Prob Models Crit Pheno

8/13/2019 Prob Models Crit Pheno

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Princeton Companion to Mathematics Proof 1

Probabilistic Models of

Critical Phenomena

By Gordon Slade

1 Critical Phenomena

1.1 Examples

A population can explode if its birth rate exceedsits death rate, but otherwise it becomes extinct.The nature of the populations evolution dependscritically on which way the balance tips betweenadding new members and losing old ones.

A porous rock with randomly arranged micro-scopic pores has water spilled on top. If there are

few pores, the water will not percolate through therock, but if there are many pores, it will. Surpris-ingly, there is a critical degree of porosity thatexactly separates these behaviours. If the rocksporosity is below the critical value, then water can-not flow completely through the rock, but if itsporosity exceeds the critical value, even slightly,then water will percolate all the way through.

A block of iron placed in a magnetic field willbecome magnetized. If the magnetic field is extin-guished, then the iron will remain magnetized ifthe temperature is below the Curie temperature770C, but not if the temperature is above this

critical value. It is striking that there is a specifictemperature above which the magnetization of theiron does not merely remain small, but actuallyvanishes.

The above are three examples of critical phe-nomena. In each example, global properties of thesystem change abruptly as a relevant parameter(fertility, degree of porosity, or temperature) is var-ied through a critical value. For parameter values

just below the critical value, the overall organiza-tion of the system is quite different from how it isfor values just above. The sharpness of the transi-tion is remarkable. How does it occur so suddenly?

1.2 Theory

The mathematical theory of critical phenomenais currently undergoing intense development. In-tertwined with the science ofphase transitions, itdraws on ideas from probability theory and statis-tical physics. The theory is inherently probabilis-tic: each possible configuration of the system (e.g. a

particular arrangement of pores in a rock, or of themagnetic states of the individual atoms in a blockof iron) is assigned a probability, and the typicalbehaviour of this ensemble of random configura-tions is analyzed as a function of parameters ofthe system (e.g. porosity or temperature).

The theory of critical phenomena is now guidedto a large degree by a profound insight from physicsknown as universality, which, at present, is moreof a philosophy than a mathematical theorem.The notion of universality refers to the fact thatmany essential features of the transition at a crit-ical point depend on relatively few attributes ofthe system under consideration. In particular, sim-ple mathematical models can capture some of thequalitative and quantitative features of critical be-

haviour in real physical systems, even if the mod-els dramatically oversimplify the local interactionspresent in the real systems. This observation hashelped to focus attention on particular mathemati-cal models, among both physicists and mathemati-cians.

This essay discusses several models of criticalphenomena that have attracted much attentionfrom mathematicians, namely branching processes,the model of random networks known as the ran-dom graph, the percolation model, the Ising modelof ferromagnetism, and the random cluster model.As well as having applications, these models are

mathematically fascinating. Deep theorems havebeen proved, but many problems of central impor-tance remain unsolved and tantalizing conjecturesabound.

2 Branching Processes

Branching processes provide perhaps the simplestexample of a phase transition. They occur nat-urally as a model of the random evolution of apopulation that changes in time due to births anddeaths. The simplest branching process is definedas follows.

Consider an organism that lives for a unit timeand that reproduces immediately before death.The organism has two potential offspring, which wecan regard as the left offspring and the rightoffspring. At the moment of reproduction, the or-ganism has either no offspring, a left but no rightoffspring, a right but no left offspring, or both aleft and a right offspring. Assume that each of


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2 Princeton Companion to Mathematics Proof

Figure 1.1: A possible family tree, with probability

p10(1p)12.

the potential offspring has a probability p of be-ing born and that these two births occur indepen-dently. Here, the number p, which lies between 0and 1, is a measure of the populations fecundity.

Suppose that we start with a single organism attime zero, and that each descendant of this organ-ism reproduces independently in the above man-ner.

A possible family tree is depicted in Figure 1.1,showing all births that occurred. In this familytree, 10 offspring were produced in all, but 12 po-tential offspring were not born, so the probabilityof this particular tree occurring is p10(1p)12.

If p = 0, then no offspring are born, and thefamily tree always consists of the original organ-ism only. Ifp = 1, then all possible offspring are

born, the family tree is the infinite binary tree, andthe population always survives forever. For inter-mediate values ofp, the population may or may notsurvive forever: let (p) denote the survival prob-ability, that is, the probability that the branchingprocess survives forever when the fecundity is setat p. How does (p) interpolate between the twoextremes(0) = 0 and (1) = 1?

2.1 The Critical Point

Since an organism has each of two potential off-spring independently with probabilityp, it has, onaverage, 2p offspring. It is natural to suppose that

survival for all time will not occur ifp < 12 , sincethen each organism, on average, produces less than1 offspring. On the other hand, ifp > 12 , then, onaverage, organisms more than replace themselves,and it is plausible that a population explosion canlead to survival for all time.

Branching processes have a recursive nature, notpresent in other models, that facilitates explicit

0

1

p = 0

q

11

2

Figure 1.2: The survival probability versus p.

computation. Exploiting this, it is possible to showthat the survival probability is given by

(p) =

0 ifp 12 ,

1

p2(2p 1) if p 12 .

The value p= pc = 12 is a critical value, at which

the graph of(p) has a kink (see Figure 1.2). Theintervalp < pc is referred to as subcritical, whereas

p > pc is supercritical.Rather than asking for the probability (p) that

the initial organism has infinitely many descen-

dants, one could ask for the probability Pk(p) thatthe number of descendants is at leastk. If there areat leastk + 1 descendants, then there are certainlyat least k, so Pk(p) decreases as k increases. Inthe limit ask increases to infinity, Pk(p) decreasesto (p). In particular, when p > pc, Pk(p) ap-proaches a positive limit as k approaches infinity,whereas Pk(p) goes to zero when p pc. When pis strictly less thanpc, it can be shown that Pk(p)goes to zero exponentially rapidly, but at the crit-ical value itself we have

Pk(pc)

2

k.

