arXiv subjects

Francesco Mezzadri

Publications and source records attributed to Francesco Mezzadri.

32 records · Page 2Linked to original sources

arXiv2016

Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities

We consider the $1/N$-expansion of the moments of the proper delay times for a ballistic chaotic cavity supporting $N$ scattering channels. In the random matrix approach, these moments correspond to traces of negative powers of Wishart matrices. For systems with and without broken time reversal symmetry (Dyson indices $β=1$ and $β=2$) we obtain a recursion relation, which efficiently generates the coefficients of the $1/N$-expansion of the moments. The integrality of these coefficients and their possible diagrammatic interpretation is discussed.

math-ph↗

arXiv2016

Large deviations of radial statistics in the two-dimensional one-component plasma

The two-dimensional one-component plasma is an ubiquitous model for several vortex systems. For special values of the coupling constant $βq^2$ (where $q$ is the particles charge and $β$ the inverse temperature), the model also corresponds to the eigenvalues distribution of normal matrix models. Several features of the system are discussed in the limit of large number $N$ of particles for generic values of the coupling constant. We show that the statistics of a class of radial observables produces a rich phase diagram, and their asymptotic behaviour in terms of large deviation functions is calculated explicitly, including next-to-leading terms up to order 1/N. We demonstrate a split-off phenomenon associated to atypical fluctuations of the edge density profile. We also show explicitly that a failure of the fluid phase assumption of the plasma can break a genuine $1/N$-expansion of the free energy. Our findings are corroborated by numerical comparisons with exact finite-N formulae valid for $βq^2=2$.

math-ph↗

arXiv2016

Correlators for the Wigner-Smith time-delay matrix of chaotic cavities

We study the Wigner-Smith time-delay matrix $Q$ of a ballistic quantum dot supporting $N$ scattering channels. We compute the $v$-point correlators of the power traces $\mathrm{Tr} Q^κ$ for arbitrary $v\geq1$ at leading order for large $N$ using techniques from the random matrix theory approach to quantum chromodynamics. We conjecture that the cumulants of the $\mathrm{Tr} Q^κ$'s are integer-valued at leading order in $N$ and include a MATHEMATICA code that computes their generating functions recursively.

math-ph↗

arXiv2015

Fluctuations in the two-dimensional one-component plasma and associated fourth-order phase transition

We study the distribution of the mean radial displacement of charges of a 2D one-component plasma in the thermodynamic limit $N\to\infty$ at finite temperature $β>0$. We compute explicitly the large deviation functions showing the emergence of a fourth-order phase transition as a consequence of a change of topology in the plasma distribution. This weak phase transition occurs exactly at the ground state of the plasma. These results have been compared with the integrable case (finite $N$) of plasma parameter $βq^2=2$. In this case the problem can be mapped to the stationary properties of 2D Dyson Brownian particles and to a non-Hermitian matrix model.

math-ph↗

arXiv2015

A unified fluctuation formula for one-cut $β$-ensembles of random matrices

Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut $β$-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut $β$-ensemble. As particular cases of the main result we consider the classical $β$-Gaussian, $β$-Wishart and $β$-Jacobi ensembles, for which we derive previously available results as well as new ones within a unified simple framework. We also discuss the connections between the problem of trace fluctuations for the Gaussian Unitary Ensemble and the enumeration of planar maps.

math-ph↗

arXiv2015

Random matrix ensembles with singularities and a hierarchy of Painlevé III equations

We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order $k$ at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hierarchy of higher order analogues to the Painlevé III equation, which reduces to the Painlevé III equation itself when the pole is simple.

