Three theorems in discrete random geometry

0.55 MB·

by  Geoffrey Grimmett

Probability Surveys
Vol.8(2011)294–332
ISSN:1549-5787
DOI:10.1214/11-PS185
Three theorems in
discrete random geometry
2 Geoffrey Grimmett
1
0 Centre for Mathematical Sciences
2 University of Cambridge
Wilberforce Road
n
Cambridge CB3 0WB, UK
a e-mail: [email protected]
J

7 Abstract: Thesenotesarefocusedonthreerecentresultsindiscreteran-
2 domgeometry,namely:theproofbyDuminil-CopinandSmirnovthatthe
connective constant ofthe hexagonal lattice isp2+√2;theproof bythe

] author andManolescuoftheuniversalityofinhomogeneous bondpercola-
R tiononthesquare,triangular,andhexagonallattices;theproofbyBeffara
P andDuminil-Copinthat thecriticalpointofthe random-clustermodelon
Z2 is √q/(1+√q). Background information on the relevant random pro-
.
h cesses is presented on route to these theorems. The emphasis is upon the
t communicationofideasandconnectionsaswellasuponthedetailedproofs.
a
m AMS 2000 subject classifications:Primary60K35;secondary82B43.
Keywords and phrases: Self-avoiding walk, connective constant, per-
[
colation, random-clustermodel,Isingmodel,star–triangletransformation,

2 Yang–Baxter equation,criticalexponent, universality,isoradiality.
v
5 ReceivedOctober 2011,correctedbytheauthor27January2012.
9
3
Contents
2
.
0
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
1
1 2 Self-avoiding walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
1 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
v: 2.2 Hexagonal-lattice connective constant . . . . . . . . . . . . . . . 298
i 3 Bond percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
X
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
r
a 3.2 Power-lawsingularity. . . . . . . . . . . . . . . . . . . . . . . . . 306
3.3 Box-crossing property . . . . . . . . . . . . . . . . . . . . . . . . 309
3.4 Star–triangle transformation. . . . . . . . . . . . . . . . . . . . . 311
3.5 Universality for bond percolation . . . . . . . . . . . . . . . . . . 313
4 Random-cluster model . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
4.2 Critical point on the square lattice . . . . . . . . . . . . . . . . . 320
4.3 Proof of the box-crossing property . . . . . . . . . . . . . . . . . 322
4.4 Proof of the critical value . . . . . . . . . . . . . . . . . . . . . . 325
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
294
Three theorems in discrete random geometry 295
1. Introduction
These notes are devoted to three recent rigorous results of significance in the
areaofdiscreterandomgeometryintwodimensions.Theseresultsareconcerned
with self-avoiding walks, percolation, and the random-cluster model, and may
be summarized as:
(a) the connective constant for self-avoiding walks on the hexagonallattice is
2+√2, [15].
(b) the universality of inhomogeneous bond percolation on the square, trian-
p
gular and hexagonal lattices, [24],
(c) the critical point of the random-cluster model on the square lattice with
cluster-weighting factor q 1 is √q/(1+√q), [7].
≥
In each case, the background and context will be described and the theorem
stated. A complete proof is included in the case of self-avoiding walks, whereas
reasonably detailed outlines are presented in the other two cases.
If the current focus is on three specific theorems, the general theme is two-
dimensionalstochasticsystems.Inanexciting areaofcurrentresearchinitiated
by Schramm [55, 56], connections are being forged between discrete models
and conformality; we mention percolation [60], the Ising model [14], uniform
spanning trees and loop-erased random walk [47], the discrete Gaussian free
field[57],andself-avoidingwalks[16].Ineachcase,ascalinglimitleads (orwill
lead) to a conformal structure characterized by a Schramm–L¨owner evolution
(SLE).Inthesettingsof(a),(b),(c)above,therelevantscalinglimitsareyetto
be proved, and in that sense this article is about three ‘pre-conformal’ families
of stochastic processes.
Therearenumeroussurveysandbookscoveringthehistoryandbasicmethod-
ology of these processes, and we do not repeat this material here. Instead, we
presentcleardefinitions ofthe processesinquestion,andweoutline those parts
of the general theory to be used in the proofs of the above three theorems.
