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Papers by Giovanni Cicuta

arXiv (Cornell University), Jun 19, 2022

We study ensembles of sparse random block matrices generated from the adjacency matrix of a Erdös... more We study ensembles of sparse random block matrices generated from the adjacency matrix of a Erdös-Renyi random graph with N vertices of average degree Z, inserting a real symmetric d × d random block at each non-vanishing entry. We consider some ensembles of random block matrices with rank r < d and with maximal rank, r = d. The spectral moments of the sparse random block matrix are evaluated for N → ∞, d finite or infinite, and several probability distributions for the blocks (e.g. fixed trace, bounded trace and Gaussian). Because of the concentration of the probability measure in the d → ∞ limit, the spectral moments are independent of the probability measure of the blocks (with mild assumptions of isotropy, smoothness and sub-gaussian tails). The Effective Medium Approximation is the limiting spectral density of the sparse random block ensembles with finite rank. Analogous classes of universality hold for the Laplacian sparse block ensemble. The same limiting distributions are obtained using random regular graphs instead of Erdös-Renyi graphs.

Journal of Physics A: Mathematical and Theoretical, 2021

The spectral moments of ensembles of sparse random block matrices are analytically evaluated in t... more The spectral moments of ensembles of sparse random block matrices are analytically evaluated in the limit of large order. The structure of the sparse matrix corresponds to the Erdös-Renyi random graph. The blocks are i.i.d. random matrices of the classical ensembles GOE or GUE. The moments are evaluated for finite or infinite dimension of the blocks. The correspondences between sets of closed walks on trees and classes of irreducible partitions studied in free probability together with functional relations are powerful tools for analytic evaluation of the limiting moments. They are helpful to identify probability laws for the blocks and limits of the parameters which allow the evaluation of all the spectral moments and of the spectral density.

The Oxford Handbook of Random Matrix Theory, 2018

This article considers phase transitions in matrix models that are invariant under a symmetry gro... more This article considers phase transitions in matrix models that are invariant under a symmetry group as well as those that occur in some matrix ensembles with preferred basis, like the Anderson transition. It first reviews the results for the simplest model with a nontrivial set of phases, the one-matrix Hermitian model with polynomial potential. It then presents a view of the several solutions of the saddle point equation. It also describes circular models and their Cayley transform to Hermitian models, along with fixed trace models. A brief overview of models with normal, chiral, Wishart, and rectangular matrices is provided. The article concludes with a discussion of the curious single-ring theorem, the successful use of multi-matrix models in describing phase transitions of classical statistical models on fluctuating two-dimensional surfaces, and the delocalization transition for the Anderson, Hatano-Nelson, and Euclidean random matrix models.

Journal of Statistical Physics, 2019

We study the large N limit of a sparse random block matrix ensemble. It depends on two parameters... more We study the large N limit of a sparse random block matrix ensemble. It depends on two parameters: the average connectivity Z and the size of the blocks d, which is the dimension of an euclidean space. In the limit of large d with Z d fixed, we prove the conjecture that the spectral distribution of the sparse random block matrix converges in the case of the Adjacency block matrix to the one of the effective medium approximation, in the case of the Laplacian block matrix to the Marchenko-Pastur distribution. We extend previous analytical computations of the moments of the spectral density of the Adjacency block matrix and the Lagrangian block matrix, valid for all values of Z and d.

Journal of Physics A: Mathematical and Theoretical, 2016

The ensemble of antagonistic matrices is introduced and studied. In antagonistic matrices the ent... more The ensemble of antagonistic matrices is introduced and studied. In antagonistic matrices the entries A i,j and 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.

Thesis University of California Santa Barbara 1970 Source American Doctoral Dissertations Source Code X1970 Page 0257, 1970

Phase transitions generically occur in random matrix models as the parameters in the joint probab... more Phase transitions generically occur in random matrix models as the parameters in the joint probability distribution of the random variables are varied. They affect all main features of the theory and the interpretation of statistical models. In this paper a brief review of phase transitions in invariant ensembles is provided, with some comments to the singular values decomposition in complex non-hermitian ensembles.

