# Probability

## Interacting Particle Systems 8

Speaker:
Omer Angel
Date:
Fri, Jun 15, 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.

## Limit theorems for conditioned non-generic Galton-Watson trees

Speaker:
Igor Kortchemski
Date:
Thu, Jun 14, 2012
Location:
PIMS, University of British Columbia
Conference:
PIMS-MPrime Summer School in Probability
Abstract:
We are interested in a particular type of subcritical Galton-Watson trees, which are called non-generic trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in large conditioned non-generic trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. We investigate this phenomenon by studying scaling limits of such trees. In particular, we show that the height of such trees grows logarithmically in their size.

## 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.