Past Random Matrix Theory Seminars

The critical Fortuin–Kasteleyn random planar map with parameter q>0 is a model of random (discretised) surfaces decorated by loops, related to the q-state Potts model. For q<4, Sheffield established a scaling limit result for these discretised surfaces, where the limit is described by a so-called Liouville quantum gravity surface decorated by a conformal loop ensemble. At q=4 a phase transition occurs, and the correct rescaling needed to obtain a limit has so far remained unclear. I will talk about joint work with William Da Silva, XinJiang Hu, and Mo Dick Wong, where we identify the right rescaling at this critical value and prove a number of convergence results.

We present a new dynamical way of establishing local laws for sparse random matrices, the Bernoulli flow method. It is based on a Markovian jump process, where the entries of the matrix jump independently from 0 to 1 at rate one. As an application, we show optimal (up to a constant) isotropic delocalisation for bulk eigenvectors of Erdös-Rényi graphs with edge probability p \geq (log N)^2/N. In the same regime, we obtain a local law with optimal (up to a constant) error bounds. Joint work with Antti Knowles.

By using the ratios conjecture, we study the asymptotic behaviour of the mean square of long truncations of the Dirichlet series for \(\bigl(\zeta'/\zeta\bigr)^{k}\) near the critical line. We explain the connection between this problem and the variance of the convoluted von Mangoldt function in short intervals. We obtain an explicit leading piecewise polynomial in the length parameter which is consistent with the microscopic-shift results of Fan Ge. We also discuss other RMT results for moments of the logarithmic derivative of characteristic polynomials and their relation to trace-average problems over U(N). 

 I will talk about recent progress on (a) a conjecture of Fyodorov and Keating on supercritical asymptotics of moments of moments of characteristic polynomials of the circular beta ensemble and (b) on the correlation functions of the sine beta point process. This is joint work with Joseph Najnudel.

We discuss the asymptotic local behavior of the second correlation functions of the characteristic polynomials of a certain class of Gaussian N X N non-Hermitian random band matrices with a bandwidth W. Given W,N → ∞, we show that this behavior near the point in the bulk of the spectrum exhibits the crossover at W ∼√N: it coincides with those for Ginibre ensemble for W ≫√N, and factorized as 1 ≪ W ≪√N. The behavior of the correlation function near the threshold (W/√N →C) will be also discussed.

I will present a generalisation of Mirzakhani’s recursion for the volumes of moduli spaces of bordered Klein surfaces, including non-orientable surfaces. On these moduli spaces, the top form introduced by Norbury diverges as the lengths of one-sided geodesics approach zero. However, integrating this form over Gendulphe’s regularised moduli space—where the systole of one-sided geodesics is bounded below by epsilon—yields a finite volume. Using Norbury’s extension of the Mirzakhani–McShane identities to the non-orientable setting, we derive an explicit formula for the volume of the moduli space of one-bordered Klein bottles, as well as a recursion for arbitrary topologies that fully captures the dependence on the geometric regularisation parameter epsilon. I will conclude with remarks on the relation to refined topological recursion, which leads us to a refinement of the Witten–Kontsevich recursion and of the Harer–Zagier formula for the orbifold Euler characteristic of the moduli space of curves of genus g with n marked points. Based on joint work with P. Gregori and K. Osuga; the final part reflects ongoing work with N. Chidambaram, A. Giacchetto, and K. Osuga.

I will present a construction and characterization of the (massive) sine-Gordon EQFT up to 6π in the full space.  The construction relies on a systematic study of the renormalization flow equation and a forward backward stochastic differential equation (FBSDE) which give good control of the EQFT and allows to derive various additional properties.

This is based on joint work with Massimiliano Gubinelli.