Self-avoiding walks (SAWs) are the subject of Section 2, bond percolation of
Section3,andthe random-clustermodelofSection4.Moreexpositorymaterial
about these three topics may be found, for example, in [22], as well as: SAWs
[49];percolation[10,20,66];therandom-clustermodel[21,67].Therelationship
between SAWs, percolation, and SLE is sketched in the companion paper [46].
Full references to original material are not invariably included.
A balanceis attempted inthese notes betweenprovidingenoughbut nottoo
much basic methodology. One recurring topic that might delay readers is the
theory of stochastic inequalities. Since a sample space of the form Ω = 0,1 E
{ }
is a partially ordered set, one may speak of increasing random variables. This
in turn gives rise to a partial order on probability measures1 on Ω by: µ µ′
st
≤
if µ(X) µ′(X) for all increasing random variables X. Holley’s theorem [31]
≤
providesa useful sufficientcriterionfor suchaninequality inthe contextof this
1Theexpectation ofarandomvariableX underaprobabilitymeasureµiswrittenµ(X).
296 G. Grimmett
Fig 1.1. The square lattice L2 and its dual square lattice. The triangular lattice T and its
dual hexagonal (or ‘honeycomb’) lattice H.
article. The reader is referred to [21, Chap. 2] and [22, Chap. 4] for accounts of
Holley’s theorem, as well as of ‘positive association’ and the FKG inequality.
A variety of lattices will be encountered in this article, but predominantly
the square, triangular, and hexagonal lattices illustrated in Figure 1.1. More
generally,alattice inddimensionsisaconnectedgraph withboundedvertex-
degrees, together with an embedding in Rd such that:Lthe embedded graph is
locally-finite and invariant under shifts of Rd by any of d independent vectors
τ ,τ ,...,τ . We shall sometimes speak of a lattice without having regard to
1 2 d
its embedding. A lattice is vertex-transitive if, for any pair v, w of vertices,
there exists a graph-automorphism of mapping v to w. For simplicity, we
L
shall consider only vertex-transitive lattices. We pick some vertex of a lattice
and designate it the origin, denoted 0, and we generally assume that 0 is
eLmbedded at the origin of Rd. The degree of a vertex-transitive lattice is the
numberofedgesincidenttoanygivenvertex.WewriteLd forthed-dimensional
(hyper)cubic lattice, and T, H for the triangular and hexagonallattices.
2. Self-avoiding walks
2.1. Background
Let be a lattice with origin 0, and assume for simplicity that is vertex-
L L
transitive. A self-avoiding walk (SAW) is a lattice path that visits no vertex
more than once.
How many self-avoiding walks of length n exist, starting from the origin?
Whatisthe‘shape’ofsuchaSAWchosenatrandom?Inparticular,whatcanbe
saidaboutthedistancebetweenitsendpoints?Theseandrelatedquestionshave
attracted a great deal of attention since the notable paper [26] of Hammersley
and Morton, and never more so than in recent years. Mathematicians believe
buthavenotprovedthatatypicalSAWonatwo-dimensionallattice ,starting
L
at the origin, converges in a suitable manner as n to a SLE curve. See
8/3
→∞
[16, 49, 56, 61] for discussion and results.
Three theorems in discrete random geometry 297
Paper [26] contained a number of stimulating ideas, of which we mention
here the use of subadditivity in studying asymptotics. This method and its
elaborations have proved extremely fruitful in many contexts since. Let be
n
S
the set of SAWs with length n starting at the origin, with cardinality σ =
n
σ ( ):= .
n n
L |S |
Lemma 2.1 ([26]). We have that σ σ σ , for m,n 0.
m+n m n
≤ ≥
Proof. Letπandπ′befiniteSAWsstartingattheorigin,anddenotebyπ π′the
∗
walkobtainedby followingπ from0 to its otherendpoint x, andthen following
the translated walk π′+x. Everyν may be written in a unique way as
m+n
∈S
ν =π π′ for some π and π′ . The claim of the lemma follows.
m n
∗ ∈S ∈S
Theorem 2.2. Let be a vertex-transitive lattice in d 2 dimensions with
L ≥
degree ∆. The limit κ = κ( ) = lim (σ )1/n exists and satisfies 1 < κ
n→∞ n
L ≤
∆ 1.
−
Proof. By Lemma 2.1, x =logσ satisfies the ‘subadditive inequality’
m m
x x +x .
m+n m n
≤
By the subadditive-inequality theorem2 (see [20, App. I]), the limit
1
λ= lim x
n
n→∞n
exists, and we write κ=eλ.
