Events

Title: Lie Symmetries and Bi-Hamiltonian structures of the Pais-Uhlenbeck Oscillator

Abstract:

In this talk I will give a pedagogical introduction to the problems usually associated with theories where the Lagrangian has a time derivative of order higher than two, using the Pais-Uhlenbeck model as an illustrative example. I will then discuss the approach we have taken in two recent research projects (see [2, 1]) to try and find a way to mitigate these problems. 

In this approach, the Lie symmetries of the Pais-Uhlenbeck Oscillator (PUO) are identified. They are then used to generate the Bi-Hamiltonian structure of this system. We then study how we might leverage this Bi-Hamiltonian structure to mitigate the pathologies associated with theories where, as in the case of the PUO, the Lagrangian admits time derivatives of order two or higher. Theories of this nature are usually thought to admit classical Hamiltonians that are unbounded from below, and either unbounded energy eigenvalue spectra or negative norm ”ghost” states at the quantum level. For this reason they are often generally referred to as ”ghost ridden”. However the appearance of a Bi-Hamiltonian structure can be shown to allow for the generation of positive definite Hamiltonians with appropriate symplectic structure in the case of the free theory. In addition, we leverage the Bi-Hamiltonian structure to construct families of transforms between the higher derivative Pais-Uhlenbeck oscillator and a two dimensional system where the dynamics and symplectic structure are preserved. We will also discuss the implications of including interactions, which provides an important caveat to this argument.

Title: On the evolution of the mean-field random cluster model and its free energy

Abstract: In this talk I will give an overview of the random cluster (or FK-percolation) model of random graphs. I will explain how this generalizes the Erdos-Renyi, Ising and Potts models and give a picture of the phase transition in the mean-field case.

I will then sketch ongoing work into understanding the structure and phase transition of the model in the complete bipartite case, determining its limiting free energy. If time permits, I will show how this can be used to obtain the rate function for the large deviation event of the supercritical mean-field Potts model being bipartite.

Scattering experiments are still one of the best ways for physicists to understand and probe nature. While for classical systems there is no conceptual problem, if the probed system is inherently quantum, it is very hard to understand the scattering process. For instance, what is the correct wave function to describe the system with? And what precisely do we measure?

To solve these mysteries, we first need to turn the Schrödinger equation into something useful. This is the essence of scattering theory, a beautiful theory where mathematical peculiarities acquire precise physical meaning. It showcases nicely that operators on infinite dimensional vector spaces are way more than just a matrix, and that proper mathematical understanding is the key to dealing with them. Therefore, I will first give a “poor man’s” introduction to infinite dimensional calculus. Then, we are going to use this to derive the Lippmann-Schwinger equation of scattering theory. The derivation and the resulting equation seem simple, but are actually packed with an incredibly amount of physical and mathematical interpretation.

The Bethe Ansatz is a very powerful technique with applications in various fields of theoretical physics that ranges from condensed matter to quantum field theory and string theory. This method is widely used to obtain exact solutions in the context of integrable systems but even in real cold atom lab experiments certain features of these models can nowadays  be reproduced. In this talk we will see, without going in deep details, how a simple 1-dimensional discrete system of atoms, can be described by a "fictitious spin fluid" whose excitations/waves completely and exactly describe the dynamics of the system.

In many areas of statistics we care more about large and rarely seen events than those closer to the mean. For rivers we look at river flows large enough to cause flooding, for stock markets we look for market crashes, for insurance data we look for extra large loses. By their nature these extreme events don't happen often and so we need to take care when doing analysis with them. This lack of data is made worse by the fact that often we need to extrapolate, i.e. finding what a 1-in-500 year event is with only 100 years of data. To try and deal with these issues we turn to extreme value theory, which gives asymptotic distributions to maxima similarly to the central limit theorem giving limiting distributions to sample means. In this talk a background of extreme value theory will be given, first for the univariate case and then how to expand into the multivariate setting, with application to river flow data.

They will be reviewing the paper 'The Asymmetric Simple Exclusion Process : An Integrable Model for Non-Equilibrium Statistical Mechanics' by Oliver Golinelli and Kirone Mallick from 2006.

