# Luca Guido MolinariUniversity of Milan | UNIMI · Department of Physics

Luca Guido Molinari

PhD

## About

133

Publications

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2,179

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Citations since 2017

Introduction

Additional affiliations

January 1983 - present

## Publications

Publications (133)

I reconsider the approximation of Bessel functions with finite sums of trigonometric functions, in the light of recent evaluations of Neumann-Bessel series with trigonometric coefficients. A proper choice of the angle allows for an efficient choice of the trigonometric sum. Based on these series, I also obtain straightforward non-standard evaluatio...

We study the geometric properties of certain Codazzi tensors for
their own sake, and for their appearance in the recent theory of Cotton gravity.
We prove that a perfect-fluid tensor is Codazzi if and only if the metric is a generalized Stephani universe. A trace condition restricts it to a warped space-time, as proven by Merton and Derdzi´nski. We...

Spherically symmetric spacetimes are ambient spaces for models of stellar collapse and inhomogeneous cosmology. We obtain results for the Weyl tensor and the covariant form of the Ricci tensor on general doubly warped (DW) spacetimes. In a spherically symmetric metric, the Ricci and electric tensors become rank-2, built with the metric tensor, a ve...

I conjecture three identities for the determinant of adjacency matrices of graphene triangles and trapezia with Bloch (and more general) boundary conditions. For triangles, the parametric determinant is equal to the characteristic polynomial of the symmetric Pascal matrix. For trapezia it is equal to the determinant of a sub-matrix. Finally, the de...

I reconsider the approximation of Bessel functions with finite sums of trigonometric functions, in the light of recent evaluations of Neumann-Bessel series with trigonometric coefficients. A proper choice of the angle allows for an efficient choice of the trigonometric sum. Based on these series, I also obtain straightforward non-standard evaluatio...

Sufficient conditions for a Lorentzian generalized quasi-Einstein manifold M,g,f,μ to be a generalized Robertson–Walker spacetime with Einstein fibers are derived. The Ricci tensor in this case gains the perfect fluid form. Likewise, it is proven that a λ,n+m-Einstein manifold M,g,w having harmonic Weyl tensor, ∇jw∇mwCjklm=0 and ∇lw∇lw<0 reduces to...

Spherically symmetric spacetimes are ambient spaces for models of stellar collapse and inhomogeneous cosmology. We obtain results for the Weyl tensor and the covariant form of the Ricci tensor on general doubly warped (DW) spacetimes. In a spherically symmetric metric, the Ricci and electric tensors become rank-2, built with a velocity vector field...

A main issue in cosmology and astrophysics is whether the dark sector phenomenology originates from particle physics, then requiring the detection of new fundamental components, or it can be addressed by modifying General Relativity. Extended Theories of Gravity are possible candidates aimed in framing dark energy and dark matter in a comprehensive...

A main issue in cosmology and astrophysics is whether the dark sector phenomenology originates from particle physics, then requiring the detection of new fundamental components, or it can be addressed by modifying General Relativity. Extended Theories of Gravity are possible candidates aimed in framing dark energy and dark matter in a comprehensive...

Several sums of Neumann series with Bessel and trigonometric functions are evaluated, as finite sums of trigonometric functions. They arise from a generalization of the Neumann expansion of the eigenstates of the Laplacian in regular polygons. A simple accurate approximation of J0(x) is found on the interval [0, 2].

Given the Riemann, or the Weyl, or a generalized curvature tensor K, a symmetric tensor bij is called compatible with the curvature tensor if bi m K jklm + bj m K kilm + b k m K ijlm = 0. In addition to establishing some known and some new properties of such tensors, we prove that they form a special Jordan algebra, i.e. the symmetrized product of...

The simple structure of doubly torqued vectors allows for a natural characterization of doubly twisted down to warped spacetimes, as well as Kundt spacetimes down to PP waves. For the first ones the vectors are timelike, for the others they are null. We also discuss some properties, and their connection to hypersurface orthogonal conformal Killing...