The symbol denotes asymptotic behaviour,and means that the ratio of the left- and right-handsides in the above formula goes to 1 as k goes toinfinity. In other words,Pk(pc) behaves essentiallylike 2/

k when k is large.

There is a pronounced difference between theexponential decay of Pk(pc) for p < pc and the


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0

1

p = 0 112

Figure 1.3: The average family size versus p.

square-root decay atpc. When p = 14 , family treeslarger than 100 are sufficiently rare that in prac-tical terms they do not occurthe probability isless than 1014. However, when p = pc, roughlyone in every ten trees will have size at least 100,and roughly one in a thousand will have size atleast 1 000 000. At the critical value, the process ispoised between extinction and survival.

Another important attribute of the branchingprocess is the average size of a family tree, denoted(p). A calculation shows that

(p) =

1

12p ifp < 1

2 , ifp 12 .

In particular, the average family size becomes infi-nite at the same critical value pc =

12 above which

the probability of an infinite family ceases to bezero. The graph of is shown in Figure 1.3. At

p = pc, it may seem at first sight contradictorythat family trees are always finite (since (pc) = 0)and yet the average family size is infinite (since(pc) =). However, there is no inconsistency,and this combination, which occurs only at thecritical point, reflects the slowness of the square-

root decay ofPk(pc).

2.2 Critical Exponents and Universality

Some aspects of the above discussion are specificto twofold branching, and will change for a branch-ing process with higher-order branching. For exam-ple, if each organism has not two but m potentialoffspring, again independently with probability p,

then the average number of offspring per organ-ism is mp and the critical probability pc changesto 1/m. Also, the formulas written above for thesurvival probability, for the probability of at leastkdescendants, and for the average family size mustall be modified and will involve the parameter m.

However, the way that (p) goes to zero at thecritical point, the way that Pk(pc) goes to zero ask goes to infinity, and the way that (p) divergesto infinity asp approaches the critical pointpc willall be governed by exponents that are independentofm. To be more specific, they behave in the fol-lowing manner:

(p)C1(p pc) , as pp+c ,Pk(pc)C2k1/, ask ,

(p)C3(pcp)

, as ppc .

Here, the numbersC1,C2, andC3 are constantsthat depend on m. By contrast, the exponents ,, and take on the same values for every m 2.Indeed, those values are = 1, = 2, and = 1.They are called critical exponents, and they areuniversalin the sense that they do not depend onthe precise form of the law that governs how the in-dividual organisms reproduce. Related exponentswill appear below in other models.

3 Random Graphs

An active research field in discrete mathematicswith many applications is the study of objectsknown as graphs. These are used to model sys-tems such as the internet, the World Wide Web,and highway networks. Mathematically, a graph isa collection ofvertices(which might represent com-puters, web pages, or cities) joined in pairs byedges(physical connections between computers, hyper-links between web pages, highways). Graphs arealso called networks, vertices are also called nodesor sites, and edges are also called linksor bonds.

3.1 The Basic Model of aRandom Graph

A major subarea of graph theory, initiated byErdos and Renyi in 1960, concerns the propertiesthat a graph typically has when it has been gener-ated randomly. A natural way to do this is to taken vertices and for each pair to decide randomly(by the toss of a coin, say) whether it should be


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linked by an edge. More generally, one can choosea numberp between 0 and 1 and let p be the prob-ability that any given pair is linked. (This wouldcorrespond to using a biased coin to make the deci-sions.) The properties of random graphs come intotheir own whennis large, and of particular interestis the fact that there is a phase transition.

3.2 The Phase Transition

Ifx and y are vertices in a graph, then a pathfromx to y is a sequence of vertices that starts with xand ends with y in such a way that neighbouringterms of the sequence are joined by edges. (If thevertices are represented by points and the edges bylines, then a path is a way of getting from x to yby travelling along the lines.) Ifx and y are joined

by a path then they are said to be connected. Acomponent, or connected cluster, in a graph is whatyou obtain if you take a vertex together with all theother vertices that are connected to it.

Any graph decomposes naturally into its con-nected clusters. These will, in general, have differ-ent sizes (as measured by the number of vertices),and given a graph it is interesting to know the sizeof its largest cluster, which we shall denote by N.If we are considering a random graph with n ver-tices, then the value ofNwill depend on the multi-tude of random choices made when the graph wasgenerated, and thusNis itself a random variable.

The possible values of N are everything from 1,the value it takes when no edges are present andevery cluster consists of a single vertex, to n, whenthere is just one connected cluster consisting of allthe vertices. In particular,N= 1 when p = 0, andN = n when p = 1. At a certain point betweenthese extremes,Nundergoes a dramatic jump.

It is possible to guess where the jump might takeplace, by considering the degree of a typical ver-texx. This means the number ofneighbours ofx,that is, other vertices that are directly linked tox by a single edge. Each vertex has n1 poten-tial neighbours, and for each one the probability

that it is an actual neighbour is p, so the expecteddegree of any given vertex is p(n 1). When p isless than 1/(n1), each vertex has, on average,less than one neighbour, whereas when p exceeds1/(n 1), it has, again on average, more than one.This suggests that pc= 1/(n 1) will be a criticalvalue, withNbeing small whenp is below pc, andlarge when p is abovepc.

