Generalized Divide and Color models
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Steif, J., Tykesson, J. (2019). Generalized Divide and Color models. Alea, 16(2): 899-955.
http://dx.doi.org/10.30757/ALEA.V16-33
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ALEA, Lat. Am. J. Probab. Math. Stat. 16, 1–57 (2019)
DOI: 10.30757/ALEA.v16-
Generalized Divide and Color Models
Jeffrey E. Steif and Johan Tykesson
Department of Mathematical Sciences, Chalmers University of Technology and Gothen-
burg University,
Chalmers tvärgata 3, 41258 Göteborg, Sweden.
E-mail address: steif@chalmers.se
URL:
http://www.math.chalmers.se/∼steif/
Department of Mathematical Sciences, Chalmers University of Technology and Gothen-
burg University,
Chalmers tvärgata 3, 41258 Göteborg, Sweden.
E-mail address: johant@chalmers.se
URL:
https://www.chalmers.se/en/Staff/Pages/johant.aspx
Abstract. In this paper, we initiate the study of “Generalized Divide and Color
Models”. A very interesting special case of this is the “Divide and Color Model”
(which motivates the name we use) introduced and studied by Olle Häggström.
In this generalized model, one starts with a finite or countable set V , a random
partition of V and a parameter p ∈ [0, 1]. The corresponding Generalized Divide
and Color Model is the {0, 1}-valued process indexed by V obtained by indepen-
dently, for each partition element in the random partition chosen, with probability
p, assigning all the elements of the partition element the value 1, and with proba-
bility 1 − p, assigning all the elements of the partition element the value 0.
Some of the questions which we study here are the following. Under what sit-
uations can different random partitions give rise to the same color pro cess? What
can one say concerning exchangeable random partitions? What is the set of prod-
uct measures that a color process stochastically dominates? For random partitions
which are translation invariant, what ergodic properties do the resulting color pro-
cesses have?
The motivation for studying these processes is twofold; on the one hand, we
believe that this is a very natural and interesting class of processes that deserves
investigation and on the other hand, a number of quite varied well-studied processes
actually fall into this class such as (1) the Ising model, (2) the fuzzy Potts model,
(3) the stationary distributions for the Voter Model, (4) random walk in random
scenery and of course (5) the original Divide and Color Model.
Received by the editors March 6th, 2017; accepted June 22th, 2019.
2010 Mathematics Subject Classification. 60K99, 28D99.
Key words and phrases. Exchangeable processes; ergodic theory; random partitions; stochastic
domination.
Research supported by the Swedish research council and the Knut and Alice Wallenberg foun-
dation (JS and JT).
1
2 J. E. Steif and J. Tykesson
The first author dedicates this paper to the memory of Jonathan Kaminsky
(1978-2016).
Contents
1. Introduction 2
2. The finite case 7
3. Color processes associated to infinite exchangeable random partitions 17
4. Connected random partitions on Z 31
5. Stochastic domination of product measures 35
6. Ergodic results in the translation invariant case 42
7. Questions and further directions 52
Acknowledgements 54
References 54
1. Introduction
1.1. Overview. In this paper, we initiate the study of a large class of processes
which we call “Generalized Divide and Color Models”. The name is motivated by
a model, intro d uced and studied by Olle Häggström (
Häggström, 2001), called the
“Divide and Color Model”, which is a special case of the class we look at here; this
special case will be described later in this section.
We believe that this general class of models warrants investigation, partly be-
cause it seems to be a very natural class and partly because a number of very
different processes studied in probability theory fall into this class, as described in
Subsection
1.3.
We now describe this class somewhat informally; formal definitions will be given
in Subsection
1.2. We start with a finite or countable set V . In the first step, a
random partition of V (with an arbitrary distribution) is chosen and in the second
step, independently, for each partition element in the random partition chosen in the
first step, with probability p, all the elements of the partition element are assigned
the value 1 and with probability 1 − p, all the elements of the partition element
are assigned the value 0. This yields in the end a {0, 1}-valued process indexed
by V , which we call a “Generalized Divide and Color Model” and it is this process
which will be our focus. Note that this process depends on, in addition of course to
the set V , the distribution of the random partition and the parameter p. A trivial
example is when the random partition always consists of singletons, in which case
we simply obtain an i.i.d. process with parameter p.
1.2. Definitions and notation. Let V be a finite or countable set and let Part
V
be the set of all partitions of V . Elements of V will be referred to as vertices.
Elements of a partition will be referred to either as equivalence classes or clusters.
If π ∈ Part
V
and v ∈ V , we let π(v) denote the partition element of π containing v.
For any measurable space (S, σ(S)), let P(S) denote the set of probability mea-
sures on (S, σ(S)). If π ∈ Part
V
and K ⊆ V , let π
K
denote the partition of K
induced from π in the obvious way. On Part
V
we consider the σ-algebra σ(Par t
V
)
generated by {π
K
}
K⊂V, |K|<∞
.
Generalized Divide and Color Models 3
We denote the set of all probability measures on (Part
V
, σ(Part
V
)) by RER
V
where RER stands for “random equivalence relation”. When V has a natural set of
translations (such as Z
d
), we let RER
stat
V
(“stat” for stationary) denote the elements
of RER
V
which are invariant under these translations. When V is a graph (such
as Z
d
with nearest neighbor edges), we let RER
conn
V
denote the subset of RER
V
which are supported on partitions for which each cluster is connected in the induce d
graph. Finally, we let RER
exch
V
(“exch” for exchangeable) denote the elements of
RER
V
which are invariant under all permutations of V which fix all but finitely
many elements.
