Disorder in Physical Systems Disorder in Physical Systems A volume in honour of John M. Hammersley on the oc c a sion of his 70th birthday Edited by G.R. Grimmett and D.J.A. Welsh Preface On 21 March 1990 John Hammersley celebrates his seventieth birth- day. A number of his colleagues and friends wish to pay tribute on this occasion to a mathematician whose exce ptional inventiveness has greatly enriched mathematical science. The breadth and versatility of Hammersley’s interests are remarkable, doubly so in an age of increased specialisation. In a range of highly individ- ual papers on a variety of topics, he has theorised, and posed (and solved) problems, thereby laying the foundations for many subjects currently un- der study. By his e vident love for mathematics and an affinity for the hard problem, he has be e n an inspiration to many. If one must single out one particular area where Hammersley’s con- tribution has proved especially vital, it would probably be the study of random processes in space. He was a pioneer in this field of recognised impo rtance, a field abounding in apparently simple questions whose res- olutions usually require new ideas and methods. This area is not just a mathematician’s playground, but is of fundamental importance for the un- derstanding of physical phenomena. The principal theme of this volume reflects various aspects of Hammersley’s work in the area, including disor- dered media, subadditivity, numerical methods, and the like. The authors of these papers join with those unable to contribute in wishing John Hammersley many further years of fruitful mathematical ac- tivity. August 1989 G.R. Grimmett D.J.A. Welsh Contents Contributors ix Speech Propos ing the Toast to John Hammersley — 1 October 1987 1 David Kendall Jakimovski Methods and Almost-Sure Convergence 5 N.H. Bingham and U. Stadtm¨uller Markov Random Fields in Statistics 19 Peter Clifford On Hammersley’s Method for One-Dimensional Covering Problems 33 Cyril Domb On a Problem of Straus 55 P. Erd˝os and A. S´ark¨ozy Directed Compact Percolation II: Nodal Points, Mass Distribution, and Scaling 67 J.W. Essam and D. Tanlakishani Critical Points, Large-Dimensionality Expansions, and the Ising Spin Glass 87 Michael E. Fisher and Rajiv R.P. Singh Bistability in Communication Networks 113 R.J. Gibbens, P.J. Hunt, and F.P. Kelly A Quantal Hypothesis fo r Ha drons and the Judging of Physical Numero logy 129 I.J. Good Percolation in ∞ + 1 Dimensions 167 G.R. Grimmett and C.M. Newman Monte Carlo Methods Applied to Quantum-Mechanical Order-Disorder Phenomena in Crystals 191 D.C. Handscomb The Diffusion of Euclidean Shape 203 Wilfred S. Kendall Asymptotics in High Dimensions for Percolation 219 Harry Kesten viii Contents Some Random Collections of Finite Subsets 241 J.F.C. Kingman Probabilistic Analysis of Tree Search 249 C.J.H. McDiarmid Probability Densities for Some One-Dimensional Problems in Statistical Mechanics 261 J.S. Rowlinson Seedlings in the Theory of Shortest Paths 277 J. Michael Steele The Computational Complexity of Some Classical Problems from Statistical Physics 307 D.J.A. Welsh Lattice Animals: Rigorous Results and Wild Guesses 323 S.G. Whittington and C.E. Soteros Fields and Flows on Random Graphs 337 P. Whittle Bond Percolation Critical Probability Bounds for the Kagom´e Lattice by a Substitution Method 349 John C. Wierman Brownian Motion and the Riemann Zeta-Function 361 David Williams Index 373 Contributors N. H. BINGHAM, Department of Mathematics, Royal Holloway and Bedford New College, Egham Hill, Egham, Surrey TW20 0EX, UK. P. CLIFFORD, Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford OX1 3LB, UK. C. DOMB, Physics Department, Bar-Ilan University, Ramat-Gan, Israel. P. ERD ˝ OS, Mathematical Institute, Hungarian Academy of Sciences , Re´altanoda ul. 13–15, Budapest, Hungary. J. W. ESSAM, Department of Mathematics, Royal Holloway and Bedford New College, Egham Hill, Egham, Surrey TW20 0EX, UK. M. E. FISHER, Institute for Physical Science and Technology, The University of Maryland, College Park, Maryland 20742, USA. R. J. GIBBENS, Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1 SB , UK. I. J. GOOD, Department of Statistics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA. G. R. GRIMMETT, School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK. D. C. HANDSCOMB, Oxford University Computing Laboratory, 8–11 Keble Road, Oxford OX1 3QD, UK. P. J. HUNT, Statistical Lab oratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1 SB , UK. F. P. KELLY, Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1 SB , UK. D. G. KENDALL, 37 Barrow Road, Cambridge CB2 2AR, UK. W. S. KENDALL, Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. H. KESTEN, Department of Mathematics, Cornell University, Ithaca, New York 14853 , USA. J. F. C. KINGMAN, Senate House, University of B ristol, Tyndall Avenue, B ristol BS8 1TH, UK. x Contributors C. J. H. MCDIARMID, Department of Statistics, University of Oxford, Oxford OX1 3TG, UK. C. M. NEWMAN, Department of Mathematics, University of Arizona, Tucson, Arizona 85721, USA. J. S. ROWLINSON, Physical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, UK. A. S ´ ARK ¨ OZY, Mathematical Institute, Hungarian Academy of Sciences, Re´altanoda ul. 13–15, Budapest, Hungary. R. R. P. SINGH, AT&T Bell Laboratories, Murray Hill, New Jersey 07974, USA. C. E. SOTEROS, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 1A1, Canada. U. STADTM ¨ ULLER, Universit¨at Ulm, Abteilung Mathematik-III, Oberer Eselsberg, 7900 Ulm, FRG. J. M. STEELE, Program in Statistics and Operations Research, School of Engineering and Applied Science, Princeton University, Princeton, New Jersey 08544, USA. D. TANLAKISHANI, Department of Mathematics, Royal Holloway and Bedford New College, Egham Hill, Egham, Surrey TW20 0EX, UK. D. J. A. WELSH, Merton College, Oxford OX1 4JD, UK. S. G. WHITTINGTON, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 1A1, Canada. P. WHITTLE, Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1 SB , UK. J. C. WIERMAN, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland 21218, USA. D. WILLIAMS, Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1 SB , UK. [...]