Since thereareatmost∆ 1choicesforeachstepofaSAW (apartfromthe
−
first), we have that σ ∆(∆ 1)n−1, giving that κ ∆ 1. It is left as an
n
≤ − ≤ −
exercise to show κ>1.
The constantκ=κ( )iscalledthe connective constant ofthe lattice .The
exact value of κ = κ(LLd) is unknown for every d 2, see [32, Sect. 7L.2, pp.
≥
481–483]. As explained in the next section, the hexagonal lattice has a special
structure which permits an exact calculation, and our main purpose here is
to present the proof of this. In addition, we include next a short discussion
of critical exponents, for a fuller discussion of which the reader is referred to
[2, 49, 59]
ByTheorem2.2,σ growsexponentiallyinthemannerofκn(1+o(1)).Itisbe-
n
lieved by mathematicians and physicists that there is a power-ordercorrection,
in that
σ A nγ−1κn,
n 1
∼
where the exponent γ depends only on the number of dimensions and not oth-
erwise on the lattice (here and later, there should be an additional logarithmic
correction in four dimensions). Furthermore, it is believed that γ = 43 in two
32
dimensions. We mention two further critical exponents. It is believed that the
(random) end-to-end distance D of a typical n-step SAW satisfies
n
E(D2) A n2ν,
n ∼ 2
2SometimesknownasFekete’s Lemma.
298 G. Grimmett
Fig2.1.TheArchimedeanlattice(3,122)isobtainedbyreplacingeachvertexofthehexagonal
lattice H by a triangle.
and furthermore ν = 3 in two dimensions. Finally, let Z (x) = σ (v)xk
4 v k k
where σ (v) is the number of k-step SAWs from the originto the vertex v. The
k
generating function Z has radius of convergence κ−1, and it is beliPeved that
v
Z (κ−1) A v −(d−2+η) as v ,
v 3
∼ | | | |→∞
in d dimensions. Furthermore, η should satisfy the so-called Fisher relation
γ =(2 η)ν, giving that η = 5 in two dimensions. The numerical predictions
− 24
for these exponents in two dimensions are explicable on the conjectural basis
that a typical n-step SAW in two dimensions converges to SLE as n
8/3
→ ∞
(see [6, 48]).
2.2. Hexagonal-lattice connective constant
Theorem 2.3 ([15]). The connective constant of the hexagonal lattice H satis-
fies κ(H)= 2+√2.
This resuplt of Duminil-Copin and Smirnov provides a rigorous and provoca-
tiveverificationofapredictionofNienhuis [50]basedonconformalfieldtheory.
Theprooffallsshortofaproofofconformalinvarianceforself-avoidingwalkson
H.TheremainderofthissectioncontainsanoutlineoftheproofofTheorem2.3,
and is drawn from [15]3.
The reader may wonder about the special nature of the hexagonal lattice.
It is something of a mystery why certain results for this lattice (for example,
Theorem 2.3, and the conformal scaling limit of ‘face’ percolation) do not yet
extend to other lattices.
WenoteanapplicationofTheorem2.3tothelatticeillustratedinFigure2.1,
namely the Archimedean lattice denoted A = (3,122) and known also as a
‘Fisher lattice’ after [17]. Edges of A lying in a triangle are called triangular.
For simplicity, we shall consider only SAWs of A that start with a triangular
3Theproofhasbeenre-workedin[41].
Three theorems in discrete random geometry 299
edge and finish with a non-triangular edge. The non-triangular edges of such a
SAWainduceaSAWhofH.Furthermore,forgivenh,thecorrespondingaare
obtained by replacing each vertex v of h (other than its final vertex) by either
a single edge in the triangle of A at v, or by two such edges. It follows that the
generating function of such walks is
ZA(a)= σ (H)(a2+a3)k =Z(a2+a3),
k
k
X
whereZ(x):= σ (H)xk.TheradiusofconvergenceofZA is1/κ(A),andwe
n n
deduce the following formula of [33]:
P
1 1 1
+ = . (2.1)
κ(A)2 κ(A)3 κ(H)
One may show similarly that the critical exponents γ, ν, η are equal for A
and H, assuming they satisfy suitable definitions. The details are omitted.
