The Weizmann Students' Seminar in Probability and Related Fields

Each week a student gives an introductory talk on a topic related to probability. Anyone is welcome to speak, but it is not mandatory.

To give a talk, send an email to Tal Orenshtein: talo at weizmann dot ac dot il

Time and Place:

Sunday 14:00-16:00, Ziskind 261.

Announcements:

1) During the first semester, the students' seminar unites with Prof. Itai Benjamini's seminar. Therefore, it will hold on Wednesdays 11:00-13:00, in the same room. The first lecture will be given Nov. 2nd by Itai. (27/10/2011)

2) Starting from Nov. 30, open problem session is added to the seminar. Lectures will be held 10:00-12:00 and open problems will be presented 12:15-12:45. (30/11/2011)

3) The seminar is going on a winter break. We will meet again in the spring semester. The first lecture will be held on Sunday, March 11 2012, 14:00-16:00. (25/1/2012)4) The seminar spring semester is currently on hold (in particular, there will be NO lecture on March 11). As agreed, we wish to have a more restrictive spectrum of subjects. We are considering a relevant material. For suggestions please email Tal. (7/3/2012)

5) We are back! The first lecture will be held on Sunday, March 25, 14:00-16:00. (20/3/2012)

6) There will be no talk tomorrow, April 15. (14/4/2012)

7) Due to the faculty retreat May 13 talk will be held on May 20. Please note the talk will begin at 13:00 and will last 3 hours. (10/5/2012)

8) The next talk will be held June 17 (there will be no talk June 10). (5/6/2012)

Here is our Google calender:

Spring 2012

March 25

Eviatar Procaccia

Poison processes and applications

Abstract. Poison processes are a fundamental concept in probability theory. They appear naturally in the theory of particle systems, Markov processes and long range percolation. We will define the Poison process, prove some key properties and see some applications.

April 2, Room 1

NOTE change of date & place!

Xiaoqin Guo

Einstein relation for random walks in random environment

Abstract. The Einstein relation for a system in equilibrium describes a relation between the response of the system to a perturbation and its diffusivity at equilibrium. It states that the derivative of the velocity (with respect to the strength of the perturbation) equals the diffusivity. In this talk, I will explain the Einstein relation in the context of random walks in a balanced uniformly elliptic iid environment. We consider the velocity of the random walks in the perturbed environment where drift is added in one direction. We will show that the derivative of the velocity with respect to the size of the perturbation converges to the diffusivity constant of the environment. This talk is based on a joint work with my advisor Ofer Zeitouni.

April 22

14:00-17:00

NOTE unusual length.

Tal Orenshtein

CLT for the tagged exclusion process

Abstract. Arratia proved in 1983 a central limit theorem for the motion of a tagged particle in a one dimensional simple symmetric exclusion process. The theorem gives a singular rate, comparing to other settings. A full proof will be supplied, based on Liggett 1985 book.

April 29

Hilary Finucane

CLT for reversible Markov chains

May 6

Ori Hirschberg

On Bessel processes of negative order

Abstract. The Bessel process of order d describes the distance of a d-dimensional Brownian motion from the origin. This definition can be generalized to any real d, including negative “dimensions”. I will present a physicist’s perspective on some of the basic properties of such Bessel processes, and of processes which from afar look like them. Part of the talk will be devoted to my work (joint with D. Mukamel and G. Schutz) on the convergence of these processes to their stationary measures.

May 20

13:00-16:00

NOTE change of time & unusual length.

Ron Rosenthal

0-1 law for random walks in random environments

Abstract. The model of random walks in random environments has been the object of intensive study in the last decades. In the talk I'll present the model and discuss one of its most basic questions, the 0-1 law. I will present the proof in dimensions 1 and 2, and will state the conjecture for higher dimensions.

June 3

14:00-17:00

NOTE unusual length.

Jonathan Hermon

Domination by product measures and applications

Abstract. Domination by product measures is a machinery for dealing with local dependencies in many cases, which allows one to transfer to a probability space in which everything is independent. This is a very strong generalization of the "Local lemma" used by combinatorialists. We will go through the proof of the theorem and state other related results and see applications to particle systems, percolation processes and statistical mechanics models (as time permits).

June 17

Ron Rosenthal

0-1 law for random walks in random environments, cont.

Past talks

Fall 2011

Thursday, September 8

Tal Orenshtein

Galton - Watson processes and applications

Abstract. In 1874 H.W. Watson attempted to solve F. Galton's question: What are the odds that a surname will survive forever. They published a joint paper but a gap in their proof was found. Fortunately it was corrected by Stefengen around 1930. We will present a solution for the problem and discuss some consequences: one is characterization of having a splitting property, and another is related to Mandelbrot's fractal percolation.

Lecture notes.

Thursday, September 15

Tal Orenshtein

Galton-Watson processes and applications, cont.

