Some geometric critical exponents for percolation and the random-cluster model

0.55 MB·English

by  Youjin Deng

Some geometric critical exponents for percolation and the random-cluster model
Youjin Deng,1 Wei Zhang,2 Timothy M. Garoni,3 Alan D. Sokal,4,5 and Andrea Sportiello6
1Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
2Department of Physics, Jinan University, Guangzhou 510632, China
3ARC Centre of Excellence for Mathematics and Statistics of Complex Systems,
Department of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia
4Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA
0
5Department of Mathematics, University College London, London WC1E 6BT, United Kingdom
1
6Dipartimento di Fisica and INFN, Universita` degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy
0
2 (Dated: April 20, 2009; revised January 12, 2010)

n We introduce several infinite families of new critical exponents for the random-cluster model
a and present scaling arguments relating them to the k-arm exponents. We then present Monte
J Carlo simulations confirming these predictions. These new exponents provide a convenient way to

5 determinek-armexponentsfromMonteCarlosimulations. Anunderstandingoftheseexponentsalso
1 leadstoaradicallyimprovedimplementationoftheSweenyMonteCarloalgorithm. Inaddition,our
MonteCarlodataallowustoconjectureanexactexpressionfortheshortest-pathfractaldimension

] ?
h dmin in twodimensions: dmin=(g+2)(g+18)/(32g) where g is theCoulomb-gas coupling, related
c tothe cluster fugacity q via q=2+2cos(gπ/2) with 2≤g≤4.
e
m PACSnumbers: 05.50.+q,05.10.Ln,64.60.ah,64.60.De
-
t
a
The random-cluster model [1] is a correlated bond-
t
s percolation model that plays a central role in the the-
.
t ory of critical phenomena, especially in two dimensions
a
m where it arises in recent developments of conformal field
theory[2]viaitsconnectionwithSchramm–Loewnerevo-
-
d lution (SLE) [3, 4]. To each bond configuration A E x
⊆ x x 3
n of a given graph G = (V,E), the random-cluster model 1 2
o assigns a weight proportional to qk(A)v|A| where k(A) is
c
the number of clusters (connected components), A is
[
| |
the number of bonds, and v,q 0 are parameters (see

2 ≥
Fig. 1). For q = 1 the random-cluster model reduces to
v
independent bondpercolation[5], while forintegerq 2 Figure1: Typicalbondconfiguration A⊆E on afinitesub-
8 ≥ graph of the square lattice: here |A| = 31 and k(A) = 4.
4 it provides a graphical representation of the q-state fer-
Also shown are the corresponding medial-lattice loops (inte-
4 romagnetic Potts model [6]. Furthermore, the q 0
4.3 ltihmaittvco∝rreqsαpownidtsht0o <unαifo<rm1s;ptahninsinapgptlireesesi,nppraorv→tiidceud- erTi1mor=ins,2xsh1=axd2Teadxn1fdo=re2c4l=a.rxiTt2yhx)e.3 HygireearlpdehsC-Mtmhmienoi,n2r,e2=ti=c|C|dxMi1s|txa=2n,ec12e1|;=(schh1oo6ro;tsaeisnntdg-
0 lar to the critical square-lattice random-cluster model, path length) between x2 and x3 is 3.
9 for which v = √q [7]. The random-cluster model thus
0
provides an extension of both percolation and the Potts
:
v modelthat allowsallpositive values ofq, integerornon- studyseveralinfinitefamiliesofcriticalexponentsforthe
i integer, to be studied within a unified framework. random-clustermodel, relatedto cluster size,hulllength
X
and shortest path. Some of these exponents have been
r When the graph G is planar, it is useful to map the
a studiedpreviouslyforspanningtreesorpercolationonly,
bond configurations to loop configurations [8], as illus-
while others appear to be entirely new. We shall give
trated in Fig. 1. The loop configurations are drawn on
scaling arguments determining all but one of these ex-
the medial graph, the vertices of which correspond to
ponents as a function of q in the two-dimensional (2D)
the edges of the original graph. The medial graph of
case, and Monte Carlo simulations confirming these pre-
the square lattice is again a square lattice, rotated 45◦.
dictions. The remaining undetermined exponent is the
Eachunoccupiededgeoftheoriginallatticeiscrossedby
shortest-path fractal dimension d [10], which relates
min
precisely two looparcs, while occupied edges are crossed
theshortest-pathlengthandEuclideandistancebetween
by none. The continuum limits of such loops are of cen-
twoverticesonacluster. However,ourMonteCarlodata
tral interest in studies of SLE [4]. The outermost loop
lead us to conjecture the exact formula
bounding a cluster defines its hull [9].
