By
conditioning on every admissible deterministic (Xe )e∼e0 :Re =1 satisfying the bound
from Lemma 3.12 and taking the essential infimum we conclude that
(3.19) ≥ P(S ∩ M )η0 /2 = const(β, ∆) > 0, (3.20)
where S is the event that a Poisson point process (conditioned to have at least one
cross) of intensity 1 on 2(∆ − 1) + 1 disjoint intervals Ii of size |Ii | = β2 2(∆−1)+1
1

drops exactly two crosses at heights a < b in the same interval Ii and M is the
event that N crosses (uniformly distributed on the intervals Ii ) do not fall between
a, b. The probability of the intersection of these events is bounded away from zero.
This concludes the proof of Proposition 3.2. 

4. Generalisations

The result may be extended in various directions. We briefly discuss some of

them.

4.1. Expander graphs and more. Instead of one fixed infinite graph G, consider
now a sequence of finite connected graphs (Gn )n = (Vn , En )n . A natural analogue
of the critical percolation parameter pc is the smallest p such that there exists
304 P. Mühlbacher

a percolation cluster of size proportional to |Vn | with positive probability; more

formally, we define
pc ((Gn )n ) := inf{p : ∃c > 0 : lim πGn ,p (|C(v)| > c|Vn |) > 0}
n
(4.1)
and βcper ((Gn )n ) := − ln(1 − pc ((Gn )n )),
where πGn ,p is the Bernoulli bond-percolation measure with parameter p on Gn .
Note that without further assumptions on the sequence of graphs (Gn )n it is not
clear whether pc ((Gn )n ) even exists. One might think that requiring macroscopic
clusters on finite subgraphs is a much stronger condition than merely infinite clus-
ters on infinite graphs, but it is conjectured (and partially proven) that they are
compatible in the sense that limn pc ((Gn )n ) = pc (G), whenever Gn converges in a
suitable sense to G and all Gn are sufficiently nice – e.g. regular expander graphs
with diverging girth; see Alon et al. (2004) and Benjamini et al. (2012, Conjec-
ture 1.2) for a more thorough discussion. Similarly, we define βc to be
βc ((Gn )n , u) := inf{β : ∃c > 0 : lim PGn ,β,u (|L(v)| > c|Vn |) > 0)}. (4.2)
n

Now note that Proposition 3.2 implies that the law of the blue edges B is stochas-
tically dominated by a Bernoulli product measure with parameter p(1 − δ), where
p = 1 − e−β and δ > 0 depends only on β and the maximal degree of G, but not
on anything else like the number of vertices or its spectral gap, etc. In particular it
is uniformly bounded away from zero when considering any sequence of uniformly
bounded degree graphs.
We now restrict ourselves to “nice" families (Gn )n for which we know that
pc ((Gn )n ) and hence βcper ((Gn )n ) exist (Alon et al., 2004); in this case it follows
from the above discussion:
Corollary 4.1. Let (Gn )n be a family of d-regular expander graphs with diverging
girth. For u ∈ (0, 1] we have
βc ((Gn )n , u) > βcper ((Gn )n ). (4.3)
In particular one observes macroscopic percolation clusters strictly before observ-
ing macroscopic loops. Depending on one’s background this might not be obvious a
priori since amongst bounded degree graphs expander graphs tend to behave most
like the complete graph. It does not come as a surprise, however, when recalling
Hammond (2015) for u = 1 since (locally) expander graphs behave like regular
trees.

