DOCSLIB.ORG

Introduction to the Gaussian Free Field and Liouville Quantum Gravity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to the Gaussian Free Field and Liouville Quantum Gravity Introduction to the Gaussian Free Field and Liouville Quantum Gravity Nathanaël Berestycki [Draft lecture notes: December 19, 2017] 1 Contents 1 Definition and properties of the GFF6 1.1 Discrete case∗ ...................................6 1.2 Continuous Green function............................ 10 1.3 GFF as a stochastic process........................... 12 1.4 Integration by parts and Dirichlet energy.................... 14 1.5 Reminders about functions spaces∗ ....................... 15 1.6 GFF as a random distribution∗ ......................... 18 1.7 Itô’s isometry for the GFF∗ ........................... 21 1.8 Markov property................................. 22 1.9 Conformal invariance............................... 23 1.10 Circle averages.................................. 24 1.11 Thick points.................................... 25 1.12 Exercises...................................... 28 2 Liouville measure 30 2.1 Preliminaries................................... 31 2.2 Convergence and uniform integrability in the L2 phase............ 32 2.3 Weak convergence to Liouville measure..................... 33 2.4 The GFF viewed from a Liouville typical point................. 34 2.5 General case∗ ................................... 37 2.6 The phase transition in Gaussian multiplicative chaos............. 39 2.7 Conformal covariance............................... 39 2.8 Random surfaces................................. 42 2.9 Exercises...................................... 42 3 Multifractal aspects and the KPZ relation 44 3.1 Scaling exponents; KPZ theorem........................ 44 3.2 Applications of KPZ to exponents∗ ....................... 46 3.3 Proof in the case of expected Minkowski dimension.............. 47 3.4 Sketch of proof of Hausdorff KPZ using circle averages............ 49 3.5 Kahane’s convexity inequality∗ .......................... 50 3.6 Proof of multifractal spectrum of Liouville measure.............. 54 3.7 Exercises...................................... 55 4 Statistical physics on random planar maps 56 4.1 Fortuin–Kesteleyn weighted random planar maps............... 56 4.2 Conjectured connection with Liouville Quantum Gravity........... 58 4.3 Mullin–Bernardi–Sheffield’s bijection in the case of spanning trees...... 61 4.4 The Loop-Erased Random Walk Exponent................... 64 4.5 Sheffield’s bijection in the general case..................... 66 4.6 Local limits and the geometry of loops..................... 68 2 4.7 Exercises...................................... 70 5 Scale-Invariant Random Surfaces∗ 71 5.1 Definition of Neumann boundary GFF..................... 71 5.2 Green function in the Neumann case...................... 72 5.3 Semicircle averages and boundary Liouville measure.............. 75 5.4 Convergence of random surfaces......................... 77 5.5 Quantum wedges................................. 78 5.6 Exercises...................................... 82 6 SLE and the quantum zipper∗ 84 6.1 SLE and GFF coupling; domain Markov property............... 84 6.2 Quantum length of SLE............................. 90 6.3 Uniqueness of the welding............................ 101 6.4 Slicing a wedge with an SLE........................... 102 6.5 Exercises...................................... 104 3 Introduction These lecture notes are intended to be an introduction to the two-dimensional continuum Gaussian Free Field and Liouville Quantum Gravity. Topics covered include the definition and main properties of the GFF, the construction of the Liouville measure, its non degeneracy and conformal covariance, and applications to the KPZ formula. An extra chapter covers Sheffield’s quantum zipper. This section is quite technical and readers are advised that this will be of most use to people who are actively working in this area. (While the arguments in that section follow the main ideas of [She10a], some are new and others have been simplified; and the structure of the proof has been rethought. The result is, I hope, that some of the main ideas are more transparent and will be helpful to others as it was to me.) The theory is in full blossom and attempting to make a complete survey of the field would be hopeless, so quickly is the theory developing. Nevertheless, as the theory grows in complexity and applicability, it has appeared useful to me to summarise some of its basic and foundational aspects in one place, especially since complete proofs of some of these facts can be spread over a multitude of papers. Clearly, the main drawback of this approach is that many of the important subsequent developments and alternative points of view are not included. For instance the theory is deeply intertwined with Kahane’s theory of Gaussian multiplicative chaos, but this will not be apparent in these notes. Likewise, topics such as random planar maps, the Brownian map, Liouville Brownian motion, critical Gaussian multiplicative chaos, as well as the beau- tiful way