?
In this Rapid Communication we shall define and d = (g+2)(g+18)/(32g) (1)
min
2
where q =2+2cos(gπ/2) and g [2,4] is the Coulomb- or algorithms: namely, perform simultaneous breadth-
∈
gas coupling [11]. To our knowledge, d has not previ- first searches starting at both endpoints x and y, and
min
ously been studied for q =0,1. stop when one of the clusters has been fully visited
6
Our original motivation for studying these exponents or the clusters merge. In the first case this takes a
aroseoutofpracticalconcernsrelatedtoMonteCarloal- time min( x , y ), and in the second case a time Bs
|C | |C |
gorithms,aswediscussbelow. However,wesubsequently (the number of sites visited in breadth-first search un-
discovered a relationship between them and the k-arm til merger). A natural question is therefore to the de-
exponents x [9], which have proved to be of fundamen- termine the critical behavior of C and B , where
k min,2 s
tal importance in critical phenomena, not least within Cmin,2 = min( x , y ) if x and y are distinct and 0
|C | |C | C C
the context of rigorous studies of percolation [12]. The otherwise. We will provide here a scaling argument sug-
exponents we study here are defined via the scaling of gesting that both scale (in mean) as LdF−x2 Lγ/ν,
≪
verynaturalgraphicalobservables,whichareeasilymea- where d =d β/ν is the cluster fractal dimension and
F
−
sured in Monte Carlo simulations. Therefore, the rela- x is the 2-arm exponent. We will then verify this pre-
2
tionship between these exponents and the k-arm expo- diction numerically.
nents provides a new and convenientway to numerically In two dimensions an even more efficient procedure
estimate x , which can be used also in three dimensions is to simultaneously follow the medial-lattice loops sur-
k
(3D) where no exact expressions for x are known. This rounding x and y. If the two loops are distinct, this
k
has allowed us, for example, to numerically verify in 3D takes a time min( , ), where is the
x,xy y,xy x,e
|M | |M | M
that x equals the thermal dimension x when q = 1, a loop on the medial lattice that winds around x through
2 t
result previously tested only in 2D. the edge e if e is unoccupied, and = ∅ other-
x,e
M
Motivation. Monte Carlo simulations are an essen- wise (see Fig. 1). This naturally leads to the question
tialtoolinstatisticalmechanics,buttheytypicallysuffer of the scaling of M , defined analogously to C
min,2 min,2
from critical slowing-down [13]: the autocorrelation (re- asM =min( , )if and
min,2 x,xy y,xy x,xy y,xy
|M | |M | M M
laxation)timeτ divergesasacriticalpointisapproached, are distinct and 0 otherwise. We will provide a scaling
most often as a power law τ ξz, where ξ is the spatial argument, and confirm numerically, that M scales
min,2
∼
correlation length and z is a dynamic critical exponent. (in mean) as LdH−x2 where dH = 1 + 2/g is the
∼
The Sweeny algorithm [14] is a local single-bond up- hull fractal dimension [9, 18]. It follows that compu-
date dynamics for the random-cluster model; for 0 < tational critical slowing-down is completely absent for
q < 1 it is the only known general algorithm for this q >4cos2(πp2/3) 2.811520.