The asymptotic large N limit of random matrices often transforms classical concepts (independence, cumulants, partitions of sets) into their free counter-parts (free independence, free cumulants, non-crossing partitions) and the limit of random matrices gives rise to interesting operator algebras. I will explain these relations, with a particular emphasis on the effect of non-linear functions on the entries of random matrices

Given a quantum Hamiltonian, represented as an $N \times N$ Hermitian matrix $H$, we derive an expression for the largest Lyapunov exponent of the classical trajectories in the phase space appropriate for the dynamics induced by $H$. To this end we associate to $H$ a graph with $N$ vertices and derive a quantum map on functions defined on the directed edges of the graph. Using the semiclassical approach in the reverse direction we obtain the corresponding classical evolution (Liouvillian) operator. Using ergodic theory methods (Sinai, Ruelle, Bowen, Pollicott\ldots) we obtain closed expressions for the Lyapunov exponent, as well as for its variance. Applications for random matrix models will be presented.

I will discuss old and new results about the distribution of zeros of modular forms, and relation to Quantum Unique Ergodicity. It is known that a modular form of weight k has about k/12 zeros in the fundamental domain . A classical question in the analytic theory of modular forms is “can we locate the zeros of a distinguished family of modular forms?”. In 1970, F. Rankin and Swinnerton-Dyer proved that the zeros of the Eisenstein series all lie on the circular part of the boundary of the fundamental domain. In the beginning of this century, I discovered that for cuspidal Hecke eigenforms, the picture is very different - the zeros are not localized, and in fact become uniformly distributed in the fundamental domain. Very recently, we have investigated other families of modular forms, such as the Miller basis (ZR 2024, Roei Raveh 2025, Adi Zilka 2026), Poincare series (RA Rankin 1982, Noam Kimmel 2025) and theta functions (Roei Raveh 2026),  finding a variety of possible distributions of the zeroes.

The capacity of a set is a classical notion in potential theory and it is a measure of the size of a set as seen by a random walk or Brownian motion. Recently Zhu defined the notion of branching capacity as the analogue of capacity in the context of a branching random walk. In this talk I will describe joint work with Amine Asselah and Bruno Schapira where we introduce a notion of capacity of a set for critical bond percolation and I will explain how it shares similar properties as in the case of branching random walks. 

The weak noise theory (WNT) provides a framework for accessing large deviations in models of the Kardar-Parisi-Zhang (KPZ) universality class, probing the regime where randomness is small, fluctuations are rare, and atypical events dominate. Historically, two methods have been available: asymptotic analysis of Fredholm determinant formulas—applicable only for special initial data—and variational or saddle-point formulations leading to nonlinear evolution equations, which were mostly accessible perturbatively.

This talk explains how these approaches can be unified: the weak-noise saddle equations of KPZ-class models form classically integrable systems, admitting Lax pairs, conserved quantities, and an inverse scattering framework. In this setting, the large-deviation rate functions arise directly from the conserved charges of the associated integrable dynamics.

The discussion will focus on three examples:

1. The scalar Strict-Weak polymer ;
2. A matrix Strict-Weak polymer driven by Wishart noise ;
3. If time permits, the continuous-time q-TASEP.

I'll discuss some of the history of the use of random matrices for studying the structure of operator algebras, starting with Voiculescu's notion free independence. We'll see that the original notions of convergence of random matrix models to certain infinite-dimensional operators is actually fairly weak, and discuss the more recent "strong convergence" phenomenon and its applications to C*-algebras. Finally, I'll touch upon some ongoing work, joint with A. Shiner and S. White, for continuing to use random matrix tools to prove structural properties of C*-algebras.

Matrix-product state (MPS) skeletons are connected networks of local one-dimensional quantum lattice models with ground states admitting an MPS representation with finite bond dimension. In this talk, I will discuss how such skeletons underlie certain families of models obeying the Onsager algebra, and how these simple ground states provide a route to explicitly computing correlation functions.

I will review how the equations of general relativity near a spacetime singularity map onto an arithmetic hyperbolic billiard dynamics. The semiclassical quantum states for this dynamics are Maaβ cusp forms on fundamental domains of modular groups. For example, gravity in four spacetime dimensions leads to PSL(2,Z) while five dimensional gravity leads to PSL(2,Z[w]), with Z[w] the Eisenstein integers. The automorphic forms can be expressed, in a dilatation (Mellin transformed) basis as L-functions. The Euler product representation of these L-functions indicates that these quantum states admit a dual interpretation as a "primon gas" partition function. I will describe some physically motivated mathematical questions that arise from these observations.

