Probability Seminar

Organisers

Giuseppe CannizzaroLink opens in a new window and Nikolaos ZygourasLink opens in a new window

The probability seminar is held on Wednesdays 16-17 in B3.02.

Seminars in Term 1.

Oct 8 - Christophoros Panagiotis (University of Bath)

Title: Geometric representations for the $\varphi^4$ model

Abstract: The $\varphi^4$ model was originally introduced in Quantum Field Theory as the simplest candidate for a non-Gaussian theory. Its importance in statistical physics was highlighted by Griffiths and Simon, who observed that the $\varphi^4$ potential arises as the scaling limit of the fluctuations of the critical Ising model on the complete graph. In this talk, I will describe how this connection to the Ising model leads to two new geometric representations of the $\varphi^4$ model, called the random tangled current expansion and the random cluster model. I will explain how these representations can be used to prove that the phase transition of the $\varphi^4$ model is continuous in dimensions three and higher, and to obtain large-deviation estimates for spin averages in the supercritical regime.

Oct 15 - Omer Angel and Daniel de la Riva Massaad (University of British Columbia)

Title: The phase transitions in the frog model.

Abstract: The frog model is an interacting particle system, where particles are of type A (asleep) until hit by a particle of type B. Despite the simplicity of the definition, many questions remain open. We prove existence and sharpness of the phase transition for the frog model on transitive graphs of either polynomial growth, or non-amenable.

Oct 22 - Jad Hadman (University of Oxford)

Title: Log-correlated fields, Gaussian mutiplicative chaos and the Riemann zeta function

Abstract: In a pair of highly influential works, Fyodorov, Hiary and Keating formulated precise conjectures on the extreme value statistics of the Riemann zeta function on typical short intervals of the critical line Re(z)=1/2. These conjectures have since seen substantial progress, and have more generally stimulated work connecting multiplicative number theory with the study of log-correlated fields and Gaussian multiplicative chaos. In this talk, I will explore these connections and present recent results that advance this program. (Based on joint work with L-P. Arguin.)

Oct 29 -

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Nov 5 - SPECIAL OPEN PROBLEM SESSION with Bálint Tóth

Nov 12 - Thierry Bodineau (Institut des Hautes Études Scientifique)

Title: A renormalisation group perspective on functional inequalities

Abstract: Functional inequalities provide information on the structure of a probability measure and on the relaxation of associated stochastic dynamics to equilibrium. In this talk, we will describe a multiscale analysis for decomposing high-dimensional measures into simpler structures and derive from it functional inequalities. The strategy is based on the renormalization group method used in statistical physics to study the distribution of interacting particle systems. We will also review other related developments and in particular show that this decomposition of measures can be interpreted in terms of measure transport.

Nov 19 - Darrick Lee (University of Edinburgh)

Title: Parameter Estimation for Signature SDEs

Abstract: The path signature is a method to characterize the entire history of paths which is a central tool in rough paths. Signature SDEs were recently developed as a rich class of path-dependent SDEs, where the vector fields depend on the path signature of the solution. In this talk, we discuss existence and uniqueness of solutions for signature SDEs by using rough paths. Furthermore, we discuss a parameter estimation method for such SDEs by extending the expected signature matching method to our setting. This is joint work with Pardis Semnani, Vince Guan, and Elina Robeva (UBC).

Nov 26 - Quentin Berger (Université Sorbonne Paris Nord)

Title: Influence of disorder for different pinning models

Abstract: The goal of this talk is to give an overview of the question of disorder relevance for physical systems, through the example of pinning models. The main question is to determine whether an arbitrarily small amount of disorder may change the properties of the system, in which case disorder is said to be relevant. I will consider in particular two closely related pinning models (pinning on a line of defect vs. pinning on a random walk) for which this question has been studied. In works with Hubert Lacoin (IMPA), we have established some criteria for disorder relevance in both models: somewhat surprisingly, the criteria are slightly different for the two models, which was not anticipated in the physics literature.

Dec 3 -Fu-Hsuan Ho (Weizmann Institute of Science)

Title: The free energy of an enriched continuous random energy model in the weak correlation regime

Abstract: Recently, there has been an attempt to rewrite the free energy of spin glass models as a solution to a Hamilton-Jacobi equation. In the talk, I will overview the efforts and the results that several people have achieved. Then, I will talk about a joint work with Alexander Alban and Justin Ko where we try to apply this framework to the branching random walk setting. In the weak correlation regime, where the CREM’s covariance function A is bounded above by the identity, we show that the free energy is given by the Hopf formula. In contrast, we provide an example outside of the weak regime, where A is a two-segment linear function exceeding the identity. Here, the Hamilton-Jacobi framework no longer applies, and a formal application of the associated variational principle fails to yield the correct free energy.

Dec 10 - Matt Roberts (University of Bath)

Title: Accessibility percolation with Rough Mount Fuji labelling

Abstract: Consider an infinite rooted connected graph where each vertex is labelled by an iid Uniform(0,1) random variable, plus a parameter theta times its distance from the root. This is known as the Rough Mount Fuji labelling. We study whether there exist paths from the root to infinity along which the vertex labels are increasing. This is known as accessibility percolation and has been studied in both biological and computer science contexts.
When the graph is a supercritical Bienaymé-Galton-Watson tree, we give an exact characterisation of the critical value of theta such that there are infinite increasing paths with positive probability if and only if theta is larger than that value. We also give more explicit bounds on the critical value; the lower bound holds for a much wider class of trees. When the graph is the d-dimensional lattice, the model depends on the class of paths allowed and the distance used in the labelling. We show that there is a non-trivial phase transition. The proof involves a coupling with oriented percolation. This is joint work with Diana De Armas Bellon.