≈
model. We have recently shown [15] that the Sweeny al- Definition of exponents. These scaling results for
gorithm has unusually weak critical slowing-down,fairly C and M can be generalized in a very natu-
min,2 min,2
close to the theoretical lower bound z α/ν [16]. Fur- ral way. We shall consider a variety of positive-integer-
≥
thermore, it exhibits (especially for small q) the surpris- valued observables ; for each one we expect that its
ing phenomenon of “critical speeding-up” [15], in which probability distributOion obeys a scaling law P( = s)
O ∼
suitable global observables exhibit significant decorrela- s−ψO (with ψ > 1) for large s at criticality in infinite
O
tion on time scales much less than one sweep (namely, volume,ormoregenerally s−ψOF (s/ξdO,s/LdO)near
O
∼
Lw hits for some w < d); this makes the algorithm po- criticality in large finite volume, where F is a scaling
O
tentially very efficient. function. Ourgoalistodetermine,foreach ,thedecay
O
The main obstacle to the use of the Sweeny algorithm exponentψ andthe fractaldimensiond . Notethatat
O O
is the need (when q = 1) to determine how updating a criticality in finite volume, n L(n+1−ψO)dO +const
6 hO i∼
bond xy affects k(A): this potentially requires travers- + corrections to scaling as L (or logL if
→ ∞ ∼
ing an entire cluster to determine whether x and y are n+ 1 ψ = 0); we shall use this fact in our Monte
O
−
connected. Given a lattice site x, we write for the Carlo determinations of ψ and d .
x O O
C
cluster containing x and for the number of sites in Fix nearby sites x ,...,x , and let C =
x 1 k min,k
|C |
it. Since has mean Lγ/ν near the critical point, min( ,..., ) if these clusters are all distinct, and
|Cx| ∼ |Cx1| |Cxk|
the connectivity check threatens to impose a “computa- 0 otherwise. We expect that all these observables have
tional critical slowing-down” that would more than out- fractal dimension d equal to the cluster fractal di-
Cmin,k
weigh the good “physical” behavior of the Sweeny dy- mensiond =d β/ν. Moreover,standardhyperscaling
F
−
namics. There do exist sophisticated algorithms in the arguments [5] give ψ = d/d (the usual notation
Cmin,1 F
computer-scienceliterature[17]forperformingsuchcon- is τ = ψ + 1). Our goal is to determine ψ
Cmin,1 Cmin,k
nectivity checking dynamically, which have been proven for k 2. To our knowledge these exponents are new,
≥
tobe(asymptotically)veryefficient,buttheircomplexity except ψ = 11 for 2D spanning trees [19].
Cmin,2 8
appears prohibitive for use in practical simulations. For two-dimensional lattices, choose for each site
Our interest in the present project began with a x a bond e incident on it, and let M =
i i min,k
simple idea for reducing this computational slowing- min( ,..., )iftheseloopsarealldistinct,
|Mx1,e1| |Mxk,ek|
down without the need for complex data structures and 0 otherwise. We expect that all these observables
3
have fractal dimension d equal to the hull fractal
Mmin,k 1
dimensiond ;andstandardhyperscalingargumentsgive
H 10−1
ψ =d/d . Weaimtodetermineψ fork 2.
MNmoinw,1let TxHdenote the maximum grapMhm-tinh,keoretic≥dis- ≥)s 10−2
tmainnc(eTxf1r,o.m..,xTxtko)aifntyhesictelusinterCsxC,xianadredaellfindeistTinmcitn,,kan=d Cmin2, 1100−−34 12684
0otherwise(seeFig.1). Weexpectthatalltheseobserv- (
ableshavefractaldimensiondTmin,k equaltotheshortest- PdF 10−5 255162
pathfractaldimensiondmin. WeaimtodetermineψTmin,k x/2 10−6 1024
for k 1, as well as dmin. s 10−7 2048
≥ 4096
Finally,considerapairofsitesseparatedbyadistance
L, say x=0 and x=αL where α Rd, and let SαL 10−8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
∼ ∈
bethelengthoftheshortestpathconnectingthesesitesif
s/LdF
oneexists,and0otherwise. Since P(0 αL) L−2β/ν,
↔ ∼
we expect that (SαL)n Lndmin−2β/ν. We will use Figure 2: (Color online) Finite-size-scaling plot showing
(SαL)n to numhericallyid∼etermine dmin in 2D, leading sx2/dFP(Cmin,2 ≥ s) versus s/LdF for q = 1 and 64 ≤ L ≤
hto the coinjecture (1). 4096.