in which Duplantier, Miller and Sheffield have made sense of and analysed the “peanosphere" (which is really the scaling limit of random planar maps equipped with a collection of loops coming from a critical FK model), imaginary geometry and QLE, are not covered. The list is endless. To help beginners, I have tried to make a distinction between what is an important calculation and what can be skipped in a first reading without too much loss. Such sections are indicated with a star. In a first reading, the reader is encouraged to skip these sections. What’s new? Beside a streamlined and essentially self-contained presentation, several arguments in these notes can be considered novel: Itô’s isometry for the GFF (see section 1.7), showing that the continuum GFF can be • integrated not just against test functions, but any distribution which integrates the Green function (this is more or less our definition of the GFF in these notes for sim- plicity, so this shows that applying the Itô isometry property with the naive definition yields an essentially optimal set of test functions against which we can integrate the GFF). a new proof for the dimension of thick points of the GFF (especially the lower bound), • see Exercise5 in Section2. 4 The proof for the convergence of Gaussian multiplicative chaos for the GFF combines • the general arguments of [Ber15] together with a few additional features of the GFF which make the proof even more simple in this case. Computation of loop-erased random walk exponent using KPZ (thanks to Xin Sun for • the suggestion to include this) A new proof based on Girsanov’s theorem for one of the most delicate steps in the • construction of the quantum zipper, which says that the reweighted interface has the law of SLEκ(ρ), see Lemma 6.16. Acknowledgements: A first draft was written in preparation for the LMS / Clay institute Research school on “Modern developments in probability" taking place in Oxford, July 2015. The draft was revised in April 2016 on the occasion of the Spring School on “Geometric models in proba- bility" in Darmstadt, and then in July 2016 when I lectured this material at the Probability Summer School at Northwestern (for which a chapter on statistical physics on random planar maps was added). In all cases I thank the organisers (respectively, Christina Goldschmidt and Dmitry Beliaev; Volker Betz and Matthias Meiners; Antonio Auffinger and Elton Hsu) for their invitations and superb organisation. Thanks also to Benoit Laslier for agreeing to run the exercise sessions accompanying the lectures at the initial Clay Research school. The notes were written towards the end of my stay at the Newton institute during the semester on Random Geometry. Thanks to the INI for its hospitality and to some of the programme participants for enlightening discussions related to aspects discussed in these notes: especially, Omer Angel, Juhan Aru, Stéphane Benoît, Bertrand Duplantier, Ewain Gwynne, Nina Holden, Henry Jackson, Benoit Laslier, Jason Miller, James Norris, Ellen Powell, Gourab Ray, Scott Sheffield, Xin Sun, Wendelin Werner, Ofer Zeitouni. I received comments on versions of this draft at various stages by Henry Jackson, Benoit Laslier, Ellen Powell, Gourab Ray, Mark Sellke, Huy Tran, Fredrik Viklund, Henrik Weber. I am grateful for their input which helped me correct some typos and other minor problems, and emphasise some of the subtle aspects of the arguments. Thanks also to Jason Miller and to Jérémie Bettinelli and Benoit Laslier for letting me use some of their beautiful simulations which can be seen on the cover and in Section ??. 5 1 Definition and properties of the GFF 1.1 Discrete case∗ The discrete case is included here only for the purpose of helping the intuition when we come to the continuous case. Consider a finite graph G = (V; E) (possibly with weights (we)e E on the edges). Let 2 @ be a distinguished set of vertices, called the boundary of the graph, and set V^ = V @. n Let (Xt)t 0 be simple random walk on G in continuous time. Let Q = (qx;y)x;y V denote ≥ 2 the Q-matrix and of X (i.e., its infinitesimal generator, so that q = w for x = y and x;y x;y 6 qx;x = y x wxy) ; and let τ be the first hitting time of @. Note that π(x) = 1 is a reversible− measure∼ for X. P Definition 1.1 (Green function). The Green function G(x; y) is defined for x; y V by putting 2 1 G(x; y) = Ex 1 Xt=y;τ>t dt : f g Z0 In other words this is the expected time spent at y, started from x, until hitting the boundary. Note that with this definition we have G(x; y) = G(y; x) for all x; y V^ , since 2 Px(Xt = y; τ > t) = Py(Xt = x; τ > t) by reversibility of X with respect to the uniform measure π(x) = 1. An equivalent expression for the Green function when working with the simple random walk in discrete time Yn is the following: 1 1 G(x; y) = Ex 1 Xn=y;τ>n : deg(y) f g n=0 ! X Indeed for each visit of Yn to y, X would stay in y for an exponential random variable of mean 1= deg(y). The Green function is a basic ingredient in the definition of the Gaussian free field, so the following elementary properties will be important to us. Proposition 1.2.
