Past Random Matrix Theory Seminars

The two-point Chowla conjecture predicts that $\sum_{x<n<2x} \lambda(n)\lambda(n+1) = o(x)$ as $x\to \infty$, where $\lambda$ is the Liouville function (a $\{\pm 1\}$-valued multiplicative function encoding the parity of the number of prime factors). While this remains an open problem, weaker versions of this conjecture are known. In this talk, we outline an approach initiated by Helfgott and Radziwill, which reformulates the problem in terms of bounding the eigenvalues of a certain matrix.

The Wasserstein metric originates in the theory of optimal transport, and among many other applications, it provides a natural way to measure how evenly distributed a finite point set is. We give a survey of classical and more recent results that describe the behaviour of some random point processes in Wasserstein metric, including the eigenvalues of some random matrix models, and explain the connection to the logarithm of the characteristic polynomial of a random unitary matrix. We also discuss a simple random walk model on the unit circle defined in terms of a quadratic irrational number, which turns out to be related to surprisingly deep arithmetic properties of real quadratic fields.

One curious little fact about the Riemann zeta function is that if you evaluate its derivatives at the zeros of zeta, then on average this is real and positive (even though the function is complex). This has been proven for some time now, but the aim of this talk is to generalise the question further (higher derivatives, complex moments) and gain insight using random matrix theory. The takeaway message will be that there are a multitude of different proof techniques in RMT, each with their own advantages

We consider two approximation schemes for the construction of a class of non-Gaussian multiplicative chaos, and show that they give rise to the same limit in the entire subcritical regime. Our approach uses a modified second moment method with the help of a new coupling argument, and does not rely on any Gaussian approximation or thick point analysis. As an application, we extend the martingale central limit theorem for partial sums of random multiplicative functions to L^1 twists. This is a joint work with Ofir Gorodetsky.

The Critical 2d Stochastic Heat Flow arises as a non-trivial solution
of the Stochastic Heat Equation (SHE) at the critical dimension 2 and at a phase transition point.
It is a log-correlated field which is neither Gaussian nor a Gaussian Multiplicative Chaos.
We will review the phase transition of the 2d SHE, describe the main points of the construction of the Critical 2d SHF
and outline some of its features and related questions. Based on joint works with Francesco Caravenna and Rongfeng Sun.

In this talk I will consider properties of the disordered elastic manifold, describing an N-dimensional field u(x) defined for sites x of a d-dimensional lattice of linear size L. This prototypical model is used to describe interfaces in a wide range of physical systems [1]. I will consider properties of the ground-state energy for this model whose optimal configuration u_0(x) results from a compromise between the disorder which tend to favour sharp variations of the field and elastic interactions that smoothen them. I will study in particular the limit of large N>>1 and finite d which has been studied extensively in the physics literature (notably using the replica approach) [1,2] and has recently been considered in a series of paper by Ben Arous and Kivimae [3,4]. For this model, we compute exactly the large deviation function of the ground-state energy E_0, showing that it displays replica-symmetry breaking transitions. As an interesting outcome of this study, we show analytically the validity of the scaling law conjectured by Mezard and Parisi [2] for the variance of the ground-state energy. The latter relates the exponent of the variance Var(E_0)\sim L^{2\theta} such that \theta=2\zeta+d-2 with \zeta the exponent characterising the transverse fluctuations of the optimal configuration u_0(x), i.e.  (u_0(x)-u_0(x+y))^2\sim |y|^{2\zeta}. This work is done in collaboration with Y.V. Fyodorov (KCL) and P. Le Doussal (LPENS, CNRS).

[1] Giamarchi, T., & Le Doussal, P. (1998). Statics and dynamics of disordered elastic systems. In Spin glasses and random fields (pp. 321-356).

[2] Mézard, M., & Parisi, G. (1991). Replica field theory for random manifolds. Journal de Physique I, 1(6), 809-836.

[3] Ben Arous, G., & Kivimae, P. (2024). The Free Energy of the Elastic Manifold. arXiv preprint arXiv:2410.19094.

[4] Ben Arous, G., & Kivimae, P. (2024). The larkin mass and replica symmetry breaking in the elastic manifold. arXiv preprint arXiv:2410.22601.

It is a remarkable property of random matrices, that their resolvents tend to concentrate around a deterministic matrix as the dimension of the matrix tends to infinity, even for a small imaginary part of the involved spectral parameter.
These estimates are called local laws and they are the cornerstone in most of the recent results in random matrix theory. 
In this talk, I will present a novel method of proving single-resolvent and multi-resolvent local laws for random matrices, the Zigzag strategy, which is a recursive tandem of the characteristic flow method and a Green function comparison argument. Novel results, which we obtained via the Zigzag strategy, include the optimal Eigenstate Thermalization Hypothesis (ETH) for Wigner matrices, uniformly in the spectrum, and universality of eigenvalue statistics at cusp singularities for correlated random matrices. 
 

Based on joint works with G. Cipolloni, L. Erdös, O. Kolupaiev, and V. Riabov.

Selberg’s CLT informs us that the logarithm of the Riemann zeta function evaluated on the critical line behaves as a complex Gaussian. It is natural, therefore, to study how far this Gaussianity persists. This talk will present conditional and unconditional results on atypically large values, and concerns work joint with Louis-Pierre Arguin and Asher Roberts.