This is indeed the case. If we set pc= 1/(n 1)and write p = pc(1 +), with a fixed numberbetween1 and +1, then = p(n1)1. Since

p(n

1) is the average degree of each vertex, isa measure of how much the average degree differsfrom 1. Erdos and Renyi showed that, in an appro-priate sense, as n goes to infinity,

N

22 log n if 0.

The A in the above formula is not a constant buta certain random variable that is independent ofn (the distribution of which we have not specifiedhere). When = 0 and n is large, the formula willtell us, for any a < b, the approximate probabilitythat N lies between an2/3 and bn2/3. To put itanother way, A is the limiting distribution of thequantity n2/3N when = 0.

There is a marked difference between the be-haviour of the functions log n, n2/3, and n, forlargen. The small clusters present forp < pccorre-spond to what is called a subcritical phase, whereasin the so-called supercritical phase, where p > pc,there is a giant cluster whose size is of the sameorder of magnitude as the entire graph (see Fig-ure 1.4).

It is interesting to consider the evolution of

the random graph, as p is increased from subcrit-ical to supercritical values. (Here one can imaginemore and more edges being randomly added to thegraph.) A remarkable coalescence occurs, in whichmany smaller clusters rapidly merge into a giantcluster whose size is proportional to the size ofthe entire system. The coalescence is thorough, inthe sense that in the supercritical phase the giantcluster dominates everything: indeed, the second-largest cluster is known to have asymptotic sizeonly 22 log n, which makes it far smaller thanthe giant cluster.

3.3 Cluster SizeFor branching processes, we defined the quantity(p) to be the average size of the family treespawned by an individual when the probability ofeach potential offspring being born was p. By anal-ogy, for the random graph it is natural to take anarbitrary vertex v and define (p) to be the av-erage size of the connected cluster containing v.


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(a)

(b)

p= pc= 0.00123

4

p= pc= 0.00205

4

Figure 1.4: The largest cluster (black) and second

largest cluster (dots) in random graphs with 625 ver-

tices. These clusters have sizes (a) 17 and 11 and

(b) 284 and 16. The hundreds of edges in the graphs

are not clearly shown.

Since all the vertices play identical roles, (p) is

independent of the particular choice ofv . If we fix

a value of , set p = pc(1 + ), and let n tend toinfinity, it turns out that the behaviour of(p) is

described by the formula

(p)

1/|| if 0,

where c is a constant. Thus the expected clustersize is independent ofnwhen 0.

To continue the analogy with branching pro-cesses, let Pk(p) denote the probability that thecluster containing the arbitrary vertex vconsists ofat least k vertices. Again this does not depend onthe particular choice ofv . In the subcritical phase,when p = (1 +)pc for some fixed negative valueof , the probability Pk(p) is essentially indepen-dent of n and is exponentially small in k. Thus,large clusters are extremely rare. However, at thecritical point p = pc, Pk(p) decays like a multi-ple of 1/

k (for an appropriate range ofk). This

much slower square-root decay is similar to whathappens for branching processes.

3.4 Other Thresholds

It is not only the largest cluster size that jumps.Another quantity that does so is the probabilitythat a random graph is connected, meaning thatthere is a single connected cluster that containsall the n vertices. For what values of the edge-probability p is this likely? It is known that the

property of being connected has a sharp thresh-old, at pconn = (1/n)log n, in the following sense.If p = pconn(1 + ) for some fixed negative ,

then the probability that the graph is connectedapproaches 0 as n . If on the other hand is positive, then the probability approaches 1.Roughly speaking, if you add edges randomly, thenthe graph suddenly changes from being almost cer-tainly not connected to almost certainly connectedas the proportion of edges present moves from justbelow pconn to just above it.

There is a wide class of properties with thresh-

olds of this sort. Other examples include the ab-sence of any isolated vertex (a vertex with no inci-dent edge), and the presence of a Hamiltonian cy-cle (a closed loop that visits every vertex exactlyonce). Below the threshold, the random graph al-most certainly does not have the property, whereasabove the threshold it almost certainly does. Thetransition occurs abruptly.


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Figure 1.5: Bond-percolation configurations on a 14 14 piece of the square lattice Z2 for p= 0.25, p= 0.45,

p= 0.55, p= 0.75. The critical value is pc= 1

2.

4 Percolation

The percolation model was introduced by Broad-bent and Hammersley in 1957 as a model of fluidflow in a porous medium. The medium contains a

network of randomly arranged microscopic poresthrough which fluid can flow. A d-dimensionalmedium can be modelled with the help of the infi-nited-dimensional lattice Zd, which consists of allpointsx of the form (x1, . . . , xd), where each xi isan integer. This set can be made into a graph in anatural way if we join each point to the 2d pointsthat differ from it by1 in one coordinate andare the same in the others. (So, for example, in Z2

the neighbours of (2, 3) are the four points (1, 3),(3, 3), (2, 2), and (2, 4).) One thinks of the edgesas representing all pores potentially present in themedium.

To model the medium itself, one first chooses aporosity parameter p, which is a number between0 and 1. Each edge (or bond) of the above graphis then retained with probability p and deletedwith probability 1p, with all choices indepen-dent. The retained edges are referred to as occu-pied and the deleted ones as vacant. The resultis a random subgraph of Zd whose edges are theoccupied bonds. These model the pores actuallypresent in a macroscopic chunk of the medium.

