Two-Dimensional Random Geometry

I will give an introductory talk on the relationships among SLE, GFF, LQG, GMC and other TLAs in random geometry using the exploration perspective.  I will describe some of the basic building blocks for those new to the area.  My perspective is somewhat novel so I hope it will be of interest to specialists as well.

The conformal loop ensemble (CLE) is a natural conformally invariant probability measure on infinite collections of non-crossing loops, where each loop looks like an SLE_kappa curve. In this talk, we will focus on nested simple CLE_kappa for 8/3<kappa<=4 on both the annulus and the disk. Extending the results in Aru, Lupu and Sepulveda (2022) and Ang, Remy and Xin (2022), we compute joint distributions of extremal distances between nested loops in CLE_kappa on both the annulus and the disk.  In particular, for CLE_kappa on the disk, we show that the reweighted joint distribution of extremal distances between CLE_kappa loops can be expressed as linear combinations of first exit times and last hitting times of a one-dimensional Brownian motion. This is based on joint work with Nina Holden and Xin Sun.

Lattice Yang-Mills theories are important models in particle physics. They are defined on the d-dimensional lattice Z^d using a group of matrices of dimension N, and Wilson loop expectations are the fundamental observables of these theories. Recently, Cao, Park, and Sheffield showed that Wilson loop expectations can be expressed as sums over certain embedded bipartite maps of any genus. Building on this novel approach, we prove in the so-called strongly coupled regime: A rigorous formula in terms of embedded bipartite planar maps of Wilson loop expectations in the large N limit, in any dimension d. An exact computation of Wilson loop expectations in the large N limit, in dimension d=2. Similar results to the two aforementioned points were previously established by Chatterjee (2019) and Basu & Ganguly (2016), respectively. Our results offer simpler proofs and provide a new perspective on these significant quantities.   This work is a collaboration with Sky Cao and Jasper Shogren-Knaak.”

I will explain a version of the Polyakov-Alvarez formula on non-smooth surfaces in terms of Brownian loops. This was motivated by earlier attempts to understand Liouville quantum gravity (LQG) surfaces from a heuristic viewpoint in physics. The result implies that weighting certain regularized LQG surfaces by the Brownian loop soup partition function changes the central charges as expected. Based on joint work with Joshua Pfeffer and Scott Sheffield.

“Hamiltonian walks are self-avoiding random walks that visit all sites of a given lattice. We consider various configuration exponents of Hamiltonian walks drawn on random planar maps. Estimates from exact enumerations are compared with predictions based on the Knizhnik-Polyakov-Zamolodchikov (KPZ) relations, as applied to exponents on the regular hexagonal lattice. Astonishingly, when the maps are bipartite, a naive use of KPZ does not reproduce all the measured exponents, but an Ansatz may possibly account for the observed discrepancies.   We further study Hamiltonian cycles on various families of bipartite planar maps, which fall into two universality classes, with respective central charges c = -1 or c = -2. The first group comprises maps of fixed vertex valency p larger than 3, whereas the second group involves maps with mixed vertex valencies, as well as a so-called rigid case. For each class, a universal configuration exponent and a novel critical exponent associated with long-distance contacts along a Hamiltonian cycle are predicted from KPZ and the corresponding exponent on regular (hexagonal or square) lattices. This time, the KPZ predictions are numerically confirmed by exact enumeration results for p-regular maps, with p = 3, 4, 5, 6, 7, and for maps with mixed valencies (2,3) and (2,4).    The scaling limit of fully-packed systems thus poses intriguing unresolved questions from both the Liouville Quantum Gravity and the Schramm-Loewner Evolution perspectives.   Based on joint works with Ph. Di Francesco, O. Golinelli and E. Guitter.”

The large-scale geometry of discrete models of random curved surfaces, like the uniform triangulation of the sphere, has been studied
intensively. In this talk I will focus instead on random flat surfaces with the topology of the disk, in which all degrees of freedom are
located at the boundary of the disk. Addressing a recent question of Frank Ferrari in the physics context of JT gravity, I will present a
pair of discrete models of flat disks with controlled boundary length whose enumeration can be established via bijective means, involving planar random walks in one case and labeled planar maps in the other. These observations hint at the existence of a universal continuous flat random disk.

