Past Random Matrix Theory Seminars

Spectral properties of Schrödinger operators are studied a lot in mathematical physics. They can give the description of trapped fermionic particles. This presentation will focus on the non-interacting case. I will explain why it is relevant to estimate L^p bounds of orthonormal families of eigenfuntions at the semiclassical regime and then, give the main ideas of the proof.

In this talk, I will firstly give asymptotic formulas for the moments of the n-th derivative of the characteristic polynomials from the CUE. Secondly, I will talk about the connections between them and a solution of certain Painleve differential equation. This is joint work with Jonathan P. Keating.
 

The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlevé equations to the case of the several Painlevé “particles” coupled via the Calogero type interactions. In 2014, I. Rumanov discovered a remarkable fact that a particular case of the Calogero– Painlevé II equation describes the Tracy-Widom distribution function for the general $\beta$-ensembles with the even values of parameter $\beta$. in 2017 work of M. Bertola, M. Cafasso , and V. Rubtsov, it was proven that all Calogero-Painlevé systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous asymptotic analysis of the Calogero-Painlevé equations which in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of $\beta$ beyond the classical $\beta =1, 2, 4$. In the talk these recent developments will be outlined with a special focus on the Calogero-Painlevé system corresponding to $\beta = 6$. This is a joint work with Andrei Prokhorov.

The resolvents of finite volume restricted Hamiltonians, G^(⍵), have long been used to describe the localization of quantum systems. More recently, projected Green's functions (pGfs) -- finite volume restrictions of the resolvent -- have been applied to translation invariant free fermion systems, and the pGf zero eigenvalues have been shown to determine topological edge modes in free-fermion systems with bulk-edge correspondence. In this talk, I will connect the pGfs to the G^(⍵) appearing in the transfer matrices of quasi-periodic systems and discuss what pGF zeros can tell us about the solutions to transfer matrix equations. Using these methods, we re-examine the critical almost-Matthieu operator and notice new guarantees on analytic regions of its resolvent for Liouville irrationals.
 

Conformal Field Theories (CFT) are believed to be exactly solvable once their primary scaling fields and their 3-point functions are known. This input is called the spectrum and structure constants of the CFT respectively. I will review recent work where this conformal bootstrap program can be rigorously carried out for the case of Liouville CFT, a theory that plays a fundamental role in 2d random surface theory and many other fields in physics and mathematics. Liouville CFT has a probabilistic formulation on an arbitrary Riemann surface and the bootstrap formula can be seen as a "quantization" of the plumbing construction of surfaces with marked points axiomatically discussed earlier by Graeme Segal. Joint work with Colin Guillarmou, Remi Rhodes and Vincent Vargas

This talk is motivated by computing correlations for domino tilings of the Aztec diamond.  It is inspired by two of the three distinct methods that have recently been used in the simplest case of a doubly periodic weighting, that is the two-periodic Aztec diamond. This model is of particular probabilistic interest due to being one of the few models having a boundary between polynomially and exponentially decaying macroscopic regions in the limit. One of the methods to compute correlations, powered by the domino shuffle, involves inverting the Kasteleyn matrix giving correlations through the local statistics formula. Another of the methods, driven by a Wiener-Hopf factorization for two- by-two matrix valued functions, involves the Eynard-Mehta theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by an LU- and UL-decomposition, based on a matrix refactorization, for the product of the transition matrices. In this talk, we present results to say that the evolution of the face weights under the domino shuffle and the matrix refactorization is the same. This is based on joint work with Maurice Duits (Royal Institute of Technology KTH).  

I will present two examples of log-correlated fields in 2 dimensions. It is well known that the log-characteristic polynomial of a uniform unitary matrix converges toward a 1 dimensional log-correlated field, and our first example will be obtained from a dynamical version of this model. The second example will be obtained from a radically different construction, based on the Brownian loop soup that we will introduce. It will lead to a whole family of log-correlated fields. We will focus on the description of the behaviour of these objects, more than on rigorous details.

Conformal blocks appear in several areas of mathematical physics from random geometry to black hole physics. A probabilistic notion of conformal blocks using gaussian multiplicative chaos measures was recently formulated by Promit Ghosal, Guillaume Remy, Xin Sun, Yi Sun (arxiv:2003.03802). In this talk, I will show that the semiclassical limit of the probabilistic conformal blocks recovers a special case of the elliptic form of Painlevé VI equation, thereby proving a conjecture by Zamolodchikov. This talk is based on an upcoming paper with Promit Ghosal and Andrei Prokhorov.