For fluid to flow through the medium there mustbe a set of pores connected together on a macro-scopic scale. This idea is captured in the model

by the existence of an infinite cluster in the ran-dom subgraph, that is, a collection of infinitelymany points all connected to one another. The ba-sic question is whether or not an infinite clusterexists. If it does, then fluid can flow through themedium on a macroscopic scale, and otherwise itcannot. Thus, when an infinite cluster exists, it issaid that percolation occurs.

Percolation on the square lattice Z2 is depictedin Figure 1.5. Percolation in a three-dimensionalphysical medium is modelled using Z3. It is instruc-tive, and mathematically interesting, to think howthe models behaviour might change as the dimen-

sion d is varied.For d = 1, percolation will not occur unless

p = 1. The simple observation that leads to thisconclusion is the following. Given any particularsequence of m consecutive edges, the probabilitythat they are all occupied is pm, and ifp pc. The exact value ofpc is notknown in general, but a special symmetry of thesquare lattice allows for a proof that pc =

12 when

d= 2.Using the fact that (p) is the probability that

anyparticular vertex lies in an infinite cluster, itcan be shown that when(p)> 0 there must be aninfinite connected cluster somewhere in Zd, whilewhen(p) = 0 there will not be one. Thus, percola-

tion occurs when p > pc but not when p < pc, andthe systems behaviour changes abruptly at thecritical value. A deeper argument shows that when

p > pc there must be exactly one infinite clusterinfinite clusters cannot coexist on Zd. This is analo-gous to the situation in the random graph, whereone giant cluster dominates when p is above thecritical value.


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Let(p) denote the average size of the connectedcluster containing a given vertex. Certainly(p) isinfinite for p > pc, since then there is a positiveprobability that the given vertex is in an infinitecluster. It is conceivable that (p) could be infi-nite also for some values of p less than pc, sinceinfinite expectation is in principle compatible with(p) = 0. However, it is a nontrivial and importanttheorem of the subject that this is not the case:(p) is finite for all p < pc and diverges to infinityasp approaches pc from below.

Qualitatively, the graphs of and have the ap-pearance depicted for the branching process in Fig-ures 1.2 and 1.3, although the critical value will beless than 12 for d 3. There is, however, a caveat.It has been proved that is continuous in p ex-

cept possibly at pc, and right-continuous for allp.It is widely believed that is equal to zero at thecritical point, so that is continuous for all p andpercolation does not occur at the critical point. Butproofs that(pc) = 0 are currently known only ford = 2, for d 19, and for certain related mod-els when d > 6. The lack of a general proof is allthe more intriguing since it has been proved for alld 2 that there is zero probability of an infinitecluster in any half-space whenp = pc. This still al-lows for an infinite cluster with an unnatural spiralbehaviour, for example, though it is believed thatthis does not occur.

4.2 Critical Exponents

Assuming that (p) does in fact approach zero asp is decreased to pc, it is natural to ask in whatmanner this occurs. Similarly, we can ask in whatmanner(p) diverges asp increases topc. Deep ar-guments of theoretical physics, and substantial nu-merical experimentation, have led to the predictionthat this, as well as other, behaviour is describedby certain powers known as critical exponents. Inparticular, it is predicted that there are asymptoticformulas

(p)C(p pc) , aspp+c,(p)C(pcp), asppc.

The critical exponents here are the powers and, which depend, in general, on the dimension d.(The letter C is used to denote a constant whoseprecise value is inessential and may change fromline to line.)

Whenp is less thanpc, large clusters have expo-nentially small probabilities. For example, in thiscase the probability Pk(p) that the size of the con-nected cluster containing any given vertex exceedsk is known to decay exponentially as k . Atthe critical point, this exponential decay is pre-dicted to be replaced by a power-law decay involv-ing a number, which is another critical exponent:

Pk(pc)Ck1/, as k .Also, for p < pc, the probability p(x, y) that

two vertices x and y are in the same connectedcluster decays exponentially like e|xy|/(p) as theseparation betweenx and y is increased. The num-ber (p) is called the correlation length. (Roughlyspeaking, p(x, y) starts to become small when the

distance between x and y exceeds (p).) The cor-relation length is known to diverge as p increasestopc, and the predicted form of this divergence is

(p)C(pcp) as ppc ,where is a further critical exponent. As before,the decay at the critical point is no longer expo-nential. It is predicted thatpc(x, y) decays insteadvia a power law, traditionally written in the form

pc(x, y)C 1

|x y|d2+ , as|x y| ,

for yet another a critical exponent .The critical exponents describe large-scale as-

pects of the phase transition and thus provide in-formation relevant to the macroscopic scale of thephysical medium. However, in most cases they havenot been rigorously proved to exist. To do so, andto establish their values, is a major open problemin mathematics, one of central importance for per-colation theory.

In view of this, it is important to be aware of aprediction from theoretical physics that the expo-nents are not independent, but are related to eachother by what are called scaling relations. Three

scaling relations are

= (2 ), + 2= (+ 1), d= + 2.

4.3 Universality

Since the critical exponents describe large-scale be-haviour, it seems plausible that they might dependonly weakly on changes to the fine structure of the


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model. In fact, it is a further prediction of theoret-ical physics, one that has been verified by numer-ical experiments, that the critical exponents areuniversal, in the sense that they depend on thespatial dimensiond but on little else.

For example, if the two-dimensional lattice Z2 isreplaced by another two-dimensional lattice, suchas the triangular or the hexagonal lattice, then thevalues of the critical exponents are believed not tochange. Another modification, for general d 2,is to replace the standard percolation model bythe so-called spread-out model. In the spread-out model, the edge set ofZd is enriched so thatnow two vertices are joined whenever they are sep-arated by a distance ofL or less, where L 1 isa fixed finite parameter, usually taken to be large.