Many “local” models of random planar maps, obtained by gluing uniformly at random the edges of a collection of polygons with given degree, in order to obtain a topological sphere, are known to converge to the so-called Brownian sphere, a canonical model of random surface. However, it is possible to escape from the wide universality class of the Brownian sphere, either by considering models of random planar maps with long-range correlations, typically by endowing them with a statistical physics model at criticality, or by considering local models in which the variance of the face degrees distribution is infinite. In this talk, we investigate this second question by showing that random Boltzmann maps whose face degree distributions belong to a stable domain of attraction converge in the scaling limit to a random fractal object, which is called a stable carpet or a stable gasket depending on the value of the stable exponent. Indeed, a phase transition for the topology of these objects occur at the value 3/2 of the stable exponent. This completes earlier results by Le Gall-Miermont, who obtained a scaling limit result only up to extraction of subsequences. This is joint work with Nicolas Curien and Armand Riera.

We study the volume of rigid loop-O(n) quadrangulations with a boundary of length 2p in the critical non-generic regime. We prove that, as the half-perimeter p goes to infinity, the volume scales in distribution to an explicit random variable. This limiting random variable is described in terms of the multiplicative cascades of Chen, Curien and Maillard, or alternatively (in the dilute case) as the law of the area of a suitable unit-boundary quantum disc, as determined by Ang and Gwynne. Our arguments go through a classification of the map into several regions, where we rule out the contribution of bad regions to be left with a tractable portion of the map. One key observable for this classification is a Markov chain which explores the nested loops around a size-biased vertex pick in the map, making explicit the spinal structure of the discrete multiplicative cascade.

We will be interested in uniform random triangulations with fixed size and genus when the genus is proportional to the size. In this regime, typical triangulations enjoy expander properties which brings them close to random graph models. For example, we will see that graph distances become logarithmic in the size. This contrasts sharply with the fractal nature of random planar triangulations. Based on joint works with Baptiste Louf and Guillaume Chapuy.

 If an infinite Euclidean graph is sufficiently regular, then random walk converges to Brownian motion. To what extent does this hold for irregular graphs, e.g., random LQG-like graphs?  I will discuss joint work with Ewain Gwynne in which we identify a family of graphs upon which the trace of random walk, with a canonical choice of conductances, converges to Brownian motion. Our assumptions are deterministic and mild and the result applies to Delaunay triangulations with vertices sampled from a d-dimensional GMC measure. 

In recent years, a technique has been developed to compute the conformal radii of random domains defined by SLE curves, which is based on the coupling between SLE and Liouville quantum gravity (LQG). Compared to prior methods that compute SLE related quantities via its coupling with LQG, the crucial new input is the exact solvability of structure constants in Liouville conformal field theory. It appears that various percolation exponents can be expressed in terms of conformal radii that can be computed this way. This includes known exponents such as the one-arm and polychromatic two-arm exponents, as well as the backbone exponents, which is unknown previously. In this talk we will review this method using the derivation of the backbone exponent as an example, based on a joint work with Nolin, Qian, and Sun.

S-embeddings of planar weighted graphs do for the nearest-neighbor Ising model the same what harmonic (or Tutte’s) embeddings do for random walks on planar graphs. This construction was suggested a few years ago as a tool to describe small mesh sizes limit of fermionic observables beyond the setup of regular grids/isoradial graphs. The goals of this talk are (a) to overview what is known/has been understood for the Ising model on general s-embeddings since then and (b) to discuss several questions on this framework that remain widely open. 

We build a conformal field theory on multiply connected domains out of correlation functions of the Gaussian free field. A key ingredient is the derivation of the generalized Eguchi-Ooguri equations, which allow us to derive an explicit form of Ward’s equations. Our version of the Ward equation relates the insertion of a stress tensor in the computation of a correlation function to the application of a Lie derivative to the correlation function, along with a differential operator that accounts for the change in the Teichmuller modular parameters. By applying an appropriate operator product expansion to the Ward equation, we also produce an infinite family of martingale observables for an associated SLE process on the multiply connected domain.