Person:
Martínez Alonso, Luis

First Name

Luis

Last Name

Martínez Alonso

URI

https://hdl.handle.net/20.500.14352/75166

Affiliation

Universidad Complutense de Madrid

Faculty / Institute

Ciencias Físicas

Department

Física Teórica

Area

Física Teórica

Identifiers

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Now showing 1 - 10 of 10

An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model
(Nuclear Physics B, 2011) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, Elena
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.
Phase space and phase transitions in the Penner matrix model with negative coupling constant
(Journal of physics A: Mathematical and theoretical, 2017) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, Elena
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.
Kinetic dominance and the wave function of the Universe
(Physical review D, 2022) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, Elena
We analyze the emergence of classical inflationary universes in a kinetic-dominated stage using a suitable class of solutions of the Wheeler-DeWitt 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-DeWitt 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 wave function and the Hartle-Hawking no-boundary wave functions are recovered as particular instances of our class of wave functions.
Complex saddle points in the Gross-Witten-Wadia matrix model
(Physical review D, 2016) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina, Elena
We give an exhaustive characterization of the complex saddle point configurations of the Gross-WittenWadia 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 with 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.
Separatrices in the Hamilton-Jacobi formalism of inflaton models.
(Journal of mathematical physics, 2020) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, Elena; Vázquez, Juan Luis
We consider separatrix solutions of the differential equations for inflaton models with a single scalar field in a zero-curvature Friedmann-Lemaitre-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.
Superpotentials, quantum parameter space and phase transitions in N=1 supersymmetric gauge theories
(Journal of high energy physics, 2013) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, Elena
We study the superpotentials, quantum parameter space and phase transitions that arise in the study of large N dualities between 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. We introduce a prepotential functional via a variational problem which determines the complex density as an extremal constrained by the period conditions. This prepotential is shown to satisfy the special geometry relations of the spectral curve. We give a system of equations for the branch points of the spectral curves and determine the appropriate branch cuts as Stokes lines of a suitable set of polynomials. 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.
Large N expansions and Painlevé hierarchies in the Hermitian matrix model
(Journal of Physics A: Mathematical and Theoretical, 2011) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, Elena
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.
Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies
(Journal of Statistical Mechanics: Theory and Experiment, 2010) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, Elena
We present a method for studying phase transitions in the large N limit of matrix models using matched solutions of Whitham hierarchies. The endpoints of the eigenvalue spectrum as functions of the temperature are characterized both as solutions of hodograph equations and as solutions of a system of ordinary differential equations. In particular we show that the free energy of the matrix model is the quasiclassical tau-function of the associated hierarchy, and that critical processes in which the number of cuts changes in one unit are third-order phase transitions described by C(1) matched solutions of Whitham hierarchies. The method is illustrated with the Bleher-Eynard model for the merging of two cuts. We show that this model involves also the birth of a cut.
Partition functions and the continuum limit in Penner matrix models
(Journal of physics A: Mathematical and Theoretical, 2014) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina, helena
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.
Phase structure and asymptotic zero densities of orthogonal polynomials in the cubic model
(Journal of computational and applied mathematics, 2015) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, Elena
We apply the method we have described in a previous paper (2013) to determine the phase structure of asymptotic zero densities of the standard cubic model of non-Hermitian orthogonal polynomials. We provide a complete description of the two phases: the one cut phase and the two cut phase, and analyze the phase transition processes of the types: splitting of a cut, birth and death of a cut. (C) 2014 Elsevier B.V. All rights reserved.