Time | Speaker | Title | Resources | |
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10:00 to 11:00 | Lionel Levine |
Abelian networks and sandpile models In this mini-course, we explore particle systems with an “abelian property”: the order of certain interactions does not matter. Deepak Dhar [D] observed that many dynamical questions (what does this particle system do?) have a computational side (what can this network of automata compute?). The computational counterpart of the abelian property is a “least action principle”: the particles conspire to solve a certain optimization problem, by reaching stability in the most efficient possible way. We are interested in the phase transition between activity and fixation, and in universal properties of the “threshold state” that separates the two phases. The dynamical question “will this system fixate?” corresponds to the computational question “will this program halt?”. Alan Turing proved in 1936 that the latter question is undecidable in general. Hence, we should expect these systems to be hard, and there will be no completely satisfactory general theory; but questions in the neighborhood of an undecidable question are where the most fruitful mathematics lies! References: |
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11:00 to 11:15 | -- | Discussion | ||
11:15 to 11:45 | -- | Tea/coffee break | ||
11:45 to 12:45 | Rajat Subhra Hazra |
A PDE approach to scaling limit of random interface models on Z^d In this talk we shall discuss the scaling limit of Gaussian interface models, where the covariance structure comes from a discrete partial differential equation. In some models of random interfaces, the explicit description of the covariance is either lacking or sometimes difficult to derive. We suggest an approach through the approximation of solutions of continuum PDEs through discrete solutions by finite difference methods. We discuss the implications of such approximation results in the cases of the discrete Gaussian free field, the Membrane model, and the mixed model containing both gradient and Laplacian interaction. We derive the weak convergence in appropriate spaces, depending on the dimension of the lattice. This talk is based on joint and on-going works with Biltu Dan (ISI, Kolkata) and Alessandra Cipriani (TU, Delft). |
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12:45 to 14:00 | -- | Lunch | ||
14:00 to 15:00 | Wioletta Ruszel | Scaling limits of odometers in sandpile models | ||
15:00 to 15:30 | -- | Tea/ Coffee break | ||
15:30 to 16:30 | -- | Colloquium |
Time | Speaker | Title | Resources | |
---|---|---|---|---|
10:00 to 11:00 | Lionel Levine |
Abelian networks and sandpile models (Lecture 2) In this mini-course, we explore particle systems with an “abelian property”: the order of certain interactions does not matter. Deepak Dhar [D] observed that many dynamical questions (what does this particle system do?) have a computational side (what can this network of automata compute?). The computational counterpart of the abelian property is a “least action principle”: the particles conspire to solve a certain optimization problem, by reaching stability in the most efficient possible way. We are interested in the phase transition between activity and fixation, and in universal properties of the “threshold state” that separates the two phases. The dynamical question “will this system fixate?” corresponds to the computational question “will this program halt?”. Alan Turing proved in 1936 that the latter question is undecidable in general. Hence, we should expect these systems to be hard, and there will be no completely satisfactory general theory; but questions in the neighborhood of an undecidable question are where the most fruitful mathematics lies! References: |
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11:00 to 11:15 | -- | Discussion | ||
11:15 to 11:45 | -- | Tea/ Coffee break | ||
11:30 to 11:45 | -- | Tea/ Coffee break | ||
11:45 to 12:45 | Christopher Hoffman |
Frogs on Trees The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. I will discuss a collection of results that are joint with Tobias Johnson and Matthew Junge. These papers all discuss the frog model on regular trees. On infinite trees we consider the question of transience and recurrence. On finite depth trees we examine the time it takes before every vertex on the graph is visited. |
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12:45 to 14:00 | -- | Lunch | ||
14:00 to 15:00 | Vladas Sidoravicius |
Random walks in growing domains - recurrence vs transience We will discuss set of models where the random walk (or Brownian motion) moves in a restricted domain D(t), which itself evolves in time. This evolution could be independent of random walk evolution, but still affecting its motion, or evolution of D itself could be affected by random walk. |
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15:00 to 15:30 | -- | Tea/ Coffee break |