... building upon models used originally in the description of physical systems and borrowing and improving upon ideas from computational physics Monte Carlo methods, in particular, have played a dominant role in dealing with problems of inference The practicalities of working with high dimensional parameter sets within a Bayesian framework, have led to the invention of refreshing and novel techniques (Geman... of colours and χ maps T into C The colouring is said to be polygonal if and only if the set of discontinuity points of χ is the union of intervals of a finite number of distinct lines, where each line contributes exactly one interval We disregard intervals of zero length Associated with each polygonal colouring χ there is the unique set of lines which contain the discontinuity points We call this set... theory The introduction of MRFs into the theory of statistics is yet another example of the continuing transfer of knowledge from the world of theoretical physics John Hammersley whose interests include both domains of study, was ideally placed to facilitate the process of cross-fertilisation Others who were involved in this instance include Neyman and Besag Neyman was responsible for bringing Hammersley... the way in which Bayesian methods are used in more general contexts and which may serve to reintegrate these methods into the main body of applied statistics Much of physics is concerned with providing an understanding of the spatial organisation of matter and it is not suprising that many of the ideas which have become central in the theory of spatial statistics should have their origins in physical. .. constant The following theorem gives a sufficient condition for this to be so Theorem 3 If |C| = 2 and λdT < 1, then LT |ΩT |µT (d ) < ∞ Before proving the theorem, we must introduce a little more notation An extended polygonal colouring is a function χ+ : R2 → C, whose discontinuity points are the union of intervals of lines in LT , but here the intervals are either semi -in nite, in nite, or of finite... sites in Y to black A partial colouring has colours assigned on only a subset of sites The partial colouring obtained by considering which colours have been assigned to sites in X by the colouring χ is denoted by χX In particular, the colour at a site z is written as χz The set of all possible colourings of Z is given by C = C1 × C2 × · · · × Cn A set Y is said to be light relative to χ if no site in. .. 3.1 Polygonal Colouring Measure The simplest building block for polygonal fields is the Poisson line process (Kendall and Moran 1963) To describe the construction we introduce the following notation Let T ⊂ R2 be a convex bounded domain Let Ln be the family of T all sets of n distinct lines which intersect T and let LT = ∪∞ Ln , with n=0 T L0 defined to be {∅}, the family consisting of the empty set... 0) cover the cases of main interest (though the result below and its proof may √ be extended to cover the case σn ∼ c µn , for constant c) In (i) below, ‘log’ in the denominator means ‘max(1, log+ )’ In Theorem 2, which gives the rates of convergence in Theorem 1, the Karamata-Stirling methods diverge from those of Euler and Borel, and one obtains an iterated logarithm, as in the classical case but... wanting to discuss what we had been saying Experiences like this convinced John that some massive effort should be made to bring before school teachers a review of the exciting and really quite simple — but new — kinds of mathematics that could easily and usefully be added to the curriculum, whether they were reflected in the examinations or not This led to an Oxford Conference inspired by John, in which... Poisson line process defined on LT To fix ideas we will assume that the process is homogeneous and isotropic with intensity λ, so that the number of lines crossing a disc of diameter d has a Poisson distribution with mean λd and the mean number of lines intersecting T is λdT , where dT is the mean diameter of T We write µT for the Poisson line measure n on LT , and we denote the conditional line measure . Disorder in Physical Systems Disorder in Physical Systems A volume in honour of John M. Hammersley on the oc c a sion of his 70th. fundamental importance for the un- derstanding of physical phenomena. The principal theme of this volume reflects various aspects of Hammersley’s work in the area, including disor- dered media, subadditivity,. of main interest (though the result below and its proof may be extended to cover the case σ n ∼ c √ µ n , for constant c). In (i) below, ‘log’ in the denominator means ‘max(1, log + )’. In Theorem