Universality suggests that the critical exponentsfor percolation in the spread-out model do not de-pend on the parameterL.

The discussion so far falls within the generalframework ofbond percolation, in which it is bonds(edges) that are randomly occupied or vacant. Amuch-studied variant issite percolation, where nowit is vertices, or sites, that are independentlyoccupied with probability p and vacant withprobability 1p. The connected cluster of a ver-tex x consists of the vertex x itself together withthose occupied vertices that can be reached by a

path that starts at x, travels along edges in thegraph, and only visits occupied vertices. Ford 2,site percolation also experiences a phase transition.Although the critical value for site percolation isdifferent from the critical value for bond percola-tion, it is a prediction of universality that site andbond percolation on Zd have the same critical ex-ponents.

These predictions are mathematically very in-triguing: the large-scale properties of the phasetransition described by critical exponents appearto be insensitive to the fine details of the model, in

contrast to features like the value of critical prob-abilitypc, which depends heavily on such details.

At the time of writing, the critical exponentshave been proven to exist, and their values rigor-ously computed, only for certain percolation mod-els in dimensions d = 2 and d > 6. And a gen-eral mathematical understanding of universalityremains an elusive goal.

4.4 Percolation in Dimensions d > 6

Using a method known as the lace expansion, ithas been proven that the critical exponents exist,

with values

= 1, = 1, = 2, = 12 , = 0,

for percolation in the spread-out model whend > 6 and L is large enough. The proof makesuse of the fact that vertices in the spread-outmodel have many neighbours. For the more con-ventional nearest-neighbour model, where bondshave length 1 and there are fewer neighbours pervertex, results of this type have also been obtained,but only in dimensionsd 19.

The above values of, , andare the same as

those observed previously for branching processes.A branching process can be regarded as percola-tion on an infinite tree rather than on Zd, andthus percolation in dimensions d > 6 behaves likepercolation on a tree. This is an extreme exampleof universality, in which the critical exponents arealso independent of the dimension, at least whend >6.

If the above values for the exponents are sub-stituted into the scaling relationd=+ 2, theresult is d = 6. Thus, the scaling relation (called ahyperscaling relation due to the presence of the di-mensiond in the equation) is false for d >6. How-

ever, this particular relation is predicted to applyonly in dimensions d 6. In lower dimensions,the nature of the phase transition is affected bythe manner in which critical clusters fit into space,and the nature of the fit is partly described by thehyperscaling relation, in whichdappears explicitly.

The critical exponents are predicted to take ondifferent values belowd = 6. Recent advances haveshed much light on the situation for d = 2, as weshall see in the next section.

4.5 Percolation in Dimension 2

4.5.1 Critical Exponents andSchrammLoewner Evolution

For site percolation on the two-dimensional trian-gular lattice it has been shown, in a major recentachievement, that the critical exponents exist andtake the remarkable values

= 536 , = 4318

, = 915, = 43

, = 524 .


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Figure 1.6: The exploration process.

The scaling relations play an important role inthe proof, but an essential additional step requiresunderstanding of a concept known as the scalinglimit.

To get some idea of what this is, let us look at theso-called exploration process, which is depicted inFigure 1.6. In Figure 1.6, hexagons represent ver-tices of the triangular lattice. Hexagons in the bot-tom row have been coloured grey on the left halfand white on the right half. The other hexagonshave been chosen to be grey or white independentlywith probability 12 , which is the critical probabil-ity for site percolation on the triangular lattice. It

is not hard to show that there is a path, also il-lustrated in Figure 1.6, which starts at the bottomand all along its length is grey to the left and whiteto the right. The exploration process is this randompath, which can be thought of as the grey/whiteinterface. The boundary conditions at the bottomforce it to be infinite.

The exploration process provides informationabout the boundaries separating large critical clus-ters of different colour, and from this it is possibleto extract information about critical exponents. Itis the macroscopic large-scale structure that is es-sential, so interest is focussed on the exploration

process in the limit as the spacing between ver-tices of the triangular lattice goes to zero. In otherwords, what does the curve in Figure 1.6 typicallylook like in the limit as the size of the hexagonsshrinks to zero? It is now known that this limitis described by a newly discovered stochastic pro-cess called the SchrammLoewner evolution (SLE)with parameter six, or SLE6 for short. The SLE

processes were introduced by Schramm in 2000,and have become a topic of intense current researchactivity.

This is a major step forward in the understand-ing of two-dimensional site percolation on the tri-angular lattice, but much remains to be done. Inparticular, it is still an unsolved problem to proveuniversality. There is currently no proof that crit-ical exponents exist for bond percolation on thesquare lattice Z2, although universality predictsthat the critical exponents for the square latticeshould also take on the interesting values listedabove.

4.5.2 Crossing Probabilities

In order to understand two-dimensional percola-tion, it is very helpful to understand the probabil-ity that there will be a path from one side of aregion of the plane to another, especially when theparameterp takes its critical value pc.

To make this idea precise, fix a simply connectedregion in the plane (i.e. a region with no holes),and fix two arcs on the boundary of the region.The crossing probability (which depends on p) isthe probability that there is an occupied path in-side the region that joins one arc to the other, ormore accurately the limit of this probability as thelattice spacing between vertices is reduced to zero.

For p < pc, clusters with diameter much largerthan the correlation length (p) (measured by thenumber of steps in the lattice) are extremely rare.However, to cross the region, a cluster needs tobe larger and larger as the lattice spacing goes tozero. It follows that the crossing probability is 0.Whenp > pc, there is exactly one infinite cluster,from which it can be deduced that if the latticespacing is very small, then with very high proba-bility there will be a crossing of the region. In thelimit, the crossing probability is 1. What ifp = pc?There are three remarkable predictions for criticalcrossing probabilities.