Load more

Recommended publications
  • Conformal Welding Problem, Flow Line Problem, and Multiple Schramm
    CONFORMAL WELDING PROBLEM, FLOW LINE PROBLEM, AND MULTIPLE SCHRAMM–LOEWNER EVOLUTION MAKOTO KATORI AND SHINJI KOSHIDA Abstract. A quantum surface (QS) is an equivalence class of pairs (D,H) of simply connected domains D (C and random distributions H on D induced by the conformal equivalence for random metric spaces. This distribution-valued random field is extended to a QS with N + 1 marked boundary points (MBPs) with N Z≥0. We propose the conformal welding problem for it in the case of ∈ N Z≥1. If N = 1, it is reduced to the problem introduced by Sheffield, who solved∈ it by coupling the QS with the Schramm–Loewner evolution (SLE). When N 3, there naturally appears room of making the configuration of MBPs random,≥ and hence a new problem arises how to determine the probability law of the configuration. We report that the multiple SLE in H driven by the Dyson model on R helps us to fix the problems and makes them solvable for any N 3. We also propose the flow line problem for an imaginary surface with boundary≥ condition changing points (BCCPs). In the case when the number of BCCPs is two, this problem was solved by Miller and Sheffield. We address the general case with an arbitrary number of BCCPs in a similar manner to the conformal welding problem. We again find that the multiple SLE driven by the Dyson model plays a key role to solve the flow line problem. 1. Introduction Gaussian free field (GFF) [She07] in two dimensions gives a mathematically rigorous formulation of the free bose field, a model of two-dimensional conformal field theory (CFT) [BPZ84].
    [Show full text]
  • HARMONIC FUNCTIONS for BOUNDED and UNBOUNDED P(X)
    UNIVERSITY OF JYVASKYL¨ A¨ UNIVERSITAT¨ JYVASKYL¨ A¨ DEPARTMENT OF MATHEMATICS INSTITUT FUR¨ MATHEMATIK AND STATISTICS UND STATISTIK REPORT 130 BERICHT 130 EXISTENCE AND UNIQUENESS OF p(x)-HARMONIC FUNCTIONS FOR BOUNDED AND UNBOUNDED p(x) JUKKA KEISALA JYVASKYL¨ A¨ 2011 UNIVERSITY OF JYVASKYL¨ A¨ UNIVERSITAT¨ JYVASKYL¨ A¨ DEPARTMENT OF MATHEMATICS INSTITUT FUR¨ MATHEMATIK AND STATISTICS UND STATISTIK REPORT 130 BERICHT 130 EXISTENCE AND UNIQUENESS OF p(x)-HARMONIC FUNCTIONS FOR BOUNDED AND UNBOUNDED p(x) JUKKA KEISALA JYVASKYL¨ A¨ 2011 Editor: Pekka Koskela Department of Mathematics and Statistics P.O. Box 35 (MaD) FI{40014 University of Jyv¨askyl¨a Finland ISBN 978-951-39-4299-1 ISSN 1457-8905 Copyright c 2011, Jukka Keisala and University of Jyv¨askyl¨a University Printing House Jyv¨askyl¨a2011 Contents 0 Introduction 3 0.1 Notation and prerequisities . 6 1 Constant p, 1 < p < 1 8 1.1 The direct method of calculus of variations . 10 1.2 Dirichlet energy integral . 10 2 Infinity harmonic functions, p ≡ 1 13 2.1 Existence of solutions . 14 2.2 Uniqueness of solutions . 18 2.3 Minimizing property and related topics . 22 3 Variable p(x), with 1 < inf p(x) < sup p(x) < +1 24 4 Variable p(x) with p(·) ≡ +1 in a subdomain 30 4.1 Approximate solutions uk ...................... 32 4.2 Passing to the limit . 39 5 One-dimensional case, where p is continuous and sup p(x) = +1 45 5.1 Discussion . 45 5.2 Preliminary results . 47 5.3 Measure of fp(x) = +1g is positive . 48 6 Appendix 51 References 55 0 Introduction In this licentiate thesis we study the Dirichlet boundary value problem ( −∆ u(x) = 0; if x 2 Ω; p(x) (0.1) u(x) = f(x); if x 2 @Ω: Here Ω ⊂ Rn is a bounded domain, p :Ω ! (1; 1] a measurable function, f : @Ω ! R the boundary data, and −∆p(x)u(x) is the p(x)-Laplace operator, which is written as p(x)−2 −∆p(x)u(x) = − div jru(x)j ru(x) for finite p(x).
