Berg, J. (Rob) van den

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Volume     	 pp   Uniqueness and Mixing Properties of Gibbs MeasuresJ van den BergCWI Kruislaan   SJ Amsterdam The Netherlandse	mail jvdbergcwinlA short introduction to the theory of niterange Gibbs measures and phasetransitions is given	 starting with the famous Ising model Some recent resultson uniqueness and mixing	 based on percolationtype arguments	 are summarizedand compared with more classical results  The Ising modelThe basic Ising model is a simple but very useful model of a ferromagnet Inthe dimensional case it is as follows Consider a nite part  for instance ann n box centered at O of the square lattice this is the graph whose verticesare the elements of Z and where two vertices share an edge i	 their distanceis  Assign to each vertex i a variable i which can take the value  or and which represents the spin of the elementary magnet at i It is assumedthat only neighbours vertices at distance  interact namely in the followingway Each pair i j contributes an amount J to the total energy if i jand an amount J otherwise Here J   is the interaction parameter of themodel The total energy His the sum of all such contributions Furtherbased on statisticalphysical arguments it is assumed that each conguration  fgoccurs with a probability proportional to exp H  So wehave exp H Z where Z is a normalizing constant This probability distribution is called anitevolume Gibbs measureIt is clear that this probability distribution has a preference for neighbourspins to be the same It is also clear that if  is xed and we make J very largethen with very high probability there is either a vast majority of spins ora vast majority of spins More surprising and certainly not trivial is thatif we rst x J at a suciently high value then this symmetry breaking persistsfor arbitrary large  More precisely there is a critical value Jcsuch that ifJ  Jc the following related properties hold  If we let the size of  go to  then the frequency of  spins tends in astrong probabilistic sense to  Moreover the spins and spinsare typically well mixed so that no macroscopic magnetic eld is created  Suppose we have a socalled boundary condition   This means that eachvertex i on the outer boundary of  has a xed value iwhich inuencesthe energy and hence the distribution by its interaction with its neighbour inside  Then no matter which boundary condition we choose theprobability that the spin at the origin is  tends to  as the size of tends to   There is a unique innitevolume Gibbs measure This means that all probability distributions on fgZwhich arise as weak limits of nitevolume Gibbs measures are the sameOn the other hand if J  Jc these properties no longer hold In particularthere is a p   depending on J such that if  is very large then withhigh probability either the fraction of spins is close to p or the fraction ofspins is close to p The behaviour in the dimensional case can be morecomplicated This symmetry breaking causes a macroscopic magnetic eldspontaneous magnetization This behaviour for very large boxes correspondswith the nonuniqueness of innitevolume Gibbs measuresSummarizing we say that this model has a phase transition Since thesephenomena correspond reasonably well to what is observed for real feromagnetsthis model and several variations have become very popular in theoreticalphysics Finiterange Gibbs measures Markov propertyThe Ising model is an example of a niterange Gibbs model In general we havea singlesite state space S and for each A  Zdwith diameter  r the rangeof interaction we have an interaction function which assigns to each elementof SAa real number the contribution to the total energy H Then as inthe Ising model we consider the nitevolume Gibbs measure on Sdenedanalogously to Again we can introduce a boundary condition and one of the central questions is whether the inuence of the boundary condition vanishes as the sizeof  grows to  and the innitevolume Gibbs measure is uniqueGibbs measures originate from Statistical Physics More recently they became important in the context of statistical image analysis randomized optimization algorithms and certain large communication networks Kelly In the last context the notion of phase transition is important because it reects a certain instability an event very far away for instance the breakdownof a node may have a nonnegligible impact throughout the networkIn fact niterange Gibbs measures are from a mathematical point of viewquite natural objects for the following reason It is not dicult to check thatthey satisfy the following Markov property For each vertex x the conditionaldistribution of x given all other spins depends only on the spins of the verticesat distance  r from x Moreover the reverse als holds which is not trivial ifthe above Markov property holds then the distribution is a Gibbs distributionwith respect to an interaction with range r MixingAs we stated before an important question is whether the inuence of a boundary condition vanishes as the size of a box goes to  More generally we areinterested in convenient bounds for the inuence of two sets A and B of verticeson each other First we have to quantify the notion inuence This is oftendone as follows We assume that we deal with a nite box  Take two possiblespin congurations  and on A and consider the corresponding conditionalGibbs ditributions on  n A restricted to B Take their variational distanceand maximise this over all pairs   We call the result the inuence of A on Bin  denoted by fAB If the vertices x and y are neighbours notationx  y then somewhat confusing we dene the local inuence of x on y asfollows let  and be two congurations on the set of neighbours of y whichdi	er only at x Again take the variational distance of the corresponding distributons of y and maximise over all such pairs    The result is denotedby gx y It is important to note that each gx y can be explicitly calculated when the interaction functions are given It can be shown by a cleverargument that fis dominated by a function fwhich satises for arbitraryx y  fx y Xiyfx igi y	 Related to this it can be shown that if the supremum over j  ZdofPijgi jis smaller than  then there exist C and    independent of  such thatfor all nite boxes   Zd and all AB  fAB  CjAj exp distABwhere jAj is the size of A and distAB the distance between A and B Thisis an example of a mixing property In particular the boundary inuence thenvanishes and there is a unique innitevolume Gibbs measure This is a classicalresult