In this note, we characterize [Formula: see text] doubly twisted spacetimes in terms of “doubly torqued” vector fields. They extend Bang–Yen Chen’s characterization of twisted and generalized Robertson–Walker spacetimes with torqued and concircular vector fields. The result is a simple classification of [Formula: see text] doubly-twisted, doubly-wa...

The simple structure of doubly torqued vectors allows for a natural characterization of doubly twisted down to warped spacetimes, as well as Kundt spacetimes down to PP waves. For the first ones the vectors are timelike, for the others they are null. We also discuss some properties, and their connection to hypersurface orthogonal conformal Killing...

In this note we characterize 1+n doubly twisted spacetimes in terms of `doubly torqued' vector fields. They extend Bang-Yen Chen's characterization of twisted and generalized Robertson-Walker spacetimes with torqued and concircular vector fields. The result is a simple classification of 1+n doubly-twisted, doubly-warped, twisted and generalized Rob...

Several sums of Neumann series with Bessel and trigonometric functions are evaluated, as finite sums of trigonometric functions. They arise from a generalization of the Neumann expansion of the eigenstates of the Laplacian in regular polygons.

General properties of vacuum solutions of $f(R)$ gravity are obtained by the condition that the divergence of the Weyl tensor is zero and $f''\neq 0$. Specifically, a theorem states that the gradient of the curvature scalar, $\nabla R$, is an eigenvector of the Ricci tensor and, if it is time-like, the space-time is a Generalized Friedman-Robertson...

We prove that in Robertson–Walker space-times (and in generalized Robertson–Walker spacetimes of dimension greater than 3 with divergence-free Weyl tensor) all higher-order gravitational corrections of the Hilbert–Einstein Lagrangian density F(R,□R,…,□kR)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \...

Given the Riemann, or the Weyl, or a generalized curvature tensor K, a symmetric tensor $b_{ij}$ is named `compatible' with the curvature tensor if $b_i{}^m K_{jklm} + b_j{}^m K_{kilm} + b_k{}^m K_{ijlm} = 0$. Amongst showing known and new properties, we prove that they form a special Jordan algebra, i.e. the symmetrized product of K-compatible ten...

Given the Riemann, or the Weyl, or a generalized curvature tensor K, a symmetric tensor b ij is named 'compatible' with the curvature tensor if b i m K jklm + b j m K kilm + b k m K ijlm = 0. Amongst showing known and new properties, we prove that they form a special Jordan algebra, i.e. the symmetrized product of K-compatible tensors is K-compatib...

We prove that in space-times a velocity field that is shear, vorticity and acceleration-free, if any, is unique up to reflection, with these exceptions: generalized Robertson-Walker space-times whose space sub-manifold is warped, and twisted space-times (the scale function is space-time dependent) whose space sub-manifold is doubly twisted. In spac...

We prove that in space-times a velocity field that is shear, vorticity and acceleration-free, if any, is unique up to reflection, with these exceptions: generalized Robertson-Walker space-times whose space sub-manifold is warped, and twisted space-times (the scale function is space-time dependent) whose space sub-manifold is doubly twisted. In spac...

We prove that in Robertson-Walker space-times (and in generalized Robertson-Walker spacetimes of dimension greater than 3 with divergence-free Weyl tensor) all higher-order gravitational corrections of the Hilbert-Einstein Lagrangian density $F(R,\square R, ... , \square^k R)$ have the form of perfect fluids in the field equations. This statement d...

We prove that in Robertson-Walker space-times (and in generalized Robertson-Walker spacetimes of dimension greater than 3 with divergence-free Weyl tensor) all higher-order gravitational corrections of the Hilbert-Einstein Lagrangian density F (R, R, ..., k R) have the form of perfect fluids in the field equations. This statement definitively allow...

