# Columbia Probability Seminar (Fall 2016)

The seminar is organized by the Mathematics and Statistics departments and covers topics in pure and applied probability.
Organizers: Guillaume Barraquand, Ivan Corwin, Julien Dubedat, Ioannis Karatzas, Jeffrey Kuan, Marcel Nutz, Philip Protter, Hao Shen, Yi Sun, and Li-Cheng Tsai.

Meeting Time: Wednesday 5:30-6:30 pm and Friday 12:00-1:00 pm
Location: Math 520 (Directions to the Mathematics Department)
Contact: probability_seminar@math.columbia.edu

Here is a list of upcoming probability conferences and meetings.

### Schedule of Talks

Date and Time Speaker Title
Wednesday, September 7
5-6 pm Math 507
Special time/location!
Ioannis Karatzas (Columbia) Competing Brownian particle systems
We consider systems of diffusing particles, whose local characteristics are assigned in terms of their ranks. We construct the associated multidimensional diffusions, and discuss questions of strength and pathwise uniqueness for the stochastic equations that realize them. Multi-dimensional and perturbed versions of the Tanaka equation arise, and are studied in some detail – as are questions regarding triple (or higher-order) collisions, which play here an important role. When we allow for elastic collisions between particles, the ranked versions of the resulting diffusive motions arise as scaled limits of asymmetric, exclusion-type interacting particle systems. (Survey of joint works with E.R. Fernholz, T. Ichiba, S. Pal, V. Prokaj and M. Shkolnikov.)
Friday, September 16
12-1 pm
Minghan Yan (Columbia) Walsh semimartingales and diffusions: construction, stochastic calculus, and control.
We define and construct planar processes called "Walsh semimartingales", and develop for them a Freidlin-Sheu-type change-of-variable formula. These processes generalize the celebrated Walsh's Brownian motion. For diffusions of this type, we study their well-posedness and explosion times, and provide a surprisingly concrete solution to a related control problem with discretionary stopping.
Wednesday, September 21
5:30-6:30 pm
Ofer Zeitouni (NYU/Weizmann) Extremal eigenvalue correlations in the GUE minor process and a law of fractional logarithm
Let $$\lambda^{(N)}$$ be the largest eigenvalue of the $$N \times N$$ GUE matrix which is the $$N$$th element of the GUE minor process, rescaled to converge to the standard Tracy-Widom distribution. We consider the sequence $$\{\lambda^{(N)}\}_{N\geq 1}$$ and prove a law of fractional logarithm for the limsup:
$$\limsup_{N \to \infty} \frac{\lambda^{(N)}}{\left( \log N \right)^{2/3} } = \left(\frac{1}{4}\right)^{2/3}\,,\quad \text{ almost surely}.$$
For the liminf, we prove the weaker result that there are constants $$c_1, c_2 > 0$$ so that
$$-c_1 \leq \liminf_{N \to \infty} \frac{\lambda^{(N)}}{\left( \log N \right)^{1/3} } \leq -c_2\,,\quad \text{ almost surely.}$$
We conjecture that in fact, $$c_1=c_2=4^{1/3}$$.
Wednesday, September 28
5:30-6:30 pm
Amol Aggarwal (Harvard) Current fluctuations of the stationary ASEP and six-vertex model
We consider the following three models from statistical mechanics: the asymmetric simple exclusion process, the stochastic six-vertex model, and the ferroelectric symmetric six-vertex model. It had been predicted by the physics communities for some time that the limiting behavior for these models, run under certain classes of translation-invariant (stationary) boundary data, are governed by the large-time statistics of the stationary Kardar-Parisi-Zhang (KPZ) equation. The purpose of this talk is to explain these predictions in more detail and survey some of our recent work that verifies them.
Friday, September 30
12-1 pm
Andrey Sarantsev (UCSB) Rank-based Brownian particles
Consider a finite or infinite rank-based system of Brownian particles on the real line. Each particle has drift and diffusion coefficients depending on its current rank relative to other particles. We survey recent developments in this field.
Wednesday, October 5
5:30-6:30 pm
Robin Pemantle (UPenn) Evolution of one-cells on a line
At time zero, the real line is partitioned into intervals. The original partition, which may be random, evolves according to a deterministic rule whereby the interface between consecutive pair of cells moves so as to make the larger cell grow and the smaller cell shrink. This is a somewhat degenerate one-dimensional version of a two (and higher) dimensional mean-curvature flow model about which almost nothing rigorous is known.
I will discuss different versions of this problem. In one case, we have data that suggest conjectures about the limit, but no theorems. In another case, perhaps surprisingly, a Poisson limit can be proved, though questions still remain.
Joint work with Emanuel Lazar
Friday, October 7
12-1 pm
Alexander Drewitz (Köln) Random Walk Among a Poisson System of Moving Traps
We review some old and new results on the survival probability of a random walk among a Poisson system of moving traps on the lattice, which can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential. We will also report on some recent progress concerning the path measures of this random walk conditioned on survival.
As a byproduct of this line of research, some results of independent interest have emerged which we will touch upon if time admits.
Friday, October 14 Columbia-Courant Probability Day
Thursday, October 20
4:10-5:25 pm SSW 903
Special time/location!
René Carmona (Princeton) Minerva Lecture: Introduction to Mean Field Games and the Two Pronged Probabilistic Approach Slides
Since its inception about a decade ago, the theory of Mean Field Games has rapidly developed into one of the most significant and exciting sources of progress in the study of the dynamical and equilibrium behavior of large systems. The introduction of ideas from statistical physics to identify approximate equilibria for sizeable dynamic games created a new wave of interest in the study of large populations of competitive individuals with "mean field" interactions.
