Seminars in Probability Theory

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Jason Miller: Conformal Removability of SLE

6th October, 2023

We consider the Schramm-Loewner evolution (SLEκ) with κ = 4, the critical value of κ > 0 at or below which SLEκ is a simple curve and above which it is self-intersecting. We show that the range of an SLE4 curve is a.s. conformally removable. Such curves arise as the conformal welding of a pair of independent critical (γ = 2) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set X ⊆ ℂ to be conformally removable which applies in the case that X is not necessarily the boundary of a simply connected domain. We will also describe how this theorem can be applied to obtain the conformal removability of the SLEκ curves for κ ∈ (4,8) in the case that the adjacency graph of connected components of the complement is a.s. connected. This talk will assume no prior knowledge of SLE or LQG.

Sumegha Garg: Tight Space Complexity of the Coin Problem

26th July, 2023

In the coin problem we are asked to distinguish, with probability at least 2/3, between n i.i.d. coins which are heads with probability 1/2 + β from ones which are heads with probability 1/2 − β. We are interested in the streaming space complexity of the coin problem, corresponding to the width of a read-once branching program solving the problem.

The coin problem becomes more difficult as β becomes smaller. Statistically, it can be solved whenever β = Ω(n−1/2), using counting. It has been previously shown that for β =O(n−1/2), counting is essentially optimal (equivalently, width poly(n) is necessary).

On the other hand, the coin problem only requires O(log n) width for β >≈ n^{−0.3} (following low-width simulation of AND-OR tree).

In the paper, we close the gap between the bounds, showing a tight threshold between the values of β = n−c where O(log n) width suffices and the regime where poly(n) width is needed, with a transition at c = 1/3.

In this talk, we will first briefly discuss the low width construction for detecting biases β > n−1/3, which is based on recursive majority. Then we will focus on the lower bound that uses new combinatorial techniques to analyse progression of the success probabilities in read-once branching programs.

Jinyoung Park: p-smallness of Increasing Families

24th July, 2023

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.

Ilya Chevyrev: Invariant measure and universality of the 2D Yang-Mills Langevin dynamic, II

13th June, 2023

In this talk, I will present a recent work on the invariance of the 2D Yang-Mills measure for its Langevin dynamic. The Langevin dynamic both in 2D and 3D had previously been constructed in joint work with Chandra-Hairer-Shen, but it was an open problem to show the existence of an invariant measure even in 2D. In establishing this invariance, we follow Bourgain’s invariant measure argument by taking lattice approximations, but with several twists. An important one, which I will focus on, is that the approximating invariant measures require gauge-fixing, which we achieve by developing a rough version of Uhlenbeck compactness combined with rough path estimates of random walks. I will also present several corollaries of our main result, including a representation of the YM measure as a perturbation of the Gaussian free field, and a new universality result for its discrete approximations.

Hao Shen: Invariant measure and universality of the 2D Yang-Mills Langevin dynamic, I

13th June, 2023

In an earlier work with Chandra, Chevyrev and Hairer, we constructed the local solution to the stochastic Yang-Mills equation on 2D torus, which was shown to have gauge covariance property and thus induces a Markov process on a singular space of gauge equivalent classes. In this talk, we discuss a more recent work with Chevyrev, where we consider the Langevin dynamics of a large class of lattice gauge theories on 2D torus, and prove that these discrete dynamics all converge to the same limiting dynamic. A novel step in the argument is a geometric way to identify the limit using Wilson loops. This universality of the dynamics is crucial for obtaining a sequence of important results for 2D Yang-Mills, including for instance the invariance of the 2D Yang-Mills measure for its Langevin dynamic, which will be discussed by Ilya Chevyrev.

Martin Hairer: The role of symmetry in renormalisation

13th June, 2023

There are several interesting situations where the solutions to singular SPDEs exhibit a symmetry at a formal level that could in principle be broken by the renormalization procedure required to define them. We’ll discuss a relatively simple argument showing that, in many cases, the renormalization can be chosen in such a way that the symmetry does indeed hold and we’ll apply it to the stochastic quantization of the 3D Yang-Mills theory.

Nikos Zygouras: SPDEs at the critical dimension

13th June, 2023

I will make an overview of the progress on treating SPDEs at the critical dimension, the current status and further challenges. Examples will include stochastic heat equations and a more recent Allen-Cahn.

Margherita Disertori: The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

13th June, 2023

Initially introduced as toy model for quantum diffusion, the non-linear supersymmetric hyperbolic sigma model has been attracting much attention in recent years due to its connection to history dependent stochastic processes. In this talk I will present a version of the model with hierarchical interactions. The internal symmetries of the model allow to perform some block-spin renormalization steps exactly. The resulting effective action has renormalized coefficients but no additional interaction terms. I will show the corresponding derivation and some applications.

Luca Fresta: The forward-backward SDE for subcritical Euclidean fermionic field theories

12th June, 2023

In this talk, I will describe a synergy between the renormalization group (RG) in the form of Polchinski's equation and the stochastic quantisation in the form of a forward-backward stochastic differential equation (FBSDE). This approach can be used for constructing subcritical Grassmann Gibbsian measures and is based on controlling the solution of the FBSDE by means of a flow equation with respect to a scale parameter. However, unlike the standard RG approach, we only need to solve Polchinski’s equation in an approximate way, resulting in a great simplification of the analysis.

