arXiv subjects

Francesco Mezzadri

Publications and source records attributed to Francesco Mezzadri.

At least 19 recordsLinked to original sources

arXiv2026

Logarithmic Spectral Distribution of a Non-Hermitian $β$-Ensemble

We introduce a non-Hermitian $β$-ensemble and determine its spectral density in the limit of large $β$ and large matrix size $n$. The ensemble is given by a general tridiagonal complex random matrix of normal and chi-distributed random variables, extending previous work of Mezzadri and Taylor (2025). The joint distribution of eigenvalues contains a Vandermonde determinant to the power $β$ and a residual coupling to the eigenvectors. A tool in the computation of the limiting spectral density is a single characteristic polynomial for centred tridiagonal Jacobi matrices, for which we explicitly determine the coefficients in terms of its matrix elements. In the low temperature limit $β\gg1$, our ensemble reduces to such a centred matrix with vanishing diagonal. A general theorem from free probability based on the variance of the coefficients of the characteristic polynomial allows us to obtain the spectral density when additionally taking the large-$n$ limit. It is rotationally invariant on a compact disc, given by the logarithm of the radius plus a constant. The same density is obtained when starting form a tridiagonal complex symmetric ensemble, which thus plays a special role. Extensive numerical simulations confirm our analytical results and put this and the previously studied ensemble in the context of the pseudospectrum. The numerical study of the local nearest-neighbour spacing distribution shows agreement between the tridiagonal ensemble and two-dimensional Poisson statistics (independently of $β$), whereas we observe a $β$-dependence for the previously introduced ensemble.

math-ph↗

arXiv2026

Interpolating non-Hermitian universality classes A and AI$^\dagger$: eigenvalue density and transition regime

We employ the recently developed Kac-Rice formalism for non-Hermitian random matrices to derive the joint distribution of an eigenvalue and its associated normalised right eigenvector in a Gaussian ensemble that interpolates between complex Ginibre (Class A) and complex symmetric matrices (Class AI$^\dagger$). This distribution is valid at finite matrix size, $N$, for any value of the interpolation parameter $σ\in [0,1]$, with $0$ and $1$ corresponding to classes A and AI$^\dagger$ respectively. The marginal distribution for the density of the eigenvalues is derived at finite $N$ and then considered asymptotically as $N \to \infty$. When considering bulk eigenvalues, we recover the standard circular law for all $σ$. Furthermore, for edge eigenvalues we find that for fixed $σ$, the eigenvalues follow the edge density associated with matrices in Class A. However, a transitional regime is discovered for the interpolation parameter being scaled as $σ= 1 - κN^{-1/2}$, where new edge behaviour is observed for the density of eigenvalues - smoothly interpolating two previously known results. This transitional regime and the associated density of eigenvalues is conjectured to be universal for non-Gaussian matrices and we provide numerical evidence in support of this.

math-ph↗

arXiv2026

Higher order derivative moments of CUE characteristic polynomials and the Riemann zeta function

We use random matrix theory for the Circular Unitary Ensemble (CUE) to study moments of derivatives of the Riemann zeta function shifted a small distance from the critical line. The corresponding CUE moments are studied in the limit of large matrix size in two regimes: when the spectral parameter is (1) suitably far inside the unit disc, and (2) at a small distance from the unit circle. In case (1), we obtain an asymptotic formula as a combinatorial sum over contingency tables, while in case (2) we obtain a sum over certain determinants with multiplicative coefficients given by Kostka numbers. The latter result is also valid exactly on the unit circle. Then, we consider the analogous problem for mean values of derivatives of the zeta function with suitable shifts. Assuming the Lindelöf hypothesis, we show that this mean value gives rise to the same sum over contingency tables obtained in the CUE. For sufficiently low-order moments, we establish this result unconditionally.

math-ph↗

arXiv2025

Magnetic Manipulation of Spatially Confined Multiferroic Heuslers by Martensitic Microstructure Engineering