I will present recent results on the growth of entanglement between two adjacent regions in a tripartite, one-dimensional many-body system after a quantum quench. Combining a replica trick with a space-time duality transformation a universal relation between the entanglement negativity and Renyi-1/2 mutual information can be derived, which holds at times shorter than the sizes of all subsystems. The proof is directly applicable to any local quantum circuit, i.e., any lattice system in discrete time characterised by local interactions, irrespective of the nature of its dynamics. The derivation indicates that such a relation can be directly extended to any system where information spreads with a finite maximal velocity. The talk is based on Phys. Rev. Lett. 129, 140503 (2022).

The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlevé equations to the case of the several Painlevé “particles” coupled via the Calogero type interactions. In 2014, I. Rumanov discovered a remarkable fact that a particular case of the Calogero– Painlevé II equation describes the Tracy-Widom distribution function for the general $\beta$-ensembles with the even values of parameter $\beta$. in 2017 work of M. Bertola, M. Cafasso , and V. Rubtsov, it was proven that all Calogero-Painlevé systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous asymptotic analysis of the Calogero-Painlevé equations which in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of $\beta$ beyond the classical $\beta =1, 2, 4$. In the talk these recent developments will be outlined with a special focus on the Calogero-Painlevé system corresponding to $\beta = 6$. This is a joint work with Andrei Prokhorov.

Neural networks are the most practically successful class of models in modern machine learning, but there are considerable gaps in the current theoretical understanding of their properties and success. Several authors have applied models and tools from random matrix theory to shed light on a variety of aspects of neural network theory, however the genuine applicability and relevance of these results is in question. Most works rely on modelling assumptions to reduce large, complex matrices (such as the Hessians of neural networks) to something close to a well-understood canonical RMT ensemble to which all the sophisticated machinery of RMT can be applied to yield insights and results. There is experimental work, however, that appears to contradict these assumptions. In this talk, we will explore what can be derived about neural networks starting from RMT assumptions that are much more general than considered by prior work. Our main results start from justifiable assumptions on the local statistics of neural network Hessians and make predictions about their spectra than we can test experimentally on real-world neural networks. Overall, we will argue that familiar ideas from RMT universality are at work in the background, producing practical consequences for modern deep neural networks.

I will describe the interaction between a single soliton and a gas of solitons, providing for the first time a mathematical justification for the kinetic theory as posited by Zakharov in the 1970s.  Then I will explain how to use random matrix theory to introduce randomness into a large collection of solitons.

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 this talk, I will investigate the origin of Euclidean wormholes in the gravitational part integral in the context of AdS/CFT. These geometries are confusing since they prevent products of partition functions to factorize, as they should in any quantum mechanical system. I will briefly review the different proposals for the origin of these wormholes, one of which is that one should consider ensemble of average of boundary systems instead of a fixed quantum system with a fixed Hamiltonian. I will explain that it seems unlikely that one can average over CFTs and present a new idea: averaging over approximate CFTs, which I will define. I will then study the variance of the crossing equation in an ensemble relevant for 3d gravity. Based on work in progress with de Boer, Jafferis, Nayak and Sonner.

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

In this talk we will discuss the correlations of the Riemann Zeta in various ranges, and prove a new result for correlations of squares. This problem is closely related to correlations of the characteristic polynomial of CUE with a very subtle difference. We will explain where this difference comes from, and what it means for the moments of moments of the Riemann Zeta, and its maximum in short intervals.

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 characteristic polynomial of a random Hermitian matrix induces naturally a field on the real line. In the case of the Gaussian Unitary ensemble (GUE), this fields is expected to have a very special correlation structure: the logarithm of this field is log-correlated and its maximum is at the heart of a conjecture from Fyodorov and Simm predicting its asymptotic behavior.   As a first step in this direction, we obtained in collaboration with R. Butez and O. Zeitouni, a central limit theorem for the logarithm of the characteristic polynomial of the Gaussian beta Ensembles and for a certain class of random Jacobi matrices. In this talk, I will explain how the tridiagonal representation of the GUE and orthogonal polynomials techniques allow us to analyse the fluctuations of the characteristic polynomial.

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.

In this talk we cover recent work in collaboration with Diego Granziol and Steve Roberts where we study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian and derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. 

In this talk, we give an overview of recent results on the fluctuation of the statistic $\sum_{i\neq j} f(L_N(\theta_i-\theta_j))$ for the Circular Beta Ensemble in the global, mesoscopic and local regimes. This work is morally related to Johansson's 1988 CLT for the linear statistic $\sum_i f(\theta_i)$ and Lambert's subsequent 2019 extension to the mesoscopic regime. The special case of the CUE ($\beta=2$) in the local regime $L_N=N$ is motivated by Montgomery's study of pair correlations of the rescaled zeros of the Riemann zeta function. Our techniques are of combinatorial nature for the CUE and analytical for $\beta\neq2$.

I will present a collection of moment computations over the unitary, symplectic and special orthogonal matrix ensembles that I've done throughout my thesis. I will focus on the methods used, the motivation from number theory, the relationship to Painlev\'e equations, and directions for future work.