Proof of Theorem 2.3. This exploits the relationship between R2 and the Ar-
gand diagram of the complex numbers C. We embed H = (V,E) in R2 in a
natural way: edges have length 1 and are inclined at angles π/6, π/2, 5π/6 to
the x-axis,the origins of andR2 coincide, andthe line-segmentfrom(0,0)to
(0,1) is an edge of H. AnLy point in H may thus be represented by a complex
number. Let be the set of midpoints of edges of H. Rather than counting
paths betweenMvertices of H, we count paths between midpoints.
Fix a , and let
∈M
Z(x)= x|γ|, x (0, ),
∈ ∞
γ
X
where the sum is over all SAWs γ starting at a, and γ is the number of ver-
| |
tices visited by γ. Theorem 2.3 is equivalent to the assertion that the radius of
convergence of Z is χ:=1/ 2+√2. We shall therefore prove that
p
Z(χ)= , (2.2)
∞
Z(x)< for x<χ. (2.3)
∞
Towards this end we introduce a function that records the turning angle of
a SAW. A SAW γ departs from its starting-midpoint a in one of two possible
directions, and its direction changes by π/3 at each vertex. On arriving at its
±
otherendpointb,ithasturnedthroughsometotalturningangle T(γ),measured
anticlockwise in radians.
We work within some bounded region M of H. Let S V be a finite set
⊆
of vertices that induces a connected subgraph, and let M = M be the set of
S
midpoints of edges touching points in S. Let ∆M be the set of midpoints for
which the corresponding edge of H has exactly one endpoint in S. Later in the
proof we shall restrict M to a region of the type illustrated in Figure 2.2.
300 G. Grimmett
U
h,v
T− T+
h,v h,v
a
L
h
Fig 2.2. The region Mh,v has 2h+1 midpoints on the bottom side, and v at the left/right
sides. In thisillustration, we have h=2 and v=5.
Let a ∆M and σ,x (0, ), and define the so-called ‘parafermionic ob-
∈ ∈ ∞
servable’ of [15] by
Fσ,x(z)= e−iσT(γ)x|γ|, z M, (2.4)
a,M ∈
γ:a→z
X
where the summation is over all SAWs from a to z lying entirely in M. We
shallsuppresssomeofthe notationinFσ,x whennoambiguityensues.Thekey
a,M
ingredient of the proof is the following lemma, which is strongly suggestive of
discrete analyticity.
Lemma 2.4. Let σ = 5 and x=χ. For v S,
8 ∈
(p v)F(p)+(q v)F(q)+(r v)F(r)=0, (2.5)
− − −
where p,q,r are the midpoints of the three edges incident to v.
∈M
The quantities in (2.5) are to be interpreted as complex numbers.
Proof of Lemma 2.4. Let v S. We assume for definiteness that the star at v
∈
is as drawn on the left of Figure 2.34.
Let bethesetofSAWsofM startingatawhoseintersectionwith p,q,r
k
P { }
has cardinality k, for k =1,2,3.We shall show that the aggregatecontribution
to (2.5) of is zero, and similarly of .
1 2 3
P ∪P P
Considerfirst . Letγ ,andwrite b , b , b for the orderingof p,q,r
3 3 1 2 3
P ∈P { }
encountered along γ starting at a. Thus γ comprises:
– a SAW ρ from a to b ,
1
– a SAW of length 1 from b to b ,
1 2
– a SAW τ from b to b that is disjoint from ρ,
2 3
4Thisandlaterfiguresareviewedbestincolour.
Three theorems in discrete random geometry 301
τ τ
q q qq
r r rr
v v vv
p p pp
ρ ρρ
a aa
Fig 2.3. The star centred at the vertex v is on the left. Two SAWs lying in 3, visiting the
P
midpoints of edges in the respective orders pqr and prq. They follow the same SAWs ρ and
τ (in one or the other directions), and differonly within the star.
asillustratedinFigure2.3.Wepartition accordingtothepairρ,τ.Forgiven
3
P
ρ, τ, the aggregatecontribution of these two paths to the left side of (2.5) is
c θe−iσ4π/3+θeiσ4π/3 (2.6)
(cid:16) (cid:17)
where c=(b v)e−iσT(ρ)x|ρ|+|τ|+1 and
1
−
q v
θ = − =ei2π/3.
p v
−
The parenthesis in (2.6) equals 2cos 2π(2σ+1) which is 0 when σ = 5.