Scalingarguments. Letp (R)betheprobabilitythat,
k
in an annulus of inner radius r O(1) and outer radius
∼ (thesmallestLvalueincludedinthefit)untiltheχ2 was
R, the inner circle is connected to the outer one by at
reasonable. The errorbar is a subjective 68%confidence
least k distinct clusters. The k-arm exponent x char-
k
limit that includes both statistical error and systematic
acterizes the large-R asymptotics of p (R) at criticality:
k
error due to unincluded corrections to scaling.
pk(R) R−xk as R . In particular, x1 = β/ν. In
∼ → ∞ Our results for C and M are presented
two dimensions it is known [9, 11, 20] that min,2 min,2
h i h i
in Table I; complete results for Cn , Cn and
h min,2i h min,3i
x1 = (g−2)(6−g)/(8g) (2a) mhMenmntinw,2iithwitthhenp=re1d,ic2t,e3dcaenxpboenfeonutnsdisinex[2c6e]l.leTnth,eeaxgcreepet-
x = (g/8)k2 (g 4)2/(8g) for k 2 (2b)
k
− − ≥ where the exponent is very negative and hence possibly
overshadowed by correction-to-scaling terms. A finite-
Now fix nearby sites x ,...,x , and let C (s) be the
1 k k
size-scaling plot for C and q = 1 is shown in Fig. 2
probability that x ,...,x belong to k distinct clusters, min,2
1 k
and exhibits excellent collapse.
each of which contains at least s sites. The correspon-
dence s RdF suggests that Ck(s) pk(s1/dF). Since NextwestudiedSαLforα=(21,0)inordertoestimate
aCnkd(sh)e=nc∼Pe(C(mCimn,ikn,k≥)ns), wLenpdrFe−dixckt+ψ∼Ccomnins,kt.=Txhkis/daFgr+ee1s idbmlein:wsietheTGarbalsesIbaerngder[2’s6][.27O]uersrteimsualttefodrmqin==1i1s.1co3m06p(a3t)-.
h i ∼ The corrections to scaling are very strong for these ob-
with standard hyperscaling for k = 1; we will test it
for k 2. A similar argument suggests that Bn servables, and our error bars are dominated by our as-
LndF−≥x2. h si ∼ sessment of the likely systematic error from such correc-
tions. It would be very useful (but also very expensive)
Analogousreasoningpredictsψ =x /d +1and
Mmin,k k H to obtain data at larger values of L.
ψ = x /d + 1. For q = 1 this argument for
Tmin,k k min
shortest-pathscalingisduetoZiff[21]. In2Ditisknown Our results for dmin are consistent with the simple
[9, 20] that d = 1+2/g, but to our knowledge (1) is formula (1): see Table I and Fig. 3. This formula has
H
the only known conjecture for dmin, even for q =1. the nice property that dmin is monotone decreasing for
Monte Carlo simulations. We simulated the two- 2 g 6 and reaches dmin = 1 precisely at g = 6, in
≤ ≤
dimensional random-cluster model at criticality for q = accordance with the idea [28] that clusters become more
0,0.01,0.25,0.5,1,1.5,2,3,3.5[22]and4 L 1024(pe- compact as g grows. It also agrees with the known fact
riodicboundaryconditions)bytheSween≤yalg≤orithm[14] thatdmin = 54 atq =0[19]. Indeed,ifoneseeksaformula
when 0 < q < 1 and the Chayes–Machta algorithm [23] ofthe“Coulomb-gas”formdmin =F(g)=Ag+B+C/g
when q > 1. For 0.25 q 2 we also have data at [11] and imposes the constraints F(2) = 5, F(6) = 1,
4
L = 2048. For q = 1 we≤used≤cluster-growth algorithms F′(6)=0, then the unique solution is (1).
to handle 4 L 4096. For q =0 we used Wilson’s al- Our results for Tmin,1 and Tmin,2 are similar to those
≤ ≤
gorithm[24] to generatespanningtrees fromloop-erased shown in Table I and will be reported elsewhere.
