arXiv subjects

Gabriel Álvarez

Publications and source records attributed to Gabriel Álvarez.

17 recordsLinked to original sources

arXiv2026

The Schwarz function and the shrinking of the Szegő curve: electrostatic, hydrodynamic, and random matrix models

We study the deformation of the classical Szegő curve $γ_0$ given by $γ_t = \{ z\in\mathbb{C}: |z\, e^{1-z}| = e^{-t}, |z|\leq 1\}$, $t\geq 0$ from three different viewpoints: an electrostatic equilibrium problem, the dual hydrodynamic model, and a random matrix model. The common framework underlying these models is the asymptotic distribution of zeros of the scaled varying Laguerre polynomials $L^{(α_n)}_n(n z)$ in the critical regime where $\lim_{n\to\infty}α_n/n=-1$, for which the limiting zero distribution is supported on $γ_t$, where the deformation parameter $t$ encodes the exponential rate at which the sequence $α_n$ approximates the set of negative integers. We show that the Schwarz functions of these curves can be written in terms of the Lambert $W$ function, and that in this formulation the $S$-property of Stahl and Gonchar and Rachmanov can be explictly written as the Schwarz reflection symmetry. We also discuss a conformal map of the interior of the curves $γ_t$ onto the disks $D(0,e^{-t})$ and the harmonic moments of the curves.

math-ph↗

arXiv2026

Unraveling anomalous relaxation effects in the thermodynamic limit

We address two central open problems in the theory of anomalous Mpemba-like relaxations: their extension beyond one spatial dimension and their consistent formulation in the thermodynamic limit. Our framework is the antiferromagnetic Ising model on a square lattice under an externally applied magnetic field, which enables us to work in the presence of a phase transition. The rich phase diagram contains two control parameters: temperature and magnetic field. We demonstrate that the standard assumption of relaxation dominated by a single leading exponential is inconsistent for intensive observables exhibiting standard fluctuations. Instead, as the system size increases, a continuous spectrum of time scales emerges. Nevertheless, we make the ansatz that, in the vicinity of the phase transition, the spectral projector onto the slowest time scales can be effectively characterized in terms of an equilibrium thermodynamic quantity: the susceptibility associated with the order parameter of the metastable phase. Combined with the richness of the phase diagram, this ansatz yields qualitative and semi-quantitative predictions for optimal protocols leading to a variety of anomalous relaxation phenomena involving simultaneous variations of temperature and magnetic field. These include direct and inverse Mpemba effects, cooling-heating asymmetries, and faster heating induced by precooling. Careful Monte Carlo simulations validate our theoretical predictions. Furthermore, minimal post-optimization suffices to convert our analytically guided protocols into fully optimal ones that display anomalous relaxations in their most pronounced form.

cond-mat.stat-mech↗

arXiv2024

Kinetic dominance and the wavefunction of the universe

We analyze the emergence of classical inflationary universes in a kinetic-dominated stage using a suitable class of solutions of the Wheeler-De Witt equation with a constant potential. These solutions are eigenfunctions of the inflaton momentum operator that are strongly peaked on classical solutions exhibiting either or both a kinetic dominated period and an inflation period. Our analysis is based on semiclassical WKB solutions of the Wheeler-De Witt equation interpreted in the sense of Borel (to perform a correct connection between classically allowed regions) and on the relationship of these solutions to the solutions of the classical model. For large values of the scale factor the WKB Vilenkin tunneling wavefunction and the Hartle-Hawking no-boundary wavefunctions are recovered as particular instances of our class of wavefunctions.

gr-qc↗

arXiv2024

Adaptive asymptotic solutions of inflationary models in the Hamilton-Jacobi formalism: Application to T-models

We develop a method to compute the slow-roll expansion for the Hubble parameter in inflationary models in a flat Friedmann-Lemaître-Robertson-Walker spacetime that is applicable to a wide class of potentials including monomial, polynomial, or rational functions of the inflaton, as well as polynomial or rational functions of the exponential of the inflaton. The method, formulated within the Hamilton-Jacobi formalism, adapts the form of the slow-roll expansion to the analytic form of the inflationary potential, thus allowing a consistent order-by-order computation amenable to Padé summation. Using T-models as an example, we show that Padé summation extends the domain of validity of this adapted slow-roll expansion to the end of inflation. Likewise, Padé summation extends the domain of validity of kinetic-dominance asymptotic expansions of the Hubble parameter into the fast-roll regime, where they can be matched to the aforesaid Padé-summed slow-roll expansions. This matching in turn determines the relation between the expansions for the number $N$ of e-folds and allows us to compute the total amount of inflation as a function of the initial data or, conversely, to select initial data that correspond to a fixed total amount of inflation. Using the slow-roll stage expansions, we also derive expansions for the corresponding spectral index $n_s$ accurate to order $1/N^2$, and tensor-to-scalar ratio $r$ accurate to order $1/N^3$ for these T-models.

