# Scientific

## On Pólya Urn Schemes with Infinitely Many Colors.

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.

(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.

## Integral representations for the self-avoiding walk

The self-avoiding walk is a fundamental model in probability, combinatorics and statistical mechanics, for which many of the basic mathematical problems remain unsolved. Recent and ongoing progress for the four-dimensional self-avoiding walk has been based on a renormalization group analysis. This analysis takes as its starting point an exact representation of the self-avoiding walk problem as an equivalent problem for a perturbation of a Gaussian integral involving anti-commuting variables (fermions). This lecture will give a self-contained introduction to fermionic Gaussian integrals and will explain how they can be used to represent self-avoiding walks.

The talk is mainly based on the paper: D.C. Brydges, J.Z. Imbrie, G. Slade. Functional integral representations for self-avoiding walk. Probability Surveys, 6:34--61, (2009)

## Random Maps 4

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.

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## Interacting Particle Systems 4

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 5

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.

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## Seed bank models with long range dependence

In this talk I present a new model for seed banks, where individuals may obtain their type from ancestors which have lived in the near as well as the very far past. We discern three parameter regimes of the seed bank age distribution, which lead to substantially different behaviour in terms of genetic variability, in particular with respect to fixation of types and time to the most recent common ancestor. The classical Wright-Fisher model, as well as a seed bank model with bounded age distribution considered by Kaj, Krone and Lascoux (2001) are particular cases of the model. The mathematical methods are based not only on Markov chains, but also on renewal theory as well as on a Gibbsian approach introduced by Hammond and Sheffield (2011) in a different context. This talk is based in a joint work with Jochen Blath, Noemi Kurt, Dario Span`o.

## Hydrodynamic limits for a reaction diffusion system

We will investigate an interacting particle system in which two types of Brownian particles annihilate at a certain rate when they're near the interface of their respective domains. This can model the propagation of charges in a single solar cell. We will show that the particle density is the solution of a coupled PDE and obtain a probabilistic representation of the solution.

## Interacting Particle Systems 3

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 3

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.

- Read more about Random Maps 3
- 4468 reads