random walk [25]. The total CPU time used in these We aregratefultoGaryReichforadiscussionthatin-
simulations was approximately 66 years using a 3.2 GHz spiredthiswork;toWenanGuoforparticipationinearly
Xeon EM64T processor. stagesofthisproject;toRussLyons,RobinPemantleand
For each observable , we fit n for n = 1,2,3 to David Wilson for helpful correspondence; and to NYU
O hO i
theAnsa¨tzeaLp,aLp+bLp−∆ andaLp+c,varyingL ITS for use of their computer cluster. This researchwas
min
4
Quantity q= 0 0.01 0.25 0.5 1 1.5 2 3 3.5
hCmin,2i p(num.) 1.25002(4) 1.1800(1) 0.9273(2) 0.8071(2) 0.6458(1) 0.5226(2) 0.4163(3) 0.214(2) 0.108(3)
p(pred.) 1.25000 1.18007 0.92707 0.80768 0.64583 0.52298 0.41667 0.21667 0.10376
hMmin,2i p(num.) 1.25002(4) 1.1639(5) 0.8527(5) 0.7036(3) 0.4994(7) 0.3444(4) 0.2087(2) −0.0525(25) −0.195(3)
p(pred.) 1.25000 1.16439 0.85222 0.70341 0.50000 0.34420 0.20833 −0.05000 −0.19748
SL/2 dmin (num.) 1.24999(3) 1.2371(10) 1.1825(3) 1.1596(4) 1.1303(8) 1.1112(7) 1.0955(10) 1.0677(40) 1.0560(30)
2≤g≤4 dmin (conj.) 1.25000 1.23463 1.18211 1.15918 1.13021 1.10997 1.09375 1.06667 1.05343
Table I: Numerical estimates versustheoretical predictions for exponentsassociated to hCmin,2i,hMmin,2i and SL/2.
(1987).
dmin npurmedeircitceadl [10] H.J.Herrmann,D.C.Hongand H.E.Stanley,J.Phys.
A 17, L261 (1984).
1.2 [11] B. Nienhuis,J. Stat.Phys. 34, 731 (1984).
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critical (2≤g≤4)
Lett. 83, 1359 (1999); H. Kesten, Comm. Math. Phys.
109,109(1987); S.SmirnovandW.Werner,Math.Res.
Lett. 8, 729 (2001).
1.1
[13] A.D.Sokal,inFunctional Integration: Basics and Appli-
cations,ed.C.deWitt-Morette,P.CartierandA.Folacci
tricritical (6≥g≥4)
(Plenum, New York,1997), pp. 131–192.
[14] M. Sweeny,Phys.Rev.B 27, 4445 (1983).
1.0 [15] Y. Deng, T.M. Garoni and A.D. Sokal, Phys. Rev. Lett.
0 1 2 3 4 98, 230602 (2007).
q [16] X.-J.LiandA.D.Sokal,Phys.Rev.Lett.63,827(1989),
see last paragraph.
Figure 3: (Color online) Numerical estimates for dmin to- [17] M.R.HenzingerandV.King,J.ACM46,502(1999); J.
getherwiththeconjecturedexactformuladmin=? (g+2)(g+ Holm, K. de Lichtenberg and M. Thorup, J. ACM 48,
18)/(32g). 723 (2001).
[18] Ifthetwoloopsarenotdistinct,wefindthatittakes(in
mean)atime∼LdH−x2 todeterminethisalso.Ifthetwo
loops aredistinct andbothtopologically nontrivial,con-
supported in part by NSF grant PHY–0424082, NSFC
nectednessneedstobedeterminedbyafullsimultaneous
grant 10975127,and the Chinese Academy of Sciences. clustertraversal.Empiricallythiscaseariseswithproba-
bility∼L−x2 (andscalingargumentssuggestthatthisis
anywayanupperbound),sothatitmakesacontribution
∼LdF−2x2 ≪LdH−x2 to thetotal cost.