gr-qc↗

arXiv2020

Separatrices in the Hamilton-Jacobi Formalism of Inflaton Models

We consider separatrix solutions of the differential equations for inflaton models with a single scalar field in a zero-curvature Friedmann-Lema\^ıtre-Robertson-Walker universe. The existence and properties of separatrices are investigated in the framework of the Hamilton-Jacobi formalism, where the main quantity is the Hubble parameter considered as a function of the inflaton field. A wide class of inflaton models that have separatrix solutions (and include many of the most physically relevant potentials) is introduced, and the properties of the corresponding separatrices are investigated, in particular, asymptotic inflationary stages, leading approximations to the separatrices, and full asymptotic expansions thereof. We also prove an optimal growth criterion for potentials that do not have separatrices.

math-ph↗

arXiv2019

Topography effect on the seismogenic deformation of the earth's surface

A comparison of the displacements of the earth's surface after an earthquake was made, calculating with the analytical expressions coming from an infinite flat slab approximation and compared with these numerically considering the topography of the Earth. One conclusion of this work is that the flat Earth approximation, has a greater error in the lateral displacement than in the vertical one. It can also be noted that the error in the magnitude of the displacement is less or of the order of ten percent of the maximum displacement of the earth's surface.

physics.geo-ph↗

arXiv2017

A new method to sum divergent power series: educated match

We present a method to sum Borel- and Gevrey-summable asymptotic series by matching the series to be summed with a linear combination of asymptotic series of known functions that themselves are scaled versions of a single, appropriate, but otherwise unrestricted, function $Φ$. Both the scaling and linear coefficients are calculated from Padé approximants of a series transformed from the original series by $Φ$. We discuss in particular the case that $Φ$ is (essentially) a confluent hypergeometric function, which includes as special cases the standard Borel-Padé and Borel-Leroy-Padé methods. A particular advantage is the mechanism to build knowledge about the summed function into the approximants, extending their accuracy and range even when only a few coefficients are available. Several examples from field theory and Rayleigh-Schrödinger perturbation theory illustrate the method.

math-ph↗

arXiv2017

Phase space and phase transitions in the Penner matrix model with negative coupling constant

The partition function of the Penner matrix model for both positive and negative values of the coupling constant can be explicitly written in terms of the Barnes G function. In this paper we show that for negative values of the coupling constant this partition function can also be represented as the product of an holomorphic matrix integral by a nontrivial oscillatory function of n. We show that the planar limit of the free energy with 't Hooft sequences does not exist. Therefore we use a certain modification that uses Kuijlaars-McLaughlin sequences instead of 't Hooft sequences and leads to a well-defined planar free energy and to an associated two-dimensional phase space. We describe the different configurations of complex saddle points of the holomorphic matrix integral both to the left and to the right of the critical point, and interpret the phase transitions in terms of processes of gap closing, eigenvalue tunneling, and Bose condensation.

math-ph↗

arXiv2016

Complex saddles in the Gross-Witten-Wadia matrix model

We give an exhaustive characterization of the complex saddle point configurations of the Gross-Witten-Wadia matrix model in the large-N limit. In particular, we characterize the cases in which the saddles accumulate in one, two, or three arcs, in terms of the values of the coupling constant and of the fraction of the total unit density that is supported in one of the arcs, and derive an explicit condition for gap closing associated to nonvacuum saddles. By applying the idea of large-N instanton we also give direct analytic derivations of the weak-coupling and strong-coupling instanton actions.

hep-th↗

arXiv2015

Fine structure in the large n limit of the non-hermitian Penner matrix model

In this paper we apply results on the asymptotic zero distribution of the Laguerre polynomials to discuss generalizations of the standard large $n$ limit in the non-hermitian Penner matrix model. In these generalizations $g_n n\to t$, but the product $g_n n$ is not necessarily fixed to the value of the 't Hooft coupling $t$. If $t>1$ and the limit $l = \lim_{n\rightarrow \infty} |\sin(π/g_n)|^{1/n}$ exists, then the large $n$ limit is well-defined but depends both on $t$ and on $l$. This result implies that for $t>1$ the standard large $n$ limit with $g_n n=t$ fixed is not well-defined. The parameter $l$ determines a fine structure of the asymptotic eigenvalue support: for $l\neq 0$ the support consists of an interval on the real axis with charge fraction $Q=1-1/t$ and an $l$-dependent oval around the origin with charge fraction $1/t$. For $l=1$ these two components meet, and for $l=0$ the oval collapses to the origin. We also calculate the total electrostatic energy $\mathcal{E}$, which turns out to be independent of $l$, and the free energy $\mathcal{F}=\mathcal{E}-Q\ln l$, which does depend of the fine structure parameter $l$. The existence of large $n$ asymptotic expansions of $\mathcal{F}$ beyond the planar limit as well as the double-scaling limit are also discussed.