4.2. Sequences of graphs of diverging degree. Consider, as before, sequences of finite

connected graphs (Gn )n = (Vn , En )n ; for example one might think of sequences of
complete graphs or hypercubes. In order not to get unnecessarily technical, let
us restrict ourselves to Gn being vertex transitive for all n. Assume the vertex
degree ∆n of Gn diverges as n → ∞. If we chose the same definition of pc and
βc as in Subsection 4.1, then we would get pc = βc = 0. This prompts us to
look at finer scales; it turns out that Θ(∆−1n ) is the correct one for our purposes.
Instead of saying that pc and βc take a certain value for a sequence of graphs, one
usually writes pc = C/∆n , so the critical parameter depends on n. For sequences of
complete graphs, i.e. Gn being the complete graph on n + 1 vertices, one recovers
the Erdős-Rényi model for which it is well-known that pc = 1/∆n = 1/n.
Critical parameters for loop and Bernoulli percolation 305

More formally, we define pc , βc to be the formal expressions

tp tβ
pc ((Gn )n ) := and βc ((Gn )n , u) := , (4.4)
∆n ∆n
where (for a fixed sequence (Gn )n and fixed u) we define
tp := inf{t : ∃c > 0 : lim πGn ,p=t/∆n (|C(v)| > c|Vn |) > 0}, (4.5)
n
tβ := inf{t : ∃c > 0 : lim PGn ,β=t/∆n ,u (|L(v)| > c|Vn | > 0)}, (4.6)
n

whenever the limits are well-defined. Just as before define βcper ((Gn )n ) := − ln(1 −
pc ((Gn )n )). We can now state the conjecture complementing our main result, The-
orem 2.1. This addresses the question what is expected to happen if the assumption
of having uniformly bounded vertex degree is dropped:
Conjecture 4.2. Consider a sequence of finite, connected, vertex transitive graphs
(Gn )n with diverging vertex degree ∆n such that pc ((Gn )n ) and βc ((Gn )n , u) exist.
Then for u ∈ [0, 1] we expect
βc ((Gn )n , u) = βcper ((Gn )n ). (4.7)
We do not expect the vertex transitivity to be a necessary requirement, but it
allows us to easily make sense of “diverging vertex degree". While this conjecture
might extend to non-vertex transitive graphs where the vertex degree for all vertices
is bounded from below by some diverging sequence an → ∞, we do not make any
claims about more pathological graphs, where e.g. only half the vertices are of
bounded degree and the other half diverges quickly.

4.3. Weighing configurations by θ#loops . One natural generalisation is a weighted

version of the loop model presented in the introduction. Tóth showed that a variant
of this model, where permutations receive the weight θ#cycles (θ = 2), is closely
related to the quantum Heisenberg ferromagnet (Tóth, 1993). Another loop model
was introduced by Aizenman and Nachtergaele to describe spin correlations of the
quantum Heisenberg antiferromagnet (Aizenman and Nachtergaele, 1994). These
loop models were combined in order to describe a family of quantum systems that
interpolate between the two Heisenberg models, and which contains the quantum
XY model Ueltschi (2013). In order to represent a quantum model, one should
choose the weight θ = 2, 3, 4, . . . ; quantum correlations are then given in terms
of loop correlations, and magnetic long-range order is equivalent to the presence
of macroscopic loops. Notice that the parameter β plays the role of the inverse
temperature of the quantum spin system, hence the notation.
More specifically, instead of Pβ,u consider Pβ,u,θ , which is characterised by its
Radon-Nikodym derivative with respect to Pβ,u as follows:
dPβ,u,θ θ`(X)
(X) = . (4.8)
dPβ,u Zβ,u,θ
Here Zβ,u,θ is the appropriate normalisation and `(X) denotes the number of loops
in the configuration X. A priori this is only well-defined on finite G.
Note that for θ 6= 1 we introduce some global dependencies in the sense that,
even though the reference measure Pβ,u was a product measure on edges e ∈ E,
the configurations Xe and Xe0 are not independent under Pβ,u,θ anymore. This
creates additional complications which have been addressed in recent works such
306 P. Mühlbacher

as Björnberg (2015, 2016); Alon and Kozma (2018); Björnberg et al. (2020) on the
complete graph, Björnberg and Ueltschi (2018b); Betz et al. (2018a) on trees and
Adamczak et al. (2018) on the Hamming graph.
What process is the loop percolation process naturally compared to? By the lack
of independence of configurations on different edges, Bernoulli percolation gives a
crude bound, but it easy to convince oneself that this is not sharp. In Barp et al.
(2015) it is briefly discussed why the random cluster model, closely related to the
Potts and Ising model, is a potential candidate for a natural comparison giving
sharp bounds in some cases and numerical estimates are given.
We conclude this subsection by noting that our results (Proposition 3.2) are
robust enough to carry over to the θ > 1 case in an appropriate sense, but we lack
another model (like percolation for θ = 1) to naturally compare it to; hence there is
no obvious meaningful generalisation of Theorem 2.1 to this physically interesting
regime.

