The interplay between probability theory and number theory has a rich history of producing deep results and conjectures. Important instances are the works of Erdős, Kac, Selberg, Montgomery, Soundararajan and Granville, to name a few. This talk will review recent results in this spirit where the insights of probability have led to a better understanding of large values of the Riemann zeta function on the critical line.

This is based in part on joint works with Emma Bailey, and with Paul Bourgade & Maksym Radziwill.

The Kardar-Parisi-Zhang (KPZ) equation and the parabolic Airy₂ process are two central putatively universal objects believed to describe, in a limiting sense, the behavior of a large number of planar stochastic growth models in the KPZ universality class. In this talk we will discuss recent results on understanding the behavior of these processes under upper tail conditionings, eg., under the event that the value at zero is large, as well as probability asymptotics of such events. Our approach is geometric, related to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model, and utilizes a Gibbs resampling property of the processes. Time permitting, we will also see a consequence of this picture on the behavior of the geodesic in the directed landscape conditioned on having large weight, as the weight tends to infinity: appropriately rescaled, it converges weakly to a Brownian bridge.

Based on joint works with Shirshendu Ganguly and Lingfu Zhang.

We present how to construct a stochastic process on a finite interval with given roughness and finite joint moments of marginal distributions. Our construction method is based on Schauder representation along a general sequence of partitions and has two ramifications. The variation index of a process (the infimum value p such that the p-th variation is finite) may not be equal to the reciprocal of Hölder exponent. Moreover, we can construct a non-Gaussian family of stochastic processes mimicking (fractional) Brownian motions. Therefore, when observing a path of process in a financial market such as a price or volatility process, we should not measure its Hölder roughness by computing p-th variation and should not conclude that a given path is sampled from Brownian motion or fractional Brownian motion even though it exhibits the same properties of those Gaussian processes. This is based on joint work with Erhan Bayraktar and Purba Das.

For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a "threshold." Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures, with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. M. Talagrand introduced the notion of "p-smallness" as an explicit certificate to show the p-biased product measure of a given increasing family F is small. In this talk, we will introduce various problems related to "p-smallness" of increasing families, and discuss some proof techniques. Based on joint works with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.

Random band matrices provide a simple setting for studying the metal-insulator transition of the Anderson model. After giving some background on this problem I will present a simple proof that N×N Gaussian random band matrices with band width ~W exhibit dynamical Anderson localization at all energies when W ≪ N ^1/4. The proof uses the fractional moment method and an adaptive Mermin-Wagner-style shift. Joint with Cipolloni, Peled, and Schenker.

The Kardar-Parisi-Zhang (KPZ) universality class of models is characterized by non-Gaussian asymptotic fluctuations coming from random matrices. In this talk, I will introduce a number of models which are known or expected to be in the KPZ universality class, including discrete and semi-discrete models of random polymers, interacting diffusions, the stochastic six vertex model, and ASEP.

I will explain how, using an idea of Emrah-Janjigian-Seppalainen, one derives for each of these models the analogue of a formula due to Rains in the context of Last Passage Percolation. Several results on fluctuations, including upper tail bounds on the correct scale can be deduced from this formula.