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 Basicsin graphtheory ............................. ................ 1 1.2 Cyclomatic space and Euler formulas ................. ............... 6 1.3 Planargraphs.................................... ................. 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 1for thegeneralcase................................ 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 Pottsmodel 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 PG(1), PG(0) and PG( 1)underone roof ............. 86 5.3 Antiferromagnetic Fortuin-Kasteleyn expansion− . .................. 88 5.4 Antiferro F-K and acyclic orientations: a multilinearityproof............ 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 Latticenotation................................. ..................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 Wardidentities ............................... .................186 9.6 Combinatorial meaning of Ward Identities . ................189 10 The ideal of relations in Forest Algebra .............................. 195 1|2 10.1 Scalar products on RP and Rabcd =0..............................196 b 10.2 Even/odd Temperley-Lieb Algebra and Rac =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 f ......................................214 {h Ci}C∈NC[n] 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 Theresults ..................................... ..................223 11.5 Proofof the theorem.............................. .................226 11.6 Proofof 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 kernels in Clifford Algebra ................. .................253 References ................................................... ............ 257 Introduction In the novel by Danilo Kiˇs Garden, Ashes, the father of the protagonist spends all his life in the impossible goal of writing a complete timetable of the Jugoslavian train trans- port system. Although this paradigm is not making here its first appearence in literature (among the predecessors is of course Borges – is the recurrence of this theme partially autobiographic among world novelists?), and certainly not the most famous, it is emblem- atic in the self-evidence of its impossibility: train timetables change at a rate vanifying any systematisation effort. The writing of this thesis has a similar hystory. The main topic is at the interface between critical phenomena in statistical mechanics, and tools in enumerative combina- torics, both aimed to be “beyond mean-field theory”, whatever is the precise assertion of this expression. A paradigmatic system is the ground for original developments: Kirchhoff classical determinantal formula for counting the spanning trees of a given graph (i.e., from the point of view of statistical mechanics, a certain limit q 0 of the q-state Potts model). → For this system, a surprising number of connections with other fields of physics/math- ematics/computer-science arises, in such a structural way that it would be hard omitting a discussion for each. Determinantal point processes, the Abelian Sandpile Model, the XXZ quantum spin chain at ∆ = 0, Logarithmic minimal models in CFT (this one being the first member of the family, (1, 2)), Temperley-Lieb and Hecke Algebras, Invari- ant theory of classical superalgebras,LM Loop Models, SLE (at (κ, κ˜)=(8, 2)), Loop-erased Random Walks, algorithmic aspects of exact sampling (in particular Propp and Wilson algorithm), non-intersecting lattice paths, tiling problems and Gessel-Viennot formula (through Temperley bijection), complexity issues for the evaluation of Tutte polynomial (the Jaeger-Vertigan-Welsh plane), partitionability of Tutte polynomial, Schaeffer decom- position of planar maps into blossomed trees, and so on. All of this, still, to be dealt with before going to the main research topic of this report, that is, which of these results, and how, can be extended to spanning forests, and understanding that this more general problem is related to a QFT being a OSP(1 2) | Introduction supersymmetric non-linear σ-model (spanning trees being the limit