4.4. The u = 0 case. Numerics Barp et al. (2015) suggest that the result should
be true in this case as well. However, a few new ideas seem to be needed since two
subsequent double bars do not cancel each other out. Can there be an analogue of
Re satisfying (3.4) on finite graphs? By analogue we mean any function H : X →
{0, 1}E with (a) He (X) = 1 only for X such that there is a non-empty configuration
Xe on e, but in terms of loops it would not make a difference if we replaced Xe
by the empty configuration and (b) H = (He )e∈E , as a stochastic process, can be
bounded (stochastically) from below by some Bernoulli percolation measure πδ of
strictly positive parameter δ > 0.
The answer is no: For any configuration enumerate all the vertices of G in an
arbitrary way, pick the first one, say x1 , and mark it with a +; follow the loop
emanating from (x1 , 0) in the upward direction and mark every vertex it visits at
time 0 with a + or −, depending on whether we traverse it in upward or downward
direction, respectively; remove marked vertices from the list, pick the next one from
the list and continue this process until no vertex is left in the list. It is immediate
that unless all vertices are marked with +’s the configuration cannot correspond
to the identity permutation on the vertices of G, which would be obtained if said
analogue of Re was equal to 1 for all e ∈ E. Noting that there is no configuration
with at least one double bar for which the above algorithm does not assign a “−" to
at least one vertex – hence it cannot be equal to the configuration where no double
bars have been added – concludes the argument.
It appears that this is not merely a technicality. This is because even after
restricting our attention to infinite graphs, there does not seem to be an obvious
analogue of Re to satisfy (3.4) essentially by a quantified version of the above
argument when conditioning on Xe on some boundary.

Acknowledgements

I thank Daniel Ueltschi and Eleanor Archer for many useful discussions.
Critical parameters for loop and Bernoulli percolation 307

