Workshop Fall 2022

We are organizing an in-person workshop on Random Geometry and Statistical Physics at UPenn from Friday October 28 to Sunday October 30. If you are interested in participating, please register here. The event is supported by the NSF Career award 2046514.

We will try our best to provide financial support to junior researchers, especially graduate students. If you are in need of financial support please send a request with your CV to nina.holden@nyu.edu. You can submit your application for funding at any time, but it will only get full consideration if you send us your CV by September 20.

Information for parking and restaurant recommendation near Penn.

Speakers

Morris Ang (Columbia)

Sky Cao (IAS)

Hugo Falconet (NYU)

Shirshendu Ganguly (Berkeley)

Richard Kenyon (Yale)

Peter Lin (Stony Brook)

Jinwoo Sung (U Chicago)

Balint Virag (U Toronto)

Catherine Wolfram (MIT)

Baojun Wu (Aix-Marseille University)

Pu Yu (MIT)

Zijie Zhuang (U Penn)

Schedule

There will be 7 long talks (45 minutes) and 5 short talks (25 minutes), the latter by PhD students.

Friday Oct 28

Venue: Room 218, Houston Hall, 3417 Spruce St.

1:00 pm Registration (coffee&snacks)

1:30 pm Rick Kenyon (45 min)

2:45 pm Catherine Wolfram (25 min)

3:15 pm Coffee break

4:00 pm Peter Lin (45 min)

5:00 pm Pu Yu (25 min)

Saturday Oct 29

Venue: Room A2, David Rittenhouse Laboratory, 209 S 33rd St.

09:00 am Continental breakfast

10:00 am Shirshendu Ganguly (45 min)

11:15 am Hugo Falconet (45 min)

12:00 pm Lunch break

2:00 pm Jinwoo Sung (25 min)

2:45 pm Zijie Zhuang (25 min)

3:15 pm Coffee break

4:00 pm Morris Ang (45 min)

5:00 pm Baojun Wu (25 min)

Sunday Oct 30

Venue: Room A2, David Rittenhouse Laboratory, 209 S 33rd St.

09:00 am Continental breakfast

10:00 am Baliant Virag (45 min)

11:15 am Sky Cao (45 min)

Abstracts

Morris Ang: Liouville conformal field theory and the quantum zipper

Sheffield introduced a seminal coupling between Liouville quantum gravity (LQG) and Schramm-Loewner evolution (SLE), which makes sense for all three phases of SLE: simple, self-intersecting, and space-filling. The simple phase corresponds to conformal welding of LQG [Sheffield], and the self-intersecting phase arises as a mating of independent ``forested lines’’ (forests of looptrees of LQG disks) [Duplantier-Miller-Sheffield]; these phenomena are called quantum zippers. We show that in the space-filling phase the coupling is also a quantum zipper, arising from a mating of correlated continuum random trees introduced in [DMS]. We give the LQG-SLE coupling an interpretation as dynamics in Liouville conformal field theory (LCFT) and explain several consequences. Firstly, we prove the boundary BPZ equations of LCFT via mating-of-trees. This is an important ingredient in solving boundary LCFT structure constants in future work with Remy, Sun and Zhu. Secondly, it leads to further developments on conformal welding in LCFT in future work with Yu.

Sky Cao: Exponential decay of correlations in finite gauge group lattice gauge theories

Lattice gauge theories with finite gauge groups are statistical mechanical models, very much akin to the Ising model, but with some twists. In this talk, I will describe how to show exponential decay of correlations for these models at low temperatures. This is based on joint work with Arka Adhikari.

Hugo Falconet: Liouville quantum gravity from random matrix dynamics

The Liouville quantum gravity measure is a properly renormalized exponential of the 2d GFF. In this talk, I will explain how it appears as a limit of natural random matrix dynamics: if (U_t) is a Brownian motion on the unitary group at equilibrium, then the measures $|det(U_t - e^{i theta}|^gamma dt dtheta$ converge to the 2d LQG measure with parameter $gamma$, in the limit of large dimension. This extends results from Webb, Nikula and Saksman for fixed time. The proof relies on a new method for Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols, which extends to multi-time settings. I will explain this method and how to obtain multi-time loop equations by stochastic analysis on Lie groups. Based on a joint work with Paul Bourgade.

Shirshendu Ganguly: Some fractal dimensions in models of random planar geometry

In last passage percolation models predicted to lie in the Kardar-Parisi-Zhang (KPZ) universality class, geodesics are oriented paths moving through random noise accruing maximum weight. These extremal paths exhibit the key phenomenon of coalescence by typically merging into a few ''highways".

In this talk we will survey recent progress in understanding the random fractal geometry exhibited by such highways, in particular through the set of exceptional points where they bifurcate. This also provides insight about the coupling structure of the geodesic weights as their endpoints are varied.