[19] S.S.Manna,D.DharandS.N.Majumdar, Phys.Rev.A
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Field Theory (Springer, New York,1997). e.g. J. Salas and A.D. Sokal, J. Stat. Phys. 88, 567
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Some geometric critical exponents for percolation and the random-cluster model
EPAPS Appendix: Monte Carlo Data
Youjin Deng,1 Wei Zhang,2 Timothy M. Garoni,3 Alan D. Sokal,4,5 and Andrea Sportiello6
1Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
2Department of Physics, Jinan University, Guangzhou 510632, China
3ARC Centre of Excellence for Mathematics and Statistics of Complex Systems,
Department of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia
4Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA
5Department of Mathematics, University College London, London WC1E 6BT, United Kingdom
6Dipartimento di Fisica and INFN, Universit`a degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy
(Dated: January 12, 2010)
Detailed results for hCn i and hCn i with n = Results for hSn i with α = (1,0) and n = 1,2,3 are
min,2 min,3 αL 2
1,2,3arepresentedhere inTable I. The agreementwith presentedinTableIII. Thesedataareconsistentwiththe
the predicted exponents is excellent, except where the scaling hSn i ∼ Lndmin−2β/ν and with the conjectured
αL
exponent is very negative and hence possibly overshad- ?
exact formula d =(g+2)(g+18)/(32g).
min
owed by correction-to-scaling terms. (Exponents p < 0
alwayscomefromafittoaLp+c.) ForhC iatq =0,
min,3
where the predicted exponent is zero, we observe a clear
alogL+b behavior: see Fig. 1.
Detailed results for hMn i with n = 1,2,3 are re- [1] Y.Deng,unpublished.SeealsoY.Deng,H.W.J.Bl¨oteand
ported in the top half ofmTina,b2le II. The predicted val- B.Nienhuis,Phys.Rev.E69,026123 (2004) forarelated
construction.
ues are again confirmed. We also studied M for
min,2
[2] The critical O(n) loop model with 2 ≥ n ≥ 0 is known
the “dense” Fortuin–Kasteleyn (FK) clusters in which to correspond to the tricritical Potts model with q = n2
all bonds between sites in the same cluster are inserted. and hence 4 ≤ g ≤ 6 [B. Nienhuis, Physica A 177, 109
Thismodificationleavesconnectivities(hencexk anddF) (1991)]. The critical O(n) loop model with 0 ≥ n ≥ −2
unchanged but is conjectured [1] to change the hull and maycorrespondto6≤g≤8,butithasnegativeweights.
shortest-path dimensions to those of FK clusters with
′
coupling g = 16/g. The results for M are shown
min,2
in the bottom half of Table II: they are consistent with
′
our prediction over the entire range 4 ≤ g ≤ 8. We re-
′
mark that 4≤g ≤6 corresponds to the tricriticalPotts
′
model for 4≥q ≥0, but no interpretation of 6≤g ≤8
in terms of a spin model with positive weights seems to
be known [2].
1.8
1.6
1.4
1.2
3
min, 1
C
0.8
0.6
0.4
0.2
10 100 1000
L
Figure 1: Quantity Cmin,3 for q=0.
Quantity q= 0 0.01 0.25 0.5 1 1.5 2 3 3.5
Cmin,2 p(num.) 1.25002(4) 1.1800(1) 0.9273(2) 0.8071(2) 0.6458(1) 0.5226(2) 0.4163(3) 0.214(2) 0.108(3)
p(pred.) 1.25000 1.18007 0.92707 0.80768 0.64583 0.52298 0.41667 0.21667 0.10376
(Cmin,2)2 p(num.) 3.25002(4) 3.1647(2) 2.8633(1) 2.7252(2) 2.5418(2) 2.4065(2) 2.2923(3) 2.0832(7) 1.9695(8)
p(pred.) 3.25000 3.16489 2.86336 2.72492 2.54167 2.40621 2.29167 2.08333 1.96999
(Cmin,2)3 p(num.) 5.24998(5) 5.1496(2) 4.7997(2) 4.6424(2) 4.4377(2) 4.2896(3) 4.1668(6) 3.950(2) 3.838(2)
p(pred.) 5.25000 5.14970 4.79965 4.64216 4.43750 4.28943 4.16667 3.95000 3.83621
Cmin,3 p(num.) alogL+b −0.109(2) −0.532(9) −0.723(8) −0.961(8) −1.04(3) −0.98(2) −0.7(1) −0.6(1)
p(pred.) 0.00000 −0.10974 −0.52401 −0.72989 −1.02083 −1.25148 −1.45833 −1.86667 −2.10868
(Cmin,3)2 p(num.) 2.0009(5) 1.8755(12) 1.415(3) 1.184(4) 0.867(5) 0.623(5) 0.38(3) 0.01(2) −0.29(9)
p(pred.) 2.00000 1.87508 1.41229 1.18735 0.87500 0.63174 0.41667 0.00000 −0.24245
(Cmin,3)3 p(num.) 4.0012(9) 3.860(2) 3.352(4) 3.100(6) 2.739(18) 2.494(11) 2.33(7) 1.73(7) 1.56(5)
p(pred.) 4.00000 3.85990 3.34857 3.10460 2.77083 2.51497 2.29167 1.86667 1.62378
Table I: Numerical estimates versus theoretical predictions for exponentsassociated to Cmin,k with k=2,3.
