Schedule for: 22w5154 - Markov Chains with Kinetic Constraints and Applications

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

BIRS participants are welcome to gather in TCPL the day prior to workshop. Please note no BIRS staff will be present, and it's not an organized event.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this introductory talk we aim to overview several fundamental results in the study of kinetically constrained models (KCM) - one of the central themes of the workshop. We begin by treating bootstrap percolation, which can be viewed as a simpler but vital prerequisite. After duly defining both bootstrap percolation and KCM models, we review common results and techniques in the field through the lens of classical models such as j-neighbour, East and North-East in two dimensions at equilibrium. We discuss some of the basic notions of universality, in order to understand how various models position in the bootstrap percolation and KCM universe.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Bootstrap percolation and kinetically constrained models (KCMs) are two classes of models of statistical mechanics which are both defined through constraints that must be satisfied to allow a state change. These constraints can be sorted into three universality classes with different behaviors: supercritical, critical and subcritical. The most complex and interesting class is the critical class. The aim of this talk is to present the universality classification of the constraints inside the critical class for bootstrap percolation and for KCMs, as well as the differences of behavior between the two. The audience will be expected to have listened to the talk of Ivailo Hartarsky presenting the definitions of the models as well as the supercritical and subcritical classes.

In the late 20th century, statistical physicists introduced a chemical reaction model called ballistic annihilation. In it, particles are placed randomly throughout the real line and then proceed to move at independently sampled velocities. Collisions result in mutual annihilation. Many results were inferred by physicists, but it wasn’t until recently that mathematicians joined in. I will describe my trajectory through this model. Expect tantalizing open questions.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Brownian net is the diffusive scaling limit of systems of one-dimensional branching and coalescing random walks, started from each point in space and time. The branching-coalescing point set is a Markov process that can can loosely be described as coalescing and branching Brownian motions -but with infinite branching rate. A more precise definition involves the Brownian net. In my lectures, I will look at these objects motivated by the one-dimensional 1-facilitated Frederickson-Anderson model and a closely related model studied by Neuhauser and Sudbury. Heuristic arguments indicate that at high densities, the empty sites of these models effectively behave like coalescing random walks that in addition branch with a small rate. This leads to a precise conjecture about their scaling limit, which is work in progress with Rongfeng Sun.

The ALE (Aggregate Loewner Evolution) models describe growing random clusters on the complex plane, built by iterated composition of random conformal maps. A striking feature of these models is that they can be used to define natural off-lattice analogues of several fundamental discrete models, such as Eden or Diffusion Limited Aggregation, by tuning the correlation between the defining maps appropriately. In this talk I will discuss shape theorems and fluctuations of ALE clusters, which include Hastings-Levitov clusters as particular cases, in the subcritical regime. Based on joint work with James Norris (Cambridge) and Amanda Turner (Lancaster).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Kob-Anderesen is a fundamental example of a cooperative kinetically constrained lattice gas (KCLG). KCLGs are conservative particle systems, where particles jump between sites, under certain constraints which prohibit transitions from occurring when the neighborhood is dense. Such a system is called cooperative when, roughly, its evolution is determined by the collective motion of a large number of particles, diverging near criticality. We will describe the diffusive scaling of the Kob-Andersen model and the consequences of its cooperative nature.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Brownian net is the diffusive scaling limit of systems of one-dimensional branching and coalescing random walks, started from each point in space and time. The branching-coalescing point set is a Markov process that can can loosely be described as coalescing and branching Brownian motions -but with infinite branching rate. A more precise definition involves the Brownian net. In my lectures, I will look at these objects motivated by the one-dimensional 1-facilitated Frederickson-Anderson model and a closely related model studied by Neuhauser and Sudbury. Heuristic arguments indicate that at high densities, the empty sites of these models effectively behave like coalescing random walks that in addition branch with a small rate. This leads to a precise conjecture about their scaling limit, which is work in progress with Rongfeng Sun.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Brownian net is the diffusive scaling limit of systems of one-dimensional branching and coalescing random walks, started from each point in space and time. The branching-coalescing point set is a Markov process that can can loosely be described as coalescing and branching Brownian motions -but with infinite branching rate. A more precise definition involves the Brownian net. In my lectures, I will look at these objects motivated by the one-dimensional 1-facilitated Frederickson-Anderson model and a closely related model studied by Neuhauser and Sudbury. Heuristic arguments indicate that at high densities, the empty sites of these models effectively behave like coalescing random walks that in addition branch with a small rate. This leads to a precise conjecture about their scaling limit, which is work in progress with Rongfeng Sun.

Discovered by Aldous, Diaconis and Shahshahani in the context of card shuffling, the cutoff phenomenon is a remarkable phase transition in the convergence to equilibrium of certain Markov chains. Despite the accumulation of several examples, a general theory is still missing, and identifying the exact mechanisms underlying this phenomenon constitutes one of the most fundamental problems in the area of mixing times. After a self-contained introduction to this question, I will present a new approach based on entropy and curvature, and use it to establish cutoff for a broad class of Markov chains.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Markov chain Monte Carlo is a widely used class of algorithms for sampling from high-dimensional probability distributions, both in theory and in practice. While simple to implement, analyzing the rate of convergence to stationarity, i.e. the "mixing time", remains a challenging problem in many settings. We introduce a new technique to bound mixing times called "spectral independence", which says that certain pairwise correlation matrices all have bounded spectral norm. This surprisingly powerful technique originates in the emerging study of high-dimensional expanders, and has allowed us to "unify" nearly all existing approaches to approximate counting and sampling by building new connections with other areas, including statistical physics, geometry of polynomials, functional analysis, and more. Through these connections, several long-standing open problems have recently been answered, including counting bases of matroids and optimal mixing of the Glauber dynamics/Gibbs sampler up to the algorithmic phase transition threshold. Based on several joint works with Dorna Abdolazimi, Nima Anari, Zongchen Chen, Shayan Oveis Gharan, Eric Vigoda, and Cynthia Vinzant.

The facilitated exclusion process is a degenerate kinetically constrained lattice gas, in which particles can hop to an empty neighboring site provided that another neighboring site is also occupied. Because of the kinetic constraint, the system in dimension 1 exhibits two phases, a frozen phase at density less than 1/2, and a diffusive phase at density larger than 1/2. For this reason, standard entropic tools cannot be used to straightforwardly derive its macroscopic behavior, in particular its hydrodynamic limit. To workaround this issue, we exploit a classical mapping between exclusion and zero-range processes, and deduce from the hydrodynamic limit of the zero-range process that of the exclusion process. Despite the lack of smoothness of the various macroscopic limits, thanks to the attractiveness of the zero-range process, this allows to derive the hydrodynamic limits in both the symmetric and asymmetric case. Based on JW with O. Blondel, M. Sasada, M. Simon and L. Zhao

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Brownian net is the diffusive scaling limit of systems of one-dimensional branching and coalescing random walks, started from each point in space and time. The branching-coalescing point set is a Markov process that can can loosely be described as coalescing and branching Brownian motions -but with infinite branching rate. A more precise definition involves the Brownian net. In my lectures, I will look at these objects motivated by the one-dimensional 1-facilitated Frederickson-Anderson model and a closely related model studied by Neuhauser and Sudbury. Heuristic arguments indicate that at high densities, the empty sites of these models effectively behave like coalescing random walks that in addition branch with a small rate. This leads to a precise conjecture about their scaling limit, which is work in progress with Rongfeng Sun.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.