    [Show full text]
  • Arxiv:1402.0298V7 [Math.PR] 9 Jul 2020 V Uppoeson Process Jump ( Il De O Op Rmavre Oisl.Teegsaeen Are Edges the Itself
    FROM LOOP CLUSTERS AND RANDOM INTERLACEMENTS TO THE FREE FIELD TITUS LUPU Laboratoire de Mathématiques, Université Paris-Sud, Orsay Abstract. It was shown by Le Jan that the occupation field of a Poisson 1 ensemble of Markov loops ("loop soup") of parameter 2 associated to a tran- sient symmetric Markov jump process on a network is half the square of the Gaussian free field on this network. We construct a coupling between these loops and the free field such that an additional constraint holds: the sign of the free field is constant on each cluster of loops. As a consequence of our 1 coupling we deduce that the loop clusters of parameter 2 do not percolate on periodic lattices. We also construct a coupling between the random interlace- ment on Zd, d ≥ 3, introduced by Sznitman, and the Gaussian free field on the lattice such that the set of vertices visited by the interlacement is contained in a one-sided level set of the free field. We deduce an inequality between the critical level for the percolation by level sets of the free field and the critical parameter for the percolation of the vacant set of the random interlacement. 1. Introduction Here we introduce our framework, some notations, state our main results and outline the layout of the paper. We consider a connected undirected graph = (V, E) where the set of vertices V is at most countable and every vertex has finiteG degree. We do not allow mul- tiple edges nor loops from a vertex to itself. The edges are endowed with positive conductances (C(e))e E and vertices endowed with a non-negative killing measure ∈ (κ(x))x V .
    [Show full text]
  • Berestycki, Introduction to the Gaussian Free Field and Liouville
    Introduction to the Gaussian Free Field and Liouville Quantum Gravity Nathanaël Berestycki [Draft lecture notes: July 19, 2016] 1 Contents 1 Definition and properties of the GFF6 1.1 Discrete case∗ ...................................6 1.2 Green function..................................8 1.3 GFF as a stochastic process........................... 10 1.4 GFF as a random distribution: Dirichlet energy∗ ................ 13 1.5 Markov property................................. 15 1.6 Conformal invariance............................... 16 1.7 Circle averages.................................. 16 1.8 Thick points.................................... 18 1.9 Exercises...................................... 19 2 Liouville measure 22 2.1 Preliminaries................................... 23 2.2 Convergence and uniform integrability in the L2 phase............ 24 2.3 The GFF viewed from a Liouville typical point................. 25 2.4 General case∗ ................................... 27 2.5 The phase transition in Gaussian multiplicative chaos............. 29 2.6 Conformal covariance............................... 29 2.7 Random surfaces................................. 32 2.8 Exercises...................................... 32 3 The KPZ relation 34 3.1 Scaling exponents; KPZ theorem........................ 34 3.2 Applications of KPZ to exponents∗ ....................... 36 3.3 Proof in the case of expected Minkowski dimension.............. 37 3.4 Sketch of proof of Hausdorff KPZ using circle averages............ 39 3.5 Proof of multifractal spectrum of Liouville
    [Show full text]
  • The First Passage Sets of the 2D Gaussian Free Field: Convergence and Isomorphisms Juhan Aru, Titus Lupu, Avelio Sepúlveda
    The first passage sets of the 2D Gaussian free field: convergence and isomorphisms Juhan Aru, Titus Lupu, Avelio Sepúlveda To cite this version: Juhan Aru, Titus Lupu, Avelio Sepúlveda. The first passage sets of the 2D Gaussian free field: convergence and isomorphisms. Communications in Mathematical Physics, Springer Verlag, 2020, 375 (3), pp.1885-1929. 10.1007/s00220-020-03718-z. hal-01798812 HAL Id: hal-01798812 https://hal.archives-ouvertes.fr/hal-01798812 Submitted on 24 May 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THE FIRST PASSAGE SETS OF THE 2D GAUSSIAN FREE FIELD: CONVERGENCE AND ISOMORPHISMS. JUHAN ARU, TITUS LUPU, AND AVELIO SEPÚLVEDA Abstract. In a previous article, we introduced the first passage set (FPS) of constant level −a of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path along which the GFF is greater than or equal to −a. This description can be taken as a definition of the FPS for the metric graph GFF, and it justifies the analogy with the first hitting time of −a by a one- dimensional Brownian motion.