Magnetic shape-memory (MSM) Heuslers show a strong coupling between magnetic and structural characteristics, evidencing a correlation between magnetic, thermal, and mechanical properties through a magnetostructural martensitic transformation. This functional aspect makes MSM Heuslers promising for integration into smart micro/nanodevices, including sensors, energy harvesters, and actuators. Controlling the martensitic microstructure, which determines the magnetic characteristics, is among the key points for optimization of the magnetic functional properties of these materials at different length scales. Here, we report a strategy for manipulating the magnetic properties of spatially confined epitaxial Ni-Mn-Ga films grown on Cr(001)//MgO(001) by twinning configuration engineering in the low-temperature ferromagnetic (martensitic) phase. We show how the twinning configurations in the continuous films and the micropatterned structures can be switched from Y-type (showing negligible magnetic stray field) into X-type (presenting significant magnetic stray field) by a post-annealing process. Advanced characterization techniques enable us to analyze the atomic structure and the surface quality of the annealed samples and to disentangle the twin-switching phenomenon. The martensitic microstructure engineering reported in this study introduces a simple method for promoting the magnetic stray-field contribution at the surface of Ni-Mn-Ga epitaxial thin films and micropatterns initially showing a negligible magnetic stray field.

cond-mat.mtrl-sci↗

arXiv2024

A matrix model of a non-Hermitian $β$-ensemble

We introduce the first random matrix model of a complex $β$-ensemble. The matrices are tridiagonal and can be thought of as the non-Hermitian analogue of the Hermite $β$-ensembles discovered by Dumitriu and Edelman (J. Math. Phys., Vol. 43, 5830 (2002)). The main feature of the model is that the exponent $β$ of the Vandermonde determinant in the joint probability density function (j.p.d.f.) of the eigenvalues can take any value in $\mathbb{R}_+$. However, when $β=2$, the j.p.d.f. does not reduce to that of the Ginibre ensemble, but it contains an extra factor expressed as a multidimensional integral over the space of the eigenvectors.

math-ph↗

arXiv2023

High pressure bulk synthesis of InN by solid state reaction of binary oxide in a multi-anvil apparatus

We present a new method to synthesize bulk indium nitride by means of a simple solid-state chemical reaction carried out under hydrostatic high pressure/high temperature conditions in a multi-anvil apparatus, not involving gases or solvents during the process. The reaction occurs between the binary oxide $In_2O_3$ and the highly reactive $Li_3N$ as nitrogen source, in powder form. The formation of the hexagonal phase of InN, occurring at 350 °C and P $\geq$ 3 GPa, was successfully confirmed by powder X-ray diffraction, with the presence of $Li_2O$ as unique byproduct. A simple washing process in weak acidic solution followed by centrifugation, allowed to obtain pure InN polycrystalline powders as precipitate. With an analogous procedure it was possible to obtain pure bulk GaN, from $Ga_2O_3$ and $Li_3N$ at T $\geq$ 600°C and P $\geq$ 2.5 GPa. These results point out, particularly for InN, a clean, and innovative way to produce significant quantities of one of the most promising nitrides in the field of electronics and energy technologies.

cond-mat.mtrl-sci↗

arXiv2022

Universal characteristics of deep neural network loss surfaces from random matrix theory

This paper considers several aspects of random matrix universality in deep neural networks. Motivated by recent experimental work, we use universal properties of random matrices related to local statistics to derive practical implications for deep neural networks based on a realistic model of their Hessians. In particular we derive universal aspects of outliers in the spectra of deep neural networks and demonstrate the important role of random matrix local laws in popular pre-conditioning gradient descent algorithms. We also present insights into deep neural network loss surfaces from quite general arguments based on tools from statistical physics and random matrix theory.

math-ph↗

arXiv2021

Temperature induced Neutral to Ionic phase Transition of the charge transfer crystal Tetrathiafulvalene-Fluoranil

