arXiv subjects

Elena Medina

Publications and source records attributed to Elena Medina.

At least 19 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↗

arXiv2025

The role of the initial distribution in population survival within a bounded habitat

In this paper, we analyze the role of initial conditions in population persistence. Specifically, we consider the reaction-diffusion equation $u_t\,=\,D\,(u^{ν-1}\,u_x)_x\,+\,a\,u^μ$, with $μ,ν>0$, accompanied by hostile boundary conditions and examine two families of one-parametric initial distributions, including homogeneous distributions. The model was previously studied by Colombo and Anteneodo (2018). They determined appropriate habitat sizes $l$ for the survival of a population, whose individuals are initially placed homogeneously within the full habitat domain with a total initial population $n_0$. We show that the survival condition can be naturally formulated in terms of the parameter $Q:=\frac{a}{D}l^{-μ+ν+2}n_0^{μ-ν}$. Indeed, there exists a critical value $Q_c$ determined by $μ$, $ν$ and the initial distribution parameter such that the survival condition can always be written as $Q\geq Q_c$. Notably, from this point of view, one can derive a condition for $Q$ that holds universally for our model under conditional persistence ($μ\geqν$). It applies, in particular, to the case $μ=ν+2$, which was not addressed in the previously mentioned work. Nevertheless, in this case $Q=\frac{a}{D}n_0^2$, therefore survival depends solely on the total population, not on the habitat size. We apply a finite-difference scheme to estimate $Q_c$. Conversely, given a population whose evolution is determined by $μ$, $ν$, $l$, $n_0$, and the growth and diffusion coefficients $a$ and $D$ (and consequently the value of $Q$) we use the numerical algorithm to estimate the initial distribution to ensure population survival.

math.AP↗

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

Kinetic dominance and psi series in the Hamilton-Jacobi formulation of inflaton models

Single-field inflaton models in the kinetic dominance period admit formal solutions given by generalized asymptotic expansions called psi series. We present a method for computing psi series for the Hubble parameter as a function of the inflaton field in the Hamilton-Jacobi formulation of inflaton models. Similar psi series for the scale factor, the conformal time and the Hubble radius are also derived. They are applied to determine the value of the inflaton field when the inflation period starts and to estimate the contribution of the kinetic dominance period to calculate the duration of inflation. These psi series are also used to obtain explicit two-term truncated psi series near the singularity for the potentials of the Mukhanov-Sasaki equation for curvature and tensor perturbations. The method is illustrated with wide families of inflaton models determined by potential functions combining polynomial and exponential functions as well as with generalized Starobinsky 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

Generalised asymptotic solutions for the inflaton in the oscillatory phase of reheating

We determine generalised asymptotic solutions for the inflaton field, the Hubble parameter, and the equation-of-state parameter valid during the oscillatory phase of reheating for potentials that close to their global minima behave as even monomial potentials. For the quadratic potential we derive a generalised asymptotic expansion for the inflaton with respect to the scale set by inverse powers of the cosmic time. For the quartic potential we derive an explicit, two-term generalised asymptotic solution in terms of Jacobi elliptic functions, with a scale set by inverse powers of the square root of the cosmic time. Finally, in the general case, we find similar two-term solutions where the leading order term is defined implicitly in terms of the Gauss' hypergeometric function. The relation between the leading terms of the instantaneous equation-of-state parameter and different averaged values is discussed in the general case.

gr-qc↗

arXiv2018

Gravitational lensing by eigenvalue distributions of random matrix models

We propose to use eigenvalue densities of unitary random matrix ensembles as mass distributions in gravitational lensing. The corresponding lens equations reduce to algebraic equations in the complex plane which can be treated analytically. We prove that these models can be applied to describe lensing by systems of edge-on galaxies. We illustrate our analysis with the Gaussian and the quartic unitary matrix ensembles.

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

Spectral curves in gauge/string dualities: integrability, singular sectors and regularization

We study the moduli space of the spectral curves $y^2=W'(z)^2+f(z)$ which characterize the vacua of $\mathcal{N}=1$ U(n) supersymmetric gauge theories with an adjoint Higgs field and a polynomial tree level potential $W(z)$. It is shown that there is a direct way to associate a spectral density and a prepotential functional to these spectral curves. The integrable structure of the Whitham equations is used to determine the spectral curves from their moduli. An alternative characterization of the spectral curves in terms of critical points of a family of polynomial solutions $\mathbb{W}$ to Euler-Poisson-Darboux equations is provided. The equations for these critical points are a generalization of the planar limit equations for one-cut random matrix models. Moreover, singular spectral curves with higher order branch points turn out to be described by degenerate critical points of $\mathbb{W}$. As a consequence we propose a multiple scaling limit method of regularization and show that, in the simplest cases, it leads to the Painlevè-I equation and its multi-component generalizations.

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↗

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↗