Lettere Al Nuovo Cimento Series 2, 1974

The eikonal approximation has been extensively studied and usefully applied in many areas of phys... more The eikonal approximation has been extensively studied and usefully applied in many areas of physics, including quantum field theory (1). In some investigations perturbative expansions of the exact solution were obtained with the first term of the expansion being the eikonal approximation (2). The eikonal approximation arises when the recoil of a massive or fast particle is neglected. This is often a relevant approximation in the high-energy regime. We derive an exact expansion in the large-mass limit for a relativistic particle in arbitrary external field. It is known (3) that the two-points Green function of a particle in arbitrary external field determines, by functional derivatives, all Green's functions for a system of interacting particles. The asymptotic expansion we obtain is defined for the amputated propagator on the mass shell and this is sufficient to derive an approximation for the elastic two-body scattering amplitude (4). The relativistic propagator in an external field is studied with the Feynman parametric representation by a technique very similar to the recent Swift work in the nonrelativistic case (5) and we find that the eikonal approximation results as the limit for large mass of the relativistic Born series. Furthermore we solve a well-known difficulty which arises in the relativistic scattering amplitude for the scalar case, because of the existence

Lettere Al Nuovo Cimento Series 2, 1976

Physical Review D, 1984

In a model with global SU(N) invariance with a scalar field that transforms as the (Nsup2-1)-dime... more In a model with global SU(N) invariance with a scalar field that transforms as the (Nsup2-1)-dimensional representation (symmetric adjoint) and interacts with cubic coupling, we study Regge poles and Regge cuts. The interplay between Regge behavior and the group of internal symmetry is analyzed with care.

Physica A: Statistical Mechanics and its Applications, 1993

In a two dimensional square lattice, where all sites are covered by loops, in the past few years ... more In a two dimensional square lattice, where all sites are covered by loops, in the past few years a classical statistical mechanics model of dense loop gas was considered for its relevance for the evaluation of observables of the quantum antiferromagnetic Heisenberg model on the resonating valence bond state. By use of transfer matrices we evaluate several observables for narrow strips with a width of two or three sites. A simple extrapolation, consistent with rigorous bounds, is conjectured for its possible use in a variational ansatz in the Hubbard model.

Il Nuovo Cimento A, 1984

The large-iV limit is found in closed form for a class of matrix models in a zero-dimensional spa... more The large-iV limit is found in closed form for a class of matrix models in a zero-dimensional space-time. A method to sum in an approximate way planar graphs in every dimension is proposed and tested in dimension one.

Lettere Al Nuovo Cimento Series 2, 1977

The two besl-known examples of soliton solutions: the Abrikosov-Nielsea-Olesen vortex (x) and the... more The two besl-known examples of soliton solutions: the Abrikosov-Nielsea-Olesen vortex (x) and the 't Hooft-Polyakov magnetic monopole (2) may also be considered as instantons in an appropriate space-time (the euclidean R and R a, respectively). It was emphasized (2) the important ...

Physical Review, 1968

The natural assumption of Fermi statistics for quarks is studied in the framework of a nonrelativ... more The natural assumption of Fermi statistics for quarks is studied in the framework of a nonrelativistic model of the nucleon as a bound state of three quarks. Quark dynamics is assumed to be dominated by a three-body attractive force. Different statistical assumptions lead to nucleon form factors (FF) which are compared with experimental data. The FF depend only on the quark mass as a free parameter. We argue that the nucleon FF do not provide specific evidence against Fermi statistics for quarks. The possibility of Fermi statistics appears to be related to the quark mass M and is found consistent with experimental data for M ^ 3m (^-nucleon mass).

The study of the convergence of power series expansions of energy eigenvalues for anharmonic osci... more The study of the convergence of power series expansions of energy eigenvalues for anharmonic oscillators in quantum mechanics differs from general understanding, in the case of quasi-exactly solvable potentials. They provide examples of expansions with finite radius and suggest techniques useful to analyze more generic potentials.

Physical Review D, 1971

A detailed study is made of the scattering of two high-energy particles which interact via the ex... more A detailed study is made of the scattering of two high-energy particles which interact via the exchange of ladders. The legs of the ladders are attached to the world lines of the incident particles in all possible ways. Working to lowest order in the coupling constant for the Regge trajectory and residue functions, we find that the leading Regge-pole term arising from single-ladder-exchange exponentiates, and we obtain the Regge eikonal model.