Seminars in Term 2.

Jan 14 - Samuel Johnston (King's College London)

Title: On creating convexity in high dimensions

Abstract: Which probability measures occur as the law of a random variable λ_1 Z_1 + ... + λ_k Z_k where λ_i ≥ 0, and λ_1 + ... + λ_k = 1, and each Z_i has a Gaussian marginal, but the (Z_1,...,Z_k) can depend on one another in any way possible? It turns out the question above is related to a problem in convex geometry. Given a subset A of R^n, let conv_k(A) denote the set of all k-fold convex combinations of points in A; that is, vectors of the form λ_1 a_1 + ... + λ_k a_k with a_i in A, λ_i ≥ 0, and λ_1 + ... + λ_k = 1. Carathéodory’s theorem in convex geometry states that the convex hull of A is precisely conv_{n+1}(A): in other words, every point in the convex hull can be written as a convex combination of at most n+1 points from A. In the 1990s, Talagrand asked whether large convex subsets can already be constructed from large sets using only a bounded number k < n of convex combinations, independent of the dimension. In a Gaussian formulation: if g_n denotes the standard Gaussian measure on R^n, does there exist a universal k such that for every set A with g_n(A) > 0.9, the set conv_k(A) contains a convex subset B with g_n(B) > 0.1? Although this problem lies in convex geometry, I will discuss how in the preprint https://arxiv.org/abs/2502.10382Link opens in a new window, ideas from probability and optimal transport provide new perspectives and some initial inroads on Talagrand's question.

Jan 21 -Maria Chiara Ricciuti (Imperial College London)

Title: Scaling Limits of a Weakly Perturbed Random Interface Model

Abstract: In this talk, we consider a random interface model on the discrete torus with $2n$ sites, obtained from the classical corner flip dynamics but with a weak global perturbation, namely an asymmetry of order $n^{-\gamma}$ of the direction of growth that switches direction based on the sign of the total area under the interface. The slopes of this model can be viewed as a non-simple exclusion process at half filling with globally dependent rates. We show that, for $\gamma=1$, the hydrodynamic equation of the empirical density is given by a time concatenation of the viscous Burgers equation and the heat equation. Moreover, for $n$ prime and $\gamma>\frac{6}{7}$, we establish convergence in law of the equilibrium fluctuations to an infinite-dimensional Ornstein-Uhlenbeck process.

Based on joint work with Patrícia Gonçalves and Martin Hairer.

Jan 28 - Ivailo Hartasky (Université Claude Bernard Lyon 1)

Title: Constrained-degree percolation

Abstract: In constrained-degree percolation, we consider the $d$-dimensional hypercubic lattice and supply each edge with a clock. Each clock rings only once at an independent and identically distributed time. When the clock of an edge rings, it becomes open, provided that there are currently at most $k-1$ open edges incident with each of its vertices, where $2\leq k\leq 2d$ is a parameter of the model. The study is made difficult by the lack of most key properties of more conventional percolation models such as monotonicity in the underlying graph, finite energy, FKG inequality, finite dependence range, etc. We will start with the first non-perturbative argument for this model, which allows proving that the transition is non-trivial for most values of $k$ for $d\geq 3$, as well as to determine its high-dimension asymptotics. Then, we will show exponential decay in the subcritical regime. The talk is based on joint works with Bernardo de Lima and with Roger Silva available at https://arxiv.org/abs/2010.08955 and https://arxiv.org/abs/2509.16162.

Feb 04 - Paul Dario (Cergy Paris Université)

Title: Delocalization for integer-valued height functions in the presence of a random disorder.

Abstract: In this talk, we will discuss the properties of a model of random interfaces known as the integer-valued Gaussian free field. One of the fundamental features of this model is the existence of a localisation/delocalisation phase transition in two dimensions. From a mathematical perspective, this result was first established by Fröhlich and Spencer in 1981 and has recently been the subject of renewed activity following the works of Lammers, van Engelenburg and Lis, and Aizenman, Harel, Peled and Shapiro. We will present the model and some of its properties. We will then address the following question: does the phase transition persist when the integer-valued Gaussian free field is subject to a random disorder? Specifically, is it observed when the following random constraint is incorporated to the model: we sample a supercritical Bernoulli bond percolation on $\mathbb{Z}^2$ and, for each closed edge of the percolation configuration, we constrain the integer-valued Gaussian free field to take the same values on both ends of the edge? This is joint work with Diederik van Engelenburg and Christophe Garban.

Feb 11 - Maurizia Rossi (Università di Milano-Bicocca)

Feb 18 - Arnout van Enter (University of Groeningen)

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Feb 25 - Federico Sau (Università di Milano)

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Feb 27 - Patricia Gonçalves (Instituto Superior Tecnico Lisboa) - MATHEMATICS COLLOQUIUMLink opens in a new window 

Mar 04 - El Mehdi Haress (University of Leeds)

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Mar 06 - Erwin Bolthausen (University of Zurich) - MATHEMATICS COLLOQUIUMLink opens in a new window 

Mar 11 - Replaced by Maths Colloquium (Mar 6)

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Mar 18 - Replaced by Mar 25

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Mar 11 - William Da Silva (University of Vienna)

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