The Temperature Induced Neutral to Ionic phase Transition (TI-NIT) is a rare phenomenon occurring in mixed stack charge transfer (CT) crystals made up of alternating $π$-electron Donor (D) and Acceptor (A) molecules. We were able to grow crystals of Tetrathiafulvalene-Fluoranil (TTF-FA), and to show that it undergoes TI-NIT like the prototype CT crystal TTF-Chloranil. We characterized both room and low T phases through IR and Raman spectroscopy and XRD, demonstrating that while TTF-FA is quasi-neutral at room T, its ionicity jumps from 0.15 to 0.7 at low T, therefore crossing the Neutral-Ionic borderline. The transition, occuring around 150K, is first order, with large thermal hysteresis and accompanied by crystal cracking. In the high T phase D and A molecules lie on inversion center, i.e. the stacks are regular, whereas the low T phase is characterized by the loss of the inversion symmetry along the stack as the stacks are strongly dimerized and by the doubling of the unit cell.

cond-mat.mtrl-sci↗

arXiv2021

Symmetric Function Theory and Unitary Invariant Ensembles

Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices drawn from the Circular Unitary Ensemble and other Circular Ensembles related to the classical compact groups. The reason is that they enable the derivation of exact formulae, which then provide a route to calculating the large-matrix asymptotics of these quantities. We develop a parallel theory for the Gaussian Unitary Ensemble of random matrices, and other related unitary invariant matrix ensembles. This allows us to write down exact formulae in these cases for the joint moments of the traces and the joint moments of the characteristic polynomials in terms of appropriately defined symmetric functions. As an example of an application, for the joint moments of the traces we derive explicit asymptotic formulae for the rate of convergence of the moments of polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity.

math-ph↗

arXiv2021

On the moments of characteristic polynomials

We examine the asymptotics of the moments of characteristic polynomials of $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, in the limit as $N\to\infty$. We focus in particular on the Gaussian Unitary Ensemble, but discuss other Hermitian ensembles as well. We employ a novel approach to calculate asymptotic formulae for the moments, enabling us to uncover subtle structure not apparent in previous approaches.

math-ph↗

arXiv2021

The Loss Surfaces of Neural Networks with General Activation Functions

The loss surfaces of deep neural networks have been the subject of several studies, theoretical and experimental, over the last few years. One strand of work considers the complexity, in the sense of local optima, of high dimensional random functions with the aim of informing how local optimisation methods may perform in such complicated settings. Prior work of Choromanska et al (2015) established a direct link between the training loss surfaces of deep multi-layer perceptron networks and spherical multi-spin glass models under some very strong assumptions on the network and its data. In this work, we test the validity of this approach by removing the undesirable restriction to ReLU activation functions. In doing so, we chart a new path through the spin glass complexity calculations using supersymmetric methods in Random Matrix Theory which may prove useful in other contexts. Our results shed new light on both the strengths and the weaknesses of spin glass models in this context.

math.PR↗

arXiv2021

On the number of real eigenvalues of a product of truncated orthogonal random matrices

Let $O$ be chosen uniformly at random from the group of $(N+L) \times (N+L)$ orthogonal matrices. Denote by $\tilde{O}$ the upper-left $N \times N$ corner of $O$, which we refer to as a truncation of $O$. In this paper we prove two conjectures of Forrester, Ipsen and Kumar (2020) on the number of real eigenvalues $N^{(m)}_{\mathbb{R}}$ of the product matrix $\tilde{O}_{1}\ldots \tilde{O}_{m}$, where the matrices $\{\tilde{O}_{j}\}_{j=1}^{m}$ are independent copies of $\tilde{O}$. When $L$ grows in proportion to $N$, we prove that $$ \mathbb{E}(N^{(m)}_{\mathbb{R}}) = \sqrt{\frac{2m L}π}\,\mathrm{arctanh}\left(\sqrt{\frac{N}{N+L}}\right) + O(1), \qquad N \to \infty. $$ We also prove the conjectured form of the limiting real eigenvalue distribution of the product matrix. Finally, we consider the opposite regime where $L$ is fixed with respect to $N$, known as the regime of weak non-orthogonality. In this case each matrix in the product is very close to an orthogonal matrix. We show that $\mathbb{E}(N^{(m)}_{\mathbb{R}}) \sim c_{L,m}\,\log(N)$ as $N \to \infty$ and compute the constant $c_{L,m}$ explicitly. These results generalise the known results in the one matrix case due to Khoruzhenko, Sommers and Życzkowski (2010).

