arXiv subjects

Domingos H. U. Marchetti

Publications and source records attributed to Domingos H. U. Marchetti.

At least 19 recordsLinked to original sources

arXiv2025

The majorant method for the fermionic effective action

We revisit the problem of controlling Polchinski's equation by the solution of an associate Hamilton-Jacobi equation which determines a norm majorant for the fermionic effective action. This method, referred to as the majorant method, was first introduced by D. Brydges and J. Wright in 1988, but its original formulation contains a gap which has never been addressed. We overcome this gap and show that the majorant equation and its existence condition are analogous to the ones originally obtained by Brydges and Wright. As an application of the method, we investigate a fermion model with a local quartic interaction.

math-ph↗

arXiv2024

A mathematical theory of the critical point of ferromagnetic Ising systems

We develop a theory of the critical point of the ferromagnetic Ising model, whose basic objects are the ergodic (pure) states of the infinite system. It proves the existence of anomalous critical fluctuations, for dimension $ν=2$ and, under a standard assumption, for $ν=3$, for the model with nearest neighbor interaction, in a way which is consistent with the probabilistic approach of Cassandro, Jona-Lasinio and several others, reviewed in Jona-Lasinio's article in Phys. Rep. 352, 439 (2001). We propose to single out the state at the critical temperature $T_{c}$ among the ergodic thermal states associated to temperatures $0 \le T \le T_{c}$, by a condition of non-summable clustering of the connected two-point function. The analogous condition for the connected $(2r)-$ point functions for $r \ge 2$, together with a scaling hypothesis, proves that the (macroscopic) fluctuation state is quasi-free, also at the critical temperature, after a proper rescaling, also at the critical temperature, in agreement with a theorem by Cassandro and Jona-Lasinio, whose proof is, however, shown to be incomplete. Other subjects include topics related to universality, including spontaneous breaking of continuous symmetries and violations of universality in problems of energetic and dynamic stability.

math-ph↗

arXiv2019

On the Mayer series of two-dimensional Yukawa gas at inverse temperature in the interval of collapse

A Theorem on the minimal specific energy for a system with \pm 1 charged particles interacting through the Yukawa pair potential v is proved which may stated as follows. Let v be represented by scale mixtures of d-dimensional Euclid's hat (cutoff at short scale distances) with d\geq 2. For any even number of particles n, the interacting energy U_{n} divided by n, attains an n-independent minimum at a configuration with zero net charge and particle positions collapsed altogether to a point. For any odd number of particles n, the ratio U_{n}/(n-1) attains its minimum value, the same of the previous cases, at the configuration with \pm 1 net charge and particle positions collapsed to a point. This Theorem is then used to resolve an obstructive remark of an unpublished paper (Remark 7.5 of \cite{Guidi-Marchetti}) which, whether the standard decomposition of the Yukawa potential into scales were adopted, would impede a direct proof of the convergence of the Mayer series of the two-dimensional Yukawa gas for the inverse temperature in the whole interval [4π,8π) of collapse. In the present paper, it is proven convergence up to the second threshold 6πand its given explanations on the mechanism that allow it to be extend up to 8π. The paper distinguishes the matters concerning stability from those related to convergence of the Mayer series. In respect to the latter the paper dedicates to the Cauchy majorante method applied to the density function of Yukawa gas in the interval of collapses. It also dedicates to the proof of the main Theorem and estimates of the modified Bessel functions of second kind involved in both representations of two-dimensional Yukawa potential: standard and scale mixture of the Euclid's hat function.

math-ph↗

arXiv2018

Contact process under renewals I

Motivated by questions regarding long range percolation, we investigate a non-Markovian analogue of the Harris contact process in $\mathbb{Z}^d$: an individual is attached to each site $x \in \mathbb{Z}^d$, and it can be infected or healthy; the infection propagates to healthy neighbors just as in the usual contact process, according to independent exponential times with a fixed rate $λ$; nevertheless, the possible recovery times for an individual are given by the points of a renewal process with heavy tail; the renewal processes are assumed to be independent for different sites. We show that the resulting processes have a critical value equal to zero.

math.PR↗

arXiv2015

Layered systems at the mean field critical temperature

We consider the Ising model on $\mathbb Z\times \mathbb Z$ where on each horizontal line $\{(x,i), x\in \mathbb Z\}$, the interaction is given by a ferromagnetic Kac potential with coupling strength $J_γ(x,y)\sim γJ(γ(x-y))$ at the mean field critical temperature. We then add a nearest neighbor ferromagnetic vertical interaction of strength $ε$ and prove that for every $ε>0$ the systems exhibits phase transition provided $γ>0$ is small enough.

