# Mathematics

## Interacting Particle Systems 7

Speaker:
Omer Angel
Date:
Tue, Jun 19, 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 7

Speaker:
Gregory Miermont
Date:
Thu, Jun 14, 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.

## Local relaxation for FA-1f out of equilibrium

Speaker:
Oriane Blondel
Date:
Tue, Jun 12, 2012
Location:
PIMS, University of British Columbia
Conference:
PIMS-MPrime Summer School in Probability
Abstract:
We consider the Fredrickson and Andersen one spin facilitated model (FA1f)on Z^d. Each site with rate one refreshes its occupation variable to a filled or to an empty state with probability p or q=1-p respectively, provided that at least one of its nearest neighbours is empty. We study the non-equilibrium dynamics started from an initial distribution $\nu$ different from the stationary product p-Bernoulli measure $\mu$, which has enough zeros. We then prove local convergence to equilibrium when the vacancy density q is above a proper threshold. The convergence is exponential (d=1) or stretched exponential (d>1). Joint work with N. Cancrini, F. Martinelli, C. Roberto and C. Toninelli.

## Random Maps 6

Speaker:
Gregory Miermont
Date:
Tue, Jun 12, 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 6

Speaker:
Omer Angel
Date:
Tue, Jun 12, 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

## Finite range decomposition of Gaussian fields

Speaker:
Roland Bauerschmidt
Date:
Tue, Jun 12, 2012
Location:
PIMS, University of British Columbia
Conference:
PIMS-MPrime Summer School in Probability
Abstract:
I will show how to decompose the Gaussian free field on a (weighted) graph into a sum of finite range Gaussian fields, which are smoother than the original field and have spatially localized correlations.

## Interacting Particle Systems 5

Speaker:
Omer Angel
Date:
Mon, Jun 11, 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.

## Longest convex chains

Speaker:
Gergely Ambrus
Date:
Mon, Jun 11, 2012
Location:
PIMS, University of British Columbia
Conference:
PIMS-MPrime Summer School in Probability
Abstract:
A classical problem in probability is to determine the length of the longest increasing subsequence in a random permutation. Geometrically, the question can be formulated as follows: given n independent, uniform random points in the unit square, find the longest increasing chain (polygonal path through the given points) connecting two diagonally opposite corner of the square, where "length" means the number of points on the chain. The variant of the problem I am going to talk about asks for the length of the longest convex chain connecting two vertices. We determine the asymptotic expectation up to a constant factor, and derive strong concentration and limit shape results. We also prove an ergodic result as well as giving a heuristic argument for the exact asymptotics of the expectation. Some of these results are joint with Imre Barany.

## On Pólya Urn Schemes with Infinitely Many Colors.

Speaker:
Debleena Thacker
Date:
Mon, Jun 11, 2012
Location:
PIMS, University of British Columbia
Conference:
PIMS-MPrime Summer School in Probability
Abstract:
In this talk, we extend the mutlicolor P/'olya urn schemes to countably infinitely many colors. We index the colors by \mathbb{Z}. Throughout the talk, we discuss mainly replacement matrices arising out of random walks. We show that the proportion of colors with suitable centering and scaling show central tendencies. Also the centering and scaling are fairly general. This behavior is in sharp contrast with the finite color case, where the asypmtotic behavior of the proportion of colors are determined by the qualitative properties (transience or recurrence) of the Markov chain underlying the replacement matrix. We also extend the infinite color case to fairly general graphs on \mathbb{R}^{d} and show that the proportion of colors show central tendencies similar to that in the case for \mathbb{Z}. Even the centering and scaling remains same.

## Distributional fixed points and attractors in queueing theory.

Speaker:
Sergio Ivan Lopez Ortega
Date:
Mon, Jun 11, 2012
Location:
PIMS, University of British Columbia
Conference:
PIMS-MPrime Summer School in Probability
Abstract:
(Joint work with Pablo Ferrari) In 1956, Burke proved that the departure process of a stationary queue with Poisson arrivals is a Poisson process. In other words, Poisson process is a distributional fixed point for ./M/1 operation. Mountford and Prabhakar's theorem ('95) showed that Poisson process is not only a fixed point but an attractive distribution over some wide class of ergodic point processes on the line. Some extensions to non-markovian servers were made by Mairesse and Prabhakar ('03) for the existence of fixed points, and Prabhakar ('03) for the attractiveness. In 2001 O'Connell and Yor showed some brownian analogues to Burke's theorem. For a Brownian queue, Brownian motion is a distributional fixed point. In this talk we will show some results about the attractiveness of Brownian motion under the Brownian queue operation.