The paper can be found here:

 https://arxiv.org/pdf/cond-mat/0611701

Some Models in Percolation Theory

The mathematical model of percolation was introduced by Broadbent and Hammersley in 1957 as a probabilistic model of porosity. Since then, percolation theory has become one of the most studied subfields of probability theory, both on its own and as a tool for studying phase transitions in statistical mechanics.

In this talk, I will introduce (hopefully) 4 interesting percolation models with non-trivial phase transitions, three of which exhibit interesting but complicated dependencies.

Presentation here.

This is an elementary introduction to Free Probability, which, in the simplest sense, is the study of random variables that do not commute.  Without assuming any previous knowledge of free probability or random matrix theory, I will introduce the elementary results including the idea of Free Cumulants and the Free Central Limit Theorem. If time permits, I will try to demonstrate an application by deriving the Marchenko-Pastur Law for the eigenvalue density of Wishart matrices.

Title: Towards and on the Variational Formulation of the Fokker-Planck Equation

Abstract: The seminal paper “The variational formulation of the Fokker-Planck equation” from R. Jordan, D. Kinderlehrer, and F. Otto establishes - via a time discretization scheme - a connection between the Fokker-Planck Equation of the overdamped Langevin Dynamics and the gradient flow of relative entropy with respect to the Wasserstein-2 metric. In this talk, we want to understand this result, which is interesting from both a physical and mathematical perspective, step-by-step. If time permits, we will also discuss examples of subsequent work that builds upon this foundational result.

Trouble with Renormalization: Gibbsianity for Infinite Volumes, Quasilocality and Fuzzy Gibbs Measures  

Abstract: Renormalization Group (RG) is a powerful technique for studying statistical mechanical systems in the proximity of critical points via coarse-graining. However, in the late 70s and early 80s, some of its principle tenets started being questioned, due to certain pathologies appearing in applications. Fast forward to early 90s, a rigorous framework was established and the source of pathologies was identified to be the loss of Gibbs property.

In this talk, I will introduce the general notion of Gibbs measures in Z^d and identify what can go wrong under renormalization. I will also introduce a notion of fuzzy Gibbs measures, generalization of both renormalized Gibbs measures (under a deterministic RG transformation) as well as hidden Markov models. I will state a conjecture about the equivalent condition for Gibbsianity of fuzzy Gibbs measures and a very practical ("semi-published") sufficient condition for it. If time permits, I will show you two nice fuzzy Gibbs models, where the latter can be used.

Title: Introduction to Algebraic Quanstum Statistical Mechanics and Lieb-Robinson bounds

Abstract:

In this talk I will present an introduction to the C* algebra framework of statistical mechanics. To start, we will see two main motivations for adapting such a framework. Firstly, one sees that there are inequivalent representations of quantum systems when taking the thermodynamic (infinite volume) limit. Secondly, we will see how Gibbs (equilibrium) states aren’t necessarily well-defined in that limit. We will then discuss how C* algebras solve these problems by essentially dealing with physical systems directly in the thermodynamic limit. Finally, we will show one of the most important results of algebraic quantum mechanics: the Lieb-Robinson bound, which is an upper bound on the velocity with which information can propagate in quantum (non-relativistic) systems, giving rise to a light-cone effect: when the space-time separation of two observables is large enough, then their commutator is exponentially small.

Random matrix theory techniques for analysing artificial neural networks

Authors: J. Peltier, C. Genete-Visan, H. Alston and T. Bertrand

Abstract: 

Pattern formation is ubiquitous in biological systems. Oftentimes, patterns which emerge from the nonequilibrium dynamics of the underlying physical system are critical for the organism to perform certain functions. An example is that of membraneless organelles which rely on liquid-liquid phase separation (LLPS) to exist in cells. Mathematically, LLPS is described by the Cahn-Hilliard equation. Recently, this equilibrium model was extended to couple liquids with interaction terms that drive it out of thermodynamic equilibrium, such as chemical reactions or nonreciprocal interactions. The latter drives the emergence of travelling waves, leading to the nonreciprocal Cahn-Hilliard model. 

