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The Swendsen-Wang Process Does Not Always Mix Rapidly

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The Swendsen-Wang Process Does Not Always Mix Rapidly The SwendsenWang pro cess do es not always mix rapidly Vivek K Gore and Mark R Jerrum Department of Computer Science University of Edinburgh st Octob er Abstract The SwendsenWang pro cess provides one p ossible dynamics for the Q state Potts mo del in statistical physics Computer simulations of this pro cess are widely used to estimate the exp ectations of various observ ables random variables of a Potts system in the equilibrium or Gibbs distribution The legitimacy of such simulations dep ends on the rate of convergence of the pro cess to equilibrium often known as the mixing rate Empirical observations suggest that the SwendsenWang pro cess mixes rap idly in many instances of practical interest In spite of this we show that there are o ccasions on which the SwendsenWang pro cess requires exp o nential time in the size of the system to approach equilibrium Address for corresp ondence Department of Computer Science University of Edinburgh The Kings Buildings Edinburgh EH JZ United Kingdom Introduction The Potts mo del is a natural generalisation of the Ising mo del to an arbitrary number Q of states or spins Congurations in the Potts mo del can b e viewed as Qcolourings not in general prop er of the vertices of an undirected nvertex graph One is interested in sampling congurations from a certain distri bution known as the Gibbs distribution with the aim of obtaining estimates for certain random variables on congurations For the ferromagnetic Potts mo del the fo cus of this article the Gibbs distribution assigns greater probability to congurations in which a larger number of pairs of adjacent spins are alike A precise denition of the Potts mo del is provided in Section In the absence of eective direct metho ds the usual approach to sampling congurations is via the Markov chain Monte Carlo metho d The idea is to provide the mo del with a dynamics by dening an ergo dic random walk on congurations whose stationary distribution is the required Gibbs distribution Provided the walk is rapidly mixing ie converges rapidly to equilibrium con gurations may b e eciently sampled by simulating the walk for a sucient but not excessive number of steps A number of dierent dynamics are p ossible The simplest is to move b etween congurations by changing one spin at a time with transition probabilities de termined by the Metrop olis rule It is fairly easy to demonstrate situations in which this random walk takes exp onential time in n the size of the graph to approach equilibrium even in the ferromagnetic case A more complicated dy namics which allows many spins to change in one step was prop osed by Swendsen and Wang and is now widely used in computer simulations The SwendsenWang pro cess as we shall call it app ears to converge rapidly to equilibrium in many instances of practical interest This empirical observation might encourage us to attempt to prove that the mixing time of the pro cess grows not to o quickly as a function of n sp ecically that it is b ounded by a xed p olynomial in n indep endent of the other parameters of the system Such a result would establish the existence of an ecient approximation algorithm more precisely a fully p olynomial randomised approximation scheme or fpras for computing the partition function of a Qstate ferromagnetic Potts system Such an algorithm is only known to exist in the case Q Our main result see Prop osition for a precise statement demonstrates that this is a vain hop e For a certain particularly simple family of Potts systems based on the complete graph K on n vertices the socalled CurieWeiss mo del the n SwendsenWang pro cess is still far from equilibrium after exp onentially many steps This counterexample is valid for all Q and for a suitably chosen coupling constant It is an op en question whether rapid mixing obtains when 1 The antiferromagnetic mo del in which adjacent spins tend to b e unlike includes graph colouring as a limit so rapid convergence cannot b e exp ected for any reasonable dynamics Q the Ising mo del or if the negative result can b e extended to more physically realistic instances of the Potts mo del for example or dimensional lattices The Potts mo del The Potts mo del was introduced by R B Potts in has b een a fo cus of much attention in the physics and mathematics communities ever since Rather than present a detailed historical account of the mo del here we refer the interested reader to Baxter Chap The problem is easily stated Consider a collection of sites f ng de noted by n each pair i j of which has an asso ciated interaction energy which for simplicity we assume takes on one of just two values either or J In most cases of physical interest the set E of unordered pairs of sites with non zero interaction energy forms a regular lattice graph n E A conguration is an assignment of spins to sites where denotes the spin at n i site i The number of spins is denoted by Q where Q individual spins may simply b e denoted by the numbers in Q The energy of a conguration is given by the Hamiltonian X H J i j ij E where is the Kronecker function which is if its arguments are equal and otherwise The central problem is to compute the partition function X Z exp H where is what is called the inverse temperature to b e precise k T where T is the temp erature and k is Boltzmanns constant and the sum is over all p ossible congurations Many of the physical prop erties of the system can b e computed from the knowledge of Z Essentially Z is the normalising factor in the calculation of probabilities according to the fundamental theory of statistical mechanics the probability that the system in equilibrium is found in state the steady state probability is Z exp H Moreover certain logarithmic derivatives of Z corresp ond to quantities such as mean energy and mean magnetic moment Singularities in these derivatives in the limit as n generally corresp ond to phase transitions when a small change in a parameter has an observable eect on the macroscopic prop erties of the system If a small change in temp erature causes a phase transition then that temp erature is called the critical temperature Consider the eect of the parameter J in the Hamiltonian The high probab ility congurations are those for which H is low Let d denote the number of edges in E that connect sites with dierent spins in the conguration It is easy to see that H J d and that X Z exp J d We let K J K is usually known as the coupling constant in the statistical physics community By the denition of H if J or K since is p ositive then congurations in which neighbouring spins spins asso ciated with a pair of sites with nonzero interaction energy are the same are preferred this is the ferromagnetic attractive case On the other hand if J is negative the neighbouring spins will tend to b e dierent and this is the antiferromagnetic repulsive case The search for ecient computational solutions to these problems has proved extremely hard and has generated a vast b o dy of literature The reader is re ferred to Chapter of Baxter where some sp ecial cases have b een considered A huge amount of computational eort has also b een p oured into numerical solu tions esp ecially for regular lattices Although this problem arises in statistical physics it has a very interesting connection with theoretical computer science It is another example of a signi cant combinatorial enumeration problem which is Pcomplete and is hence apparently intractable in exact form This is an intriguing class of problems and includes the problems of computing the volume of a convex b o dy and the p ermanent of a matrix The Potts mo del also turns out to b e one of the many sp ecialisations of the famous Tutte p olynomial in graph theory The reader is referred to Welsh p for more on this interesting connection A lot of research eort has b een devoted to nding ecient approximation algorithms for Pcomplete problems where by ecient we mean that the al gorithm is guaranteed to run in time p olynomial in the number of sites n Ran domness has played a ma jor role in this area and ecient randomised approx imation algorithms have b een given for computing the volume of a convex b o dy and estimating the p ermanent of a dense matrix as well as for many other problems Each of these algorithms is a fully p olynomial randomised approx imation scheme fpras ie one that pro duces solutions which with very high probability fall within arbitrarily small error b ounds sp ecied by the user the price of greater accuracy b eing a mo dest increase in runtime Most of these algorithms use Markov chain simulation This approach has b een used extensively over the years in the eld of physics and in the last ten years or so it has b een used by researchers in computer science to provide fully p olynomial randomised approximation schemes for many problems For further information on this approach and its applications refer to the surveys by Kan nan and Jerrum and Sinclair Single spin ip pro cess It is in fact very easy to use the Markov chain approach to approximate the partition function of a Potts system The state space is simply the set of all p ossible congurations Let denote the steady state probability of the conguration as describ ed earlier Transition probabilities from the current state are mo delled by the following pro cedure choose a site i n and a spin s Q uniformly at random assign spin s to site i to get a new conguration and let the probability of
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