This is the homepage for the UC Berkeley Student Probability Seminar, a venue for graduate students in the departments of mathematics, statistics, and others to study aspects of modern probability theory. We meet on Wednesdays from 2:00 PM - 3:00 PM, typically in Evans 939.

Organizers: Zachary McNulty and Daniel Raban.

The topic of this seminar for Fall 2023 will be **Superconcentration **. Our main reference will be the book Superconcentration and Related Topics by Chatterjee [C].You can
find short summaries of each talk and their corresponding references below! You can find the list of speakers here.

- 6 September, Vilas Winstein,
**Introductory talk**, C Chapter 1 - In this first talk of the semester, we will introduce the relationship between the following three phenomena which can arise in the context of an optimization problem: (1) Superconcentration - the optimal value concentrates more tighly than one can prove using classical concentration inequalities. (2) Chaos - the location of the optimizer is highly sensitive to small changes in the noise field. (3) Multiple Valleys - there are many near-optimal points. We will also define a few different models which will be running examples for the semester where these phenomena occur, and we will see how exactly each of these phenomena manifest in each model. This will be a whirlwind tour of results and ideas that we will encounter for the rest of the semester, and as such there will be no proofs.
- 13 September, Adam Jaffe,
**Some fundamental principles: Markov semi-groups and Poincare inequalities**, C Chapter 2 - In this talk, we will introduce two fundamental concepts that will be used through the semester. The first is that of Markov semi-groups, which provide a useful analytic tool for proving results about (continuous-time) stochastic processes. The second is the Poincare inequality, which is one of the simplest but most important (dimension-free!) concentration inequalities. Time permitting, we will also give applications to Gaussian polymers.
- 20 September, Vilas Winstein,
**Superconcentration and Chaos**, C Chapter 3 - In this talk we will define both superconcentration and chaos in the general framework of Markov processes that was introduced last time. We will then see why they are equivalent, and see an application of the equivalence theorem to Gaussian Polymers.
- 27 September, Ella Hiesmayr,
**Chaos implies the Multiple Valley Property**, C Chapter 4 - We will show in detail that chaos implies the multiple valley property for the Gaussian polymer model. This basically follows because under small perturbations the optimal path changes but still has a similar energy.
- 4 October, Zoe McDonald,
**Talagrand's Method**, C Chapter 5 - As noted at the start of Chatterjee's text, techniques used in proving superconcentration improve classical upper bounds. Talagrand's L^1-L^2 method for proving superconcentration is a sharpening of Poincare's inequality. In this talk, we will provide a treatment of Talagrand's method. Time permitting, we will consider its application to Gaussian polymers.
- 11 October, Mriganka Basu Roy Chowdhury,
**Two examples of Superconcentration**, C Chapter 5 - This week we will discuss two examples of superconcentration, proved using two different methods. The first is the famous BKS theorem, which proves superconcentration for the first passage time on Z^d. This will involve a clever averaging trick allowing us to apply the Talagrand L1-L2 bound discussed last time. The second example is the free energy of the Sherrington-Kirkpatrick model, for which we will establish superconcentration using spectral machinery.

Website for past semesters' seminars: