Table Of ContentReversiblefirst-ordertransitioninPaulipercolation
Mykola Maksymenko,1 Roderich Moessner,1 and Kirill Shtengel1,2
1Max-Planck-Institutfu¨rPhysikkomplexerSysteme, No¨thnitzerStraße38, 01187Dresden, Germany
2DepartmentofPhysicsandAstronomy,UniversityofCaliforniaatRiverside,Riverside,CA92521,USA
(Dated:January27,2014)
Percolationplaysanimportantroleinfieldsandphenomenaasdiverseasthestudyofsocialnetworks,the
dynamicsofepidemics,therobustnessofelectricitygrids,conductionindisorderedmedia,andgeometricprop-
ertiesinstatisticalphysics. Weanalyseanewpercolationprobleminwhichthefirstordernatureofanequi-
libriumpercolationtransitioncanbeestablishedanalyticallyandverifiednumerically. Therulesforthissite
4 percolationmodelarephysicalandverysimple,requiringonlytheintroductionofaweightW(n)=n+1for
1 aclusterofsizen.Thisestablishesthatadiscontinuouspercolationtransitioncanoccurwithqualitativelymore
0 localinteractionsthaninallcurrentlyconsideredexamplesofexplosivepercolation; andthat,unlikethese,it
2 canbereversible. Thisgreatlyextendsboththeapplicabilityofsuchpercolationmodelsinprinciple,andtheir
reachinpractice.
n
a
J PACSnumbers:71.10.Fd,64.60.De
3
2
Introduction. The percolation transition involves funda-
= + + + +
mentally geometric properties, manifest in non-local observ-
]
h ablessuchasanonsetofconductivityinadirtymetal,abreak-
c down of an electrical grid or an epidemic disease outbreak
e
m [1–4]. This is at odds with the more standard phase tran-
sitions in statistical physics which are described by a local W =5 W =6 W =4 W =4
-
t order parameter, such as the magnetisation in a bar magnet. (a)
a
It thus involves a conceptually fundamentally distinct set of
t
s issues. Its wide applicability coupled with this fundamental
.
t importance have generated much interest in defining various
a
m types of percolation problems and analysing their concomi-
tantphasetransitions. Oneenterprisehasbeenthequestfora
-
d first-orderpercolationtransition,wherethepercolatingcluster
n
setsindiscontinuously,correspondingtoaparticularlyviolent
o
transition, which can qualitatively amplify desirable proper-
c
[ ties in applications. Such a transition has remained remark-
ably elusive, but the development which has taken place un-
1 (b)
der the heading of explosive percolation has finally yielded
v
2 one,viaamechanisminwhichaninfinitenumberofnonlocal
FIG. 1. (a) In Pauli percolation, weight W =n+1 of a cluster
7 interactionsneedtooccursimultaneously[5–13].
canbereproducedbyimposingasimpletwo-color‘contagion’rule
1
shownhere:thewholeclusterofoccupiedsitescanbeeitherhealthy
6 Here,westudyPaulipercolation–asitepercolationprob-
(green)orhaveasingleinfectedsite(red). Differentclusterconfig-
. lemwithitsoriginincorrelatedquantummagnetism,charac-
1 urationsappearwithdifferentstatisticalweights. (b)Theexplosive
0 terizedbyanumberofnovelstrikinganddesirableproperties. natureofaPauli-percolationtransitiononaregularrandomgraphof
4 First of all, it exhibits a first-order phase transition invoking N =400sites:tworepresentativeconfigurations,withoutandwitha
1 only a minimal amount of non-locality, in the form of an in- giantclusteratthesamesitefugacitycorrespondingtop˜=0.45are
v: teractionsolelybetweenadjacentclusters,dependingonlyon shownsidebyside. Thelargestclusteriscoloredblue;unoccupied
i their respective sizes. Secondly, such an interaction can be sitesarenotshown.
X
very easily generated from perfectly local ones, for instance
r
a eitherviaasimpleclassicalcolouringrule,orviaaquantum-
statistical interaction between Fermionic particles. Thirdly, Paulipercolation. Themodelweconsiderfirstaroseina
it describes an equilibrium phase transition, and is hence re- quantummany-bodyproblemofitinerantelectronsonlattices
versible; at the same time, it can be thought of and analysed with flat energy bands. Such a system can exhibit flat-band
asastochasticdynamicalprocessandthusmay–butneednot ferromagnetism: the Pauli exclusion principle mandates that
– exhibit hysteresis. Finally, Pauli percolation lends itself to in the ground state the electron spins in a cluster order fer-
investigations using the toolbox of equilibrium classical sta- romagnetically in order to minimize the energy of repulsive
tisticalmechanics;wearethusabletosolveitspropertiesan- on-site interactions [14]. This leads to a weight of (n+1),
alyticallyonaregularrandomgraph,andverifythissolution reflectingthenumberofpossibleorientationsofthetotalspin
vianumericalMonteCarlosimulations. ofaferromagneticclusterofnelectrons[15].