Rick Kenyon: Dimers and 3-webs

This is joint work with Haolin Shi. Webs are bipartite, trivalent, planar graphs. They appear naturally in the representation theory of SL_3: Greg Kuperberg showed that “reduced” 3-webs form a basis for invariant functions in tensor products of SL_3-representations. Webs and reduced webs also occur naturally in the triple-dimer model. We show how various topological types of reduced webs in the triple-dimer model on a (potentially large) planar graph can be computed using Postnikov’s boundary measurement matrix. In the scaling limit of the triple dimer model, we compute probabilities of reduced webs as conformally invariant functions of the boundary points. Conjectural connections with level curves of the complex-valued GFF will be discussed.

Peter Lin: Conformal Structures on Stochastic Subdivision Rules

We consider "conformal" parameterizations of random fractal spaces $X$ arising as limits of certain stochastic subdivision rules. One motivation comes from the field of random geometry, where it is an important and difficult problem to understand this parameterization when $X$ arises from limits of random planar maps. Deterministic versions of our model, and the analogous questions relating to conformal parameterizations, are closely related to Thurston's topological characterization of rational maps.

In all the settings mentioned above, a key difficulty is in proving that the fractal approximations do not degenerate in the complex analytic sense. We overcome this difficulty in our setting by proving a contraction inequality for probabilistic iteration on a variant of the universal Teichmuller space. This inequality also provides a different perspective on random quasiconformal map models considered by Astala-Rohde-Saksman-Tao and Ivrii-Markovic.

Jinwoo Sung: The Minkowski content measure for the Liouville quantum gravity metric

A Liouville quantum gravity (LQG) surface is a natural random two-dimensional surface, initially formulated as a random measure space and later as a random metric space. In this talk, I will discuss how the LQG measure can be recovered as the Minkowski content with respect to the LQG metric, answering a question of Gwynne and Miller (2019). Our primary tool is the continuum mating-of-trees theory for space-filling SLE. This proof also leads to a Hölder continuity result for space-filling SLE with respect to the LQG metric. This is joint work with Ewain Gwynne.

Balint Virag: The planar stochastic heat equation and the directed landscape

The planar SHE describes heat flow or random polymers on an inhomogeneous surface. It is a finite-temperature version of planar first passage percolation such as the Eden growth model. It is the first model with plane symmetries for which we can show convergence to the directed landscape. The methods use a Skorokhod integral representation and Gaussian multiplicative chaos on path space. Joint work with Jeremy Quastel and Alejandro Ramirez.

Catherine Wolfram: Large deviations for the 3D dimer model

A dimer tiling of $\m Z^d$ is a collection of edges such that every vertex is covered exactly once. A lot has been understood about the dimer model in dimension $d=2$ using tools and exact formulas (e.g. the height function representation of a tiling or the Kasteleyn determinant formula) that are specific to dimension 2. In 2000, Cohn, Kenyon, and Propp showed that 2D dimer tilings satisfy a large deviations principle. Loosely speaking, this means that given a simply connected region $R$ in the plane and a sequence of finer and finer grid regions $R_n$ approximating $R$, random tilings of $R_n$ converge to a deterministic “limit shape.” In joint work with Nishant Chandgotia and Scott Sheffield, we prove an analogous large deviations principle for dimers in 3D, with new methods and without exact solvability. In this talk, I will explain how to formulate the large deviations principle in 3D, show simulations, and time permitting describe some (seemingly simple) related open problems for dimers in higher dimensions.

Baojun Wu: Liouville quantum gravity and matrix model

In this talk, I will define a quantum sphere with n holes. Conditioning on its quantum boundary length, the area distribution can be computed explicitly by the Levy process strategy. I will illuminate the relationship between this object and the matrix model, and give a geometric interpretation of genus 0 string equation. This is a joint work with Sun and Xu.

Pu Yu: Reversibility of chordal $SLE_\kappa(\underline{\rho})$

Chordal SLE$_\kappa(\underline{\rho})$ is a natural variant of chordal SLE curve. Recently, Zhan gave an explicit description of the law of the time reversal of SLE$_\kappa(\underline{\rho})$ when all force points lies on the same sides of the origin, and conjectured that a similar result holds in general. In a recent work with Xin Sun we prove this conjecture. For the non-boundary hitting case, we use the martingale observable from Schramm-Wilson, while for the boundary touching case, which appears hard to access with existing tools, we rely on the conformal welding of LQG surfaces. In particular, the proof is based on methods from And-Holden-Sun and an extension of a recent work on quantum triangles jointly with Morris Ang.

Zijie Zhuang: Crossing probabilities in critical planar percolation

In this talk, we consider critical planar site percolation on the triangular lattice. I will review some recent progress in improving the estimates of crossing probabilities. I will also talk about a proof of the annulus crossing probabilities predicted by Cardy. This is based on a joint work with Du, Gao, Li, and a work in progress with Sun and Xu.