math.PR↗

arXiv2014

Phase transitions in layered systems

We consider the Ising model on the two-dimensional square lattice where on each horizontal line, called "layer", the interaction is given by a ferromagnetic Kac potential with coupling strength $J_γ(x,y)=γJ(γ(x-y))$, where $J(\cdot)$ is smooth and has compact support; we then add a nearest neighbor ferromagnetic vertical interaction of strength $γ^{A}$ (where $A\ge 2$ is fixed) and prove that for any $β$ (inverse temperature) larger than the mean field critical value there is a phase transition for all $γ$ small enough.

math.PR↗

arXiv2014

On the Virial Series for a Gas of Particles with Uniformly Repulsive Pairwise Interaction

The pressure of a gas of particles with a uniformly repulsive pair interaction in a finite container is shown to satisfy (exactly as a formal object) a "viscous" Hamilton-Jacobi (H-J) equation whose solution in power series is recursively given by the variation of constants formula. We investigate the solution of the H-J and of its Legendre transform equation by the Cauchy-Majorant method and provide a lower bound to the radius of convergence on the virial series of the gas which goes beyond the threshold established by Lagrange's inversion formula. A comparison between the behavior of the Mayer and virial coefficients for different regimes of repulsion intensity is also provided.

math-ph↗

arXiv2012

Singular perturbation of nonlinear systems with regular singularity

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form \varepsilon zf^{\prime} = F(\varepsilon,z,f) with F a \mathbb{C}^ν-valued function, holomorphic in a polydisc \bar{D}_ρ\times \bar{D}_ρ\times \bar{D}_ρ^ν. We show that its unique formal solution in power series of \varepsilon, whose coefficients are holomorphic functions of z, is 1-summable under a Siegal-type condition on the eigenvalues of F_{f}(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple Lemma is developed to tame convolutions that appears in the power series expansion of nonlinear equations.

math.CV↗

arXiv2012

Anderson-like Transition for a Class of Random Sparse Models in d >= 2 Dimensions

We show that the Kronecker sum of d >= 2 copies of a random one-dimensional sparse model displays a spectral transition of the type predicted by Anderson, from absolutely continuous around the center of the band to pure point around the boundaries. Possible applications to physics and open problems are discussed briefly.

math-ph↗

arXiv2011

Asymptotic Integral Kernel for Ensembles of Random Normal Matrix with Radial Potentials

We use the steepest descents method to study the integral kernel of a family of normal random matrix ensembles with eigenvalue distribution P_{N}(z_{1},...,z_{N}) = Z_{N}^{-1} e^{-NΣ_{i=1}^{N}V_α(z_{i})} Π_{1\leqi<j\leqN}|z_{i}-z_{j}|^{2} where V_α(z)=|z|^α, z \in C and α\in ]0,\infty[. Asymptotic analysis with error estimates are obtained. A corollary of this expansion is a scaling limit for the n-point function in terms of the integral kernel for the classical Segal--Bargmann space.

math-ph↗

arXiv2009

Conformal Universality in Normal Matrix Ensembles

A remarkable property of Hermitian ensembles is their universal behavior, that is, once properly rescaled the eigenvalue statistics does not depend on particularities of the ensemble. Recently, normal matrix ensembles have attracted increasing attention, however, questions on universality for these ensembles still remain under debate. We analyze the universality properties of random normal ensembles. We show that the concept of universality used for Hermitian ensembles cannot be directly extrapolated to normal ensembles. Moreover, we show that the eigenvalue statistics of random normal matrices with radially symmetric potential can be made universal under a conformal transformation.

math-ph↗

arXiv2009

Conformal Deformation from Normal to Hermitian Random Matrix Ensembles

We investigate the eigenvalues statistics of ensembles of normal random matrices when their order N tends to infinite. In the model the eigenvalues have uniform density within a region determined by a simple analytic polynomial curve. We study the conformal deformations of normal random ensembles to Hermitian random ensembles and give sufficient conditions for the latter to be a Wigner ensemble.

math.PR↗

arXiv2008

Hierarchical Spherical Model as a Viscosity Limit of Corresponding O(N)Heisenberg Model

The O(N) Heisenberg and spherical models with interaction given by the long range hierarchical Laplacean are investigated. Two classical results are adapted. The Kac--Thompson solution [KT] of the spherical model, which holds for spacially homogeneous interaction, is firstly extended to hierarchical model whose interaction fails to be translation invariant. Then, the convergence proof of O(N) Heisenberg to the spherical model by Kunz and Zumbach [KZ] is extended to the long range hierarchical interaction. We also examine whether these results can be carried over as the size of the hierarchical block L goes to 1. These extensions are considered a preliminary study prior the investigation of the model by renormalization group given in [MCG] where central limit theorems for the spherical (N=\infty) model on the local potential approximation (L\downarrow 1) are then established from an explicit solution of the associate nonlinear first order partial differential equation.