Thursday, September 22

Hilary Finucane

Amenability via random walks I

Abstract. My first talk will focus on simple random walks on infinite Cayley graphs. I will define amenability, asymptotic entropy, and the spectral radius, and prove that asymptotic entropy = 0 implies spectral radius = 1 implies amenability. In my second talk, I will present the main result from the paper "Amenability via random walks" by Bartholdi and Virag: that the Basilica group has asymptotic entropy = 0, and is therefore amenable.

Thursday, September 29

Rosh Ashana

Thursday, October 6

Hilary Finucane

Amenability via random walks II

Thursday, October 13

Sukot

Thursday, October 20

Simhat Torra

Thursday, October 27

Yair Daon

Subadditive ergodic theorem

Abstract. I will prove Ligget's version of Kingman's Subadditive Ergodic theorem and present some applications.

Note change of day! We are meeting Wednesdays 11:00-13:00, room 261.

Wednesday, November 9

Yair Hartman

Poisson boundary theory I

Lecture notes.

Wednesday, November 16

Yair Hartman

Poisson boundary theory II

Wednesday, November 23

Eviatar Procaccia

Talagrand's inequality and applications

Abstract. We will begin with Fourier-Walsh expansion for functions f:\{0,1\}^n \rightarrow \mathbb{R}, give a proof of Talagrand's inequality appearing in a paper by Benjamini, Kalai and Schramm. And if time allows us show some applications.

Note change of time! We are meeting Wednesdays 10:00-13:00, room 261.

Wednesday, November 30

Omer Tamuz

Interacting Bayesian agents

Abstract. I will give an introduction and talk a little about this classical paper: Banerjee 1992: "A simple model of herd behavior". And then spend about a lecture and a half proving our result: Mossel, Sly and Tamuz 2011: " From Agreement to Asymptotic Learning".

Wednesday, December 7

Omer Tamuz

Interacting Bayesian agents, cont.

Wednesday, December 14

Jonathan Hermon

The Strange Logic of Random Graphs

Abstract. A rich combination of combinatorics, logic, probability and even games on graphs is used to prove 0-1 laws for random graphs.

Presentation.

Wednesday, December 21

Hanuka

Wednesday, December 28

Yakir Reshef

Thompson's groups

Abstract. Thompson's groups are three related groups that are built out of certain transformations on binary trees. One of these, the Thompson group T, was the first known finitely presented infinite simple group. Thompson's group F is non-Abelian and torsion free and contains a free semigroup on two generators, yet it contains no non-Abelian free subgroup. It is still open whether F is amenable. We will define Thompson's groups and discuss and prove some of the properties that make them so enigmatic.

Wednesday, January 4

Jacob Kagan

Optimality considerations in cell regulation

Abstract. I intend to talk about discrimination energy and information considerations in cell regulatory mechanisms. I will give a schematic presentation of a cell structure and present a simple statistical model that gives some quantitative insights on cell regulation.

Wednesday, January 11

Yuri Lima

Z^d - actions with prescribed topological and ergodic properties

Abstract. We will develop notions of total minimality and total unique ergodicity for Z^d-actions and construct examples with these properties. The examples are shift maps of {0,1}^{Z^d} , obtained by an inductive construction of increasing sequences of ?nite alphabets/words with controlled combinatorial and statistical properties.

You can find more info in the following post (including a file with some slides) about the topic above.

Wednesday, January 18

Igor Shinkar

Invariance principle for multilinear polynomials

Abstract. I'll discuss the "Majority is Stablest" theorem due to Mossel O'Donnell and Oleszkiewicz (see http://arxiv.org/abs/math.PR/0503503). The theorem says that among all balanced boolean functions each of whose inputs has low influence, Majority is the most stable, i.e. its has the smallest probability to change its value when flipping a small fraction of input coordinates. I'll discuss the relation of the theorem to voting schemes and (maybe) to hardness of approximations.

Although the theorem is stated in discrete setting, the proof uses results from "continuous spaces". The connection to the discrete setting is done via some sort of CTL for low degree polynomials.

Wednesday, January 25

Itai Dinur

Cryptanalytic time-memory trade-offs

Abstract. Inverting a pseudo-random cryptographic function, that maps n input bits to n output bits, can be done in time N=2^n and constant memory using exhaustive search. Alternatively, it is possible to invert the function in constant time using a look-up table of size N=2^n that is built during a long pre-processing phase. Cryptanalytic time-memory trade-offs deal with the problem of reducing the memory from N=2^n, at the cost of increasing the computational effort. Since its introduction by Hellman in 1980, several algorithms for this problem were proposed, all achieving the same asymptotic curve. In 2006, Barkan, Biham and Shamir proved that this curve is essentially optimal. In this talk, I will present the classical Hellman scheme and the alternative Rainbow scheme proposed by Oechslin in 2003. Then, I will discuss parts of the proof by Barkan, Biham and Shamir.

No prior knowledge is assumed.