2
The corresponding statistical-mechanical problem de- method widelyusedinspinglassandoptimizationproblems
scribes M particles occupying random sites of a lattice. Ev- [18, 20–23]. In the cavity method, adding a site or an edge
ery configuration = appears with statistical weight to a z-regular random graph is equivalent to connecting z or
i i
(cid:81) C ∪ C
W = (n( )+1),withn( )beingthesizeofcluster . z 1 roots (here referred to as cavity sites) of independent
C i Ci Ci (cid:80) Ci −
ThepartitionfunctionisthereforeZ = W . Cayley trees (see Figure 2(a)) via that site or edge. To com-
{C} C
Mergingtwoclustersofsizemandnreducestheiroverall plete the correspondence and get the correct set of solutions
weight from (m+1)(n+1) to (m+n+1) – a dramatic weintroduce‘wired’boundaryconditionswhichconnectthe
reductionforlargeclustersresultinginaneffectiverepulsive outer sites (‘leaves’) with one another thus allowing the for-
interactionbetweenthem. Thisisreminiscentofthe‘product mationofloops.
rule’leadingtoexplosivepercolationsuggestedbyAchlioptas The recursive structure of calculations on Cayley trees
[5] and developed in [12, 13, 16] but there are fundamental makesthemean-fieldtreatmentexactinthesesystems. Care
differences,seediscussionbelow. mustbe taken tocorrectly calculatethe bulkthermodynamic
Rather then fixing the number of occupied sites, we can potentials on such structures [22, 24–26]. For instance, the
studythegrandcanonicalensemblebylettingeachsiteofthe bulk free energy is computed as a change in free energy due
latticebeoccupiedwithanaprioriprobabilityporleftempty totheadditionofasiteandthecorrespondinglinksemanating
wtioitnhfaunncatipornioirsitphreonbability1−p. Thegrandcanonicalparti- from this site F =limk→∞(cid:104)−lnZ/Zk3+(z/2)lnZ˜/Zk2(cid:105),
where and ˜arethepartitionfunctionsforauniformBethe
(cid:18) (cid:19)n(C) Z Z
(cid:88) p lattice obtained by connecting either z or z 1 root sites of
Z = 1 p WC (1) independent trees via a new site or edge. Zk−is the partition
{C} − functionforalevel-ktree[18,20–23].
where ln[p/(1 p)] plays the role of a chemical potential In the first instance we are interested in the existence of
controlling the−density of occupied sites and letting it fluc- a giant cluster (i.e. a cluster occupying a finite fraction of
tuate. Note that a priori probability p, unlike a regular site the lattice), in the simplest case of z = 3. We define ∞
P
percolation,isnotequaltothedensityofoccupiedsites. to be the probability that a given site belongs to such giant
This model also has a simple representation as a particu- cluster;seeSupplementaryMaterialfordetails. Forp˜<4/9,
lar classical two-color, or contagion, percolation problem. It the only solution of the resulting equations is 0 = 0 – in
P∞
is a mild variation of regular percolation: sites can come in other words, there is no percolation. For p˜ 4/9 two more
≥
twocolors,green(uninfected)orred(infected). Specifically, solutionsappearwith ± = 0asshowninFigure2(b)(with
P∞ (cid:54)
each site of a lattice is occupied and colored either green or thelowerbrunchbeingunphysical). Notethatthereisnever
redwithanaprioriprobabilityp˜each, orleftemptywithan a percolating uninfected cluster: the probability that a given
a priori probability 1 2p˜. Only configurations where clusterofsizenremainsuninfectedis1/(n+1).
− {G}
everyclustercontainsnomorethanoneredsitearetakeninto The topology of the plot for already demonstrates the
∞
P
account. Thepartitionfunctionofthismodelisthensimply firstordernatureofthepercolationtransition:thecurvewhich
yields the solution + = 1 (i.e. all sites are occupied) for
(cid:88)(cid:18) p˜ (cid:19)n(G) p˜ 1/2nevercrosPse∞sthenon-percolatingsolution 0 =0,
Zgr= 1 2p˜ (2) wh→ichinturnisuniqueforp˜ 0. ThetransitionfroPm∞oneto
{G} − →
theotherthereforeimpliesajumpin ! Todeterminewhen
∞
P
It is straightforward to see that tracing over all possible site the actual transition takes place we analyze the bulk free en-
colors consistent with fixed site occupations renders Eq. (2) ergyoftheproblem.Thesolutionwhichminimizesthisquan-
identicaltoEq.(1)(withtheidentificationofp˜=p/(p+1)): titymaximizesthepartitionfunctionandthusisselected.This
eachclustermayhaveeitherallsitesgreen(uninfected),orat selectsthesolutionofP∞+ (cid:54)= 0atp˜c = 0.451606...(SeeFig-
mostonered(infected)site. Thereforeaclusterofnsiteshas ure2(b))indicatingadiscontinuousjumpassoonasp˜= p˜c.
weight (n+1) after the sum over possible locations of red Wenotethatthisisinagreementwithotherquantitiessuchas
sites is taken into account. The utility of the formulation as cluster size distribution and average cluster size which show
atwo-colorpercolationproblemliesinthefactthattheneed no signature of power-law distribution or divergence at the
evertocomputeclustersizesisobviated: thechoiceofloca- transitionpoint. These,togetherwithdetailsofthecomputa-
tionoftheinfectedsitetakescareofthat. tion,areshownintheSupplementalMaterial.