For each finite or countable set V and for each p ∈ [0, 1], we now introduce a
mapping Φ
p
from RER
V
to probability measures on {0, 1}
V
. The image of some
ν ∈ RE R
V
will be called the “color process” or “Generalized Divide and Color
Model” associated to ν with parameter p and is defined as follows. Let π ∈ Part
V
be picked at random according to ν. For each partition element φ of π, we assign all
vertices in φ the value 1 with probability p and the value 0 with probability 1 −p,
independently for different partition elements. This yields for us a {0, 1}
V
-valued
random object, X
ν,p
, whose distribution is denoted by Φ
p
(ν). (Clearly Φ
p
(ν) is
affine in ν.) We will also refer to X
ν,p
as the color process associated to ν with
parameter p. This clearly corresponds, in a more formal way, to the generalized
divide and color model introduced in Subsection
1.1. Finally, we let CP
V,p
(CP
for “color process”) be the image of RER
V
under Φ
p
and we also let CP
∗
V,p
be the
image under Φ
p
of the relevant subset RER
∗
V
of RER
V
(∗ is stat, conn or exch.)
We usually do not consider the cases p = 0 or 1 for they are of course trivial.
We let | · |
1
denote the L
1
norm on Z
d
.
We end this section with the following elementary observation. For any ν ∈
RER
V
, p ∈ [0, 1] and u, v ∈ V , we have, letting E denote the event that u and v
are in the same cluster,
P(X
ν,p
(u) = X
ν,p
(v) = 1) = pP(E) + p
2
P(E
c
)
≥ p
2
= P(X
ν,p
(u) = 1)P(X
ν,p
(v) = 1)
(1.1)
and hence X
ν,p
has nonnegative pairwise correlations. Note trivially that X
ν,p
is
pairwise independent if and only if it is i.i.d.
1.3. Examples of color processes. It turns out that a number of random processes
which have been studied in probability theory have representations as color pro-
cesses. In this subsection, we give five such key examples. There is a slight difference
between the first two examples and the last three examples. In the first two exam-
ples, the known model corresponds to a color pro cess with respect to a particular
RER at a specific value of the parameter p but not for other values of p, while in
the last three examples, the known model corresponds to all the color processes
with respect to a particular RER as p varies over all values.
1.3.1. The Ising Model. For simplicity, we stick to finite graphs here. While the
results here are essentially true also for infinite graphs as well, there are some issues
which arise in that case but they will not concern us here. Let G = (V, E) be a
finite graph.
4 J. E. Steif and J. Tykesson
Definition 1.1. The Ising model on G = (V, E) with coupling constant J ∈ R and
external field h ∈ R is the probability measure µ
G,J,h
on {−1, 1}
V
given by
µ
G,J,h
({η(v)}
v ∈ V
) := e
J
P
{v,w}∈E
η (v)η(w)+h
P
v
η (v)
/Z
where Z = Z(G, J, h) is a normalization constant.
It turns out that µ
G,J,0
is a color process when J ≥ 0; this corresponds to
the famous FK (Fortuin-Kasteleyn) or so-called random cluster representation. To
explain this, we first need to introduce the following model.
Definition 1.2. The FK or random cluster model on G = (V, E) with parameters
α ∈ [0, 1] and q ∈ (0, ∞) is the probability measure ν
RCM
G,α,q
on {0, 1}
E
given by
ν
RCM
G,α,q
({η(e)}
e∈E
) := α
N
1
(1 − α)
N
2
q
C
/Z
where N
1
is the number of edges in state 1, N
2
is the number of edges in state 0, C
is the resulting number of connected clusters and Z = Z(G, α, q) is a normalization
constant.
Note, if q = 1, this is simply an i.i.d. process with parameter α. We think of
ν
RCM
G,α,q
as an RER on V by looking at the clusters of the percolation realization;
i.e., v and w are in the same partition if there is a path from v to w using edges in
state 1.
The following theorem from
Fortuin and Kasteleyn (1972) tells us that the Ising
Model with J ≥ 0 and h = 0 is indeed a color process. We however must identify
−1 with 0. See also
Edwards and Sokal (1988).
Theorem 1.3. (
Edwards and Sokal, 1988, Fortuin and Kasteleyn, 1972) For any
graph G and any J ≥ 0,
µ
G,J,0
= Φ
1/2
(ν
RCM
G,1−e
−2J
,2
).
See
Häggström (1998) for a nice survey concerning various random cluster rep-
resentations. We remark that while for all p, Φ
p
(ν
RCM
G,α,2
) is of course a color process,
we do not know if this corresponds to anything natural when p 6=
1
2
. We mention
that, if G is the complete graph, then an alternative way to see that the Ising
model with J ≥ 0 and 0 external field is a color process is to combine Theorem
3.16
later in this paper with the fact that the Ising model on the complete graph can
be extended to an infinitely exchangeable process. This latter fact was proved in
Papangelou (1989) where the technique is credited to Kac and Thompson (1966);
see also Theorem 1.1 in
Liggett et al. (2007). We end by mentioning that for the
Ising model on the complete graph on 3 vertices, there are other RERs, besides the
random cluster model, that generate it and that in some sense, the random cluster
model is not the most natural generating RER; see remark (iii) after Question
7.7.
1.3.2. The Fuzzy Potts Model. Again for simplicity, we stick to finite graphs here
and so let G = (V, E) be a finite graph.
Definition 1.4. For q ∈ {2, 3, . . . , }, the q-state Potts model on G = (V, E) with
coupling constant J (and no external field) is the probability measure µ
Potts
G,J,q
on
{1, . . . , q}
V
given by
µ
Potts
G,J,q
({η(v)}
v ∈ V
) := e
J
P
{v,w}∈E
I
{η(v)=η(w)}
/Z
where Z = Z(G, J, q) is a normalization constant.