Quantity q= 0 0.01 0.25 0.5 1 1.5 2 3 3.5
Mmin,2 p(num.) 1.25002(4) 1.1639(5) 0.8527(5) 0.7036(3) 0.4994(7) 0.3444(4) 0.2087(2) −0.0525(25) −0.195(3)
p(pred.) 1.25000 1.16439 0.85222 0.70341 0.50000 0.34420 0.20833 −0.05000 −0.19748
(Mmin,2)2 p(num.) 3.25002(4) 3.1333(4) 2.7141(5) 2.5164(4) 2.2498(4) 2.0488(2) 1.8749(3) 1.5485(15) 1.372(2)
p(pred.) 3.25000 3.13353 2.71364 2.51638 2.25000 2.04864 1.87500 1.55000 1.36751
(Mmin,2)3 p(num.) 5.24998(5) 5.1023(4) 4.5756(6) 4.3293(5) 4.0000(2) 3.7531(4) 3.5415(3) 3.1484(16) 2.938(3)
2≤g≤4 p(pred.) 5.25000 5.10267 4.57507 4.32935 4.00000 3.75308 3.54167 3.15000 2.93250
Mm′in,2 p(num.) 0.496(3) 0.4532(6) 0.2818(10) 0.1973(5) 0.085(3) −0.006(5) −0.099(5) −0.24(3) −0.46(15)
p(pred.) 0.50000 0.45322 0.28100 0.19795 0.08333 −0.00535 −0.08333 −0.23333 −0.31998
(Mm′in,2)2 p(num.) 1.748(3) 1.7116(6) 1.5718(4) 1.5050(8) 1.4165(13) 1.351(2) 1.296(4) 1.11(4) 1.02(7)
p(pred.) 1.75000 1.71118 1.57121 1.50546 1.41667 1.34955 1.29167 1.18333 1.12250
(Mm′in,2)3 p(num.) 2.995(4) 2.9688(10) 2.8612(20) 2.8136(8) 2.749(4) 2.709(10) 2.667(2) 2.59(4) 2.55(4)
4≤g′≤8 p(pred.) 3.00000 2.96914 2.86143 2.81297 2.75000 2.70444 2.66667 2.60000 2.56499
Table II: Numerical estimates versus theoretical predictions for exponents associated to Mmin,2. Top half of table refers to
ordinary FK clusters; bottom half of table referes to “dense” FK clusters in which all bonds between sites in the same cluster
are inserted.
Quantity q= 0 0.01 0.25 0.5 1 1.5 2 3 3.5
SL/2 dmin (num.) 1.24999(3) 1.2371(10) 1.1825(3) 1.1596(4) 1.1303(8) 1.1112(7) 1.0955(10) 1.0677(40) 1.0560(30)
(SL/2)2 dmin (num.) 1.25000(3) 1.2364(10) 1.1826(3) 1.1596(4) 1.1304(10) 1.1108(6) 1.0950(9) 1.0668(30) 1.0538(25)
(SL/2)3 dmin (num.) 1.24999(4) 1.2359(10) 1.1825(3) 1.1595(3) 1.1304(10) 1.1106(3) 1.0949(9) 1.0662(30) 1.0525(23)
2≤g≤4 dmin (conj.) 1.25000 1.23463 1.18211 1.15918 1.13021 1.10997 1.09375 1.06667 1.05343
TableIII:Numericalestimatesfortheshortest-pathexponentdminasdeducedfromh(SL/2)niforn=1,2,3,versusconjectured
exact formula.