In an-dimensional Friedmann-Robertson-Walker metric, it is rigorously shown that any analytical theory of gravity f(R,G), where R is the curvature scalar and G is the Gauss-Bonnet topological invariant, can be associated to a perfect-fluid stress-energy tensor. In this perspective, dark components of the cosmological Hubble flow can be geometricall...

In an $n$-dimensional Friedmann-Robertson-Walker metric, it is rigorously shown that any smooth theory of gravity $f(R,{\cal G})$, where $R$ is the curvature scalar and $\cal G$ is the Gauss-Bonnet topological invariant, can be associated to a perfect-fluid stress-energy tensor. In this perspective, dark components of the cosmological Hubble flow c...

We prove that in space-times a velocity field that is shear, vorticity and acceleration-free is unique, up to reflection, with these exceptions: warped space-times whose space sub-manifold is warped, and twisted space-times (the scale function is space-time dependent) whose space sub-manifold is doubly twisted. In space-time dimension n = 4, the Ri...

The Ricci tensor of a Robertson-Walker space-time is here specified by requiring constancy of the scalar curvature and a vanishing spatial curvature. By entering this Ricci tensor in Einstein's equations (without cosmological constant), the cosmological fluid shows a transition from a pure radiation to a Lambda equation of state. In other words, th...

We give new necessary and sufficient conditions on the Weyl tensor for generalized Robertson-Walker (GRW) space-times to be perfect-fluid space-times. For GRW space-times, we determine the form of the Ricci tensor in all the O(n)-invariant subspaces provided by Gray’s decomposition of the gradient of the Ricci tensor. In all but one, the Ricci tens...

In this note we show that Lorentzian Concircular Structure manifolds (LCS)n coincide with Generalized Robertson-Walker space-times.

The Ricci tensor of a Robertson-Walker space-time is here specified by requiring constancy of the scalar curvature and a vanishing spatial curvature. By entering this Ricci tensor in Einstein's equations (without cosmological constant), the cosmological fluid
shows a transition from a pure radiation to a Lambda equation of state. In other words,the...

The Ricci tensor of a Robertson-Walker space-time is here specified by requiring constancy of the scalar curvature and a vanishing spatial curvature. By entering this Ricci tensor in Einstein's
equations (without cosmological constant), the cosmological fluid shows a transition from a pure radiation to a Lambda equation of state. In other words, th...

We give new necessary and sufficient conditions on the Weyl tensor for generalized Robertson-Walker (GRW) space-times to be perfect-fluid space-times. For GRW space-times, we determine the form of the Ricci tensor in all the O(n)-invariant subspaces provided by Gray's decomposition of the gradient of the Ricci tensor. In all but one, the Ricci tens...

We give new necessary and sufficient conditions on the Weyl tensor for generalized Robertson-Walker (GRW) space-times to be perfect-fluid space-times. For GRW space-times, we determine the form of the Ricci tensor in all the O(n)-invariant subspaces provided by Gray's decomposition of the gradient of the Ricci tensor. In all but one, the Ricci tens...

We show that an n-dimensional generalized Robertson-Walker (GRW) space-time with divergence-free conformal curvature tensor exhibits a perfect fluid stress-energy tensor for any f(R) gravity model. Furthermore we prove that a conformally flat GRW space-
time is still a perfect fluid in both f(R) and quadratic gravity where other curvature invariant...

We show that an n-dimensional generalized Robertson-Walker (GRW) space-time with divergence-free conformal curvature tensor exhibits a perfect fluid stress-energy tensor for any f(R) gravity model. Furthermore we prove that a conformally flat GRW space- time is still a perfect fluid in both f(R) and quadratic gravity where other curvature invariant...

We show that an n-dimensional generalized Robertson-Walker (GRW) space-time with divergence-free conformal curvature tensor exhibits a perfect fluid stress-energy tensor for any f (R) gravity model. Furthermore we prove that a conformally flat GRW space-time is still a perfect fluid in both f (R) and quadratic gravity where other curvature invarian...