We study analytically and computationally a model involving two coupled fields governed by Cahn-Hilliard-like physics supplemented by nonreciprocal couplings and transmutation. Strikingly, we reveal the existence of an interface instability leading to travelling waves along the fields’ interfaces which we coin chainsaws. This is a prime novel example of the spectacular interfacial dynamics that can arise in nonequilibrium systems. A field level and an interface level linear stability analysis allows to prove the existence of chainsaws and locate them in the phase space. Additional perturbative simulations allow us to understand how the wavelengths with which we perturb the fields influence the nature of steady state. 

Introduction to 

Abstract: Simulating quantum systems in the thermodynamic limit may not require large quantum computers. Tensors networks have produced classical state-of-the-art algorithms to probe quantum critical systems by constructing efficient representations of states in a restricted domain of Hilbert space. Given the natural mapping between tensor networks and quantum circuits, tensor network-inspired quantum circuits present a promising framework for NISQ devices to investigate large quantum systems. 

In this talk, I outline this framework and give recent results when applying translationally invariant tensor network circuits on current hardware.  These show that existing hardware can be used to quench across the quantum critical point in the 1D transverse field Ising model, capturing subtle features such as the dynamical phase transition.

Abstract: A line of work in theoretical machine learning uses statistical physics tools to analyse simple models which help in understanding some unexpected behaviors of large and complex architectures. I will tell you about some work I did in this direction, where, using the replica method from statistical physics of disordered systems and spin glasses, we characterise high-dimensional asymptotic performance of classification and robust regression estimators. The non-trivial extension in our work is from the usual convenient Gaussian data assumption to heavy-tailed data distributions.

For classification, we characterise the learning of a mixture of two clouds by studying the generalisation performance of the obtained estimator, we analyse the role of regularisation and analytically derive the data separability transition. 

For robust regression, we consider contamination of both the covariates and labels, providing a sharp asymptotic characterisation of M-estimators trained on data with heavy-tailed covariate and label noise data distributions. We show that, despite being consistent, the Huber loss with optimally tuned location parameter is suboptimal in the high-dimensional regime in the presence of heavy-tailed noise, highlighting the necessity of regularisation to achieve optimal performance, and we derive the decay rates for the excess risk of ridge regression.

Classification paper: https://arxiv.org/abs/2304.02912

Robust regression paper: https://arxiv.org/abs/2309.16476

Abstract: Understanding the dynamics of many-body quantum or classical systems is a very complicated problem. One of the most powerful approaches is the hydrodynamic approximation. It describes the evolution of conserved quantities in terms of coupled PDE equations (known as the Euler and Navier-Stokes equation). While the equations are known for centuries and are experimentally incredibly well established (literally) all across the universe, our theoretical understanding of both their emergence* and their behaviour remains surprisingly poor**.

I will present a physically motivated derivation of these equations: As you will see it is straight forward and based only on few basic assumptions, which is also the reason why it applies to so many different systems. After a short moment of triumph, we will quickly encounter a whole bunch of problems, some more obvious than others. I will complement the general discussion with new results in a specific model (hard-rods), which challenge various aspects of the derivation. Looking forward to an interesting open discussion with you.

* Win eternal fame

** Win eternal fame + 1e6$

Abstract: Solving for dynamical quantities of locally interacting many-body quantum systems is, in most cases, a hard problem - both numerically and analytically. Integrability is a tool that helps us get round this, but one which relies on a large amount of structure (specifically, the existence of an extensive set of simple conserved quantities); many interesting systems - particularly chaotic ones - do not possess it. A great deal of effort in many-body physics is hence being devoted to developing other means to analytically and numerically study many-body dynamics. Over the last decade, a number of toy models of many-body quantum dynamics have emerged in the framework of brickwork quantum circuits that realise this, marrying analytical tractability with generically chaotic dynamics. One particularly interesting example are dual-unitary quantum circuits, where a space-time symmetry encoded into the dynamics can be leveraged to attain a high degree of solvability. In this talk, I will provide an introduction to dual-unitary circuits and how interesting dynamical quantities, such as local correlation functions, can be exactly calculated in them. These calculations will, however, reveal some slightly non-universal features of dual-unitary dynamics; with this in mind, I will in the latter part of my talk discuss recent explorations of generalised and perturbed dual-unitarity circuits, that are both being utilised as starting points for analytical studies of more universal quantum many-body dynamics.

Presentation PDF.