math-ph↗

arXiv2014

Partition functions and the continuum limit in Penner matrix models

We present an implementation of the method of orthogonal polynomials which is particularly suitable to study the partition functions of Penner random matrix models, to obtain their explicit forms in the exactly solvable cases, and to determine the coefficients of their perturbative expansions in the continuum limit. The method relies on identities satisfied by the resolvent of the Jacobi matrix in the three-term recursion relation of the associated families of orthogonal polynomials. These identities lead to a convenient formulation of the string equations. As an application, we show that in the continuum limit the free energy of certain exactly solvable models like the linear and double Penner models can be written as a sum of gaussian contributions plus linear terms. To illustrate the one-cut case we discuss the linear, double and cubic Penner models, and for the two-cut case we discuss theoretically and numerically the existence of a double-branch structure of the free energy for the gaussian Penner model.

math-ph↗

arXiv2013

Determination of S-curves with applications to the theory of nonhermitian orthogonal polynomials

This paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex equilibrium potential can be written as a combination of Abelian integrals on a suitable Riemann surface whose branch points can be taken as the main parameters of the problem. Equations for these branch points can be written in terms of periods of Abelian differentials and are known in several equivalent forms. We select one of these forms and use a combination of analytic an numerical methods to investigate the phase structure of asymptotic zero densities of orthogonal polynomials and of asymptotic eigenvalue densities of random matrix models. As an application we give a complete description of the phases and critical processes of the standard cubic model.

math-ph↗

arXiv2013

Superpotentials, quantum parameter space and phase transitions in N=1 supersymmetric gauge theories

We study the superpotentials, quantum parameter space and phase transitions that arise in the study of large N dualities between $\mathcal{N}=1$ SUSY U(N) gauge theories and string models on local Calabi-Yau manifolds. The main tool of our analysis is a notion of spectral curve characterized by a set of complex partial 't Hooft parameters and cuts given by projections on the spectral curve of minimal supersymmetric cycles of the underlying Calabi-Yau manifold. In particular we show how prepotentials and superpotentials can be associated to spectral curves without relying on any holomorphic matrix model. As an application, we use a combination of analytical and numerical methods to study the cubic model, determine the analytic condition satisfied by critical one-cut spectral curves, and characterize the transition curves between the one-cut and two-cut phases both in the space of spectral curves and in the quantum parameter space.

math-ph↗

arXiv2013

Phase structure and critical processes of spectral curves in large N dualities

We examine the phase structure and the critical processes of the spectral curves that arise in the study of large N dualities between supersymmetric Yang-Mills theories and string models on local Calabi-Yau manifolds. These spectral curves are determined by a set of complex partial 't Hooft parameters and a system of cuts given by projections on the spectral curve of minimal supersymmetric cycles of the underlying Calabi-Yau manifold. Using a combination of analytical and numerical methods we give a complete description of the one-cut phase in the cubic model, determine the analytic condition satisfied by critical one-cut spectral curves, and give an algorithm to calculate the two-cut spectral curves of the cubic model for generic values of the partial 't Hooft parameters.

math-ph↗

arXiv2012

Critical role of two-dimensional island-mediated growth on the formation of semiconductor heterointerfaces

We experimentally demonstrate a sigmoidal variation of the composition profile across semiconductor heterointerfaces. The wide range of material systems (III-arsenides, III-antimonides, III-V quaternary compounds, III-nitrides) exhibiting such a profile suggests a universal behavior. We show that sigmoidal profiles emerge from a simple model of cooperative growth mediated by two-dimensional island formation, wherein cooperative effects are described by a specific functional dependence of the sticking coefficient on the surface coverage. Experimental results confirm that, except in the very early stages, island growth prevails over nucleation as the mechanism governing the interface development and ultimately determines the sigmoidal shape of the chemical profile in these two-dimensional grown layers. In agreement with our experimental findings, the model also predicts a minimum value of the interfacial width, with the minimum attainable value depending on the chemical identity of the species.

cond-mat.mtrl-sci↗

arXiv2011

Large N expansions and Painlevé hierarchies in the Hermitian matrix model

We present a method to characterize and compute the large N formal asymptotics of regular and critical Hermitian matrix models with general even potentials in the one-cut and two-cut cases. Our analysis is based on a method to solve continuum limits of the discrete string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. This method also leads to an explicit formulation, in terms of coupling constants and critical parameters, of the members of the Painlevé I and Painlevé II hierarchies associated with one-cut and two-cut critical models respectively.

math-ph↗

arXiv2011

An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model

We present a method to compute the genus expansion of the free energy of Hermitian matrix models from the large N expansion of the recurrence coefficients of the associated family of orthogonal polynomials. The method is based on the Bleher-Its deformation of the model, on its associated integral representation of the free energy, and on a method for solving the string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. As a byproduct we obtain an efficient algorithm to compute generating functions for the enumeration of labeled k-maps which does not require the explicit expressions of the coefficients of the topological expansion. Finally we discuss the regularization of singular one-cut models within this approach.

math-ph↗