    [Show full text]
  • Math 639: Lecture 24 the Gaussian Free Field and Liouville Quantum Gravity
    Math 639: Lecture 24 The Gaussian Free Field and Liouville Quantum Gravity Bob Hough May 11, 2017 Bob Hough Math 639: Lecture 24 May 11, 2017 1 / 65 The Gaussian free field This lecture loosely follows Berestycki's `Introduction to the Gaussian Free Field and Liouville Quantum Gravity'. Bob Hough Math 639: Lecture 24 May 11, 2017 2 / 65 It^o'sformula The multidimensional It^o'sformula is as follows. Theorem (Multidimensional It^o'sformula) Let tBptq : t ¥ 0u be a d-dimensional Brownian motion and suppose tζpsq : s ¥ 0u is a continuous, adapted stochastic process with values in m d m R and increasing components. Let f : R ` Ñ R satisfy Bi f and Bjk f , all 1 ¤ j; k ¤ d, d ` 1 ¤ i ¤ d ` m are continuous t 2 E 0 |rx f pBpsq; ζpsqq| ds ă 8 then a.s.³ for all 0 ¤ s ¤ t s f pBpsq; ζpsqq ´ f pBp0q; ζp0qq “ rx f pBpuq; ζpuqq ¨ dBpuq »0 s 1 s ` r f pBpuq; ζpuqq ¨ dζpuq ` ∆ f pBpuq; ζpuqqdu: y 2 x »0 »0 Bob Hough Math 639: Lecture 24 May 11, 2017 3 / 65 Conformal maps Definition 2 Let U and V be domains in R . A mapping f : U Ñ V is conformal if it is a bijection and preserves angles. Viewed as a map between domains in C, this is equivalent to f is an analytic bijection. Bob Hough Math 639: Lecture 24 May 11, 2017 4 / 65 Conformal invariance of Brownian motion The following conformal invariance of Brownian motion may be established with It^o'sformula.
    [Show full text]
  • The Geometry of the Gaussian Free Field Combined with SLE Processes and the KPZ Relation Juhan Aru
    The geometry of the Gaussian free field combined with SLE processes and the KPZ relation Juhan Aru To cite this version: Juhan Aru. The geometry of the Gaussian free field combined with SLE processes and the KPZ relation. General Mathematics [math.GM]. Ecole normale supérieure de lyon - ENS LYON, 2015. English. NNT : 2015ENSL1007. tel-01195774 HAL Id: tel-01195774 https://tel.archives-ouvertes.fr/tel-01195774 Submitted on 8 Sep 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Ecole´ doctorale InfoMaths G´eometriedu champ libre Gaussien en relation avec les processus SLE et la formule KPZ THESE` pr´esent´eeet soutenue publiquement le le 10 juillet 2015 en vue de l'obtention du grade de Docteur de l'Universit´ede Lyon, d´elivr´epar l'Ecole´ Normale Sup´erieurede Lyon Discipline : Math´ematiques par Juhan Aru Directeur : Christophe GARBAN Apr`esl'avis de: Nathana¨elBERESTYCKI Scott SHEFFIELD Devant la commision d'examen : Vincent BEFFARA (Membre) Nathana¨elBERESTYCKI (Rapporteur) Christophe GARBAN (Directeur) Gr´egoryMIERMONT (Membre) R´emiRHODES (Membre) Vincent VARGAS (Membre) Wendelin WERNER (Membre) Unit´ede Math´ematiquesPures et Appliqu´ees| UMR 5669 Mis en page avec la classe thesul.