Motivated by a recent experiment on a superconducting quantum
information processor, I will discuss the Floquet quantum Ising model in
the presence of integrability- and symmetry-breaking random fields. The
talk will focus on the relation between boundary spin correlations,
spectral pairings, and effects of the random fields. If time permits, I
will also touch upon self-similarity in the dynamic phase diagram of
Fibonacci-driven quantum Ising models.
 

We study networks of firms in which inputs for production are not easily substitutable, as in several real-world supply chains. Building on Robert May's original argument for large ecosystems, we argue that such networks generically become dysfunctional when their size increases, when the heterogeneity between firms becomes too strong, or when substitutability of their production inputs is reduced. At marginal stability and for large heterogeneities, crises can be triggered by small idiosyncratic shocks, which lead to “avalanches” of defaults. This scenario would naturally explain the well-known “small shocks, large business cycles” puzzle, as anticipated long ago by Bak, Chen, Scheinkman, and Woodford. However, an out-of-equilibrium version of the model suggests that other scenarios are possible, in particular that of `turbulent economies’.

Quantum computers are becoming a reality and current generations of machines are already well beyond the 50-qubit frontier. However, hardware imperfections still overwhelm these devices and it is generally believed the fault-tolerant, error-corrected systems will not be within reach in the near term: a single logical qubit needs to be encoded into potentially thousands of physical qubits which is prohibitive.
 
Due to limited resources, in the near term, hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage but these need to resort to quantum error mitigation techniques. I will explain the basic concepts and introduce hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage. These have the potential to solve real-world problems---including optimisation or ground-state search---but they suffer from a large number of circuit repetitions required to extract information from the quantum state. I will detail a range of application areas of randomised quantum circuits, such as quantum algorithms, classical shadows, and quantum error mitigation introducing recent results that help lower the barrier for practical quantum advantage.

 

We will present some of the remarkable properties of eigenvalue trajectories for rank-one perturbations of random matrices, with an emphasis on two models of particular interest, namely weakly non-Hermitian and weakly non-unitary matrices. In both cases, precise estimates can be obtained for the critical timescale at which an outlier can be observed with high probability. We will outline the proofs of these results and highlight their significance in connection with quantum chaotic scattering. (Based on joint works with L. Erdös and J. Reker)

I will present joint work with Alessandro Fazzari in which we prove precise conditional estimates for the third (non-absolute) moment of the logarithm of the Riemann zeta function, beyond the Selberg central limit theorem, both for the real and imaginary part. These estimates match predictions made in work of Keating and Snaith. We require the Riemann Hypothesis, a conjecture for the triple correlation of Riemann zeros and another ``twisted'' pair correlation conjecture which captures the interaction of a prime power with Montgomery's pair correlation function. This conjecture can be proved on a certain subrange unconditionally, and on a larger range under the assumption of a variant of the Hardy-Littlewood conjecture with good uniformity.

The ground state of N noninteracting Fermions in a rotating harmonic trap enjoys a one-to-one mapping to the complex Ginibre ensemble. This setup is equivalent to electrons in a magnetic field described by Landau levels. The mean, variance and higher order cumulants of the number of particles in a circular domain can be computed exactly for finite N and in three different large-N limits. In the bulk and at the edge of the spectrum the result is universal for a large class of rotationally invariant potentials. In the bulk the variance and entanglement entropy are proportional and satisfy an area law. The same universality can be proven for the quaternionic Ginibre ensemble and its corresponding generalisation. For the real Ginibre ensemble we determine the large-N limit at the origin and conjecture its universality in the bulk and at the edge.

The scattering matrix in quantum mechanics must be unitary to ensure the conservation of the number of particles, hence their 
eigenvalues are unimodular.  In systems with fully developed Quantum Chaos  the statistics of those unimodular 
eigenvalues  is well described by  the Poisson kernel.
However, in real experiments  the associated scattering matrix is sub-unitary due to intrinsic losses,  and
 the moduli of S-matrix eigenvalues become non-trivial,  yet the corresponding theory is not well-developed in general.  
 I will present some results for the mean density of those moduli in the framework of random matrix models for the case of broken time-reversal invariance,
and discuss a way to get a generalization of the Poisson kernel to systems with uniform losses.

Let X and H be large, and consider n ranging from 1 to X. For an arithmetic function f(n), what is the best mean square approximation of f(n) by a restricted divisor sum (a function of the sort sum_{d|n, d < H} a_d)? I hope to explain how for a wide variety of arithmetic functions, when X grows and H grows like a power of X, a solution of this problem is connected to the evaluation of random matrix integrals. The problem is connected to some combinatorial formula for computing high moments of traces of random unitary matrices and I hope to discuss this also.

We will present a matching upper and lower bound for the right tail probability of the maximum of a random model of the Riemann zeta function over short intervals.  In particular, we show that the right tail interpolates between that of log-correlated and IID random variables as the interval varies in length. We will also discuss a new normalization for the moments over short intervals. This result follows the recent work of Arguin-Dubach-Hartung and is inspired by a conjecture by Fyodorov-Hiary-Keating on the local maximum over short intervals.