Gaussian free field, Liouville quantum gravity and Gaussian multiplicative chaos Nathanaël Berestycki Ellen Powell [Draft lecture notes: March 16, 2021] 1 Contents 1 Definition and properties of the GFF7 1.1 Discrete case...................................7 1.2 Continuous Green function............................ 11 1.3 GFF as a stochastic process........................... 14 1.4 Integration by parts and Dirichlet energy.................... 17 1.5 Reminders about functions spaces........................ 18 1.6 GFF as a random distribution.......................... 22 1.7 Itô’s isometry for the GFF............................ 25 1.8 Cameron–Martin space of the Dirichlet GFF.................. 26 1.9 Markov property................................. 28 1.10 Conformal invariance............................... 29 1.11 Circle averages.................................. 30 1.12 Thick points.................................... 31 1.13 Exercises...................................... 34 2 Liouville measure 36 2.1 Preliminaries................................... 37 2.2 Convergence and uniform integrability in the L2 phase............ 38 2.3 Weak convergence to Liouville measure..................... 40 2.4 The GFF viewed from a Liouville typical point................. 41 2.5 General case.................................... 44 2.6 The phase transition for the Liouville measure................. 46 2.7 Conformal covariance............................... 46 2.8 Random surfaces................................. 48 2.9 Exercises...................................... 49 3 Properties of Gaussian multiplicative chaos 50 3.1 Setup for Gaussian multiplicative chaos..................... 50 3.2 Construction of Gaussian multiplicative chaos................. 52 3.2.1 Uniform integrability........................... 52 3.2.2 Convergence................................ 56 3.3 Shamov’s approach to Gaussian multiplicative chaos.............. 58 3.4 Kahane’s convexity inequality.......................... 61 3.5 Scale-invariant fields............................... 63 3.5.1 One-dimensional cone construction................... 63 3.5.2 Higher dimensional construction..................... 65 3.6 Multifractal spectrum............................... 68 3.7 Positive moments of Gaussian multiplicative chaos............... 70 3.8 Positive moments for general reference measures................ 74 3.9 Negative moments of Gaussian multiplicative chaos.............. 76 3.10 KPZ theorem................................... 80 2 3.11 Proof in the case of expected Minkowski dimension.............. 82 3.12 Duplantier–Sheffield’s KPZ theorem....................... 83 3.13 Applications of KPZ to critical exponents.................... 86 3.14 Exercises...................................... 87 4 Statistical physics on random planar maps 89 4.1 Fortuin–Kasteleyn weighted random planar maps............... 89 4.2 Conjectured connection with Liouville quantum gravity............ 94 4.3 Mullin–Bernardi–Sheffield’s bijection in the case of spanning trees...... 96 4.4 The loop-erased random walk exponent..................... 101 4.5 Sheffield’s bijection in the general case..................... 103 4.6 Infinite volume limit............................... 107 4.7 Scaling limit of the two canonical trees..................... 108 4.8 Exponents associated with FK weighted random planar maps........ 111 4.9 Exercises...................................... 114 5 Scale-invariant random surfaces 117 5.1 The Neumann boundary GFF as a random distribution............ 117 5.2 Covariance formula: the Neumann Green function............... 123 5.3 Neumann GFF as a stochastic process..................... 126 5.4 Relationship with other boundary conditions.................. 130 5.5 Semicircle averages and boundary Liouville measure.............. 134 5.6 Convergence of random surfaces......................... 137 5.7 (Thick) quantum wedges............................. 140 5.8 Exercises...................................... 145 6 SLE and the quantum zipper 148 6.1 SLE and GFF coupling; domain Markov property............... 148 6.2 Quantum length of SLE............................. 155 6.3 Proof of Theorem 6.9............................... 157 6.3.1 The capacity zipper............................ 161 6.3.2 The quantum zipper........................... 162 6.4 Uniqueness of the welding............................ 