# Probability Seminar

# Fall 2020

**Thursdays in 901 Van Vleck Hall at 2:30 PM**, unless otherwise noted.
**We usually end for questions at 3:20 PM.**

** IMPORTANT: ** In Fall 2020 the seminar is being run online. ZOOM LINK

If you would like to sign up for the email list to receive seminar announcements then please join our group.

## September 17, 2020, Boris Hanin (Princeton and Texas A&M)

**Pre-Talk: (1:00pm)**

**Neural Networks for Probabilists**

Deep neural networks are a centerpiece in modern machine learning. They are also fascinating probabilistic models, about which much remains unclear. In this pre-talk I will define neural networks, explain how they are used in practice, and give a survey of the big theoretical questions they have raised. If time permits, I will also explain how neural networks are related to a variety of classical areas in probability and mathematical physics, including random matrix theory, optimal transport, and combinatorics of hyperplane arrangements.

**Talk: (2:30pm)**

**Effective Theory of Deep Neural Networks**

Deep neural networks are often considered to be complicated "black boxes," for which a full systematic analysis is not only out of reach but also impossible. In this talk, which is based on ongoing joint work with Sho Yaida and Daniel Adam Roberts, I will make the opposite claim. Namely, that deep neural networks with random weights and biases are exactly solvable models. Our approach applies to networks at finite width n and large depth L, the regime in which they are used in practice. A key point will be the emergence of a notion of "criticality," which involves a finetuning of model parameters (weight and bias variances). At criticality, neural networks are particularly well-behaved but still exhibit a tension between large values for n and L, with large values of n tending to make neural networks more like Gaussian processes and large values of L amplifying higher cumulants. Our analysis at initialization has many consequences also for networks during after training, which I will discuss if time permits.

## September 24, 2020, Neil O'Connell (Dublin)

**Some new perspectives on moments of random matrices**

The study of `moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.

## October 1, 2020, Marcus Michelen (UIC)

**Roots of random polynomials near the unit circle**

It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle. Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei. We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle. Based on joint work with Julian Sahasrabudhe.

## October 8, 2020, Subhabrata Sen (Harvard)

**Large deviations for dense random graphs: beyond mean-field**

In a seminal paper, Chatterjee and Varadhan derived an Erdős-Rényi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained.

In this talk, we will explore large deviations for dense random graphs, beyond the “mean-field” setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdős-Rényi random graphs.

Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.

## October 15, 2020, Philippe Sosoe (Cornell)

Title: **Concentration in integrable polymer models**

I will discuss a general method, applicable to all known integrable stationary polymer models, to obtain nearly optimal bounds on the central moments of the partition function and the occupation lengths for each level of the polymer system. The method was developed for the O'Connell-Yor polymer, but was subsequently extended to discrete integrable polymers. As an application, we obtain localization of the OY polymer paths along a straight line on the scale O(n^{2/3+o(1)}).

Joint work with Christian Noack.

## October 22, 2020, Balint Virag (Toronto)

Title: **The heat and the landscape**

Abstract: The directed landscape is the conjectured universal scaling limit of the most common random planar metrics. Examples are planar first passage percolation, directed last passage percolation, distances in percolation clusters, random polymer models, and exclusion processes. The limit laws of distances of objects are given by the KPZ fixed point.

We show that the KPZ fixed point is characterized by the Baik Ben-Arous Peche statistics well-known from random matrix theory.

This provides a general and elementary method for showing convergence to the KPZ fixed point. We apply this method to two models related to random heat flow: the O'Connell-Yor polymer and the KPZ equation.

Note: there will be a follow-up talk with details about the proofs at 11am, Friday, October 23.

## October 29, 2020, Yun Li (UW-Madison)

Title: **Operator level hard-to-soft transition for β-ensembles**

Abstract: It was shown that the soft and hard edge scaling limits of β-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. By tuning the parameter of the hard edge process one can obtain the soft edge process as a scaling limit. In this talk, I will present the corresponding limit on the level of the operators. This talk is based on joint work with Laure Dumaz and Benedek Valkó.

## November 5, 2020, Sayan Banerjee (UNC at Chapel Hill)

Title: **TBA**

Abstract: TBA

## November 12, 2020, Alexander Dunlap (NYU Courant Institute)

Title: **A forward-backward SDE from the 2D nonlinear stochastic heat equation**

Abstract: I will discuss a two-dimensional stochastic heat equation in the weak noise regime with a nonlinear noise strength. I will explain how pointwise statistics of solutions to this equation, as the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor, can be related to a forward-backward stochastic differential equation (FBSDE) depending on the nonlinearity. In the linear case, the FBSDE can be explicitly solved and we recover results of Caravenna, Sun, and Zygouras. Joint work with Yu Gu (CMU).

## November 19, 2020, Jian Ding (University of Pennsylvania)

Title: **TBA**

Abstract: TBA

## December 3, 2020, Tatyana Shcherbina (UW-Madison)

Title: **TBA**

Abstract: TBA

## December 10, 2020, Erik Bates (UW-Madison)

Title: **TBA**

Abstract: TBA