Analytic and numerical results. We show that Pauli per- WesupportouranalyticresultsbyMonte-Carlosimulations
colationexhibitsadiscontinuouspercolationtransitioninin- ofEq.(1)onaregularrandomgraph. Weanalyzedensityof
finitedimensionsbystudyingitanalyticallyandnumerically occupiedsitesρaswellashistogramsofitsdistributionalong
onaregularrandomgraphofN sites. Suchgraphsareoften with the fourth-order Binder cumulant as standard indica-
U
usedtoapproximaterandomnetworks[17]. Theyhaveavan- torsofphasetransitions. Inallthequantitiestheextrapolation
ishingdensityofshortcyclesandmostlycontainloopsofsize to N is consistent with the exact solution. Below and
→ ∞
lnN; hencetheyarelocallytree-like[18,19]. Thisproperty above the transition the numerical data follows the branches
enablesustodevelopanexactsolutionviaaso-calledcavity of the exact solution for the uniform Bethe lattice. The his-
3
ρ
0.8 exact (a)
k 1 ρ
k 1 � MC
�
k 0.7
k
k k 0.6
k
k�1 k�1 k 1 ∼ 0.41 0.42 0.43 0.44 0.45 p∼
� p
c (b) P(ρ)
0.45 U
0.448
(a)
0.446
1 P0 0.444
0.8 P+∞ 0.442
0.6 P∞- 0 0.001 0.002 0.003 0.004 1/N
∞
0.4
0.2 FIG. 3. (a) Density ρ versus a priori probability p˜. Red
0.7 F line indicates the exact solution while the dots represent Monte-
0
F Carlo results for 3-regular random graphs of sizes N =
+
0.6 F 50,100,200,300,400,600,800. (b)Finitesizeextrapolationofthe
-
minimumofthefourth-orderBindercumulantofdensityUandpoint
0.5 of equally weighed peaks of histograms of density P(ρ). The red
∼ pentagonmarksthepointatwhichtheinfiniteclusterappearsinthe
0.4 0.42 0.44 0.46 0.48 p thermodynamiclimit,atadensitydistinctfromwherethesolutions
P± firstappear,indicatedbythebluediamond.
(b) ∞
FIG. 2. (a) Properties of a regular random graph make it locally
equivalenttotheneighborhoodofaninternalsiteofaBethelattice met with much excitement, yet the initial approach proved
obtainedbyconnectingrootsofindependentCayleytreesviasiteor to be deficient [7]. A discontinuous transition has finally
edge addition. Here the sites colors (red/dark grey and green/light
been found in several variants of explosive percolation mod-
grey)representoneinstanceallowedbythetwo-colorcontagionper-
els which, however, require a very elevated degree of non-
colation rules. (b) The upper pane shows the probability P that
∞
locality: adynamicalprocessdefiningthesemodelsinvolves
a site belongs to a giant cluster as a function of p˜. The blue dia-
mondmarksp˜ = 4/9atwhichthenonzerosolutionappears. The acomparisonbetweenanextensivenumberofdegreesoffree-
lowerpaneshowsthe‘bulk’freeenergypersiteofa3-regularran- dombeforeaconfigurationchangeoccurs[6,7,10,12]. Re-
domgraphcorrespondingtoeachofthesesolutions(seetextforde- cent studies also considered suppressing the onset of perco-
tails). Theredpentagonindicatesthetransitionpoint. Boldpartsof lation through a rule explicitly inhibiting bond addition if it
thelinesinbothpanelsindicatetheactualsolution.
leads to the formation of a spanning cluster [13]. Not only
suchaprocessinvolvesanextensivenumberoflocaldegrees
of freedom, it also makes the ‘microscopic’ dynamics of the
model–aplacementofaparticularbond–explicitlydepend
tograms of the density distribution give a clear double peak
ontheonsetofaglobalphenomenon,percolation.
structure – the hallmark of a discontinuous transition – and
inFigure3weprovideanextrapolationofthepointatwhich Paulipercolation,bycontrast,considersonesiteatatime,
thesetwopeaksareofequalweight. Thisnicelyextrapolates with a minimal amount of non-locality entering only via the
to the analytic result for the transition point p˜ = 0.452(3). sizes of the clusters impinging on the site in question. In
c
Finally the density Binder cumulant develops a minimum otherwords,Paulipercolationisnon-localonlyuptothesize
U
at the transition point – a typical behavior for a discontinu- of the clusters present locally. The Pauli principle of quan-
oustransition;itsextrapolationtothermodynamiclimitisalso tummechanicspresentsastraightforwardphysicaloriginfor
ingoodagreementwiththetransitionpointobtainedanalyti- such a weight: quantum statistical interactions are intrinsi-
cally. cally non-local on this level. A classical route to the same
Discussion. TheattractivenessofPaulipercolationisman- weightsinvolvespermittingatmostonesiteofeachclusterto
ifold. Firstly, it is underpinned by a simple and transparent beinfected,againasimpleandintuitivedescriptioninvolving
physical mechanisms. Secondly, it is amenable to detailed clustersonlylocallyandindividually.
numerical and analytical analyses. Thirdly, and crucially, it NordoesPaulipercolationrequiretheirreversibilityofex-
exhibits a remarkable phenomenology featuring a reversible plosive percolation. Based on statistical weights of configu-
first-orderpercolationtransition. Inthefollowing,wediscuss rations rather than rules for cluster growth, Pauli percolation
theimportoftheseitems,andembedtheminabroaderzool- provides an equilibrium first-order transition. It in particular
ogyofpercolationproblems. allowsforshrinking,aswellasgrowing,clusters. Ittherefore
Thenotionofexplosivefirst-orderpercolation[5]hasbeen naturally accommodates healing/repairing processes, in e.g.
