Online Schedule

• April 27

  Per von Soosten (Harvard)

  Localization and delocalization for ultrametric random matrices

  Abstract: We consider a Dyson-hierarchical analogue of power-law random band matrices with Gaussian entries. The model can be constructed recursively by alternating between averaging independent copies of the matrix and running Dyson Brownian motion. We use this to map out the localized regime and a large part of the delocalized regime in terms of local statistics and eigenfunction decay. Our method extends to a part of the delocalized regime in which the model has a well-defined infinite-volume limit with Holder-continuous spectral measures. This talk is based on joint work with Simone Warzel.

• May 4

  Amol Aggarwal (Harvard)

  Pure States in the Ferroelectric Six-Vertex Model

  Abstract: The classification and analysis of pure states (translation-invariant, ergodic Gibbs measures) for statistical mechanical systems is a fundamental question in mathematical physics. In this talk we investigate this question for the six-vertex model in its ferroelectric phase. We will see that the situation here differs considerably from its more well-known counterpart for dimer models. In particular, for the ferroelectric six-vertex model there now exist non-trivial regions of non-existence and new families of highly anisotropic pure states exhibiting Kardar-Parisi-Zhang (KPZ) fluctuations.

• May 11

  Marianna Russkikh (MIT)

  Dimers and embeddings

  Abstract: One of the main questions in the context of the universality and conformal invariance of a critical 2D lattice model is to find an embedding which geometrically encodes the weights of the model and that admits "nice" discretizations of Laplace and Cauchy-Riemann operators. We establish a correspondence between dimer models on a bipartite graph and circle patterns with the combinatorics of that graph. We describe how to construct a 't-embedding' (or a circle pattern) of a dimer planar graph using its Kasteleyn weights, and develop a relevant theory of discrete holomorphic functions on t-embeddings; this theory unifies Kenyon's holomorphic functions on T-graphs and s-holomorphic functions coming from the Ising model. We discuss a concept of 'perfect t-embeddings' of weighted bipartite planar graphs. We believe that these embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model. Based on: joint work with R. Kenyon, W. Lam, S. Ramassamy; and joint work with D. Chelkak, B. Laslier.

  Slides

Schedule

• February 10

  Two speakers:

  3:00-4:00 in 2-132:

  Guillaume Remy (Columbia)

  A probabilistic construction of conformal blocks for Liouville CFT

  Abstract: Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced by A. Polyakov in the context of string theory. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will present a probabilistic construction of the conformal blocks of Liouville CFT on the torus. These are the fundamental objects that allow to understand the integrable structure of CFT using the conformal bootstrap equation. We will also mention the connection with the AGT correspondence. Based on joint work with Promit Ghosal, Xin Sun, and Yi Sun.

  ---

  4:15-5:15 in 2-147:

  Nishant Chandgotia (Hebrew University)

  The Space of Invariant Probability Measures on Dimer tilings of Z^d

  Abstract: Given a finite set of rectangular parallelepipeds, T, let us denote by X(T), tilings of Z^d by elements of T. In this talk we will investigate the space of probability measures on X(T) which are invariant under translations. In recent work by the speaker and Tom Meyerovitch, it was realised that the complexity of this space is intimately linked with the growth rate of the number of tilings of boxes with closed boundary conditions as compared to tilings with free boundary conditions (also called the entropy). In this talk I will present some recent work which resolves this question for dimer tilings and develops tools in the direction of a more comprehensive theory for dimer tilings in higher dimensions.

• February 17

  President's day, no seminar

• February 24

  Nathanael Berestycki (Vienna)

  The monomer-dimer model and the Neumann GFF

  Abstract: We study the dimer model in which particles on the boundary are allowed to form monomers with some fixed weight z > 0 called the monomer fugacity. A natural question is to show conformal invariance in the scaling limit of the associated height function. We prove that on the upper half plane and on infinite strips the scaling limit is given by a Gaussian free field with Neumann (or free) boundary conditions on the part of the boundary containing the monomers, for arbitrary fixed fugacity z > 0.

  The starting point of the proof is a bijection due to Giuliani, Jauslin and Lieb to a dimer configuration on a graph which is however nonbipartite. Applying classical Kasteleyn theory we are led to the analysis of a "random walk" on a graph but where some transitions have negative rates. Nevertheless, using fine properties of an auxiliary random walk (which boil down to describing precisely the oscillations of its potential kernel) we are able to give a meaning and analyse the scaling limit of its Green function, eventually leading us to the result.

• March 2

  No speaker

• March 9

  Richard Kenyon (Yale)

  The five-vertex model

  Abstract: The five vertex model is a probability measure on monotone nonintersecting lattice paths, where each corner of each path gets weight r. It generalizes the lozenge tiling model which is the case r=1. There is a rigorous "Bethe Ansatz" solution, leading to explicit limit shapes for tilings of bounded regions such as the boxed plane partition.

  Fluctuations around the limit shape are conjecturally given by a Gaussian free field with a spatially-varying stiffness.

  We discuss a recent approach to getting local statistics in the model.

The in-person seminars below were cancelled but remain on the website for archival purposes. The current online schedule appears at the top of the page.

• March 16

  ***Postponed***

  Per von Soosten (Harvard)

  Localization and delocalization for ultrametric random matrices

  Abstract: We consider a Dyson-hierarchical analogue of power-law random band matrices with Gaussian entries. The model can be constructed recursively by alternating between averaging independent copies of the matrix and running Dyson Brownian motion. We use this to map out the localized regime and a large part of the delocalized regime in terms of local statistics and eigenfunction decay. Our method extends to a part of the delocalized regime in which the model has a well-defined infinite-volume limit with Holder-continuous spectral measures. This talk is based on joint work with Simone Warzel.

• March 23

  Spring break, no seminar

• March 30

  ***Postponed***

  Remi Rhodes (Aix-Marseille)

• April 6

  ***Postponed***

  Jacobo Borga (Zurich)

• April 13

  ***Postponed***

  Antonio Auffinger (Northwestern)

• April 20

  Patriot's day, no seminar

• April 27

  ***Postponed***

  Amol Aggarwal (Harvard)

• May 4

  ***Postponed***

  Marianna Russkikh (MIT)

• May 11

  ***Postponed***

  Vadim Kaloshin (Maryland)