The first prediction is that critical crossing prob-abilities are universal, which is to say that they arethe same for all finite-range two-dimensional bond-or site-percolation models. (As always, we are talk-ing about the limiting probabilities as the latticespacing goes to zero.)

The second prediction is that the critical cross-ing probabilities are conformally invariant. A con-


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Figure 1.7: The two regions are related by a conformal

transformation, depicted in the upper figures. In the

lower figures, the limiting critical crossing probabilitiesare identical.

formal transformation is a transformation that lo-cally preserves angles, as shown in Figure 1.7. Theremarkable Riemann mapping theoremstates thatanytwo simply connected regions that are not theentire plane are related by a conformal transfor-mation. The statement that the critical crossingprobability is conformally invariant means that ifone region with two specified boundary arcs ismapped to another region by a conformal trans-

formation, then the critical crossing probability be-tween the images of the arcs in the new region isidentical to the crossing probability of the origi-nal region. (Note that the underlying lattice is nottransformedthis is what makes the prediction sostriking.)

The third prediction is Cardys explicit formulafor critical crossing probabilities. Assuming con-formal invariance, it is only necessary to give theformula for one region. For an equilateral trian-gle, Cardys formula is particularly simple (see Fig-ure 1.8).

In 2001, in a celebrated achievement, Smirnov

studied critical crossing probabilities for site per-colation on the triangular lattice. Using the spe-cial symmetries of this particular model, Smirnovproved that the limiting critical crossing probabili-ties exist, that they are conformally invariant, andthat they obey Cardys formula. To prove univer-sality of the crossing probabilities remains a tan-talizing open problem.

side = 1

s

Figure 1.8: For the equilateral triangle of unit side

length, Cardys formula asserts that the limiting criti-

cal crossing probability shown is simply the length s.

5 The Ising Model

In 1925, Ising published an analysis of a mathe-matical model of ferromagnetism which now bearshis name (although it was in fact Isings doctoralsupervisor Lenz who first defined the model). TheIsing model occupies a central position in theoret-ical physics, and is of considerable mathematicalinterest.

5.1 Spins, Energy, and Temperature

In the Ising model, a block of iron is regarded asa collection of atoms whose positions are fixed ina crystalline lattice. Each atom has a magneticspin, which is assumed for simplicity to point up-

wards or downwards. Each possible configurationof spins has an associated energy, and the greaterthis energy is, the less likely the configuration is tooccur.

On the whole, atoms like to have the same spinas their immediate neighbours, and the energy re-flects this: it increases according to the number ofpairs of neighbouring spins that are not alignedwith each other. If there is an external magneticfield, also assumed to be directed up or down, thenthere is an additional contribution: atoms like tobe aligned with the external field, and the energyis greater the more spins there are that are not

aligned with it. Since configurations with higherenergy are less likely, spins have a general tendencyto align with each other, and also to align with thedirection of the external magnetic field. When alarger fraction of spins points up than down, theiron is said to have a positive magnetization.

Although energy considerations favour configu-rations with many aligned spins, there is a compet-


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ing effect. As the temperature increases, there aremore random thermal fluctuations of the spins, andthese diminish the amount of alignment. Wheneverthere is an external magnetic field, the energy ef-fects predominate and there is at least some mag-netization, however high the temperature. How-ever, when the external field is turned off, the mag-netization persists only if the temperature is belowa certain critical temperature. Above this temper-ature, the iron will lose its magnetization.

The Ising model is a mathematical model thatcaptures the above picture. The crystalline latticeis modelled by the lattice Zd. Vertices of Zd rep-resent atomic positions, and the atomic spin at avertexxis simply modelled by one of the two num-bers +1 (representing spin up) or1 (representingspin down). The particular number chosen at xis denoted x, and a collection of choices, one foreach x in the lattice, is called a configuration ofthe Ising model. The configuration as a whole isdenoted simply as . (Formally, a configuration is a function from the lattice to the set{1, 1}.)

Each configuration comes with an associatedenergy, defined as follows. If there is no externalfield, the energy of consists of the sum, takenover all pairs of neighbouring verticesx, y, of thequantityxy. This quantity is1 if x = y,and is +1 otherwise, so the energy is indeed largerthe more non-aligned pairs there are. If there is

a nonzero external field, modelled by a real num-ber h, then the energy receives an additional con-tributionhx, which is larger the more spinsthere are with a different sign from that ofh. Thus,in total, the energyE() of a spin configuration is defined by

E() =x,y

xy hx

x,

where the first sum is over neighbouring pairs ofvertices, the second sum is over vertices, and h isa real number that may be positive, negative, or

zero.The sums defining E() actually make sense

only when there are finitely many vertices, butone wishes to study the infinite lattice Zd. Thisproblem is handled by restricting Zd to a large fi-nite subset and later taking an appropriate limit,the so-called thermodynamic limit. This is a well-understood process that will not be described here.