Xiangchan Zhu: A class of singular SPDEs via convex integration

12th June, 2023

In this talk I will talk about our recent work on a class of singular SPDEs via convex integration method. In particular, we establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier–Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most −1/2 − κ for any κ > 0. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions. Our result applies to any divergence free initial condition in L2 ∪ B−1+κ∞,∞, κ > 0, and implies also non-uniqueness in law. Finally I will show the existence, non-uniqueness, non-Gaussianity and non-unique ergodicity for singular quasi geostrophic equation in the critical and supercritical regime.

Francesco De Vecchi: Non-commutative Lp-spaces and Grassmann stochastic analysis

12th June, 2023

We introduce a theory of non-commutative Lp spaces suitable for non-commutative probability in a non-tracial setting and use it to develop stochastic analysis of Grassmann-valued processes, including martingale inequalities, stochastic integrals with respect to Grassmann Itô processes, Girsanov’s formula and a weak formulation of Grassmann SDEs. We apply this new setting to the construction of several unbounded random variables, including a Grassmann analog of the φ24 Euclidean QFT in a bounded region.

Sky Yang Cao: Recent Results on Finite Group Lattice Gauge Theories

20th March, 2023

The rigorous study of spin systems such as the Ising model is currently one of the most active research areas in probability theory. In this talk, I will introduce one particular class of such models, known as lattice gauge theories (LGTs), and go over its origins, motivations, and then some recent results. Along the way, I will also try to highlight some of the key differences between LGTs and the usual spin systems. The general theme is that LGTs are spin systems with topological considerations.

Scott Sheffield: An Introduction to Random Surfaces

28th November, 2022

The theory of 'random surfaces' has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, and mathematical physics. I will give a friendly (I hope) colloquium-level overview of the subject with lots of pictures. Topics will include random planar maps (interpreted as discrete random surfaces), Liouville quantum gravity surfaces, conformal field theory. and the random fractal curves produced from the Schramm-Loewner evolution.  Many of these topics are motivated by physics (statistical physics, string theory, quantum field theory, etc.) but they also have simple mathematical definitions that can be understood without a lot of physics background.

Rafał Kulik: Blocks estimators in Extreme Value Theory

10th November, 2022

Extreme value theory deals with large values and rare events. These large values tend to cluster in case of temporal dependence. This clustering behaviour is widely observed in practice. I will start with a mild introduction to extreme value theory, discussing probabilistic and statistical issues. This part will be accessible to a broader audience.

Then, I will talk about a more specific problem of statistical theory for cluster functionals and rare events. Two types of estimators are of a primary importance: disjoint and sliding blocks estimators. It has been conjectured that sliding blocks estimators are “better” (to be made precise in the talk). We proved in a recent series of papers that this is not the case and in fact both disjoint and sliding blocks estimators are asymptotically equivalent. This part will be aimed at probabilistic and statisticians.

I will conclude with recent directions in extreme value theory, such as extremes in high dimension, extremes of graphs and networks.

Victor Guerassimov: Random walks on groups. An introduction.

10th March, 2022

Geometric methods proved to be useful in the study of some groups. However the geometry of the Cayley graph of a group is rather different from the geometry of classical geometric objects such as homogeneous spaces of Lie groups. The similarity between these two geometries grows as the scale of observation increases. And the asymptototic behavior of them shows surprising similarity. Random walks is an essential tool in studying large-scale geometry of groups. On the other hand it is an interesting object for probabilists since many properties of general stochastic processes are manifested here in a rather simple form. In my talk, I will provide an elementary introduction to this vast area. No special knowledge beyond the usual university mathematics is required.

László Székelyhidi: Morrey’s conjecture

14th February, 2022

Morrey’s conjecture arose from a rather innocent-looking question in 1952: is there a local condition characterizing 'ellipticity' in the calculus of variations? Morrey was not able to answer the question, and indeed, it took 40 years until first progress was made with V. Sverak’s ingenious counterexample. Nevertheless, the case pertaining to planar maps remains open despite much progress, and has fascinated many through its interesting connections to complex analysis, geometric function theory, harmonic analysis, probability and martingales, differential inclusions and the geometry of matrix space. In the talk, I will give an overview of some of these connections and some of the recent progress.

Jimmy He: Random walks on finite fields with deterministic jumps

1st March, 2021

Recently, Chatterjee and Diaconis showed that most bijections, if applied between steps of a Markov chain, cause the resulting chain to mix much faster. However, explicit examples of this speedup phenomenon are rare. I will discuss recent work studyingsuch walks on finite fields where the bijection is algebraically defined. This work gives a large collection of examples where this speedup phenomenon occurs. These walks can be seen as a non-linear analogue of the Chung-Diaconis-Graham process, where the bijectionis multiplication by a non-zero element of the finite field. This work is partially joint with Huy Pham and Max Xu.