In our paper “A simple property of the Weyl tensor for a shear, vorticity and acceleration-free velocity field”.

We show that an n-dimensional generalized Robertson-Walker space-time with divergence-free conformal curvature tensor exhibits a perfect fluid stress-energy tensor in any f(R) theory of gravitation

We prove that, in a space-time of dimension n > 3 with a velocity
field that is shear-free, vorticity-free and acceleration-free, the covariant di-
vergence of the Weyl tensor is zero if the contraction of the Weyl tensor with
the velocity is zero. The other way, if the covariant divergence of the Weyl
tensor is zero, then the contraction of the We...

We extend to twisted space-times the following property of Generalized Robertson-Walker spacetimes: the Weyl tensor is divergence-free if and only if its contraction with the time-like unit torse-forming vector is zero. Despite the simplicity of the statement, the proof is involved. As a product of the same calculation, we introduce a new generaliz...

We review a simple mechanism for the formation of plateaux in the fractional quantum Hall effect. It arises from a map of the microscopic Hamiltonian in the thin torus limit to a lattice gas model, solved by Hubbard. The map suggests a Devil’s staircase pattern, and explains the observed asymmetries in the widths. Each plateau is a new ground state...

We show that n-dimensional perfect fluid spacetimes with divergence-
free conformal curvature tensor and constant scalar curvature are generalized
Robertson Walker (GRW) spacetimes; as a consequence a perfect
fluid Yang pure space is a GRW spacetime. We also prove that perfect
fluid spacetimes with harmonic generalized curvature tensor are, under...

We show that n-dimensional perfect fluid spacetimes with divergence-
free conformal curvature tensor and constant scalar curvature are generalized
Robertson Walker (GRW) spacetimes; as a consequence a perfect fluid
Yang pure space is a GRW spacetime. We also prove that perfect
fluid spacetimes with harmonic generalized curvature tensor are, under...

We review a simple mechanism for the formation of plateaux in the fractional quantum Hall effect. It arises from a map of the microscopic Hamiltonian in the thin torus limit to a lattice gas model, solved by Hubbard. The map suggests a Devil's staircase pattern, and explains the observed asymmetries in the widths. Each plateau is a new ground state...

I present a pedagogical derivation of Hedin's equations for the eval-uation of the propagator, the proper self-energy, the effective potential, the proper polarization and the vertex of a many-body theory with two-body interaction. Their solution in d = 0 allows to enumerate Feynman diagrams in various resummation schemes.

In these pedagogical notes I introduce the operator form of Wick's theorem, i.e. a procedure to bring to normal order a product of 1-particle creation and destruction operators, with respect to some reference many-body state. Both the static and the time- ordered cases are presented.

We obtain expressions for the shear and the vorticity tensors of perfect-fluid spacetimes, in terms of the divergence of the Weyl tensor. For such spacetimes, we prove that if the gradient of the energy density is parallel to the velocity, then either the expansion rate is zero, or the vorticity vanishes. This statement recalls the "shear-free conj...

We show that in dimension n>3 the class of simple conformally recurrent space-times coincides with the class of conformally recurrent pp-waves.

Many bosonic (fermionic) fractional quantum Hall states, such as Laughlin, Moore-Read and Read-Rezayi wavefunctions, belong to a special class of orthogonal polynomials: the Jack polynomials (times a Vandermonde determinant). This fundamental observation allows to point out two different recurrence relations for the coefficients of the permanent (S...

Robertson-Walker and Generalized Robertson-Walker spacetimes may be characterized by the existence of a time-like unit torse-forming vector field, with other constrains. We show that Twisted manifolds may still be characterized by the existence of such (unique) vector field, with no other constrain. Twisted manifolds generalize RW and GRW spacetime...

Generalized Robertson-Walker spacetimes extend the notion of Robertson-Walker spacetimes, by allowing for spatial non-homogeneity. A survey is presented, with main focus on Chen's characterization in terms of a timelike concircular vector. Together with their most important properties, some new results are presented.

