L6

The Jacobi Ensembles of random matrices have joint distribution of eigenvalues proportional to the integration measure in the Selberg integral. They can also be realised as the singular values of principal submatrices of random unitaries. In this talk we will review some old and new results concerning the distribution of the largest and smallest eigenvalues.

In 2004, Diaconis and Gamburd computed statistics of secular coefficients in the circular unitary ensemble. They expressed the moments of the secular coefficients in terms of counts of magic squares. Their proof relied on the RSK correspondence. We'll present a combinatorial proof of their result, involving the characteristic map. The combinatorial proof is quite flexible and can handle other statistics as well. We'll connect the result and its proof to old and new questions in number theory, by formulating integer and function field analogues of the result, inspired by the Random Matrix Theory model for L-functions.

Partly based on the arXiv preprint https://arxiv.org/abs/2102.11966

Characterising the statistical properties of high dimensional random functions has been one of the central focus of the theory of disordered systems, and notably spin glasses, over the last decades. Applications to machine learning via deep neural network has seen a resurgence of interest towards this problem in recent years. The simplest yet non-trivial quantity to characterise these landscapes is the annealed total complexity, i.e. the rate of exponential growth of the average number of stationary points (or equilibria) with the dimension of the underlying space. A paradigmatic model for such random landscape in the $N$-dimensional Euclidean space consists of an isotropic harmonic confinement and a Gaussian random function, with rotationally and translationally invariant covariance [1]. The total annealed complexity in this model has been shown to display a ”topology trivialisation transition”: for weak confinement, the number of stationary points is exponentially large (positive complexity) while for strong confinement there is typically a single stationary point (zero complexity).

In this talk, I will present recent results obtained for a distinct exactly solvable model of random lanscape in the $N$-dimensional Euclidean space where the random Gaussian function is replaced by a superposition of $M > N$ random plane waves [2]. In this model, we compute the total annealed complexity in the limit $N\rightarrow\infty$ with $\alpha = M/N$ fixed and find, in contrast to the scenario exposed above, that the complexity remains strictly positive for any finite value of the confinement strength. Hence, there is no ”topology trivialisation transition” for this model, which seems to be a representative of a distinct class of universality.

References:

[1] Y. V. Fyodorov, Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices, Phys. Rev. Lett. 92, 240601 (2004) Erratum: Phys. Rev. Lett. 93, 149901(E) (2004).

[2] B. Lacroix-A-Chez-Toine, S. Belga-Fedeli, Y. V. Fyodorov, Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity, arXiv preprint arXiv:2202.03815, submitted to J. Math. Phys.

The sinoatrial node (SAN) is the pacemaker region of the heart.
Recently calcium signals, believed to be crucially important in heart
rhythm generation, have been imaged in intact SAN and shown to be
heterogeneous in various regions of the SAN. However, calcium imaging
is noisy, and the calcium signal heterogeneity has not been
mathematically analyzed to distinguish meaningful signals from
randomness or to identify signalling regions in an objective way. In
this work we apply methods of random matrix theory (RMT) developed for
financial data and used for analysis of various biological data sets
including β-cell collectives and EEG data. We find eigenvalues of the
correlation matrix that deviate from RMT predictions, and thus are not
explained by randomness but carry additional meaning. We use
localization properties of the eigenvectors corresponding to high
eigenvalues to locate particular signalling modules. We find that the
top eigenvector captures a common response of the SAN to action
potential. In some cases, the eigenvector corresponding to the second
highest eigenvalue appears to yield a possible pacemaker region as its
calcium signals predate the action potential. Next we study the
relationship between covariance coefficients and distance and find
that there are long range correlations, indicating intercellular
interactions in most cases. Lastly, we perform an analysis of nearest
neighbor eigenvalue distances and find that it coincides with the
universal Wigner surmise. On the other hand, the number variance,
which captures eigenvalue correlations, is a parameter that is
sensitive to experimental conditions. Thus RMT application to SAN
allows to remove noise and the global effects of the action potential
and thereby isolate the correlations in calcium signalling which are
local. This talk is based on joint work with Chloe Norris with a
preprint found here:
https://www.biorxiv.org/content/10.1101/2022.02.25.482007v1.

The dynamics of quantum many-body systems is intricately related to random matrix theory (RMT), to such a degree that quantum chaos is even defined through random matrix level statistics. However, exact results on this connection are typically precluded by the exponentially large Hilbert space. After a short introduction to the role of RMT in many-body dynamics, I will show how dual-unitary circuits present a minimal model of quantum chaos where this connection can be made rigorous. This will be illustrated using a new kind of emergent random matrix behaviour following a quantum quench: starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design, which can be shown to hold exactly in dual-unitary circuit dynamics.