Abstract: In recent years, several authors have demonstrated that solving the Schrodinger equations for atoms and small molecules using deep neural networks within a variational Monte Carlo framework can offer accurate ground-state energies comparable to those obtained by gold-standard quantum chemistry methods. This is quite an exciting prospect, as variational Monte Carlo offers a much more palatable computational complexity with electron number than, for example, coupled-cluster methods.

In this talk I will discuss the basic principles of neural network variational Monte Carlo, some of the advances that have been made in the last few years, and the many exciting prospects for future research directions in the field. I will focus particularly on my own work demonstrating how the lack of dependence on a basis set – a well known generic advantage for neural network models in any setting – makes neural network variational Monte Carlo particularly well suited to certain problems, namely the ab initio study of quantum phase transitions and the ground-state properties of multi-component systems (positron binding to molecules, nuclear quantum effects).

Abstract: What is Econophysics? Why does it have such a silly name? Can you do some calculations? What do you do all day?’. In this talk, I will ignore most of these questions and instead give a tasting menu of Econophysics papers that summarise some important findings from the 30 years of research that make up this remarkable field. The menu is as follows:

Aperitivo: Stylised Facts

Antipasti: Two Flavours of Complexity

Primi: Multiscaling and Stock Dependence

Secondi: Identifying Structure Through Dependency Measures 

Insalata: Momentum

Dolci: Parameter-free Clustering

Digestivo: Multivariate Musy-Bacry model

Abstract: When electrons move throughout a medium, it deforms the medium that it's moving through. We call the electron together with the deformation, a polaron. There are weak and strong polarons. Within strong polarons, there are small and large ones. I'll go through calculations and approximations that allows us to say something about large polarons.

Abstract: An old question of Dedekind asks for the number of antichains (monotone Boolean functions) in the Boolean lattice on $n$ elements. After a long series of increasingly precise results, Korshunov determined this number up to a multiplicative factor of (1+o(1)). We revisit Dedekind’s problem and study the typical structure of antichains using tools from probability and statistical physics. This yields a number of results which include refinements of Korshunov’s asymptotics, asymptotics for the number of antichains of a fixed size, and a 'sparse' version of Sperner’s Theorem. 

Joint work with Matthew Jenssen and Jinyoung Park.

Presentation PDF.

Abstract: Standard Bayesian modelling involves specifying a likelihood and prior over some finite dimensional parameter space. If we suspect there is additional uncertainty that is not captured by the prior, we have two options. The first is to construct and alternative prior, observe the data, and then choose the ‘best’ model based on some criteria. The second is to enlarge the prior to account for the additional uncertainty before proceeding any further. The second option is the correct approach. We may achieve this using nonparametric priors, characterized as distributions over infinite dimensional parameter spaces, for example function spaces. In this discussion we will focus on the first nonparametric prior, the Dirichlet Process (DP), which can be thought of as a distribution over distributions. We will look at some theoretical properties of the DP which make it desirable as a nonparametric prior and demonstrate its use in nonparametric density estimation.

Presentation PDF.

• Goal: We'd like to understand how the renormalisation group in QFT and statistical mechanics are connected. 

• 2023-08-16 Wed, Renormalisation Group, 1

  • To get started, read up and including section on Gaussian fixed points for Tong's notes (p. 53-68). Alternative source are these Cambridge physics notes Ch. 4, ~ p. 39-50).

  • We left with the question: Is the critical point ( ξ = ∞ ) and fixed point (RG dynamical systems sense) the same thing?

• 2023-10-18 Wed, Friedrich Hübner, Renormalization of the delta potential

• 2024-02-07, Renormalisation Group, 2, S-2.25

  • Let's read up to connection to QFT in Tong's notes (p. 68-88), understanding the beta-functions. Let's also read these Cambridge AQFT notes to understand the beta-functions from the QFT side, and join the two up during the journal club.

Abstract: Kasteleyn theory is a tool used by probabilists to obtain explicit solutions for the partition functions and correlation functions for models of random surfaces and random tilings. In this talk, we will hone in on a specific model and highlight its rich behaviour, making connections with areas of probability, algebra and statistical physics.

A fundamental introduction to Markov processes, discussing continuous-time Markov Processes on a discrete state space, writing down the Master equation for the same case, and understanding the steady state conditions, including detailed balance, the Kolmogorov loop condition etc.