We prove theorems about the Ricci and the Weyl tensors on generalized Robertson-Walker space-times of dimension $n\ge 3$. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmonic if and only if it is annihil...

Extended recurrent pseudo-Riemannian manifolds were introduced by Mileva Prvanovic'. We reconsider her work in the light of recent results and show that the manifold is conformally flat, and it is a space of quasi-constant curvature. We also show that an extended recurrent Lorentzian manifold, with time-like associated covector, is a perfect fluid...

After more than three decades, the fractional quantum Hall effect still poses challenges to contemporary physics. Recent experiments point toward a fractal scenario for the Hall resistivity as a function of the magnetic field. Here, we consider the so-called thin-torus limit of the Hamiltonian describing interacting electrons in a strong magnetic f...

The object of the present paper is to study weakly cyclic Z symmetric spacetimes. At first we prove that a weakly cyclic Z symmetric spacetime is a quasi Einstein spacetime. Then we study \({{(WCZS)}_{4}}\) spacetimes satisfying the condition div \({C=0}\). Next we consider conformally flat \({{(WCZS)}_{4}}\) spacetimes. Finally, we characterise du...

The ensemble of antagonistic matrices is introduced and studied. In antagonistic matrices the entries $\mathcal A_{i,j}$ and $\mathcal A_{j,i}$ are real and have opposite signs, or are both zero, and the diagonal is zero. This generalization of antisymmetric matrices is suggested by the linearized dynamics of competitive species in ecology.

Conformally recurrent pseudo-Riemannian manifolds of dimension n ≥ 5 are investigated. The Weyl tensor is represented as a Kulkarni–Nomizu product. If the square of the Weyl tensor is non-zero, a covariantly constant symmetric tensor is constructed, that is quadratic in the Weyl tensor. Then, by Grycak’s theorem, the explicit expression of the trac...

A perfect-fluid space-time of dimension n ≥ 4, with (1) irrotational velocity vector field and (2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with an Einstein fiber. Condition (1) is verified whenever pressure and energy density are related by an equation of state. The contraction of the Weyl tensor with the ve...

After more than three decades the fractional quantum Hall effect still poses
challenges to contemporary physics. While many features are phenomenologically
understood, a detailed comprehension of its microscopic origin still lacks.
Here, we address this problem by showing that the Hamiltonian describing
interacting electrons in a strong magnetic fi...

Conformally recurrent pseudo-Riemannian manifolds of dimension
n � 5 are investigated. The Weyl tensor is represented as a Kulkarni-Nomizu
product. If the square of the Weyl tensor is nonzero, a covariantly constant
symmetric tensor is constructed, that is quadratic in the Weyl tensor. Then,
by Grycak’s theorem, the explicit expression of the trace...

Conformally recurrent pseudo-Riemannian manifolds of dimension n>4 are
investigated. The Weyl tensor may be represented as a Koulkarni-Nomizu product
involving a symmetric tensor and the recurrence vector. If the recurrence
vector is a closed form, the Ricci and two other tensors are Weyl compatible.
If the recurrence vector is non-null, a covarian...

We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications. In particular, it is shown that the existence of a Weyl compatible vector implies that the Weyl tensor is algeb...

Conformally quasi-recurrent (CQR)n pseudo-Riemannian manifolds
are investigated, and several new results are obtained. It is shown that
the Ricci tensor and the gradient of the fundamental vector are Weyl compatible
tensors (the notion was introduced recently by the authors and applies
to significative space-times), (CQR)n manifolds with concircula...

This paper is about analytic properties of single transfer matrices
originating from general block-tridiagonal or banded matrices. Such matrices
occur in various applications in physics and numerical analysis. The
eigenvalues of the transfer matrix describe localization of eigenstates and are
linked to the spectrum of the block tridiagonal matrix b...

Derdziński and Shen's theorem on the restrictions on the Riemann tensor imposed by existence of a Codazzi tensor holds more generally when a Riemann compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity for a new “Codazzi deviation tensor”, with a...