Two features remain to be modelled, namely, themanner in which lower-energy configurations arepreferred, and the manner in which thermal fluc-tuations can lessen this preference. Both featuresare handled simultaneously, as follows. We wish toassign to each configuration a probability that de-creases as its energy increases. According to thefoundations of statistical mechanics, the right wayto do this is to make the probability proportionalto the so-calledBoltzmann factor eE()/T, whereT is a non-negative parameter that represents thetemperature. Thus, the probability is

P() = 1

ZeE()/T,

where the normalization constant, or partitionfunction, Z, is defined by

Z=

eE()/T,

where the sum is taken over all possible configura-tions (again it is necessary to work first in a finitesubset ofZd to make this precise). The reason forthis choice ofZ is that once we divide by it thenwe have ensured that the probabilities of the con-figurations add up to one, as they must. With thisdefinition, the desired preference for low energy isachieved, since the probability of a given configura-tion is smaller when the energy of the configurationis larger. As for the effect of the temperature, note

that whenTis very large, all the numbers eE()/Tare close to 1, so all probabilities are roughly equal.In general, as the temperature increases the prob-abilities of the various configurations become moresimilar, and this models the effect of random ther-mal fluctuations.

There is more to the story than energy, however.The Boltzmann factor makes any individual low-energy configuration much more likely than anyindividual high-energy configuration. However, thelow-energy configurations have a high degree ofalignment, so there are far fewer of them than thereare of the more randomly arranged high-energy

configurations. It is not obvious which of these twocompeting considerations will predominate, and infact the answer depends on the value of the tem-perature T in a very interesting way.

5.2 The Phase Transition

For the Ising model with external field h and tem-peratureT, let us choose a configuration randomly


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with the probabilities defined above. The magneti-zationM(h, T) is defined to be the expected valueof the spin x at a given vertex x. Because of thesymmetry of the lattice Zd, this does not dependon the particular vertex chosen. Accordingly, if themagnetizationM(h, T) is positive, then spins havean overall tendency to be aligned in the positivedirection, and the system is magnetized.

The symmetry between up and down impliesthatM(h, T) =M(h, T) (i.e. reversing the ex-ternal field reverses the magnetization) for all hand T. In particular, when h = 0, the magnetiza-tion must be zero. On the other hand, if there is anonzero external field h, then configurations withspins that are aligned with h are overwhelminglymore likely (because their energy is lower), and the

magnetization satisfies

M(h, T)

0.

What happens if the external field is initiallypositive and then is reduced to zero? In particular,is the spontaneous magnetization, defined by

M+(T) = limh0+

M(h, T),

positive or zero? If M+(T) is positive, then the

magnetization persists after the external field isturned off. In this case there will be a discontinuityin the graph ofM versus h at h = 0.

Whether or not this happens depends on thetemperatureT. In the limit asTis reduced to zero,a small difference in the energies of two configu-rations results in an enormous difference in theirprobabilities. When h >0 and the temperature isreduced to zero, only the minimal energy configu-ration, in which all spins are +1, has any chanceof occurring. This is the case no matter how smallthe external field becomes, so M+(0) = 1. On theother hand, in the limit of infinitely high temper-

ature, all configurations become equally likely andthe spontaneous magnetization is equal to zero.

For dimensions d 2, the behaviour ofM+(T)when T lies between these two extremes is quitesurprising. In particular, it is not differentiable ev-erywhere: there is a critical temperature Tc, de-pending on the dimension, such that the sponta-neous magnetization is strictly positive forT < Tc

h

M(h,T)

T< Tc

T= TcT> T

c

M+

1

0T= TcT= 0

M+

(T)

Figure 1.9: Magnetization versus external field, and

spontaneous magnetization versus temperature.

and zero forT > Tc, and it is atT =Tc that differ-entiability fails. Schematic graphs of the magneti-

zation versushand the spontaneous magnetizationversus T are shown in Figure 1.9. What happensat the critical temperature itself is delicate. In alldimensions except d = 3 it has been proved thatthere is no spontaneous magnetization at the criti-cal temperature, which is to say that M+(Tc) = 0.It is believed that this is true when d= 3 as well,but it remains an open problem to prove it.

5.3 Critical Exponents

The phase transition for the Ising model is againdescribed by critical exponents. The critical expo-nent , given by

M+(T)C(Tc T), as T Tc ,indicates how the spontaneous magnetization dis-appears as the temperature increases towards thecritical temperatureTc. For T > Tc, the magneticsusceptibility, denoted (T), is defined to be therate of change of M(h, T) with respect to h, ath= 0. This partial derivative in h diverges as T


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approaches Tc from above, and the exponent isdefined by

(T)

C(T

Tc)

, as T

T+c .

Finally, describes the manner in which the mag-netization goes to zero as the external field is re-duced to zero at the critical temperature. That is,

M(h, Tc)C h1/, ash0+.These critical exponents, like those for percolation,are predicted to be universal and to obey variousscaling relations. They are now understood math-ematically in all dimensions exceptd = 3.

5.4 Exact Solution for d = 2

In 1944, Onsager published a famous paper inwhich he gave an exact solution of the two-dimensional Ising model. His remarkable computa-tion is a landmark in the development of the theoryof critical phenomena. With the exact solution asa starting point, critical exponents could be cal-culated. As with two-dimensional percolation, theexponents take interesting values:

= 18 , = 74

, = 15.

5.5 Mean-Field Theory for d 4

Two modifications of the Ising model are relativelyeasy to analyze. One is to formulate the model onthe infinite binary tree, rather than on the integerlattice Zd. Another is to formulate the Ising modelon the so-called complete graph, which is thegraph consisting ofn vertices with an edge joiningeverypair of vertices, and then take the limit as ngoes to infinity. In the latter, known as the CurieWeiss model, each spin interacts equally with allthe other spins, or, put another way, each spin feelsthe mean field of all the other spins. In each ofthese modifications, the critical exponents take onthe so-called mean-field values

= 12 , = 1, = 3.

Ingenious methods have been used to prove thatthe Ising model on Zd has these same critical ex-ponents in dimensions d 4, although in dimen-sion 4 there remain unresolved issues concerninglogarithmic corrections to the asymptotic formu-las.