3 8
Consider now . This set may be partitioned according to the point
1 2
P ∪P (cid:0) (cid:1)
b in p,q,r visited first, and by the route ρ of the SAW from a to b. For
{ }
givenb, ρ, there are exactly three such SAWs, as inFigure 2.4.Their aggregate
contribution to the left side of (2.5) is
c 1+xθeiσπ/3+xθe−iσπ/3
(cid:16) (cid:17)
where c=(b v)e−iσT(γb)x|γb|. With σ = 5, we set this to 0 and solve for x, to
− 8
find x=1/[2cos(π/8)]=χ. The lemma is proved.
We return to the proof of Theorem 2.3, and we set σ = 5 henceforth. Let
8
M = M be as in Figure 2.2, and let L , T± , U be the sets of midpoints
h,v h h,v h,v
indicated in the figure (note that a is excluded from L ). Let
h
λx = x|γ|,
h,v
γ:aX→Lh
302 G. Grimmett
q q q
r r r
v v v
p p p
ρ ρ ρ
a a a
Fig2.4.TheleftSAW,denotedγp,intersectstheset p,q,r onceonly.Theothertwopaths,
{ }
denoted γq and γr, are extensionsof the first.
where the sum is over all SAWs in M from a to some point in L . All such
h,v h
γ have T(γ) = π. The sums τ±,x and ζx are defined similarly in terms of
± h,v h,v
SAWs ending in T± and U respectively, and all such γ have T(γ)= 2π/3
h,v h,v ∓
and T(γ)=0 respectively.
In summing (2.5) over all vertices v of M , with x = χ, all contributions
h,v
cancelexceptthosefromtheboundarymidpoints.UsingthesymmetryofM ,
h,v
we deduce that
iFχ(a) i (eiσπ)λχ +iθe−iσ2π/3τ−,χ+iζχ +iθeiσ2π/3τ+,χ =0.
− − ℜ h,v h,v h,v h,v
Divide by i, and use the fact that Fχ(a)=1, to obtain
c λχ +c τχ +ζχ =1, (2.7)
l h,v t h,v h,v
where τ =τ+ +τ− , c =cos(3π/8), and c =cos(π/4).
h,v h,v h,v l t
Let x (0, ). Since λx and ζx are increasing in h, the limits
∈ ∞ h,v h,v
λx = lim λx , ζx = lim ζx ,
v h,v v h,v
h→∞ h→∞
exist. Hence, by (2.7), the decreasing limit
τχ τχ as h , (2.8)
h,v ↓ v →∞
exists also. Furthermore, by (2.7),
c λχ+c τχ+ζχ =1. (2.9)
l v t v v
We shall use (2.8)–(2.9) to prove (2.2)–(2.3) as follows.
Proof of (2.2). There are two cases depending on whether or not
τχ >0 for some v 1. (2.10)
v ≥
Three theorems in discrete random geometry 303
b U
∞,v+1
a
L
∞
Fig2.5.ASAWcontributingtoλχv+1 butnotλχv isbrokenatitsfirsthighestvertexintotwo
SAWscolouredredandblue.Byaddingtwohalf-edges,weobtaintwoSAWsfromb U∞,v+1
∈
toL∞.
Assume firstthat (2.10) holds5, and pick v 1 accordingly.By (2.8), τχ τχ
≥ h,v ≥ v
for all h, so that
∞
Z(χ) τχ = ,
≥ h,v ∞
h=1
X
and (2.2) follows.
Assume now that (2.10) is false so that, by (2.9),
c λχ+ζχ =1, v 1. (2.11)
l v v ≥
We propose to bound Z(χ) below in terms of the ζχ. The difference λχ λχ
v v+1− v
is the sum of χ|γ| over all γ from a to L whose highest vertex lies between
∞
U and U . See Figure 2.5. We split such a γ into two pieces at its first
∞,v ∞,v+1
highestvertex, andadd two half-edges to obtaintwo self-avoidingpaths froma
given midpoint, b say, of U to L a . Therefore,
∞,v+1 ∞
∪{ }
λχ λχ χ(ζχ )2, v 1.
v+1− v ≤ v+1 ≥
By (2.11),
c χ(ζχ )2+ζχ ζχ, v 1,
l v+1 v+1 ≥ v ≥
whence, by induction,
1 1
ζχ min ζχ, , v 1.
v ≥ v 1 c χ ≥
(cid:26) l (cid:27)
Therefore,
∞
Z(χ) ζχ = .
≥ v ∞
v=1
X
5Infact,(2.10)doesnothold,see[5].