math-ph↗

arXiv2013

Thermally activated magnetization reversal in bulk BiFe0.5Mn0.5O3

We report on the synthesis and characterization of BiFe0.5Mn0.5O3, a potential type-I multiferroic compound displaying temperature induced magnetization reversal. Bulk samples were obtained by means of solid state reaction carried out under the application of hydrostatic pressure at 6 GPa and 1100°C. The crystal structure is an highly distorted perovskite with no cation order on the B site, where, besides a complex scheme of tilt and rotations of the TM-O6 octahedra, large off-centering of the bismuth ions is detected. Below T1 = 420 K the compound undergoes a first weak ferromagnetic transition related to the ordering of iron rich clusters. At lower temperatures (just below RT) two distinct thermally activated mechanisms are superimposed, inducing at first an enhancement of the magnetization at T2 = 288 K, then a spontaneous reversal process centered at T3 = 250 K, finally giving rise to a negative response. The application of fields higher than 1500 Oe suppresses the process, yielding a ferromagnetic like behaviour. The complementary use of SQuID magnetometry and Mössbauer spectroscopy allowed the interpretation of the overall magnetic behaviour in terms of an uncompensated weak competitive coupling between non-equivalent clusters of interactions characterized by different critical temperatures and resultant magnetizations. PACS numbers: 75.85.+t, 75.60.Jk, 76.80.+y, 75.30.Et, 75.30.Kz

cond-mat.mtrl-sci↗

arXiv2010

Roots of the derivative of the Riemann zeta function and of characteristic polynomials

We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both cases show a surprising bimodal distribution which has yet to be explained. We show by example that the bimodality is a general phenomenon. For the unitary matrix case we prove a conjecture of Mezzadri concerning the leading order behavior, and we show that the same follows from the random matrix conjectures for the zeros of the zeta function.

math.NT↗

arXiv2007

How to generate random matrices from the classical compact groups

We discuss how to generate random unitary matrices from the classical compact groups U(N), O(N) and USp(N) with probability distributions given by the respective invariant measures. The algorithm is straightforward to implement using standard linear algebra packages. This approach extends to the Dyson circular ensembles too. This article is based on a lecture given by the author at the summer school on Number Theory and Random Matrix Theory held at the University of Rochester in June 2006. The exposition is addressed to a general mathematical audience.

math-ph↗

arXiv2006

Random polynomials, random matrices, and $L$-functions

We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function.

math-ph↗

arXiv2002

Random matrix theory and the zeros of zeta'(s)

We study the density of the roots of the derivative of the characteristic polynomial Z(U,z) of an N x N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function zeta(s), this is expected to be an accurate description for the horizontal distribution of the zeros of zeta'(s) to the right of the critical line. We show that as N --> infinity the fraction of roots of Z'(U,z) that lie in the region 1-x/(N-1) <= |z| < 1 tends to a limit function. We derive asymptotic expressions for this function in the limits x --> infinity and x --> 0 and compare them with numerical experiments.

math-ph↗

arXiv2002

On the multiplicativity of quantum cat maps

The quantum mechanical propagators of the linear automorphisms of the two-torus (cat maps) determine a projective unitary representation of the theta group, known as Weil's representation. We prove that there exists an appropriate choice of phases in the propagators that defines a proper representation of the theta group. We also give explicit formulae for the propagators in this representation.

math-ph↗

arXiv2000

Quantum cat maps with spin 1/2

We derive a semiclassical trace formula for quantized chaotic transformations of the torus coupled to a two-spinor precessing in a magnetic field. The trace formula is applied to semiclassical correlation densities of the quantum map, which, according to the conjecture of Bohigas, Giannoni and Schmit, are expected to converge to those of the circular symplectic ensemble (CSE) of random matrices. In particular, we show that the diagonal approximation of the spectral form factor for small arguments agrees with the CSE prediction. The results are confirmed by numerical investigations.

nlin.CD↗

arXiv1996

On the properties of level spacings for decomposable systems

In this paper we show that the quantum theory of chaos, based on the statistical theory of energy spectra, presents inconsistencies difficult to overcome. In classical mechanics a system described by an hamiltonian $H = H_1 + H_2$ (decomposable) cannot be ergodic, because there are always two dependent integrals of motion besides the constant of energy. In quantum mechanics we prove the existence of decomposable systems \linebreak $H^q = H^q_1 + H^q_2$ whose spacing distribution agrees with the Wigner law and we show that in general the spacing distribution of $H^q$ is not the Poisson law, even if it has often the same qualitative behaviour. We have found that the spacings of $H^q$ are among the solutions of a well defined class of homogeneous linear systems. We have obtained an explicit formula for the bases of the kernels of these systems, and a chain of inequalities which the coefficients of a generic linear combination of the basis vectors must satisfy so that the elements of a particular solution will be all positive, i.e. can be considered a set of spacings.

chao-dyn↗