18.677 Stochastic Processes

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Overview

The class will start with a survey of key objects that arise in Integrable Probability and main types of results one proves about them. Students will be asked to pick their favorite topic and prepare a presentation (which would be required for a grade). The course will also feature outside lecturers giving survey and research talks related to the class.

The class meets on Tuesdays and Thursdays, 2:30-4 pm. 

Lectures will be automatically recorded. 

Lecture Notes (handwritten) 

Lecture Notes by Andrew Lin 

Outside lectures will be posted below, and are also available here.

05/20/2021. Speaker: Cesar Cuenca (Harvard University)
Title: Global asymptotics of particle systems at high temperature
Abstract: The eigenvalue distributions of random matrix ensembles often admit a generalization involving the "inverse temperature" parameter \beta > 0. We focus on two examples: the Hermite ensemble (eigenvalue distribution of GUE) and the spectra of sums of Hermitian random matrices. For both examples, we prove a Law of Large Numbers in the high temperature regime: the size of the system tends to infinity, while the inverse temperature \beta tends to zero. In our second example, we discover a new binary operation between probability measures, which interpolates between convolution and free additive convolution. This talk is based on joint work with Florent Benaych-Georges and Vadim Gorin.
05/13/2021.

 

Speaker: Carina (Letong) Hong (MIT)

Title: A survey on dense O(n) loop models.

Abstract: The dense loop models is a field of interest to probabilists and statistical physicists. When n=1, the loop O(1) model on the hexagonal graph is equivalent to the Ising model on triangular lattice. We will present the setup and survey important results of the dense O(n) loop models. We will then discuss the special cases and limits, as well as relations/ applications to/in other topics like quantum KZE. If time allows, I also hope to talk a bit about the dense O(1) model on the Lx\infty region of Z^2, and showcase some interesting combinatorial-flavored proofs.
05/11/2021.

 

Speaker: Andrew Lin (MIT)

Title: Domino Tilings and the Arctic Circle Theorem.

05/04/2021. Speaker: Korina Digalaki (MIT)

Title: Evaluating Littlewood-Richardson coefficients via Integrable Tilings.

04/29/2021. Speaker: Roger Van Peski (MIT)

Title: Lozenge tilings and the Gaussian free field on a cylinder.

Abstract: In this talk I will first give an expository survey of some of what is known and conjectured about limit shapes and Gaussian free field fluctuations for height functions of random lozenge tilings. I will then discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin, and which exhibit interesting behaviors not present for tilings of simply connected domains. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured by Gorin for tiling models on planar domains with holes. This is joint work with Andrew Ahn and Marianna Russkikh.

04/22/2021. Speaker: Jonas Arista (Center for Mathematical Modeling, University of Chile)

Title: Exit distributions associated with loop-erased walks and random matrices.

Abstract: Non-intersecting processes in one dimension have long been an integral part of random matrix theory, at least since the pioneering work of F. Dyson in the 1960s. For example, if one considers n independent one-dimensional Brownian particles, started at the origin and conditioned not to intersect up to a fixed time T, then the locations of the particles at time T have the same distribution as the eigenvalues of a random real symmetric n×n matrix with independent centred Gaussian entries, with variance T on the diagonal and T/2 above the diagonal, this is known as the Gaussian Orthogonal Ensemble (or GOE). For planar (two-dimensional) state space processes, it is not clear how to generalise the above connections since the paths under consideration are allowed to have self-intersections (or loops). In this talk, we address this problem and consider systems of random walks in planar graphs constrained to a certain type of non-intersection between their loop-erased parts (this is closely related to connectivity probabilities of branches of the uniform spanning tree). We show that in a suitable scaling limit in terms of independent planar Brownian motions, certain exist distributions have also connections with random matrices, mainly Cauchy-type and circular ensembles. This is joint work with Neil O’Connell.

04/20/2021. No lecture because of student holiday. The next lecture is on Thursday 4/22. 

04/15/2021. Speaker: Mustazee Rahman (Durham University)

Title: A random growth model and its time evolution

Abstract: Planar random growth models are irreversible statistical mechanical systems with a notion of time evolution. It is of some interest to understand this evolution by studying joint distribution of points along the time-like direction. One of these models, polynuclear growth or last passage percolation, has allowed exact statistical calculations due to a close relation to determinantal processes. I will discuss recent works with Kurt Johansson where we calculate its time-like  distribution using ideas surrounding determinantal processes. One can take a scaling limit of this distribution, which is then expected to be universal for the  time distribution of many random growth models.

 

04/08/2021. Speaker: Mackenzie Simper (Stanford University)

Title: Induced Probability Distributions on Double Cosets

Abstract: Suppose $H$ and $K$ are subgroups of a finite group $G$ and consider the $H-K$ double cosets. The uniform distribution on $G$ induces a probability distribution on this space of double cosets. I will discuss several examples where things are nice: the double cosets are indexed by classical combinatorial objects, and the induced distributions are well-known measures. When $G = S_n$ and $H$ and $K$ are parabolic subgroups, the double cosets are contingency tables with fixed row and column sums. The induced distribution is the Fisher-Yates distribution, commonly used in statistical tests of independence. The random transpositions Markov chain on $S_n$ induces a natural Markov chain on contingency tables, for which we can study the eigenvalues and eigenfunctions. Joint work with Persi Diaconis.

 

03/25/2021. Speaker: Harriet Walsh (ENS de Lyon)
Title: Schur measures, unitary matrix models, and multicriticality
Abstract: Schur measures on integer partitions define a class of determinantal point processes, or free fermion models, by way of symmetric functions. They generalise the Plancherel measure, and generically have edge fluctuations with a 1/3 exponent characteristic of the KPZ class. We introduce “multicritical” Hermitian Schur measures with 1/(2n+1) critical exponents. Their asymptotic edge distributions are higher-order analogues of the Tracy-Widom GUE distribution first observed by Le Doussal, Majumdar and Schehr for the edge momenta of trapped fermions. We find explicit examples of these measures, and compute limit shapes. By relating Schur measures to unitary matrix models, we study the phase transitions corresponding to these new edge statistics, and explain a connection with certain models of string theory. Based on joint work with Dan Betea and Jérémie Bouttier.

03/09/2021. No lecture because of Monday schedule. The next lecture is on Thursday 3/11. 

02/25/2021. Speaker: Jimmy He (Stanford University)
Title: Limit theorems for descents of Mallows permutations. 
Abstract: The Mallows measure on the symmetric group gives a way to generate random permutations which are more likely to be sorted than not. There has been a lot of recent work to try and understand limiting properties of Mallows permutations. I'll discuss some work on the joint distribution of descents, a statistic counting the number of "drops" in a permutation, and descents in its inverse, generalizing work of Chatterjee and Diaconis, and Vatutin. The proof uses Stein's method with a size-bias coupling as well as a regenerative representation of Mallows permutations due to Gnedin and Olshanski.

 

 

 

 

Course Summary:

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