4
networkapplicationswhich,notably,canremovepercolation (Taylor&Francis,1994).
discontinuously. Thegrowthprocessencodedbythe“product [3] C.Moore,M.E.J.Newman,Phys.Rev.E61,5678(2000).
rule”inexplosivepercolationisreminiscentoftheweightsof [4] G. Bizhani, M. Paczuski, P. Grassberger, Phys. Rev. E 86,
011128(2012).
Paulipercolation: thelatter,however,providesanaturalpre-
[5] D.Achlioptas, R.M.D’Souza, J.Spencer, Science323, 1453
scriptionforremovingparticlesaswell. Weshouldnotethat
(2009).
anotherroutetoareversiblefirst-orderpercolationtransition,
[6] R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, J. F. F.
although not normally thought of in these terms, is provided Mendes,Phys.Rev.Lett.105,255701(2010).
by the Fortuin–Kasteleyn (FK) representation of the q-state [7] O.Riordan,L.Warnke,Science333,322(2011).
Potts model [27]. In this mapping, the ordering transition of [8] R.M.Ziff,Phys.Rev.Lett.103,045701(2009).
thePottsmodelcorrespondstoacorrelatedbondpercolation [9] R.K.Pan,M.Kivela¨,J.Sarama¨ki,K.Kaski,J.Kerte´sz,Phys.
Rev.E83,046112(2011).
problem. Forq >q (d)(withq =2ford 4),theordering
c c
≥ [10] P. Grassberger, C. Christensen, G. Bizhani, S.-W. Son,
transitionofthePottsmodelisoffirstorder, andhencesois
M.Paczuski,Phys.Rev.Lett.106,225701(2011).
theconcomitantFKbondpercolationtransition.Anothertype
[11] K. Panagiotou, R. Spo¨hel, A. Steger, H. Thomas, Combin.
of percolation models with a known first order transition are Probab.Comput.22,133(2013).
so-calledk-coreandcloselyrelatedrigiditypercolationprob- [12] S.Manna,A.Chatterjee,PhysicaA390,177(2010).
lems[28–30].Here,despitelocalupdaterules,thepercolation [13] Y.S.Cho,S.Hwang,H.J.Herrmann,B.Kahng,Science339,
phenomenon itself cannot be detected without “postprocess- 1185(2013).
[14] A.Mielke,H.Tasaki,Commun.Math.Phys.158,341(1993).
ing”,whichbothrequiresanextensivenumberofchecksand
[15] M. Maksymenko, A. Honecker, R. Moessner, J. Richter,
complicatesareversibledynamicalprocessinterpretation.
O.Derzhko,Phys.Rev.Lett.109,096404(2012).
Pauli percolation can be easily generalized to a non- [16] R.M.Ziff,Science339,1159(2013).
equilibriumgrowthprocess,e.g. bysimplyremovingthede- [17] S.N.Dorogovtsev,LecturesonComplexNetworks(Clarendon
tailed balance implied by the configuration weights, and re- Press,Oxford,2010).
[18] M.Me´zard,G.Parisi,Eur.Phys.J.B20,217(2001).
tainingonlytherelativeratesforparticleaddition. Ingeneral,
[19] S. Janson, T. Luczak, A. Rucinski, Random Graphs (Wiley,
thereisahugefamilyofnon-equilibriumprescriptionswhich
NewYork,2000).
“generalize”agivenequilibriumdistribution.Theequilibrium
[20] C. Laumann, A. Scardicchio, S. L. Sondhi, Phys. Rev. B 78,
process–besideswideningthepurviewofapplicationsfrom 134424(2008).
the exclusivelynon-equilibrium cases– leadsto a greatsim- [21] O.Rivoire,J.Stat.Mech.2005,P07004(2005).
plificationintheanalysis. Itcanbeefficientlystudiednumer- [22] J.Barre´,B.Goncalves,PhysicaA386,212(2007).
ically on a wide range of graphs and lattices, and therefore [23] B. Hsu, C. Laumann, A. Laeuchli, R. Moessner, S. Sondhi,
Phys.Rev.A87,062334(2013).
incorporatesgeometricstructuresandinhomogeneitieswhich
[24] S.Hoory,N.Linial,A.Wigderson,Bull.Am.Math.Soc.43,439
maybecalledforinreal-lifeapplications.Onsufficientlyreg-
(2006).
ulargraphs,itcanbestudiedexactlywithstandardanalytical
[25] C.R.Laumann,S.A.Parameswaran,S.L.Sondhi,Phys.Rev.
methods. Thisinparticularobviatesworriesaboutcrossovers B80,144415(2009).
on absurdly long lengthscales or anomalously small critical [26] J. Barre´, D. Mukamel, S. Ruffo, Phys. Rev. Lett. 87, 030601
exponents[6,10].Approximationssuchasgeometry-freepre- (2001).
scriptions for product-rule percolation also become unneces- [27] C.M.Fortuin,P.W.Kasteleyn,Physica57,536(1972).
[28] C. Moukarzel, P. M. Duxbury, P. L. Leath, Phys. Rev. E 55,
sary.
5800(1997).
Insummary,Paulipercolationisasimple,physical,natural, [29] P.M.Duxbury,D.J.Jacobs,M.F.Thorpe,C.Moukarzel,Phys.
transparent and tractable novel percolation problem exhibit- Rev.E59,2084(1999).
inganintriguingphenomenology. Itholdsgreatpromiseasa [30] J.M.Schwarz,A.J.Liu,L.Q.Chayes,Europhys.Lett.73,560
benchmarkproblemacrosstherangeofdisciplinesinterested (2006).
in percolation problems, ranging from condensed matter via
biological systems and real-world networks to epidemic dis-
easeoutbreaks.
WearegreatfultoR.D’Souza,P.Grassberger,M.Hastings
andL.Chayesforhelpfuldiscussions. Wewouldalsoliketo
acknowledge our collaboration with O. Derzhko, J. Richter
and A. Honecker on closely related work. This work was
supported in part by the NSF through grant DMR-0748925
(K.S.).
[1] M.B.Isichenko,Rev.Mod.Phys.64,961(1992).
[2] D. Stauffer, A. Aharony, Introduction To Percolation Theory
5
SUPPLEMENTARYMATERIAL p˜ 4/9,however,twoadditionalsolutionsemerge:
≥
(cid:114)
1 3 4
S1.ExactsolutionforPaulipercolationonaCayleytree P± =Pi,± = 1 , Pu =0. (S4)
∞ ∞ 2 ± 2 − 9p˜ ∞
Wewillusethefollowingdefinitions: alevel-kCayleytree Ashasbeenpointedinthemaintext,thefactthatPu remains
∞
of coordination number z is constructed recursively by con- zeroevenaftertheonsetofpercolationisratherobvioussince
nectingarootsitetoz 1identicallevel-(k 1)trees–until theprobabilityofalargeclustertoremainuninfectedtendsto
− −
level 0 is reached. We will refer to level-0 sites as leaves; zerowithitssize.
they constitute the outer boundary of the Cayley tree. The In the same manner we can compute other quantities
so-called wired boundary conditions which we will consider such as the probability of a given site being empty P =
e
hereareequivalenttoestablishingadditionalconnectionsbe- lim E /Z , aswellwellasbeingoccupiedandbelong-
k→∞ k k
tweenallboundarysites[S1,S2]. Ontheotherhand,thefree ing to either an uninfected or infected finite cluster, Pu,i =
f
boundary conditions correspond to the leaves having only a lim Fu,i/Z . Ifthereisnopercolation(Pu =Pi =0),
k→∞ k k ∞ ∞
singleneighbor,theoneatthenextlevel.
theseexpressionsaregivenby
Wewritethepartitionfunctionofthetwo-colorpercolation
(cid:112)
problemforalevel-kCayleytree(herewepresentthecaseof 1 4p˜2
P = −
z = 3) as a sum of contributions corresponding to the ‘fate’ e 2p˜+1
ofitsrootsite: 4p˜2+(cid:112)1 4p˜2 1
Pu = − −
f 2p˜(2p˜+1)
Z =E +Fu+Fi +U +I , (S1)
k k k k k k 1 (cid:16) (cid:112) (cid:17)
Pi = 1 1 4p˜2 . (S5)
where E , Fu/i, U and I account for all configurations in f 2p˜ − −
k k k k
whichtherootsiteatlevelkis,respectively,emptyorbelongs Above the percolation threshold, Pi = 0, the expressions
to a finite uninfected/infected, giant uninfected (U) or giant ∞ (cid:54)
becomerathercumbersome:
infected(I)cluster. Wecallaclusterinfected ifitcontainsa
single red site; a cluster is referred to as giant if it contains √2(1 2p˜)
P = −
e
both the root and a boundary site, or as finite otherwise. By θ±(p˜)
attachingtwolevel-ktreestoanewrootsiteatlevelk+1and 1 (cid:104)
Pu = 9p˜ 14p˜2+3√2(p˜ 1)θ (p˜)
denotingHk = Ek+Fku wearriveatthefollowingrecursion f 4p˜(14p˜ 9) − − ±
−
relations (cid:112) (cid:112) (cid:16) (cid:17)(cid:105)
p˜ 9p˜ 4 9 √2θ (p˜) 14p˜
±
± − − −
EFkku++11 ==p(˜1H−k2,2p˜)Zk2F,ki+1 =p˜(Hk2+2FkiHk), Pfi = 3p˜∓(cid:112)p˜(9p˜−4p˜4)−√2θ±(p˜) (S6)
(S2)
U =p˜(cid:2)2U H +U2(cid:3),
k+1 k k k with
Ik+1 =p˜(cid:2)2IkHk+2UkHk+2UkFki +Uk2+Ik2(cid:3), (cid:114) (cid:16) (cid:112) (cid:17)
θ (p˜)= p˜ 6 11p˜ p˜(9p˜ 4) .