6 The Random-Cluster Model

The percolation and Ising models appear to bequite different. A percolation configuration con-

sists of a random subgraph of a given graph (usu-ally a lattice as in the examples earlier), with edgesincluded independently with probability p. A con-figuration of the Ising model consists of an assign-ment of values1 to spins at the vertices of agraph (again usually a lattice), with these spinsinfluenced by energy and temperature.

In spite of these differences, in around 1970 For-tuin and Kasteleyn had the insight to observe thatthe two models are in fact closely related to eachother, as members of a larger family of modelsknown as the random-cluster model. The random-

cluster model also includes a natural extension ofthe Ising model known as the Potts model.

In the Potts model, spins at the vertices of agiven graph G may take on any one ofqdifferentvalues, whereqis an integer greater than or equalto 2. Whenq= 2 there are two possible spin valuesand the model is equivalent to the Ising model. Forgeneralq, it is convenient to label the possible spinvalues as 1, 2, . . . , q . As before, a configuration ofspins has an associated energy that is smaller whenmore spins are aligned. The energy associated withan edge is1 if the spins at the vertices joined bythe edge are identical, and 0 otherwise. The total

energyE() of a spin configuration, assuming noexternal field, is the sum of the energies associatedwith all edges. The probability of a particular spinconfiguration is again taken to be proportionalto a Boltzmann factor, namely

P() = 1

ZeE()/T,

where the partition function Zis once again thereto ensure that the probabilities add up to 1.

Fortuin and Kasteleyn noticed that the partitionfunction of the Potts model on a finite graphGcan

be recast as SG

p|S|(1p)|G\S|qn(S).

In this formula, the sum is over all subgraphs Sthat can be obtained by deleting edges fromG,|S|is the number of edges in S,|G \ S| is the numberof edges deleted from G to obtain S, n(S) is the


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number of distinct connected clusters ofS, and pis related to the temperature by

p= 1e1/T.The restriction that q be an integer greater thanor equal to 2 is essential for the definition of thePotts model, but the above sum makes good sensefor any positive real value ofq.

The random-cluster model has the above sumas its partition function. Given any real numberq >0, a configuration of the random-cluster modelis a set Sof occupied edges of the graph G, ex-actly like a configuration of bond percolation. How-ever, in the random-cluster model we do not sim-ply associate p with each occupied edge and 1pto each vacant edge. Instead, the probability as-

sociated with a configuration is proportional top|S|(1p)|G\S|qn(S). In particular, for the choiceq = 1, the random-cluster model is the same asbond percolation. Thus the random-cluster modelprovides a one-parameter family of models, in-dexed by q, which corresponds to percolation forq= 1, to the Ising model for q = 2, and to thePotts model for integerq 2. The random-clustermodel has a phase transition for general q 1,and provides a unified setting and a rich family ofexamples.

7 Conclusion

The science of critical phenomena and phase tran-sitions is a source of fascinating mathematicalproblems of real physical significance. Percolationis a central mathematical model in the subject. Of-ten formulated on Zd, it can also be defined insteadon a tree or on the complete graph, as a result ofwhich it encompasses branching processes and therandom graph. The Ising model is a fundamentalmodel of the ferromagnetic phase transition. Atfirst sight unrelated to percolation, it is in factclosely connected within the wider setting of therandom-cluster model. The latter provides a uni-

fied framework and a powerful geometric represen-tation for the Ising and Potts models.

Part of the fascination of these models is due tothe prediction from theoretical physics that large-scale features near the critical point are univer-sal. However, proofs often rely on specific detailsof a model, even when universality predicts thatthese details should not be essential to the results.

For example, the understanding of critical crossingprobabilities and the calculation of critical expo-nents has been carried out for site percolation onthe triangular lattice, but not for bond percolationon Z2. Although the progress for the triangularlattice is a triumph of the theory, it is not the lastword. Universality remains a guiding principle butit is not yet a general theorem.

In the physically most interesting case of dimen-sion 3, a very basic feature of percolation and theIsing model is not understood at all: it has not yetbeen proved that there is no percolation at the crit-ical point and that the spontaneous magnetizationis zero.

Much has been accomplished but much remainsto be done, and it seems clear that further inves-

tigation of models of critical phenomena will leadto highly important mathematical discoveries.

The figures were produced by Bill Casselman, Depart-ment of Mathematics, University of British Columbia,and Graphics Editor ofNotices of the American Math-ematical Society.

Bibliography

1. G. Grimmett. 1999. Percolation, 2nd edn (Berlin,Springer).

2. G. Grimmett. 2004. The random-cluster model. InProbability on discrete structures(ed. H. Kesten),pp. 73124 (Berlin, Springer).

3. S. Janson, T. Luczak, and A. Rucinski. 2000.Ran-dom graphs(New York, John Wiley and Sons).

4. C. J. Thompson. 1988. Classical equilibrium sta-tistical mechanics(Oxford University Press).

5. W. Werner. 2004. Random planar curves andSchrammLoewner evolutions. In Lectures onprobability theory and statistics. Ecole dete deprobabilites de SaintFlour XXXII-2002 (ed. J.Picard). Lecture Notes in Mathematics, vol. 1840(Berlin, Springer).

Contributor Biography

Gordon Slade has been professor of mathematicsat the University of British Columbia since 1999.

He held positions previously at McMaster Univer-sity and the University of Virginia. His researchinterests are probability theory and statistical me-chanics.

Keywords: phase transition; critical exponent;branching process; random graph; percolation;Ising model; random-cluster model.

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