of Dobrushin A few years ago Van den Berg and Maes have obtained an alternative resultsee also Van den Berg  and Van den Berg  Steif  The idea wasto compare the spread of inuence with a percolation process In percolationmodels each vertex x is independent of the others open with probability pxand closed with probability   px One is interested in the probability thatthere is an open path between vertices at large distance of each other and inparticular the existence of innite open paths If all pxare smaller than somecritical pcwhich depends on the graph then with probability  only niteopen paths exist Van den Berg and Maes have shown the following Let foreach vertex xpx fNxNx x with f as above Nxthe set of vertices at distance  r from x andNxNxn fxg Like gx y pxcan be explicitly calculated when the interactionfunctions are given ThenfAB  PfpxgA	 B where the right hand side is the probability of an open path from A to B inthe percolation model with parameters px x  Zdand path refers to thegraph whose vertices are the elements of Zdand where two di	erent verticesshare an edge i	 their distance is at most r As a consequence using resultsfrom percolation theory if supxpx pc then there exist Cand   suchthat fAB  CjAj exp  distAB and the innitevolume Gibbs measureis uniqueIt appears that for some models in particular lowrange dimensional hardcore models and antiferromagnetic Ising models where the interaction parameter is negative and hence there is a tendency for neighbours to disagree this alternative condition gives a better result than Dobrushins condition Moreoverthis percolationlike approach appears to be more robust with respect to nonhomogeneity If a nonzero fraction of the vertices j of ZdhasPijgi j  then the inequality  is quite useless However if a nonzero fraction of thevertices have large px then the right hand side of  may still be exponentiallysmall in distAB when the other vertices have suciently small px Random interactionsThe above mentioned robustness was exploited by Gielis  Maes  for thecase of random interactions ie when the interaction functions are not xedbut rst chosen according to some random mechanism after which the Gibbsmeasures for that particular choice are studied Random interactions are usedto model disordered materials and networks for instance a dilute ferromagnetwhere not every vertex but a fraction of the vertices contains an elementarymagnet or a communication network which does not correspond with a regulargrid but which does have much regularity in a statistical sense Usuallyto make these models mathematically tractable a lot of spatial indepence isassumed as well as translation invariance of the interaction processRelated to what we said at the end of the previous section if the verticeshave a positive probability no matter how small to have bad local interactions then  is not useful Bassalygo  Dobrushin  give an elementarybut complicated method which does give useful results in many of these situations The approach of Gielis and Maes which uses  seems to be moreintuitively appealing and elegant In fact the exibility towards randomness ofthe interactions of the percolation approach is related to the following a percolation model with random independent parameters px x  Zdis essentiallyequivalent to one with xed parameters equal to the expectations of the pxsAlthough Gibbs models with random interactions lead via  to percolationmodels with random but locally dependent parameters this local dependencecreates only minor diculties As a result Gielis and Maes obtain a theoremof the form that if for each x the expectation of the right hand side of  issuciently small then for almost all realizations of the interactions there isa unique Gibbs measure and some kind of mixing holds	 RescalingDobrushin  Shlosman  present a socalled constructive version of Dobrushins uniqueness result Instead of local inuences on single sites it involveslocal inuences on boxes eg cubes of size l which we will not dene hereprecisely They obtain a result of the form if for some l each cube of size lsatises a local inuence condition then the system is in some sense mixingand in particular there is a unique Gibbs measureThis naturally leads to the question if the percolation approach also has aconstructive extension RecentlyVan den Berg  has for the dimensionalcase obtained a rescaled version of the Van den BergMaes result and combined this with the above mentioned idea of Gielis and Maes The result forrandom interactions is of the following form if for some n the expected inuence of the boundary of a n   n square on the n   n square in its middleas well as the expected inuence of the horizontal sides of an n   n squareon the horizontal strip of width r   in its middle and the analog for vertical instead of horizontal are suciently small then almost surely there is aunique innitevolume Gibbs measure and some mixing property holds Wewrite inuence because it is not exactly the same notion of inuence denedbefore Moreover an extension to random interactions is obtained of theinteresting result of Martinelli Olivieri  Schonmann  that fordimensional spin systems two mixing properties of which one is seeminglystronger than the other are actually equivalentReferences LA Bassalygo RL Dobrushin  Uniqueness of a Gibbs eldwith random potential  an elementary approach Theory Prob Appl   J van den Berg  A uniqueness condition for Gibbs measures withapplication to the twodimensional Ising antiferromagnet Comm MathPhys    J van den Berg J Steif  Percolation and the hardcore latticegas model Stochastic Process Appl    J van den Berg C Maes  Disagreement percolation in thestudy of Markov elds Ann Probab    J van den Berg  A constructive mixing condition for D Gibbsmeasures with random interactions preprint RL Dobrushin  The problem of uniqueness of a Gibbs randomeld and the problem of phase transition Funct Anal Appl    RL Dobrushin SB Shlosman  Constructive criterion for theuniqueness of a Gibbs eld J Fritz A Jaffe D Szasz eds Statistical mechanics and dynamical systems Birkh!auser Boston etc   HO Georgii  Gibbs measures and phase transitions de GruyterBerlinNew York G Gielis C Maes  The uniqueness regime of Gibbs elds withunbounded disorder J Stat Phys    G Grimmett  Percolation SpringerVerlag FP Kelly  Loss networks Ann Appl Probab    F Martinelli E Olivieri R Schonmann  For D latticespin systems weak mixing implies strong mixing Comm Math Phys	  

Uniqueness and mixing properties of Gibbs measures

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