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Ising Model and Field Theory: Lattice Local Fields and Massive Scaling Limit

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Ising Model and Field Theory: Lattice Local Fields and Massive Scaling Limit Ising Model and Field Theory: Lattice Local Fields and Massive Scaling Limit Thèse N° 7741 Présentée le 9 septembre 2019 à la Faculté des sciences de base Chaire de théorie des champs statistiques Programme doctoral en mathématiques pour l’obtention du grade de Docteur ès Sciences par Sung Chul PARK Acceptée sur proposition du jury Prof. M. Troyanov, président du jury Prof. C. Hongler, directeur de thèse Prof. F. Viklund, rapporteur Prof. D. Chelkak, rapporteur Prof. R. Dalang, rapporteur 2019 Abstract This thesis is devoted to the study of the local fields in the Ising model. The scaling limit of the critical Ising model is conjecturally described by Conformal Field Theory. The explicit predictions for the building blocks of the continuum theory (spin and energy density) have been rigorously established [HoSm13, CHI15]. We study how the field-theoretic description of these random fields extends beyond the critical regime of the model. Concretely, the thesis consists of two parts: The first part studies the behaviour of lattice local fields in the critical Ising model. A lattice local field is a function of a finite number of spins at microscopic distances from a given point. We study one-point functions of these fields (in particular, their asymptotics under scaling limit and conformal invariance). Our analysis, based on discrete complex analysis methods, results in explicit computations which are of interest in applications (e.g. [HKV17]). The second part considers the behaviour of the massive spin field. In the subcritical massive scaling limit regime first considered by Wu, McCoy, Tracy, and Barouch [WMTB76], we show that the correlations of the massive spin field in a bounded domain have a scaling limit. Fur- thermore, to this end we generalise the notions and methods of discrete complex analysis in the critical case to the massive regime, and give a new derivation of the formula for the two-point correlation in the full plane in terms of a Painlevé III transcendent. Keywords: Ising Model, Statistical Mechanics, Probability Theory, Conformal Field Theory, Isomonodromy, Discrete Complex Analysis. i Résumé Cette thèse est consacrée à l’étude des champs locaux du modèle d’Ising. Il est conjecturé que la limite d’échelle du modèle d’Ising critique est décrite par une théorie conforme des champs. Les prédictions explicites pour les éléments de base (le spin et la densité d’énergie) de la théo- rie continue ont été démontrées rigoureusement dans [HoSm13, CHI15]. On étudie comment la description du point de vue de la théorie des champs s’étend au-delà du comportement critique de ces champs aléatoires. Concrètement, cette thèse se compose de deux parties : La première partie étudie le comportement des champs locaux de réseau. Un champ local au niveau du réseau est une fonction d’un nombre fini des spins à distances microscopiques d’un point donné. On étudie les fonctions de corrélation à 1 point de ces champs (en particu- lier, comment ils se comportent dans la limite d’échelle). L’analyse, basée sur des méthodes d’analyse complexe discrète, a pour résultat des calculs explicits d’intérêt pour applications (e.g. [HKV17]). La seconde partie examine le comportement du champ de spin. Dans le régime de limite d’échelle sous-critique considéré pour la première fois par Wu, McCoy, Tracy et Barouch [WMTB76], on démontre que les corrélations du champ de spin massif dans un domaine à borne possèdent une limite d’échelle. Dans ce but, on généralise les notions et méthodes d’analyse complexe discrète du cas critique au régime massif, et on donne une nouvelle dérivation la formule pour la corrélation à 2 points dans le plan en termes de la fonction transcendante de Painlevé III. Mots-clés : modèle d’Ising, mécanique statistique, théorie des probabilités, théorie conforme des champs, déformation isomonodromique, analyse complexe discrète. iii Acknowledgements I could not possibly start this thesis without thanking my advisor Clément Hongler. Ever since we first met nine years ago, he has been the most significant and influential teacher in my mathematical life; becoming his doctoral student meant that I was lucky enough to benefit from his care and support in both mathematical and practical aspects for the past four years. I feel grateful from the bottom of my heart and shall always be indebted to him for the unending generosity and patience he has shown me throughout this ever-changing ride. Francesco Spadaro, my thesis brother, or maybe frère d’armes, has been a great mathematical presence in my day-to-day life at EPFL and is simply a lovely sport all-around. I have never been bored of discussing just about anything with him, from complex