We introduce a new kind of Riemannian manifold that includes weakly-, pseudo- and pseudo projective Ricci symmetric manifolds.
The manifold is defined through a generalization of the so called Z tensor; it is named weakly
Z-symmetric and is denoted by (WZS)
n
. If the Z tensor is singular we give conditions for the existence of a proper concircular...

Quantum models with fields in matrix representations of classical groups exhibit a high order phase transition in the limit of large order of the group. In zero dimension of space-time the Green functions of the newly found branch are those of a Gaussian model; in one dimension, the physical mass vanishes at the critical point.

Abstract: We extend a remarkable theorem of Derdziński and Shen, on the restrictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor. We show that the Codazzi equation can be replaced by a more general algebraic condition. The resulting extension applies both to the Riemann tensor and to generalized curvature tensors.

The eigenvalue equation of a band or a block tridiagonal matrix, the tight
binding model for a crystal, a molecule, or a particle in a lattice with random
potential or hopping amplitudes: these and other problems lead to three-term
recursive relations for (multicomponent) amplitudes. Amplitudes n steps apart
are linearly related by a transfer matri...

We extend a classical result by Derdzinski and Shen, on the restrictions
imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor.
The new conditions of the theorem include Codazzi tensors (i.e. closed 1-forms)
as well as tensors with gauged Codazzi condition (i.e. "recurrent 1-forms"),
typical of some well known differential s...

A second-order differential identity for the Riemann tensor is obtained on a manifold with a symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors are derived from it. Applications to manifolds with recurrent or symmetric structures are discussed. The new structure of K -recurrency naturally emerge...

Eigenvalues and localization of eigenvectors of non-Hermitian tridiagonal periodic random matrices are studied by means of the Hatano-Nelson deformation. The support of the spectrum undergoes a disk to annulus transition, with inner radius measured by the complex Thouless formula. The inner bounding circle and the annular halo are stuctures that co...

The spectrum of exponents of the transfer matrix provides the localization lengths of Anderson's model for a particle in a lattice with disordered potential. I show that a duality identity for determinants and Jensen's identity for subharmonic functions give a formula for the spectrum in terms of eigenvalues of the Hamiltonian with non-Hermitian bo...

Time-dependent density-functional theory (TDDFT) is widely used in the study of linear response properties of finite systems. However, there are difficulties in properly describing excited states, which have double- and higher-excitation characters, which are particularly important in molecules with an open-shell ground state. These states would be...

An identity is proven that evaluates the determinant of a block tridiagonal matrix with (or without) corners as the determinant of the associated transfer matrix (or a submatrix of it).

The calculation of self-energy corrections to the electron bands of a metal requires the evaluation of the intraband contribution to the polarizability in the small-q limit. When neglected, as in standard GW codes for semiconductors and insulators, a spurious gap opens at the Fermi energy. Systematic methods to include intraband contributions to th...

The theorem by Gell-Mann and Low is a cornerstone in quantum field theory and zero-temperature many-body theory. The standard proof is based on Dyson’s time-ordered expansion of the propagator; a proof based on exact identities for the time propagator is here given.

The many-body dynamics of interacting electrons in condensed matter and quantum chemistry is often studied at the quasiparticle level, where the perturbative diagrammatic series is partially resummed. Based on Hedin's equations for self-energy, polarization, propagator, effective potential, and vertex function in zero dimension of space-time, dress...

Vibrational spectra of polyatomic molecules are often obtained from a polynomial expansion of the adiabatic potential around a minimum. For several molecules, we show that such an approximation displays an unphysical saddle point of comparatively small energy, leading to a region where the potential is negative and unbounded. This poses an upper li...

A method to implement the many-body Green function formalism in the GW approximation for infinite nonperiodic systems is presented. It is suitable to treat systems of known “asymptotic” properties which enter as boundary conditions, while the effects of the lower symmetry are restricted to regions of finite volume. For example, it can be applied to...