The lectures will rely on examples from economic growth theory, flocking, herding and congestion models for crowd behavior, systemic risk, cyber security, bank runs and liquidity crises, information percolation on social networks, ... to introduce the mathematical challenges raised by the intractability of most of these large scale equilibrium problems.
We shall quickly review the original partial differential equations approach to the solution of these stochastic games, and introduce a probabilistic approach based on analysis on spaces of probability measures, the theory of forward/backward stochastic differential equations, and the optimal control of McKean-Vlasov stochastic differential equations.
We shall show how these tools can be brought to bear in an effort to solve some of these challenging equilibrium problems.
Tuesday, October 25
4:10-5:25 pm Math 507
Special time/location!
René Carmona (Princeton) Minerva Lecture: Calculus over Wasserstein Space and Control of McKean-Vlasov Equations Slides
Wednesday, October 26
5:30-6:30 pm
Senya Shlosman (CPT) Spin glass transition as interface roughening phenomenon
The talk is about the statistical mechanics on Cayley trees and Lobachevsky plane. The rich structure of the Gibbs states in these settings allows for simple understanding of the spin glass transition. (No knowledge about the latter is supposed.)
Joint work with Ch. Maes, D. Gandolfo and J. Ruiz.
Thursday, October 27
4:10-5:25 pm SSW 903
Special time/location!
René Carmona (Princeton) Minerva Lecture: Master Equations, Games with Common Noise, and with Major and Minor Players Slides
Friday, October 28
12-1 pm Math 507
Special time/location!
René Carmona (Princeton) Minerva Lecture: Finite State Space Games and Games of Timing Slides
Wednesday, November 2
2:30-4:30 pm Math 507
Special time/location!
Robin Pemantle (UPenn) Minerva Lecture: Polynomials, their coefficients, and the locations of their zeros: applications to probability, combinatorics and computer science
Univariate Theory I: Zeros of random polynomials and power series Slides
Univariate Theory II: Zeros and coefficients of polynomials in one variable Slides
Friday, November 4
12-1 pm
Yi Sun (Columbia) Laguerre and Jacobi analogues of the Warren process
We define Laguerre and Jacobi analogues of the Warren process. That is, we construct local dynamics on a triangular array of particles so that the projections to each level recover the Laguerre and Jacobi eigenvalue processes of König-O'Connell and Doumerc and the fixed time distributions recover the joint distribution of eigenvalues in multilevel Laguerre and Jacobi random matrix ensembles. Our techniques extend and generalize the framework of intertwining diffusions developed by Pal-Shkolnikov. One consequence is the construction of particle systems with local interactions whose fixed time distribution recovers the hard edge of random matrix theory.
Wednesday, November 9
2:30-4:30 pm Math 507
Special time/location!
Robin Pemantle (UPenn) Minerva Lecture: Polynomials, their coefficients, and the locations of their zeros: applications to probability, combinatorics and computer science
Multivariate Theory I: Boolean variables and the strong Rayleigh property Slides
Multivariate Theory II: Applications: random trees, determinantal measures and sampling Slides
Friday, November 11
12-1 pm
Nicolas Perkowski (HU Berlin) Universality of martingale solutions to the KPZ equation
The Kardar-Parisi-Zhang (KPZ) equation is a stochastic PDE which is conjectured to universally describe the fluctuations in slow random interface growth. The equation is very singular and has resisted a mathematical analysis until Hairer understood in 2013 how to solve it using pathwise arguments. Recently we also obtained a probabilistic understanding and were able to prove that the corresponding martingale problem is well posed. The martingale problem is a powerful tool for proving convergence to KPZ, and in particular for establishing the universality conjecture. I will discuss the uniqueness result and how to use it to prove that various models converge to KPZ. Based on joint works with Giuseppe Cannizzaro, Joscha Diehl and Massimiliano Gubinelli.
Wednesday, November 16
2:30-4:30 pm Math 507
Special time/location!
Robin Pemantle (UPenn) Minerva Lecture: Polynomials, their coefficients, and the locations of their zeros: applications to probability, combinatorics and computer science
Geometry I: Hyperbolic polynomials Slides Handout
Geometry II: Multivariate generating functions Slides Handout
Thursday, November 17
and Friday, November 18
CUNY
Northeast Probability Seminar
Wednesday, November 30
5:30-6:30 pm
Scott Armstrong (NYU) Quantitative homogenization and regularity
The talk will be about linear, divergence-form elliptic equations with random coefficients. This is also sometimes called "the random conductance model" and is essentially equivalent to the study of "(reversible) random walks/diffusions in random environments." In the last several years there has been a lot of progress in obtaining very precise results: we can now characterize the precise size of the fluctuations of the solutions and even prove their scaling limit to a variant of the Gaussian Free Field. The analytic core of the proofs of these results comes from elliptic regularity theory and ideas from the calculus of variations. I will give a broad overview of these results and try to explain the structure of the arguments.
Friday, December 2
12-1 pm
Ramis Movassagh (IBM) Eigenvalue Attraction
Much work has been devoted to the understanding of the motion of eigenvalues in response to randomness. The folklore of random matrix analysis, especially in the case of Hermitian matrices, suggests that the eigenvalues of a perturbed matrix repel. We give a simple proof that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract. We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. We apply the results to the Hatano-Nelson model, random perturbations of a fixed matrix, real stochastic processes with zero mean and independent intervals and discuss open problems.
References: Journal of Statistical Physics, February 2016, Volume 162, Issue 3, pp 615-643; “Eigenpairs of Toeplitz and disordered Toeplitz matrices with a Fisher-Hartwig symbol”, J Stat Phys (2016). [joint work with late Leo. P. Kadanoff]

