# Topics in Probability Theory (4+3 Points) First Meeting is Thursday January 18.

## Times: Thursdays 13:15-15:00 and Fridays 10:15-12:00. Place: MVL15

This course has been modified and will consist now of two parts.

• #### I will hand out material for the first part of the course.

Part 1:
In the first part, I will cover 2 or 3 of the following topics.
(These topics are in some sense not so advanced and could be placed in an undergraduate
course but they are topics not usually covered in courses).

1. The arc-sine laws for simple random walk.
2. Optimal stopping for Markov chains. (This will include the famous "secretary problem").
3. Intersection properties for simple random walk. (This will mainly consist of the fundamental
difference between random walk in 4 dimensions and random walk in 5 dimensions, not to be
confused with the transience/recurrence difference between 2 and 3 dimension). This will also
introduce and demonstrate the very useful "2nd moment method".
There will be some material handed out for 1 and 2.
For 3, here are two handouts. Handout 1 Handout 2

Some new topics will be added to this first part. A few things concerning random permutations, a proof
of the Stone-Weierstrass theorem from probability and the basic recurrence/transience dichotomy for
random walk in random enviroment.

Part 2:
The second part of the course will consist of graduate students reading through papers
in probability theory and presenting them. Below are possible papers to present. The different
options might vary a fair amount in terms of difficulty and so you need to hurry to have the
pleasure of taking the more challenging ones!

1. Oded Schramm, Compositions of random transpositions. [arXiv:math.PR/0404356]

2. (TAKEN). The critical random graph, with martingales. Asaf Nachmias, Yuval Peres. arXiv:math.PR/0512201

3. (TAKEN). First Passage Percolation Has Sublinear Distance Variance Authors: Itai Benjamini, Gil Kalai, Oded Schramm,
http://front.math.ucdavis.edu/math.PR/0203262
You might choose instead either of the papers
(a) Pemantle, R.; Peres, Y. Planar first-passage percolation times are not tight.
Probability and phase transition (Cambridge, 1993), 261--264, NATO Adv. Sci.
Inst. Ser. C Math. Phys. Sci., 420, Kluwer Acad. Publ., Dordrecht, 1994.
or
(b) Newman, Charles M.; Piza, Marcelo S. T. Divergence of shape fluctuations in two dimensions.
Ann. Probab. 23 (1995), no. 3, 977--1005.

4. Longest increasing sequences in random partititions. See
http://www.ams.org/jams/1999-12-04/S0894-0347-99-00307-0/S0894-0347-99-00307-0.pdf
for the description of this problem and theorem.
Read and present some/all of the proofs of the fact that E[\ell_n]/\sqrt(n) converges to 2 (mentioned at the bottom of page 1).
Relevant stuff in Durrett's book also.

5. (TAKEN). Random-Turn Hex and other selection games (An interesting mix of probability, game theory and percolation). http://front.math.ucdavis.edu/math.PR/0508580

6. A transient Markov chain without cutpoints by Russ Lyons and Yuval Peres.
See http://mypage.iu.edu/~rdlyons/#cuts

7. (TAKEN). Random walk in random environment. See http://www-ee.technion.ac.il/~zeitouni/ps/notesrio.ps
and in particular Theorem 2.3.3 on page 18. The goal would be to present this Theorem but not in the
general situation given by Assumption 2.3.1 but in the specific case of an i.i.d. environment.

8. Central and non-central limit theorems for the Curie Weiss model. The Curie Weiss model is a mean
field model where you do an Ising model on the complete graph. Below the critical value, the distribution
of the spins satisfies a usual Central limit Theorem (Gaussian limit) but at the critical value, the distribution
of the spins has a limiting distribution which is nonGaussian.
Pages 187-191 in the book "Entropy, Large Deviations, and Statistical Mechanics" by Richard Ellis.

9. Pemantle, R. (1990). Vertex-reinforced random walk. Prob. Theor. and Rel. Fields, 92, 117 - 136. See http://www.math.upenn.edu/~pemantle//papers/Papers.html

10. Pemantle, R. (1990). A time-dependent version of Polya's urn. Jour. Theor. Prob., 3, 627 - 637. See http://www.math.upenn.edu/~pemantle//papers/Papers.html

11. (TAKEN). Long range percolation in 1 dimension. Here it turns out that the critical "power" of decay is 2 and there is a very important so-called "first order phase transition" in the case where the power is taken to be 2. This is contained in the following two papers, parts of which will be presented. Newman, C. M.; Schulman, L. S. One-dimensional $1/\vert j-i\vert \sp s$ percolation models: the existence of a transition for $s\leq 2$. Comm. Math. Phys. 104 (1986), no. 4, 547--571. Aizenman, M.; Newman, C. M. Discontinuity of the percolation density in one-dimensional $1/\vert x- y\vert \sp 2$ percolation models. Comm. Math. Phys. 107 (1986), no. 4, 611--647.

12-\infty. Make your own proposal to me or tell me what you are interested in and maybe I can find a topic.