170 6.5 Slicing a wedge with an SLE........................... 171 A Appendix 177 A.1 SLE........................................ 177 3 Introduction These lecture notes are intended to be an introduction to the two-dimensional continuum Gaussian free field, Liouville quantum gravity and the general theory of Gaussian multiplica- tive chaos. Topics covered include • Chapter 1: the definition and main properties of the GFF; • Chapter 2: the construction of the Liouville measure, its non degeneracy and confor- mal covariance, and applications to the KPZ formula; • Chapter 3: a comprehensive exposition of the construction and properties of general Gaussian multiplicative chaos measures; • Chapter 4: an introduction to random planar maps - the discrete counterparts of Liouville quantum gravity - and bijections with trees; • Chapter 5: the Neumann GFF, the general notion of quantum surfaces, and in par- ticular the so-called "thick quantum wedge"; • Chapter 6: Sheffield’s quantum zipper theorem. The final topic above is quite technical, and readers are advised that it will be of most use to people who are actively working in this area. (While the arguments in Chapter 6 follow the main ideas of [She16a], some are new or simplified, and the overall structure of the proof has been rethought. The result is, we hope, that some of the key ideas are more transparent and will be helpful to others.) The theory is in full blossom and attempting to make a complete survey of the field would be hopeless, so quickly is it developing. Nevertheless, as the theory grows in complexity and applicability, it has appeared useful to summarise some of its basic and foundational aspects in one place, especially since complete proofs of some 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 expansive body of work on random planar maps and their connection with Liouville quantum gravity, the Brownian map, imaginary geometry, and so on. Having said that, a future version of these notes will also touch upon the critical regime for Gaussian multiplicative chaos, the rigorous construction of Liouville conformal field theory, the peanosphere or "mating of trees" description of Liouville quantum gravity surfaces, and Liouville Brownian motion. 4 What’s new? Beside a streamlined and essentially self-contained presentation, several arguments in these notes can be considered novel. • A new proof for the dimension of thick points of the GFF (especially the lower bound), see Exercise4 in Chapter2. • The construction of Liouville measure which combines the general arguments of [Ber17] together with a few additional features of the GFF - giving an even simpler proof of convergence in this case. • A reworking of Shamov’s approach to GMC, see Section 3.3, in a unified language with the rest of the chapter. • A somewhat simplified and explicit construction (inspired by [RV10a]) of scale-invariant log-correlated fields in all dimensions. This is used in particular to prove the existence of moments, positive and negative, in the most general set-up of Gaussian multiplicative chaos (with a general base measure). • Computation of loop-erased random walk exponent using KPZ (thanks to Xin Sun for the suggestion to include this). • A detailed definition of the Neumann GFF and thorough treatment of its analytic properties. Building on this, an explicit derivation of relationships between different variants of the GFF, and alternative Markov properties, see Section 5.4. • A new proof based on Girsanov’s theorem for one of the most delicate steps in the construction of the quantum zipper, see Lemma 6.20 5 Acknowledgements An original version of these notes was written by the first author, 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 Probability in Darmstadt, then in July 2016 for the Probability Summer School at Northwestern (for which the chapter on statistical physics on random planar maps was added), and one more time in December 2017 for the Lectures on Probability and Statistics (LPS) at ISI Kolkata. In all cases we thank the organisers (respectively, Christina Goldschmidt and Dmitry Beliaev; Volker Betz and Matthias Meiners; Antonio Auffinger and Elton Hsu; Arijit Chakrabarty, Manjunath Krishnapur, Parthanil Roy and especially Rajat Subhra Hazra), for their invitations and superb organisation. Thanks also to Benoit Laslier for agreeing to run the exercise sessions accompanying the lectures at the initial school. The Isaac Newton institute’s semester on Random Geometry, was another important influence and motivation for these notes, and we would like to thank the INI for its hospitality. In fact, this semester served as the second author’s initiation into the world of the Gaussian free field and Liouville quantum gravity. The resulting years of discussions together with the first author has led to the present expanded and revised