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.
This course has been modified and will consist now of two parts.
The first part of the course is worth 4 points and the second part
You can do either part or both (or none!).
The course will go on for (at least) the 3rd and 4th quarters.
I will hand out material for the first part of the course.
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
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.
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.
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
of taking the more challenging ones!
1. Oded Schramm,
Compositions of random transpositions.
The critical random graph, with martingales. Asaf Nachmias,
3. (TAKEN). First Passage Percolation Has Sublinear Distance Variance
Authors: Itai Benjamini, Gil Kalai, Oded Schramm,
You might choose instead either of the papers
(a) Pemantle, R.; Peres, Y. Planar first-passage percolation times are not
Probability and phase transition (Cambridge, 1993), 261--264,
NATO Adv. Sci.
Inst. Ser. C Math. Phys. Sci., 420, Kluwer Acad. Publ.,
(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.
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).
6. A transient Markov chain without cutpoints
by Russ Lyons and Yuval Peres.
Random walk in random environment.
and in particular Theorem 2.3.3 on page 18.
The goal would be to present this Theorem but not in the
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
spins satisfies a usual Central limit Theorem (Gaussian limit) but
at the critical value, the distribution
spins has a limiting distribution which is nonGaussian.
Pages 187-191 in the book "Entropy, Large Deviations, and Statistical
Mechanics" by Richard Ellis.
Pemantle, R. (1990). Vertex-reinforced random walk. Prob. Theor. and
Rel. Fields, 92, 117 - 136.
Pemantle, R. (1990). A time-dependent version of Polya's urn.
Jour. Theor. Prob., 3, 627 - 637.
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
This is contained in the following two papers, parts of which will be
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.