    [Show full text]
  • [Math.PR] 23 Nov 2006 Gaussian Free Fields for Mathematicians
    Gaussian free fields for mathematicians Scott Sheffield∗ Abstract The d-dimensional Gaussian free field (GFF), also called the (Eu- clidean bosonic) massless free field, is a d-dimensional-time analog of Brownian motion. Just as Brownian motion is the limit of the sim- ple random walk (when time and space are appropriately scaled), the GFF is the limit of many incrementally varying random functions on d-dimensional grids. We present an overview of the GFF and some of the properties that are useful in light of recent connections between the GFF and the Schramm-Loewner evolution. arXiv:math/0312099v3 [math.PR] 23 Nov 2006 ∗Courant Institute. Partially supported by NSF grant DMS0403182. 1 Contents 1 Introduction 3 2 Gaussian free fields 4 2.1 StandardGaussians........................ 4 2.2 AbstractWienerspaces. 6 2.3 Choosingameasurablenorm. 7 2.4 GaussianHilbertspaces . 10 2.5 Simpleexamples ......................... 12 2.6 FieldaveragesandtheMarkovproperty . 13 2.7 Field exploration: Brownian motion and the GFF . 15 2.8 Circleaveragesandthickpoints . 17 3 General results for Gaussian Hilbert spaces 17 3.1 Moments and Schwinger functions . 17 3.2 Wiener decompositions and Wick products . 19 3.3 Otherfields ............................ 19 4 Harmonic crystals and discrete approximations of the GFF 20 4.1 Harmoniccrystalsandrandomwalks . 20 4.2 Discrete approximations: triangular lattice . ... 22 4.3 Discrete approximations: other lattices . 23 4.4 Computer simulations of harmonic crystals . 23 4.5 Central limit theorems for random surfaces . 24 Bibliography 25 2 Acknowledgments. Many thanks to Oded Schramm and David Wilson for helping to clarify the definitions and basic ideas of the text and for helping produce the computer simulations.
    [Show full text]
  • A Characterisation of the Gaussian Free Field
    A characterisation of the Gaussian free field Nathanaël Berestycki∗ Ellen Powell y Gourab Rayz August 19, 2019 Abstract We prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption. 1 Introduction 1.1 Setup and main result The Gaussian free field (abbreviated GFF) has emerged in recent years as an object of central importance in probability theory. In two dimensions in particular, the GFF is conjectured (and in many cases proved) to arise as a universal scaling limit from a broad range of models, including the Ginzburg–Landau ' interface model ([20, 29, 26]), the height function associated to planar domino tilings and the dimerr model ([21, 12,5,6, 13, 23]), and the characteristic polynomial of random matrices ([18, 32, 19]). It also plays a crucial role in the mathematically rigourous description of Liouville quantum gravity; see in particular [17,1] and [14] for some recent major developments (we refer to [30] for the original physics paper). Note that the interpretations of Liouville quantum gravity in the references above are slightly different from one another, and are in fact more closely related to the GFF with Neumann boundary conditions than the GFF with Dirichlet boundary conditions treated in this paper. As a canonical random distribution enjoying conformal invariance and a domain Markov prop- erty, the GFF is also intimately linked to the Schramm–Loewner Evolution (SLE). In particular SLE4 and related curves can be viewed as level lines of the GFF ([36, 35, 11, 31]).
    [Show full text]
  • Properly-Weighted Graph Laplacian for Semi-Supervised Learning
    PROPERLY-WEIGHTED GRAPH LAPLACIAN FOR SEMI-SUPERVISED LEARNING JEFF CALDER Department of Mathematics, University of Minnesota DEJAN SLEPČEV Department of Mathematical Sciences, Carnegie Mellon University Abstract. The performance of traditional graph Laplacian methods for semi-supervised learning degrades substantially as the ratio of labeled to unlabeled data decreases, due to a degeneracy in the graph Laplacian. Several approaches have been proposed recently to address this, however we show that some of them remain ill-posed in the large-data limit. In this paper, we show a way to correctly set the weights in Laplacian regularization so that the estimator remains well posed and stable in the large-sample limit. We prove that our semi-supervised learning algorithm converges, in the infinite sample size limit, to the smooth solution of a continuum variational problem that attains the labeled values continuously. Our method is fast and easy to implement. 1. Introduction For many applications of machine learning, such as medical image classification and speech recognition, labeling data requires human input and is expensive [13], while unlabeled data is relatively cheap. Semi-supervised learning aims to exploit this dichotomy by utilizing the geometric or topological properties of the unlabeled data, in conjunction with the labeled data, to obtain better learning algorithms. A significant portion of the semi-supervised literature is on transductive learning, whereby a function is learned only at the unlabeled points, and not as a parameterized function on an ambient space. In the transductive setting, graph based algorithms, such the graph Laplacian-based learning pioneered by [53], are widely used and have achieved great success [3,26,27,44–48,50, 52].
    [Show full text]
  • A Simple Discretization of the Vector Dirichlet Energy
    Eurographics Symposium on Geometry Processing 2020 Volume 39 (2020), Number 5 Q. Huang and A. Jacobson (Guest Editors) A Simple Discretization of the Vector Dirichlet Energy Oded Stein1, Max Wardetzky2, Alec Jacobson3 and Eitan Grinspun3;1 1Columbia University, USA 2University of Göttingen, Germany 3University of Toronto, Canada Abstract We present a simple and concise discretization of the covariant derivative vector Dirichlet energy for triangle meshes in 3D using Crouzeix-Raviart finite elements. The discretization is based on linear discontinuous Galerkin elements, and is simple to implement, without compromising on quality: there are two degrees of freedom for each mesh edge, and the sparse Dirichlet energy matrix can be constructed in a single pass over all triangles using a short formula that only depends on the edge lengths, reminiscent of the scalar cotangent Laplacian. Our vector Dirichlet energy discretization can be used in a variety of applications, such as the calculation of Killing fields, parallel transport of vectors, and smooth vector field design. Experiments suggest convergence and suitability for applications similar to other discretizations of the vector Dirichlet energy. 1. Introduction The covariant derivative r generalizes the gradient of scalar func- tions to vector fields defined on surfaces. As the gradient does for scalar functions, the covariant derivative measures the infinitesimal change of a vector field in every direction. As with the gradient’s 1 R 2 scalar Dirichlet energy, Escalar(u) := 2 W kruk dx for a smooth scalar function u and a smooth surface W, the covariant derivative has a corresponding vector Dirichlet energy, Z 1 2 E(u) := krukF dx , (1) transporting the 2 W red vector (enlarged) denoising a vector field by smoothing across the surface where u is a smooth vector-valued function on W, and k·kF is the Frobenius norm.
    [Show full text]
  • The Laplacian in Its Different Guises
    U.U.D.M. Project Report 2019:8 The Laplacian in its different guises Oskar Tegby Examensarbete i matematik, 15 hp Handledare: Martin Strömqvist Examinator: Jörgen Östensson Februari 2019 Department of Mathematics Uppsala University Contents 1 p = 2: The Laplacian 3 1.1 In the Dirichlet problem . 3 1.2 In vector calculus . 6 1.3 As a minimizer . 8 1.4 In the complex plane . 13 1.5 As the mean value property . 16 1.6 Viscosity solutions . 17 1.7 In stochastic processes . 18 2 2 < p < and p = : The p-Laplacian and -Laplacian 20 2.1 Ded∞uction of th∞e p-Laplacian and -Laplaci∞an . 20 2.2 As a minimizer . ∞. 22 2.3 Uniqueness of its solutions . 24 2.4 Existence of its solutions . 25 2.4.1 Method 1: Weak solutions and Sobolev spaces . 25 2.4.2 Method 2: Viscosity solutions and Perron’s method . 27 2.5 In the complex plane . 28 2.5.1 The nonlinear Cauchy-Riemann equations . 28 2.5.2 Quasiconformal maps . 29 2.6 In the asymptotic expansion . 36 2.7 As a Tug of War game with or without drift . 41 2.8 Uniqueness of the solutions . 44 1 Introduction The Laplacian n ∂2u ∆u := u = ∇ · ∇ ∂x2 i=1 i � is a scalar operator which gives the divergence of the gradient of a scalar field. It is prevalent in the famous Dirichlet problem, whose importance cannot be overstated. It entails finding the solution u to the problem ∆u = 0 in Ω �u = g on ∂Ω. The Dirichlet problem is of fundamental importance in mathematics and its applications, and the efforts to solve the problem has led to many revolutionary ideas and important advances in mathematics.
    [Show full text]