Title: Scaling limit of the directed polymer and SHE in dimension 2+1
Speaker: Rongfeng Sun (NUS, Singapore)
Abstract: In this mini-course, I will discuss recent results (in joint work with Francesco Caravenna and Nikos Zygouras) on the scaling limit of the directed polymer model in dimension 2+1 and the implications for the stochastic heat equation (SHE) and the KPZ equation in dimension 2. This includes the existence of a phase transition on an intermediate disorder scale, the identification of sub-critical scaling limits, as well as the recent result that the directed polymer partition functions admit a unique scaling limit in the critical window, called the critical 2d stochastic heat flow, which can be interpreted as the solution of the 2-dimensional SHE in the critical window.
Some references: The first is an old survey article on scaling limits of disorder relevant models. The second and third are lecture notes for mini-courses that covers some subsequent (to the first survey) results on scaling limit of directed polymer in dimension 2+1, for which disorder is marginally relevant. The last is the most recent paper on the scaling limit of the directed polymer in dimension 2+1 in the critical window.
Scaling limits of disordered systems and disorder relevance by Francesco Caravenna, Rongfeng Sun, Nikos Zygouras.
Discrete stochastic analysis by Nikos Zygouras.
The 2d KPZ as a marginally relevant disordered system by Nikos Zygouras.
The Critical 2d Stochastic Heat Flow by Francesco Caravenna, Rongfeng Sun, Nikos Zygouras.
Title: Markov chains for sampling: some recent developments
Speaker: Piyush Srivastava (TIFR, Mumbai)
Abstract: This short course will cover techniques leading to a proof (by Anari, Liu, Oveis-Gharan and Vinzant) of the fast mixing of the natural Markov chain on the bases of a matroid. A special case of this problem is the problem of efficiently sampling spanning forests of a given size. These involve a synthesis of ideas from the study of high dimensional expansion and of geometry of polynomials (specifically, strongly log-concave/completely log-concave/Lorentzian polynomials).
If time permits, we will also look at some of the other applications of this method, and also at the more recent method of localization schemes by Chen and Eldan, which provides a different approach leading to optimal results for some of these applications.
Organization:
Lecture notes from this LPS (will be updated in due course of time)
Depending upon interest/need, we might have a few extra "evening sessions" introducing basic Markov chain theory.
Some (rough and preliminary) lecture notes are available here: (Lectures from 2020-04-01 to 2020-04-15)
Some references:
Basic Markov Chain Theory: Markov chains and Mixing Times, Levin, Peres and Wilmer. PDF available from David Levin's webpage.
Papers:
Anari, Nima, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. ‘Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid’. Proc. ACM STOC 2018. Available at http://arxiv.org/abs/1811.01816.
Brändén, Petter, and June Huh. ‘Lorentzian Polynomials’. Ann. Math. Available at https://arxiv.org/abs/1902.03719.
Chen, Yuansi, and Ronen Eldan. ‘Localization Schemes: A Framework for Proving Mixing Bounds for Markov Chains’. Proc. IEEE FOCS 2022. Available at http://arxiv.org/abs/2203.04163.
Cryan, Mary, Heng Guo, and Giorgos Mousa. ‘Modified Log-Sobolev Inequalities for Strongly Log-Concave Distributions’. Proc. IEEE FOCS 2019 Available at http://arxiv.org/abs/1903.06081.