References

Adamczak, R., Kotowski, M., and Miłoś, P. Phase transition for the interchange
and quantum heisenberg models on the hamming graph. ArXiv Mathematics
e-prints (2018). arXiv: 1808.08902.
Aizenman, M. and Grimmett, G. Strict monotonicity for critical points in per-
colation and ferromagnetic models. J. Statist. Phys., 63 (5-6), 817–835 (1991).
MR1116036.
Aizenman, M. and Nachtergaele, B. Geometric aspects of quantum spin states.
Comm. Math. Phys., 164 (1), 17–63 (1994). MR1288152.
Alon, G. and Kozma, G. The mean-field quantum heisenberg ferromagnet via
representation theory. ArXiv Mathematics e-prints (2018). arXiv: 1811.10530.
Alon, N., Benjamini, I., and Stacey, A. Percolation on finite graphs and isoperi-
metric inequalities. Ann. Probab., 32 (3A), 1727–1745 (2004). MR2073175.
Angel, O. Random infinite permutations and the cyclic time random walk. In Dis-
crete random walks (Paris, 2003), Discrete Math. Theor. Comput. Sci. Proc., AC,
pp. 9–16. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2003). MR2042369.
Balister, P., Bollobás, B., and Riordan, O. Essential enhancements revisited. ArXiv
Mathematics e-prints (2014). arXiv: 1402.0834.
Barp, A., Barp, E. G., Briol, F.-X., and Ueltschi, D. A numerical study of the 3D
random interchange and random loop models. J. Phys. A, 48 (34), 345002, 12
(2015). MR3376038.
Benjamini, I., Boucheron, S., Lugosi, G., and Rossignol, R. Sharp threshold for
percolation on expanders. Ann. Probab., 40 (1), 130–145 (2012). MR2917769.
Berestycki, N. Emergence of giant cycles and slowdown transition in random
transpositions and k-cycles. Electron. J. Probab., 16, no. 5, 152–173 (2011).
MR2754801.
Betz, V., Ehlert, J., and Lees, B. Phase transition for loop representations of
quantum spin systems on trees. J. Math. Phys., 59 (11), 113302, 13 (2018a).
MR3877248.
Betz, V., Ehlert, J., Lees, B., and Roth, L. Explicit phase conditions for random
loop models on trees using an exploration scheme. ArXiv Mathematics e-prints
(2018b). arXiv: 1812.03937.
Björnberg, J. E. Large cycles in random permutations related to the Heisenberg
model. Electron. Commun. Probab., 20, 11 pp. (2015). DOI: 10.1214/ECP.v20-
4328.
Björnberg, J. E. The free energy in a class of quantum spin systems and inter-
change processes. Journal of Mathematical Physics, 57 (7), 073303 (2016). DOI:
10.1063/1.4959238.
Björnberg, J. E., Fröhlich, J., and Ueltschi, D. Quantum spins and random
loops on the complete graph. Comm. Math. Phys., 375 (3), 1629–1663 (2020).
MR4091504.
Björnberg, J. E., Kotowski, M., Lees, B., and Miłoś, P. The interchange process
with reversals on the complete graph. Electron. J. Probab., 24, Paper No. 108,
43 (2019). MR4017126.
Björnberg, J. E. and Ueltschi, D. Critical parameter of random loop model on
trees. Ann. Appl. Probab., 28 (4), 2063–2082 (2018a). MR3843823.
Björnberg, J. E. and Ueltschi, D. Critical temperature of Heisenberg models on
regular trees, via random loops. J. Stat. Phys., 173 (5), 1369–1385 (2018b).
308 P. Mühlbacher

MR3878347.
Friedli, S. and Velenik, Y. Statistical mechanics of lattice systems. A concrete
mathematical introduction. Cambridge University Press, Cambridge (2018). ISBN
978-1-107-18482-4. MR3752129.
Hammond, A. Infinite cycles in the random stirring model on trees. Bull. Inst.
Math. Acad. Sin. (N.S.), 8 (1), 85–104 (2013). MR3097418.
Hammond, A. Sharp phase transition in the random stirring model on trees. Probab.
Theory Related Fields, 161 (3-4), 429–448 (2015). MR3334273.
Hammond, A. and Hegde, M. Critical point for infinite cycles in a random loop
model on trees. Ann. Appl. Probab., 29 (4), 2067–2088 (2019). MR3983335.
Harris, T. E. Nearest-neighbor Markov interaction processes on multidimensional
lattices. Advances in Math., 9, 66–89 (1972). MR307392.
Kotecký, R., Miłoś, P., and Ueltschi, D. The random interchange process on the
hypercube. Electron. Commun. Probab., 21, Paper No. 4, 9 (2016). MR3485373.
Liggett, T. M., Schonmann, R. H., and Stacey, A. M. Domination by product
measures. Ann. Probab., 25 (1), 71–95 (1997). MR1428500.
Miłoś, P. and Şengül, B. Existence of a phase transition of the interchange process
on the Hamming graph. Electron. J. Probab., 24, Paper No. 64, 21 (2019).
MR3978214.
Schramm, O. Compositions of random transpositions. Israel J. Math., 147, 221–243
(2005). MR2166362.
Tóth, B. Improved lower bound on the thermodynamic pressure of the spin 1/2
Heisenberg ferromagnet. Lett. Math. Phys., 28 (1), 75–84 (1993). MR1224836.
Ueltschi, D. Random loop representations for quantum spin systems. J. Math.
Phys., 54 (8), 083301, 40 (2013). MR3135479.