Time permitting we will discuss a disjoint work. The method of "Isotropic Entanglement" (IE), inspired by Free Probability Theory and Random Matrix Theory, which predicts the eigenvalue distribution of quantum many-body systems with generic interactions. At the heart is a "Slider", which interpolates between two extrema by matching fourth moments. The first extreme treats the non-commuting terms classically and the second treats them isotropically. Isotropic means that the eigenvectors are in generic positions. We prove that the interpolation is universal. We then show that free probability theory also captures the density of states of the Anderson model with an arbitrary disorder and with high accuracy. Theory will be illustrated by numerical experiments. [Joint work with Alan Edelman]
References: Phys. Rev. Lett. 107, 097205 (2011), Phys. Rev. Lett. 109, 036403 (2012)
Monday, December 5
5:30-7:30 pm Math 520
Special time/location!
Nicolai Reshetikhin (Berkeley) Minerva Lecture: Limit shapes in integrable models in statistical mechanics
The lectures will start with a discussion of local models in statistical mechanics. Main examples are dimer models and the 6-vertex model. Other models will be mentioned as well.

Then we will proceed to the discussion of details of dimer models. They are one of the oldest integrable models in statistical mechanics, the Ising model being a particular case. The integrability in dimer models and the 6-vertex model has a slightly different nature which we will see in Lectures 4, 5, and 6. In Lectures 1 and 2, details of the Kasteleyn solution will be given and the discussion of the thermodynamic limit will begin. The limit shape phenomenon, also known as an "arctic circle", will be introduced at the end of Lecture 2. Lecture 3 will be focused on the structure of limit shapes and correlation functions in dimer models in the thermodynamic limit.

The 6-vertex model on a cylinder will be defined and studied in Lectures 4 and 5. Here the spectrum of the corresponding transfer-matrix and the Bethe ansatz will be presented, followed by the exposition of what is known about this spectrum in the thermodynamic limit. Lecture 5 will conclude with conjectures about the limit shape emergence in the 6-vertex model and its description by a variational principle. In Lecture 6 we will see that there is a strong indication about the integrability of the Euler-Lagrange PDE describing the limit shape of the 6-vertex model. We will also see that in the special case of the 6-vertex model, known as the stochastic point, these PDE become the Burgers equation. The stochastic point of the 6-vertex model is closely related to the ASEP model in stochastic processes. This part of lectures is based on joint results with Ananth Sridhar. At the end some speculations about correlation functions in the 6-vertex model and about other models will be given.

Lecture 1 Slides Lecture 2 Slides
Wednesday, December 7
5:30-7:30 pm Math 520
Special time/location!
Nicolai Reshetikhin (Berkeley) Minerva Lecture: Limit shapes in integrable models in statistical mechanics
Lecture 3 Slides Lecture 4 Slides
Friday, December 9
11:45 am - 1:45 pm Math 520
Special time/location!
Nicolai Reshetikhin (Berkeley) Minerva Lecture: Limit shapes in integrable models in statistical mechanics
Lecture 5 Slides Lecture 6 Slides

Seminar archive: Old Page