analysis to everything Italian: grazie per il guanciale! In fact, everyone at our chair happens to be similarly awesome, and I am obliged to thank Franck, Arthur, Berfin, and Laurie and Nicolas for their interesting mathematics and more. In addition, I am grateful for the caring attention that Marie has given to me and any of my issues. Many mathematicians have kindly lent their ears and offered their thoughts about the subject matter of this thesis. Kalle Kytölä and Konstantin Izyurov, besides having me at their base in Helsinki, graciously gave their time and invaluable feedback. Nam-gyu Kang has welcomed me back to my hometown of Seoul, and I truly appreciate and cherish the opportunities I have been given to first present and now continue my work there. I would also like to thank the thesis jury, Dmitry Chelkak, Robert Dalang, Marc Troyanov, and Fredrik Viklund, for their time, attention, and many interesting questions, as well as Julien Dubédat, Béatrice de Tilière, and Klaus Widmeyer for substantive discussions. Last but not least, I would not be here, in many senses, without the teachings and support of my former advisor Pengfei Guan. Schliässlech muess i au die erwähnen, won i furtfahre tank ine. Mis Müeti, min Vatti u min Brüeder, wo d’Unterstützig von ere zwar niä um minere Liischtig ggange isch; de Hoonsik, de Mooyeol (u öisi oft synchronisiärtä Schlaafzyklä), de Mungi u de Ki-Hwan, di Unuufhaltsame; d’Alice, d’Ashley’s, de Jorge, de Louis, d Ro, de Sergio u d Stephanie, di amerikanische Tröime; Mihai, de Verantwortlächer; Reid, won i ooni em Paris nöd psuecht hett. Je remercie également l’équipe O&C et the Village Players de m’avoir aidé à rester normal à divers moments. Lausanne, August 22, 2019 S. P. v To Jackson and Liam Contents Abstract (English/Français)i Acknowledgementsv 1 Introduction 1 1.1 The Ising Model......................................1 1.1.1 Definition.....................................1 1.1.2 Phase Transition.................................1 1.1.3 Relation to Other Models............................2 1.2 Scaling Limit and Conformal Invariance.......................3 1.2.1 Scaling Limit and Universality.........................3 1.2.2 Conformal Invariance..............................3 1.2.3 Schramm-Loewner Evolutions.........................4 1.2.4 Conformal Field Theory.............................4 1.3 Massive Regimes.....................................5 1.3.1 Magnetic Perturbation..............................5 1.3.2 Thermal Perturbation..............................6 1.3.3 Massive Ising Correlations............................6 1.4 Methods of Analysis...................................6 1.4.1 Field Theory and Combinatorics........................6 1.4.2 Fermionic Observables.............................7 1.4.3 Criticality and Discrete Holomorphicity....................7 1.4.4 Massive Discrete Complex Analysis......................9 1.5 Objectives.........................................9 1.6 Main Theorems...................................... 10 2 Local Correlations and Conformal Invariance 17 2.1 Introduction........................................ 18 2.1.1 Notation...................................... 23 2.1.2 Main Results.................................... 24 2.1.3 Applications.................................... 25 2.1.4 Proof Outline................................... 27 2.1.5 Extra Notation and Glossary.......................... 29 2.2 Discrete Complex Analysis................................ 33 ix Contents 2.2.1 S-holomorphicity................................. 33 2.2.2 Discrete Singularities............................... 35 2.2.3 Discrete Riemann Boundary Value Problems................. 35 2.3 Discrete Two-point Observables............................ 37 2.3.1 Bounded Domain Observables......................... 38 2.3.2 Full Plane Observables.............................. 42 2.4 Discrete Multipoint Observables............................ 51 2.4.1 Multipoint Observables............................. 52 2.4.2 Fused Observables and Ising Correlations.................. 59 2.4.3 Pfaffian Formulae................................. 62 2.4.4 Observables and Ising Correlations...................... 64 2.5 Scaling Limits of Observables.............................. 65 2.5.1 Integration of the Square............................ 66 2.5.2 Continuous Observables............................. 67 2.5.3 Observable Convergence: Statements..................... 69 2.5.4 Observable Convergence: Proofs........................ 71 2.6 Proofs of Theorems.................................... 75 2.6.1 Spin-symmetric Fields.............................. 76 2.6.2 Spin-antisymmetric Fields........................... 77 2.6.3 Spin Pattern Probabilities............................ 79 2.7 Appendix: The Harmonic Measure on C1\R 0 .................... 79 È 2.8 Appendix: Contour Weights..............................
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