±
wherep˜isaprioriprobabilityofsitebeingoccupiedandcol- − ± −
oredredorgreenasfollowsfromthemaintext. Notethatthe
Thesesolutionsareusedinthederivationofthecorresponding
termcontainingI2inthelastlineimplies,somewhatcounter-
k probabilitiesforthez-regularrandomgraph.
intuitively,thattwogiantinfectedclusterscanbemerged. In
fact,thisisaconsequenceofthe’wired’boundaryconditions:
thesearetwopartsofthesameclusterwhicharealreadycon- S2.Site/Edgeadditiontoaz-regularrandomgraph
nected via boundary sites. Essentially, wired boundaries im-
ply that there may only exist a single giant cluster. For the Usingthecavitymethodwecannowobtaintheresultsfor
samereason,noUkIk termsarepossible. (Notethatthissitu- the full-space z-regular random graph. In the cavity method
ationisreversedforfreeboundaryconditions.) Thepartition the addition of a bulk site or edge is equivalent to connect-
functionofa(k+1)-leveltreeis ingz orz 1rootsofindependentlevel-k Cayleytrees(see
−
Figure 2(a)) in the main text). In other words the bulk site
Z =(1 2p˜)Z2+2p˜Z (H +U ) 2p˜I U +p˜I2 (S3)
k+1 − k k k k − k k k oredgeofz-regularrandomgraphisequivalenttothecentral
siteoredgeofauniformBethelattice. Thequantitiesforthe
WedefinePu = lim U /Z andPi = lim I /Z tobe
∞ k k ∞ k k site-centeredcase,analogoustothosegivenbyEqs.(S2)fora
k→∞ k→∞
probabilitiesthattherootsiteofalargetreeisconnectedtoits rootedtree,canbewrittenas
boundaryviaanuninfectedandinfectedclustersrespectively.
=(1 2p˜)Z3, u =p˜H3, i =p˜(cid:0)H3+3FiH2(cid:1),
Ifp˜<4/9,theonlyrealsolutionoftheresultingequationsis E − k F k F k k k
Pu = Pi = 0 – in other words, there is no percolation. If =3p˜I H2+3p˜I2H +p˜I3 (S7)
∞ ∞ I k k k k k
6
where we have discounted all configurations where the giant doesnothaveanysensiblenotionofenergyassociatedwithit,
cluster is uninfected – we have already seen that they have hencenoderivativeswithrespecttoexternalfieldscanbeused
vanishingrelativecontribution. Thepartitionfunctionisthen to define any thermodynamic potentials here, unlike, e.g. in
thecontextofthePottsmodel[S6].Onecoulddefinethelimit
= + u+ i+ ofthefreeenergyperinternalsitefollowingtheapproachof
Z E F F I
=(1 2p˜)Z3+3p˜Z H2 p˜H3+3p˜I2H +p˜I3 (S8) Ref.S7,yetthisquantityisnothelpfuleither:notonlyissuch
− k k k − k k k k free energy always minimized in the percolating phase, it is
Anotherwayofconstructingauniformlatticeisbyadding not even continuous across the (putative) transition into this
an edge between two root sites of Cayley trees. The corre- phase. Sincetheactualfreeenergymustbecontinuousacross
spondingquantitiesinthiscasebecome: any phase transitions, it is clear that the aforementioned free
energy per internal site is not the right quantity to look at in
H˜ =Hk2, F˜i =2FkiHk, I˜ =2IkHk+Ik2. (S9) ourcase. (Naturally,thetotalfreeenergydefinedviathelog-
arithm of the partition function is a continuous function of
Themeaningofthesequantitiesinthecaseofedgeadditionis
its parameters but contains an extensive boundary contribu-
asfollows: ˜ isthenumberofallconfigurationswhereeach
tion.) Inshort, thefailureofthisapproachsignifiesasimple
H
ofthe(formerroot)sitesconnectedbythenewedgewaseither
fact that the free energy cannot be associated solely with an
emptyorbelongedtoafiniteuninfectedcluster; ˜icountsall
internal site or an internal bond of a Bethe lattice, and the
F
configurations where one of these sites belonged to a finite
presence of an extensive boundary prevents one from mean-
infectedclusterwhiletheotherwaseitheremptyorapartofa
ingfully distributing its ‘shares’ between them. Specifically,
finiteuninfectedcluster; ˜countsconfigurationswhereeither
a choice of boundary conditions (e.g. free vs. wired) dra-
I
oneorbothsitesbelongedtoagiant(infected)cluster. Once
maticallychangestheratiobetweenthenumberofbondsand
again, wediscounttheconfigurationswherethegiantcluster
thenumberofsitesinthesystem.Notethatthisissuedoesnot
isuninfected. Thecorrespondingpartitionfunctionis
ariseinthecontextofcontinuousphasetransitionssincethose
always coincide with the emergence of a non-trivial solution
˜= ˜+ ˜i+ ˜ =2Z H H2+I2. (S10)
Z H F I k k− k k fortheorderparameter.