In the first half of this talk, I will provide a brief introduction to Isomonodromic deformations with the one-point torus as my main example, and show the relation to the elliptic form of Painlevé VI equation as well as the Lamé equation. In the second half of this talk, I will present an overview of my results in the past few years concerning the associated tau-functions, conformal blocks, and accessory parameters. Finally, I will motivate how probabilistic methods in conformal field theory help us understand the data within Lamé type equations.

I will report on recent progress on influential conjectures from the 1970s and 1980s (Berry-Tabor, Bohigas-Giannoni-Schmit), which suggest that the spectral statistics of the Laplace-Beltrami operator on a given compact Riemannian manifold should be described either by a Poisson point process or by a random matrix ensemble, depending on whether the  geodesic flow is integrable or “chaotic”. This talk will straddle aspects of analysis, geometry, probability, number theory and ergodic theory, and should be accessible to a broad audience. The two most recent results presented in this lecture were obtained in collaboration with Laura Monk and with Wooyeon Kim and Matthew Welsh. 

I will describe a measure of quantum state complexity defined by minimizing the spread of the wavefunction over all choices of basis. We can efficiently compute this measure, which displays universal behavior for diverse chaotic systems including spin chains, the SYK model, and quantum billiards.  In the minimizing basis, the Hamiltonian is tridiagonal, thus representing the dynamics as if they unfold on a one-dimensional chain. The recurrent and hopping matrix elements of this chain comprise the Lanczos coefficients, which I will relate through an integral formula to the density of states. For Random Matrix Theories (RMTs), which are believed to describe the energy level statistics of chaotic systems, I will also derive an integral formula for the covariances of the Lanczos coefficients. These results lead to a conjecture: quantum chaotic systems have Lanczos coefficients whose local means and covariances are described by RMTs. 
 

Understanding the size of the partial sums of the Möbius function is one of the most fundamental problems in analytic number theory. This motivated the 1944 paper of Wintner, where he introduced the concept of a random multiplicative function: a probabilistic model for the Möbius function. In recent years, it has been uncovered that there is an intimate connection between random multiplicative functions and the theory of Gaussian Multiplicative Chaos, an area of probability theory introduced by Kahane in the 1980's. We will survey selected results and discuss recent research on the distribution of partial sums of random multiplicative functions when restricted to integers with a large prime factor.

In this talk, we consider the coefficients of the Fourier series obtained by exponentiating a logarithmically correlated holomorphic function on the open unit disc, whose Taylor coefficients are independent complex Gaussian variables, the variance of the coefficient of degree k being theta/k where theta > 0 is an inverse temperature parameter. In joint articles with Paquette, Simm and Vu, we show a randomized version of the central limit theorem in the subcritical phase theta < 1, the random variance being related to the Gaussian multiplicative chaos on the unit circle. We also deduce, from results on the holomorphic multiplicative chaos, other results on the coefficients of the characteristic polynomial of the Circular Beta Ensemble, where the parameter beta is equal to 2/theta. In particular, we show that the central coefficient of the characteristic polynomial of the Circular Unitary Ensembles tends to zero in probability, answering a question asked in an article by Diaconis and Gamburd.

Perturbative expansions in quantum theory, particularly in quantum field theory and string theory, are typically factorially divergent due to underlying non-perturbative sectors. Resurgence provides a universal toolbox to access the non-perturbative effects hidden within the perturbative series, producing a collection of exponentially small corrections. Under special assumptions, the non-perturbative data extracted via resurgent methods exhibit intrinsic number-theoretic structures that are deeply rooted in the symmetries of the theory. The framework of modular resurgence aims to formalise this observation. In this talk, I will first introduce the systematic, algebraic approach of resurgence to the problem of divergences and describe the emerging bridge between the resurgence of q-series and the analytic and number-theoretic properties of L-functions and quantum modular forms. I will then apply it to the spectral theory of quantum operators associated with toric Calabi-Yau threefolds. Here, a complete realisation of the modular resurgence paradigm is found in the study of the spectral trace of local P^2, where the asymptotics at weak and strong coupling are captured by certain q-series, and is generalised to all local weighted projective planes. This talk is based on arXiv:2212.10606, 2404.10695, 2404.11550, and work to appear soon.

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.