math.PR↗

arXiv2021

A spin-glass model for the loss surfaces of generative adversarial networks

We present a novel mathematical model that seeks to capture the key design feature of generative adversarial networks (GANs). Our model consists of two interacting spin glasses, and we conduct an extensive theoretical analysis of the complexity of the model's critical points using techniques from Random Matrix Theory. The result is insights into the loss surfaces of large GANs that build upon prior insights for simpler networks, but also reveal new structure unique to this setting.

math-ph↗

arXiv2019

Moments of random matrices and hypergeometric orthogonal polynomials

We establish a new connection between moments of $n \times n$ random matrices $X_n$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s \in \mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden et al. [F. D. Cunden, F. Mezzadri, N. J. Simm and P. Vivo, J. Math. Phys. 57 (2016)] on the time-delay matrix of chaotic cavities. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order $n\to\infty$ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials.

math-ph↗

arXiv2018

Free fermions and the classical compact groups

There is a close connection between the ground state of non-interacting fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture by constructing a finite temperature extension of the Haar measure on the classical compact groups. The eigenvalue statistics of the resulting grand canonical matrix models (of random size) corresponds exactly to the grand canonical measure of non-interacting free fermions with classical boundary conditions.

math-ph↗

arXiv2017

Density and spacings for the energy levels of quadratic Fermi operators

The work presents a proof of convergence of the density of energy levels to a Gaussian distribution for a wide class of quadratic forms of Fermi operators. This general result applies also to quadratic operators with disorder, e.g., containing random coefficients. The spacing distribution of the unfolded spectrum is investigated numerically. For generic systems the level spacings behave as the spacings in a Poisson process. Level clustering persists in presence of disorder.

math-ph↗

arXiv2016

Moments of the eigenvalue densities and of the secular coefficients of $β$-ensembles

We compute explicit formulae for the moments of the densities of the eigenvalues of the classical $β$-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In particular, the moments are linear combinations of averages of Jack polynomials, whose coefficients are related to specific examples of Jack characters.

math-ph↗

arXiv2016

On the Tracy-Widom$_β$ Distribution for $β=6$

We study the Tracy-Widom distribution function for Dyson's $β$-ensemble with $β= 6$. The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Virág equation. The latter is a linear PDE which describes the Tracy-Widom functions corresponding to general values of $β$. Using his Lax pair, Rumanov derives an explicit formula for the Tracy-Widom $β=6$ function in terms of the second Painlevé transcendent and the solution of an auxiliary ODE. Rumanov also shows that this formula allows him to derive formally the asymptotic expansion of the Tracy-Widom function. Our goal is to make Rumanov's approach and hence the asymptotic analysis it provides rigorous. In this paper, the first one in a sequel, we show that Rumanov's Lax-pair can be interpreted as a certain gauge transformation of the standard Lax pair for the second Painlevé equation. This gauge transformation though contains functional parameters which are defined via some auxiliary nonlinear ODE which is equivalent to the auxiliary ODE of Rumanov's formula. The gauge-interpretation of Rumanov's Lax-pair allows us to highlight the steps of the original Rumanov's method which needs rigorous justifications in order to make the method complete. We provide a rigorous justification of one of these steps. Namely, we prove that the Painlevé function involved in Rumanov's formula is indeed, as it has been suggested by Rumanov, the Hastings-McLeod solution of the second Painlevé equation. The key issue which we also discuss and which is still open is the question of integrability of the auxiliary ODE in Rumanov's formula. We note that this question is crucial for the rigorous asymptotic analysis of the Tracy-Widom function.

math-ph↗