Hedin's equations are solved perturbatively in zero dimension to count Feynman graphs for self-energy, polarization, propagator, effective potential and vertex function in a many-body theory of fermions with two-body interaction. Counting numbers are also obtained in the GW approximation.

The first order contribution to frictional drag in bi-layered fermion gas is examined. We discuss the relevance of single photon exchange in the evaluation of transresistance, which is usually explained by second order effects such as Coulomb and phonon drag. Since the effective e.m. interaction is unscreened, in the d.c. limit we obtain a finite (...

I consider a general block tridiagonal matrix and the corresponding transfer matrix. By allowing for a complex Bloch parameter in the boundary conditions, the two matrices are related by a spectral duality. As a consequence, I derive some analytic properties of the exponents of the transfer matrix in terms of the eigenvalues of the (non-Hermitian)...

Two results are presented for reduced Yang-Mills (YM) integrals
with different symmetry groups and dimensions: the first is a
compact integral representation in terms of the relevant
variables of the integral; the second is a method to
analytically evaluate the integrals in cases of low order. This
is exhibited by evaluating a YM integral over rea...

We present some old and new results in the enumeration of
random walks in one dimension, mostly developed in work on
enumerative combinatorics. The relation between the trace of the
nth power of a tridiagonal matrix and the enumeration of
weighted paths of n steps allows an easier combinatorial
enumeration of paths. It also seems promising for the...

The two-point resolvent is calculated in the large-n limit for the generalized fixed and bounded trace ensembles. It is shown to disagree with that of the canonical Gaussian ensemble by a nonuniversal part that is given explicitly for all monomial potentials V(M)=M(2p). Moreover, we prove that for the generalized fixed and bounded trace ensemble al...

We study a class of tridiagonal matrix models, the “q-roots of unity” models, which includes the sign (q=2) and the clock (q=∞) models by Feinberg and Zee. We find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides, in the complex plane. Furthermore, the averaged traces of M
k are integers that...

The two-point resolvent is calculated in the large-n limit for the generalized fixed and bounded trace ensembles. It is shown to disagree with the one of the canonical Gaussian ensemble by a non-universal part which is given explicitly for all monomial potentials V (M) = M 2p . Moreover, we prove that for the generalized fixed and bounded trace ens...

General Hamiltonian matrices with tridiagonal block structure and the associated transfer matrices are investigated in the cases of periodic and scattering boundary conditions. They arise from tight binding models with finite range hopping in one or more dimensions of space, in the presence of a Aharonov - Bohm flux or in multichannel scattering. A...

The localization of eigenfunctions in finite samples of the 1D Anderson and Lloyd models is quantitatively described by the information length. This quantity is numerically investigated for both models and it is found to scale with the size of the sample and the disorder according to a simple law.

The authors provide the general solution of the large-N limit of matrix models with even polynomial potential in the phase with two minimal. The solution is given both in the saddle point and in the orthogonal polynomials approaches.

In the non-perturbative regime, matrix models display a large-N phase transition. For finite but large N, the transition is anticipated by strong oscillations in some coefficients in the recurrence relations for the orthogonal polynomials that allow the calculation of the partition function. The author shows how to perform the limit, requiring the...

The Helmholtz equation for regular polygons is investigated by means of the conformal mapping from the circle, which provides an expansion parameter for the approximate evaluation of the lowest eigenvalue and the corresponding eigenvector.

The simplest matrix model which exhibits multicritical points is carefully analysed. The authors reproduce results of potential interest for the non-perturbative theory of strings in the region where the orthogonal polynomials were correctly used. However, the analysis holds for the whole parameter space.

The authors show that the spacing distribution for eigenvalues of band random matrices is described by a single parameter b2/N, where b is the band half-width and N is the size of the matrices. It is also shown that the eigenvalue's density obeys the semicircle law. The found scaling behaviour suggests that the fluctuation properties in the interme...

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