Past Probability Seminars Spring 2020: Difference between revisions
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interaction''' | interaction''' | ||
:Abstract: I will introduce a model of a finite number of competing particles on R. Real-life phenomena that could be modeled this way includes the | :Abstract: I will introduce a model of a finite number of competing particles on R. Real-life phenomena that could be modeled this way includes the evolution of stocks in a market, or herding behavior of animals. Given a particle configuration, the center of mass of the particles is computed | ||
evolution of stocks in a market, or herding behavior of animals. Given a | by simply averaging the particle locations. The evolution is a continuous time Markov jump process: given a configuration and thus the center of mass, each particle jumps with a rate that depends on the | ||
particle configuration, the center of mass of the particles is computed | particle's relative position compared to the center of mass. Those left behind have a higher jump rate than those in front of the center of mass. When a jump of a particle occurs, the jump length is chosen | ||
by simply averaging the particle locations. The evolution is a | independently of everything from a positive distribution. Hence we see that the dynamics tries to keep the particles together. | ||
continuous time Markov jump process: given a configuration and thus the | |||
center of mass, each particle jumps with a rate that depends on the | |||
particle's relative position compared to the center of mass. Those left | |||
behind have a higher jump rate than those in front of the center of | |||
mass. When a jump of a particle occurs, the jump length is chosen | |||
independently of everything from a positive distribution. Hence we see | |||
that the dynamics tries to keep the particles together. | |||
The main point of interest is the behavior of the model as the number of | The main point of interest is the behavior of the model as the number of particles goes to infinity. We first heuristically wrote up a differential equation on the evolution of particle density. I will | ||
particles goes to infinity. We first heuristically wrote up a | explain the heuristics, and show traveling wave solutions in a few cases. I will also present a surprising connection to extreme value statistics. Then I will briefly sketch a hydrodynamic argument which proves that the evolution of the system indeed converges to that governed by the differential equation. | ||
differential equation on the evolution of particle density. I will | |||
explain the heuristics, and show traveling wave solutions in a few | |||
cases. I will also present a surprising connection to extreme value | |||
statistics. Then I will briefly sketch a hydrodynamic argument which | |||
proves that the evolution of the system indeed converges to that | |||
governed by the differential equation. | |||
(Joint work with Miklós Rácz and Bálint Tóth) | (Joint work with Miklós Rácz and Bálint Tóth) | ||
Revision as of 14:05, 20 January 2011
Spring 2011
Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. If you would like to receive announcements about upcoming seminars, please visit this page to sign up for the email list.
Monday, January 24, 2:25PM, B129 Sunil Chhita (Brown University)
UNUSUAL TIME AND PLACE
- Title: Particle Systems arising from an Anti-ferromagnetic Ising Model
- Abstract: We present a low temperature anisotropic anti-ferromagnetic 2D Ising model
through the guise of a certain dimer model. This model also has a bijection with a one-dimensional particle system equipped with creation and annihilation. We can find the exact phase diagram, which determines two significant values (the independent and critical value). We also highlight some of the behavior of the model in the scaling window at criticality and at independence.
Thursday, February 10, Alex Bloemendal (Toronto)
- Title: TBA
- Abstract: TBA
Thursday, February 24, Márton Balázs (Technical University Budapest)
- Title: Modelling flocks and prices: jumping particles with an attractive
interaction
- Abstract: I will introduce a model of a finite number of competing particles on R. Real-life phenomena that could be modeled this way includes the evolution of stocks in a market, or herding behavior of animals. Given a particle configuration, the center of mass of the particles is computed
by simply averaging the particle locations. The evolution is a continuous time Markov jump process: given a configuration and thus the center of mass, each particle jumps with a rate that depends on the particle's relative position compared to the center of mass. Those left behind have a higher jump rate than those in front of the center of mass. When a jump of a particle occurs, the jump length is chosen independently of everything from a positive distribution. Hence we see that the dynamics tries to keep the particles together.
The main point of interest is the behavior of the model as the number of particles goes to infinity. We first heuristically wrote up a differential equation on the evolution of particle density. I will explain the heuristics, and show traveling wave solutions in a few cases. I will also present a surprising connection to extreme value statistics. Then I will briefly sketch a hydrodynamic argument which proves that the evolution of the system indeed converges to that governed by the differential equation.
(Joint work with Miklós Rácz and Bálint Tóth)
Wednesday, March 2, Alan Hammond (Oxford)
UNUSUAL TIME
- Title: TBA
- Abstract: TBA
Wednesday, April 6, Richard Sowers (University of Illinois at Urbana-Champaign)
UNUSUAL TIME
- Title: TBA
- Abstract: TBA
Thursday, April 14, Janos Englander (University of Colorado - Boulder)
- Title: TBA
- Abstract: TBA
Thursday, April 28, John Fricks (Penn State)
- Title: TBA
- Abstract: TBA