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Probability

Search Games and Optimal Kakeya Sets

Author: 
Yuval Peres
Date: 
Fri, Sep 6, 2013
Location: 
PIMS, University of British Columbia
Abstract: 
Search Games and Optimal Kakeya Sets: Yuval Peres Based on joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler

Random Maps 13

Speaker: 
Gregory Miermont
Date: 
Mon, Jun 25, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.

The critical points of lattice trees and lattice animals in high dimensions

Speaker: 
Yuri Mejia
Date: 
Fri, Jun 29, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
Lattice trees and lattice animals are used to model branched polymers. They are of interest in combinatorics and in the study of critical phenomena in statistical mechanics. A lattice animal is a connected subgraph of the d dimensional integer lattice. Lattice trees are lattice animals without cycles. We consider the number of lattice trees and animals with n bonds that contain the origin and form the corresponding generating functions. We are mainly interested in the radii of convergence of these functions, which are the critical points. In this talk we focus on the calculation of the first three terms of the critical points for both models as the dimension goes to infinity. This is ongoing work with Gordon Slade.

Correlation functions in the 2D Ising model via signed loops and paths

Speaker: 
Marcin Lis
Date: 
Thu, Jun 28, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
Using the combinatorial method for the 2D Ising model originating in the works of Sherman, Burgoyne and others we derive formulas for the correlation functions in terms of signed loops and paths. In the case of regular lattices we also identify the critical temperature for the phase transition in the long range behavior of these functions. Joint work with Wouter Kager and Ronald Meester.

Interacting Particle Systems 16

Speaker: 
Omer Angel
Date: 
Fri, Jun 29, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
Particles attempt to follow a simple dynamic (random walk, constant flow, etc) in some space (interval, line, cycle, arbitrary graph). Add a simple interaction between particles, and the behaviour can change completely. The resulting dynamical systems are far more complex than the ingredients suggest. These processes (interchange process, TASEP, sorting networks, etc) have diverse to many topics: growth processes, queuing theory, representation theory, algebraic combinatorics. I will discuss recent progress on and open problems arising from several models of interacting particle systems.

Random Maps 16

Speaker: 
Gregory Miermont
Date: 
Fri, Jun 29, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.

Interacting Particle Systems 15

Speaker: 
Omer Angel
Date: 
Thu, Jun 28, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
Particles attempt to follow a simple dynamic (random walk, constant flow, etc) in some space (interval, line, cycle, arbitrary graph). Add a simple interaction between particles, and the behaviour can change completely. The resulting dynamical systems are far more complex than the ingredients suggest. These processes (interchange process, TASEP, sorting networks, etc) have diverse to many topics: growth processes, queuing theory, representation theory, algebraic combinatorics. I will discuss recent progress on and open problems arising from several models of interacting particle systems.

Interacting Particle Systems 14

Speaker: 
Omer Angel
Date: 
Tue, Jun 26, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
Particles attempt to follow a simple dynamic (random walk, constant flow, etc) in some space (interval, line, cycle, arbitrary graph). Add a simple interaction between particles, and the behaviour can change completely. The resulting dynamical systems are far more complex than the ingredients suggest. These processes (interchange process, TASEP, sorting networks, etc) have diverse to many topics: growth processes, queuing theory, representation theory, algebraic combinatorics. I will discuss recent progress on and open problems arising from several models of interacting particle systems.

Laplacians and connections

Speaker: 
Rick Kenyon
Date: 
Mon, Jun 25, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
We discuss the Laplacian operator on vector bundles on graphs, in particular relating its determinant to the enumeration of "cycle-rooted spanning forests" which are combinatorial objects generalizing spanning trees.

Sharp Benefit-to-Cost Rules for the Evolution of Cooperation on Regular Graphs

Speaker: 
Yu-ting Chen
Date: 
Mon, Jun 25, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
We study two of the simple rules on finite graphs under the death-birth updating and the imitation updating discovered by Ohtsuki, Hauert, Lieberman, and Nowak [Nature, 441 (2006) 502-505]. Each rule specifies a payoff-ratio cutoff point for the magnitude of fixation probabilities of the underlying evolutionary game between cooperators and defectors. We view the Markov chains associated with the two updating mechanisms as voter model perturbations. Then we present a first-order approximation for fixation probabilities of general voter model perturbations on finite graphs subject to small perturbation in terms of the voter model fixation probabilities. In the context of regular graphs, we obtain algebraically explicit first-order approximations for the fixation probabilities of cooperators distributed as certain uniform distributions. These approximations lead to a rigorous proof that both of the rules of Ohtsuki et al. are valid and are sharp.
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