math-ph↗

arXiv2008

Hierarchical Spherical Model from a Geometric Point of View

A continuous version of the hierarchical spherical model at dimension d=4 is investigated. Two limit distribution of the block spin variable X^γ, normalized with exponents γ=d+2 and γ=d at and above the critical temperature, are established. These results are proven by solving certain evolution equations corresponding to the renormalization group (RG) transformation of the O(N) hierarchical spin model of block size L^{d} in the limit L to 1 and N to \infty . Starting far away from the stationary Gaussian fixed point the trajectories of these dynamical system pass through two different regimes with distinguishable crossover behavior. An interpretation of this trajectories is given by the geometric theory of functions which describe precisely the motion of the Lee--Yang zeroes. The large--$N$ limit of RG transformation with L^{d} fixed equal to 2, at the criticality, has recently been investigated in both weak and strong (coupling) regimes by Watanabe \cite{W}. Although our analysis deals only with N=\infty case, it complements various aspects of that work.

math-ph↗

arXiv2004

Convergence of the Mayer Series via Cauchy Majorant Method with Application to the Yukawa Gas in the Region of Collapse

We construct majorant functions $Φ(t,z)$ for the Mayer series of pressure satisfying a nonlinear differential equation of first order which can be solved by the method of characteristics. The domain $| z| <(eτ) ^{-1}$ of convergence of Mayer series is given by the envelop of characteristic intersections. For non negative potentials we derive an explicit solution in terms of the Lambert $W$ --function which is related to the exponential generating function $T$ of rooted trees as $T(x)=-W(-x)$. For stable potentials the solution is majorized by a non negative potential solution. There are many choices in this case and we combine this freedom together with a Lagrange multiplier to examine the Yukawa gas in the region of collapse. We give, in this paper, a sufficient condition to establish a conjecture of Benfatto, Gallavotti and Nicoló. For any $β\in \lbrack 4π, 8π)$, the Mayer series with the leading terms of the expansion omitted (how many depending on $β$) is shown to be convergent provided an improved stability condition holds. Numerical calculations presented indicate this condition is satisfied if few particles are involved.

math-ph↗

arXiv2004

Brownian Motion Limit of Random Walks in Symetric Non-Homogeneous Media

The phenomenon of macroscopic homogenization is illustrated with a simple example of diffusion. We examine the conditions under which a $d$--dimensional simple random walk in a symmetric random media converges to a Brownian motion. For $d=1$, both the macroscopic homogeneity condition and the diffusion coefficient can be read from an explicit expression for the Green's function. Except for this case, the two available formulas for the effective diffusion matrix $κ$ do not explicit show how macroscopic homogenization takes place. Using an electrostatic analogy due to Anshelevich, Khanin and Sinai \cite{AKS}, we discuss upper and lower bounds on the diffusion coefficient $κ$ for $d>1$.

math-ph↗

arXiv2003

Combinatorial Solution of One-Dimensional Quantum Systems

We give a self-contained exposition of the combinatorial solution of quantum mechanical systems of coupled spins on a one-dimensional lattice. Using Trotter formula, we write the partition function as a generating function of a spanning subgraph of a two-dimensional lattice and solve the combinatorial problem by the method of Pfaffians provided the weights satisfy the so-called free fermion condition. The free energy and the ground state energy as a function of the inverse temperature, couplings J and magnetic fields h, for the XY model in a transverse field with period p=1 and 2, is then obtained.

cond-mat.stat-mech↗

arXiv2002

Rakib-Sivashinsky and Michelson-Sivashinsky Equations for Upward Propagating Flames: A Comparison Analysis

We establish a comparison between Rakib--Sivashinsky and Michelson-Sivashinsky quasilinear parabolic differential equations governing the weak thermal limit of upward flame front propagating in a channel. For the former equation, we give a complete description of all steady solutions and present their local and global stability analysis. For the latter, multi-coalescent unstable steady solutions are introduced and shown to be exponentially more numerous than the previous known coalescente solutions. This fact is argued to be responsible for the disagreement of the observed dynamics in numerical experiments with the exact (linear) stability analysis and also gives the ingredients to describe the quasi-stable behavior of parabolic steadily propagating flame with centered tip.

math.AP↗