Using these quantities, we can calculate various probabil- The problem with a meaningful definition of the free en-
ities in the same fashion as in the previous section for the ergy is cured by considering a z-regular random graph in-
root site – see Eqs. (S5,S6). Note that the value of parame- stead of a Bethe lattice. The two are locally equivalent to
ter p˜ = 4/9 at which the percolation solution = 0 first one another, yet the random graph lacks a distinct bound-
∞
emergesisnotaffectedbysuchcalculation,albePitthe(cid:54) valueof ary. This in turn fixes the bond to site ratio of in the
thepercolationprobabilityitselfchanges: =P . system to z/2. We can then use the prescription out-
∞ ∞
The importance of merging the rooted PCayl(cid:54)ey trees into a lined in Ref. [S9] to write the free energy per added site
(cid:104) (cid:105)
Bethelatticeinthesetwodifferentwayswillbecomeclearin as =lim ln /Z3+z/2ln ˜/Z2 wherethefirst
F k→∞ − Z k Z k
thenextsectiondedicatedtocalculatingthefreeenergy. This term corresponds to the the free energy of an internal site of
will allow us to circumvent the inherent problem of evaluat- aBethelatticedefinedsimilarlytoRef.[S7]whilethesecond
ing extensive thermodynamic potentials on the Bethe lattice, termcorrectsforthefactthataddingasitetoaz-regulargraph
wherethenumberofboundarysitesisafinitefractionofthe createsz newedges,andhencez/2existingedgesshouldbe
totalsystem. removedtomaintainthegraph’sregularity.
Usingexpressionsfor , ˜andZ givenbyEqs.(S8)and
k
Z Z
(S10)oftheprevioussection,wefindtheexpressionsforfree
S3.Bulkfreeenergy
energy corresponding to all three solutions for on a 3-
∞
P
regularrandomgraph:
In contrast with continuous phase transitions, first order
(cid:32) (cid:33)
transitions do not occur when the non-trivial solution for the 1 1 1
= ln2 lnp˜+ ln 1 (S11a)
0 (cid:112)
order parameter first appears as this normally signifies only F −2 − 2 1 4p˜2 −
the emergence of a metastable state. Therefore, in order to −
determine the actual transition point in this case, one needs 3 (cid:104) (cid:112)
to study the free energy; the transition occurs when the free ± = ln2 ln 26 6 p˜(9p˜ 4) 46p˜
F −2 − ∓ − −
energyassociatedwithanorderedstatebecomessmallerthan (cid:112) (cid:16) (cid:17)(cid:105)
+√2(11 6/p˜)θ (p˜) 9 4/p˜ 6 √2θ (p˜)
that for the disordered state. This seemingly straightforward ± ±
− ± − −
testbecomesproblematiconaBethelatticeduetotheafore- (cid:34)
3 5√2θ (p˜) 12
mentionedissueofanextensivesizeoftheboundary. While + ln 22+ ± −
2 p˜
this problem had been widely discussed in the literature –
see e.g. [S3–S8] – none of the recipes proposed there are (cid:112)9 4/p˜(cid:16)√2θ (p˜)/p˜ 2(cid:17)(cid:105). (S11b)
±
applicable (or even meaningful) for the case of percolation. ± − −
Specifically,ourpercolationmodelisacountingproblemand TheseexpressionsareplottedinFigure2(b)ofthemaintext.
7
3.5
1 0.5
3 0.1 un+in 0
2.5
0.01 -0.5 U, N=50 N=400
2
-1 N=100 N=600
1.5 1 10 100 -1.5 N=200 N=800
1 -2 N=300
0.5 χ ∼
0 0.442 0.444 0.446 0.448 0.45 p
∼
0 0.1 0.2 0.3 0.4 p
N=100 N=400
N=200 N=600
FIG.S1. Averagesizeofafiniteclusterχ(p˜). Theinsetshowsthe N=300 N=800
clustersizedistributionf(n) = u +i belowpercolation,atp˜=
n n
0.45<p˜ (red)andabovepercolation,p˜=0.452>p˜ (blue).
c c
S4.Averageclustersizeandclustersizedistribution 0.55 0.6 0.65 0.7 0.75 0.8 ρ
FIG.S2. a)Fourth-orderBindercumulantofdensity. b)Histograms
Havingobtainedthesolutionsoftherecursionrelationsfor
ofdensityatpointwherepeaksareequallyweighed; linesindicate
asinglerootedCayleytree,wehaveaccesstootherphysical
fittoapairofGaussianfunctions.
quantities of interest such as the average cluster size or the
cluster size distribution in the same recursive manner. For
example,theexpectedsizeofaclustercontainingtherootsite Heretheprobabilitiesforasitetoformanisolatedclusteror
butnotanyleavesisgivenby toremainunoccupiedare,respectively,
χk = fkFk = fkiFki +fkuFku, (S12) u1 =i1 =p˜ηPe2, u0 =i0 =Pe (S15)
Z Z
k k
andη =lim Z2/Z .