We consider close-packed tiling models of geometric objects -- a mixture of hardcore dimers and plaquettes -- as a generalisation of the familiar dimer models. Specifically, on an anisotropic cubic lattice, we demand that each site be covered by either a dimer on a z-link or a plaquettein the x-y plane. The space of such fully packed tilings has an extensive degeneracy. This maps onto a fracton-type `higher-rank electrostatics', which can exhibit a plaquette-dimer liquid and an ordered phase. We analyse this theory in detail, using height representations and T-duality to demonstrate that the concomitant phase transition occurs due to the proliferation of dipoles formed by defect pairs. The resultant critical theory can be considered as a fracton version of the Kosterlitz-Thouless transition. A significant new element is its UV-IR mixing, where the low energy behavior of the liquid phase and the transition out of it is dominated by local (short-wavelength) fluctuations, rendering the critical phenomenon beyond the renormalization group paradigm.

Recent advances in Deep Learning have enabled us to explore new application domains in molecular biology and drug discovery - including those driven by complex processes that defy analytical modelling.  However, despite the combined forces of increased data, improving compute resources and continuous algorithmic innovation all bringing previously intractable problems into the realm of possibility, many advances are yet to make a tangible impact for life science discovery.  In this talk, Dr Marcin J. Skwark will discuss the challenge of bringing machine learning innovation to tangible real-world impact.  Following a general introduction of the topic, as well as newly available methods and data, he will focus on the modelling of COVID-19 variants and, in particular, the DeepChain Early Warning System (EWS) developed by InstaDeep in collaboration with BioNTech.  With thousands of new, possibly dangerous, SARS-CoV-2 variants emerging each month worldwide, it is beyond humanities combined capacity to experimentally determine the immune evasion and transmissibility characteristics of every one.  EWS builds on an experimentally tested AI-first computational biology platform to evaluate new variants in minutes, and is capable of risk monitoring variant lineages in near real-time.  This is done by combining an AI-driven protein structure prediction framework with large, spike protein sequence-oriented Transformer models to allow for rapid simulation-free assessment of the immune escape risk and expected fitness of new variants, conditioned on the current state of the world.  The system has been extensively validated in cooperation with BioNTech, both in terms of host cell infection propensity (including experimental assays of receptor binding affinity), and immune escape (pVNT assays with monoclonal antibodies and real-life donor sera). In these assessments, purely unsupervised, data-first methods of EWS have shown remarkable accuracy. EWS flags and ranks all but one of the SARS-CoV-2 Variants of Concern (Alpha, Beta, Gamma, Delta… Omicron), discriminates between subvariants (e.g. BA.1/BA.2/BA.4 etc. distinction) and for most of the adverse events allows for proactive response on the day of the observation. This allows for appropriate response on average six weeks before it is possible for domain experts using domain knowledge and epidemiological data. The performance of the system, according to internal benchmarks, improves with time, allowing for example for supporting the decisions on the emerging Omicron subvariants on the first days of their occurrence. EWS impact has been notable in general media [2, 3] for the system's applicability to a novel problem, ability to derive generalizable conclusions from unevenly distributed, sparse and noisy data, to deliver insights which otherwise necessitate long and costly experimental assays.

Experimental biologists study diseases mostly through their abnormal molecular or cellular features. For example, they investigate genetic abnormalities in cancer, hormonal imbalances in diabetes, or an aberrant immune system in vascular diseases. Moreover, many diseases also have a mechanical component which is critical to their deadliness. Most notably, cancer kills typically through metastasis, where the cancer cells acquire the capability to remodel their adhesions and to migrate. Solid tumours are also characterised by physical changes in the extracellular matrix – the material surrounding the cells. While such physical changes are long known, only relatively recent research revealed that cells can sense altered physical properties and transduce them into chemical information. An example is the YAP/TAZ signalling pathway that can activate in response to altered matrix mechanics and that can drive tumour phenotypes such as the rate of cell proliferation.
Systems-biology models aim to study diseases holistically. In this talk, I will argue that physical signatures are a critical part of many diseases and therefore, need to be incorporated into systems-biology. Crucially, physical disease signatures bi-directionally interact with molecular and cellular signatures, presenting a major challenge to developing such models. I will present several examples of recent and ongoing work aimed at uncovering the relations between mechanical and molecular/cellular signatures in health and disease. I will discuss how blood vessel cells interact mechano-chemically with each other to regulate the passage of cells and nutrients between blood and tissue and how cancer cells grow and die in response to mechanical and geometrical stimuli.

Effectiveness of COVID-19 vaccines was first demonstrated in randomised trials, but many questions of vital importance to vaccination policies could only be addressed in subsequent observational studies. The pandemic led to a step change in the availability of population-level linked electronic health record data, analysed in privacy-protecting Trusted Research Environments, across the UK. I will discuss methodological approaches to estimating causal effects of COVID-19 vaccines, and their application in estimating vaccine effectiveness and quantifying waning vaccine effectiveness. I will present results from recent analyses using detailed linked data on up to 24 million people in the OpenSAFELY Trusted Research Environment, which was developed by the University of Oxford's Bennett Institute for Applied Data Science.