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Transcript of Combinatorial methods in Statistical Field caraccio/PhD/ Combinatorial methods in Statistical

  • Scuola Normale Superiore

    Classe di Scienze

    Andrea Sportiello

    Combinatorial methods

    in Statistical Field Theory

    Trees, loops, dimers and orientations

    vs. Potts and non-linear σ-models

    Relatore: Prof. Sergio Caracciolo

    PACS: 05.70.Fh 64.60.-i 02.10.Ox

    Tesi di Perfezionamento

    Anni Accademici 2000-2003

  • PACS: 05.70.Fh Phase transitions: general studies; 05.50.+q Lattice theory and statistics; 02.10.Ox Combinatorics; graph theory.

  • Contents

    Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

    A perspective of Combinatorics and Statistical Field Theory

    1 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Basics in graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Cyclomatic space and Euler formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Graphs vs. hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Algebraic graph theory and the Laplacian of a graph . . . . . . . . . . . . . . . . . . . 12 1.6 Subgraph enumeration: an overview of relationships . . . . . . . . . . . . . . . . . . . . 13

    2 Statistical Mechanics and Critical Phenomena . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Partition sums and equilibrium measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Spontaneous symmetry breaking in the Ising model . . . . . . . . . . . . . . . . . . . . 18 2.3 Characterization of pure phases and Cluster Property . . . . . . . . . . . . . . . . . . 20 2.4 Ising Model in two dimensions: Kramers-Wannier duality . . . . . . . . . . . . . . . 22 2.5 Ising Model in two dimensions: solution with Grassmann variables . . . . . . . 27 2.6 The OSP(2n|2m) model in the Gaussian case . . . . . . . . . . . . . . . . . . . . . . . . . 34

    Graph-enumeration problems in Statistical Mechanics

    3 Graph-enumeration problems in O(n) models . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 The O(n) non-linear σ-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 The limit n→ −1 for the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Recovering the Nienhuis Model at n = −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 The Hintermann-Merlini–Baxter-Wu Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

    4 Graph-enumeration problems in the ferromagnetic Potts Model . . . . . 57 4.1 Potts model and Fortuin-Kasteleyn expansion . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Potts model on a hypergraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 The Random Cluster Model on planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Leaf removal, series-parallel reduction and Y–∆ relation . . . . . . . . . . . . . . . . 62 4.5 Temperley-Lieb Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.6 Observables in colour and random-cluster representations . . . . . . . . . . . . . . . 71

  • ii Contents

    4.7 Positive and negative associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    5 Graph-enumeration problems in the antiferromagnetic Potts Model . 79 5.1 Chromatic polynomial and acyclic orientations . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 The polynomials P ′G(1), P

    ′ G(0) and PG(−1) under one roof . . . . . . . . . . . . . 86

    5.3 Antiferromagnetic Fortuin-Kasteleyn expansion . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 Antiferro F-K and acyclic orientations: a multilinearity proof . . . . . . . . . . . . 94 5.5 F-K for a system with antiferro and unphysical couplings . . . . . . . . . . . . . . . 95 5.6 Observables in the antiferro F-K expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

    6 The phase diagram of Potts Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.1 Potts phase diagram in D = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 Potts phase diagram in D = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.3 Potts phase diagram in infinite dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.4 A conjecture on the flow near the Spanning Tree point . . . . . . . . . . . . . . . . . 109

    Theories with supersymmetry

    7 Uniform spanning trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.1 Statistical averages in the ensemble of Uniform Spanning Trees . . . . . . . . . . 115 7.2 Coordination distribution for finite-dimensional lattices . . . . . . . . . . . . . . . . . 122 7.3 Correlation functions for edge-occupations and coordinations . . . . . . . . . . . . 129 7.4 Integration of 2-dimensional lattice integrals at finite volume . . . . . . . . . . . . 133 7.5 Forests of unicyclic subgraphs and the coupling with a gauge field . . . . . . . 137

    8 OSP(1|2) non-linear σ-model and spanning hyperforests . . . . . . . . . . . . . 145 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.2 Graphical approach to generalized matrix-tree theorems . . . . . . . . . . . . . . . . 148 8.3 A Grassmann subalgebra with unusual properties . . . . . . . . . . . . . . . . . . . . . . 156 8.4 Grassmann integrals for counting spanning hyperforests . . . . . . . . . . . . . . . . 163 8.5 Extension to correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.6 The role of OSP(1|2) symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.7 A determinantal formula for fA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

    9 Ward identities for the O(N) σ-model and connectivity probabilities 179 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 9.2 Lattice notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 9.3 Ward identities for the O(N) σ-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 9.4 Ward identities for the generating function of spanning forests . . . . . . . . . . 183 9.5 All Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.6 Combinatorial meaning of Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

    10 The ideal of relations in Forest Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 10.1 Scalar products on RP 1|2 and Rabcd = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 10.2 Even/odd Temperley-Lieb Algebra and Rbac = 0 . . . . . . . . . . . . . . . . . . . . . . . 197 10.3 Braid generators and Rab = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 10.4 Proof of the basic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 10.5 The basis of non-crossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.6 Jordan-Wigner transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 10.7 Completeness of {〈fC〉}C∈NC[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

  • Contents iii

    11 OSP(1|2n) vector models and Spanning Forests . . . . . . . . . . . . . . . . . . . . . . 217 11.1 The abstract Forest Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 11.2 A reminder on the OSP(1|2)–Spanning-Forest correspondence . . . . . . . . . . . 219 11.3 Unit vectors in R1|2n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 11.4 The results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.5 Proof of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 11.6 Proof of the lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 11.7 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 11.8 On the quotient of Forest Algebra induced by OSP(1|2n) representation . . 235

    12 OSP(2|2) non-linear σ-model and dense polymers . . . . . . . . . . . . . . . . . . . 237 12.1 The model with OSP(2|2) symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 12.2 Solution on a cycle graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 12.3 High-temperature expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

    A Grassmann-Berezin calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 A.1 Grassmann and Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 A.2 Grassmann-Berezin integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 A.3 Integral kerne