where f is the average cluster size in non-percolating con- k→∞ k k+1
k Thegeneratingfunctionsforbothsequencesare
figurations,F istheweightofcorrespondingconfigurations.
k
Aclsusbteerfso.reT, hlaebekls ‘i’ andlim‘ui’tionfditchaitseqiunafenctitteydpolrayusnitnhfeecrotelde u(x)= (cid:88)unxn = 1−√1−4xp˜ηPe (S16)
→ ∞ G 2xp˜η
of susceptibility in conventional percolation problems. With n≥0
minimaleffort,italsocanbefoundfortheuniformBethelat-
and
tice; the result is shown in Figure S1. Since in our case the
transition is first order, this quantity does not diverge at the i(x)= (cid:88)i xn = Pe+xp˜η[Gu(x)]2, (S17)
transitionp˜=p˜c. G n 1 2xp˜η u(x)
The cluster size distribution for a rooted tree can be ob- n≥0 − G
tained from the total weight of configurations where the site anditsseriesexpansionyieldsu andi .
n n
atlevelkbelongstoafinitecluster The cluster size distribution (inset of Figure S1) has an
(cid:88) (cid:88) exponential cutoff for large clusters both below and above
F =Fi +Fu = Fi(n)+ Fu(n). (S13)
k k k k k the percolation transition. After the appearance of the giant
n n
componentthecutoffdiscontinuouslyshiftstosmallercluster
HereFu,i(n)aretheweightsofconfigurationswheretheroot sizes.Asimilarclustersizedistributioncaninprinciplebeob-
k
site belongs to a cluster of size n. The corresponding prob- tainedforauniformBethelatticeyetinpracticetherecursion
abilities for a site to belong to an uninfected/infected n-site relationsbecomeextremelyunwieldy.
clusterare(u,i) = lim Fu,i(n)/Z andmaybealsofound
n k k
k→∞
fromrecursionrelations
S5.Detailsofnumericalsimulations
n−2
(cid:88)
un =p˜η ukun−k−1+2un−1Pe, (S14a) We use a Metropolis Monte-Carlo algorithm developed in
k≥1 [S10] on graphs with up to N = 820 sites. The algorithm
worksinthegrand-canonicalpicturewhereachemicalpoten-
(cid:32)n−2 n−2 tialµcontrolsthedensityofparticlesρandallowsittofluctu-
(cid:88) (cid:88) (cid:81)
in =p˜η ukun−k−1+2 ikun−k−1 ate. The statistical weight is W = exp(µn) (n(Ci)+1).
k=1 k=1 The chemical potential µ is directly related to the a priori
(cid:33) probabilityp˜viaµ = ln[p˜/(1 2p˜)]. Ateverystepweran-
−
+2P i +2P u . (S14b) domlychooseasite(orgroupofsites)andifitisempty(occu-
e n−1 e n−1
pied),proposetooccupy(empty)it. Thenewconfigurationis
8
acceptedwithaMetropolisprobability. Weuseupto2 106 tedinthemaintext.
×
steps for equilibration which are then followed by 2 106
×
steps for every measurement round. The plots presented in
thereportarebasedonaveragingoverupto30measurements
with a new realization of random graph for every measure-
[S1] J.T.Chayes,L.Chayes,J.P.Sethna,D.J.Thouless,Commun.
ment. The expander nature of the graph and the long range
Math.Phys.106,41(1986).
ofunderlyinginteractionsleadtostronghysteresis. Toreduce
[S2] L.Chayes,A.Coniglio,J.Machta,K.Shtengel,J.Stat.Phys.
hysteresis,wehaveemployedanexchangeMonte-Carlopro-
94,53(1999).
cedure by simulating the system at different values of p˜and [S3] E. Mu¨ller-Hartmann, J. Zittartz, Phys. Rev. Lett. 33, 893
allowingexchangesofconfigurationsbetweenthem. Tocon- (1974).
trol hysteresis effects, we have performed simulations start- [S4] T.P.Eggarter,Phys.Rev.B9,2989(1974).
ingfromemptyoroccupiedlatticesasinitialconditions. We [S5] Y.K.Wang,F.Y.Wu,J.Phys.A:Math.Gen.9,593(1976).
[S6] F.Peruggi,F.diLiberto,G.Monroy,J.Phys.A:Math.Gen.
presentresultsforsystemsizeswhichshownohysteresis. In
16,811(1983).
Figure S2 we present a fourth-order Binder cumulant de-
U [S7] P.D.Gujrati,Phys.Rev.Lett.74,809(1995).
finedas.
[S8] P. M. Duxbury, D. J. Jacobs, M. F. Thorpe, C. Moukarzel,
Phys.Rev.E59,2084(1999).
ρ4
=1 (cid:104) (cid:105) (S18) [S9] J.Barre´,B.Goncalves,PhysicaA386,212(2007).
U − 3 ρ2 2 [S10] M. Maksymenko, A. Honecker, R. Moessner, J. Richter,
(cid:104) (cid:105)
O.Derzhko,Phys.Rev.Lett.109,096404(2012).
Minimaofthisquantityindicatesthelocationofaphasetran-
sition,theirextrapolationtothethermodynamiclimitisplot-
