Workshop

Under suitable boundary conditions, scaling limits of random tilings present surprising geometric features: one observes definite deterministic and disordered (or frozen and liquid) limit configurations with interesting geometric properties.

In this talk we will show how with properties of the Beltrami equation it is possible to understand, for all dimer models and even beyond, the geometry of the boundary between the frozen/deterministic and liquid/random phase. It turns out that in this class the geometry frozen boundaries is universal, i.e. independent of the specific model.

The talk is based on a joint work with E. Duse, I. Prause and X. Zhong.

The chordal Loewner equation provides a correspondence between real-valued functions, called driving functions, and certain growing 2-dimensional sets, called hulls.  In this talk, we will consider the generalization to complex-valued driving functions, which was first studied by Huy Tran.  We will discuss some key differences between the hulls in the complex-valued setting and those in the real-valued setting, including the question of the phase transition from simple-curve hulls.  This is joint work with Jeffrey Utley.

I will talk about some recent progress on the problem of constructing 3D Euclidean Yang-Mills theories. This is based on joint work with Sky Cao.

We consider the Loewner evolution whose driving function is $W_t = B_t^1 + i B_t^2$, where $(B^1,B^2)$ is a pair of Brownian motions with a given covariance matrix. This model can be thought of as a generalization of Schramm-Loewner evolution (SLE) with complex parameter values. We show that our Loewner evolutions behave very differently from ordinary SLE. For example, if neither $B^1$ nor $B^2$ is identically equal to zero, then the complements of the Loewner hulls are not connected. We also show that our model exhibits three phases analogous to the phases of SLE: a phase where the hulls have zero Lebesgue measure, a phase where points are swallowed but not hit by the hulls, and a phase where the hulls are space-filling. The phase boundaries are expressed in terms of the signs of explicit integrals. These boundaries have a simple closed form when the correlation of the two Brownian motions is zero. Based on joint work with Josh Pfeffer, with simulations by Minjae Park.

We construct and study conformally invariant, log-correlated Gaussian random fields on compact Riemannian manifolds of general even dimension uniquely defined through its covariance kernel given as inverse of the Graham-Jenne-Mason-Sparling (GJMS) operator. The corresponding Gaussian Multiplicative Chaos is a generalization to the n-dimensional case of the celebrated Liouville Quantum Gravity measure in dimension two. Finally, we study the Polyakov–Liouville measure on the space of distributions on M induced by the copolyharmonic Gaussian field, providing explicit conditions for its finiteness and computing the conformal anomaly.

Liouville Brownian motion was introduced as a canonical diffusion process under Liouville quantum gravity. It is constructed as a time change of 2-dimensional Brownian motion by the continuous additive functional associated with a Liouville measure, through a regularizing approximation procedure of the Gaussian free field. In this talk, we are concerned with the question whether one can construct Liouville Brownian motion directly from the Liouville measure. We will present a discrete approximation scheme that in fact works for more general time changed Brownian motions.

We obtain exact formulae for three basic quantities in random conformal geometry that depend on the modulus of an annulus. The first is for the law of the  modulus of the Brownian annulus describing the scaling limit of uniformly sampled planar maps with annular topology, which is as  predicted from the ghost partition function in bosonic string theory. The second is for the law of the modulus of the annulus bounded by a loop of a simple conformal loop ensemble (CLE) on a disk   and the disk boundary.  The formula is as conjectured from the partition function of the $O(n)$ loop model on the annulus derived by Cardy (2006). The third is for the annulus partition function of the $\SLE_{8/3}$ loop introduced by Werner (2008). It again confirms a prediction of Cardy (2006). The physics  principle underlying  our proofs is that  2D quantum gravity coupled with conformal matters can be decomposed into three conformal field theories (CFT): the matter CFT, the Liouville CFT, and the ghost CFT. At the technical level, we rely on two types of integrability in Liouville quantum gravity, one from the scaling limit of random planar maps, the other from the  Liouville CFT. We expect our method to be applicable to a variety of questions related to the random moduli of non-simply-connected random surfaces. Joint work with Morris Ang and Guillaume Remy.

Multiple SLEs describe several random interfaces consistent with conformal symmetries. I will implement a version of conformal field theory constructed from background charge modifications of Gaussian free field and insertion of N-leg operators with screening to show that this version produces a collection of martingale-observables for commuting multiple SLEs. I will explain how this theory is related to its classical limit counterpart, "Loewner Dynamics for Real Rational Functions and the Multiple SLE(0) Process," presented by Tom Alberts in "Introductory Workshop: The Analysis and Geometry of Random Spaces."

Based on joint work with Tom Alberts and Nikolai Makarov.

We consider Jordan curves passing through a given set of n points with the property such that each edge is the hyperbolic geodesic in the complement of the rest of the curve. Those curves are related to the minimizers of the Loewner energy and give a specific complex projective structure on the n-punctured sphere with real and parabolic holonomy around the punctures. Similar to a result of Takhtajan-Zograf, we also obtain a new type of accessory parameters and show that they can be expressed as differentials of the Loewner energy.

This is based on the joint work with Don Marshall and Steffen Rohde.