Papers by LUIS MARTINEZ ALONSO
Universe, 2021
We determine generalised asymptotic solutions for the inflaton field, the Hubble parameter, and t... more 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. 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. Finally, we discuss the physical significance of the generalised as...
Classical and Quantum Gravity, 2018
We propose to use eigenvalue densities of unitary random matrix ensembles as mass distributions i... more 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.
Annals of Physics, 2015
In this paper we apply results on the asymptotic zero distribution of the Laguerre polynomials to... more 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 → 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→∞ | 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 = 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 E, which turns out to be independent of l, and the free energy F = E − Q ln l, which does depend of the fine structure parameter l. The existence of large n asymptotic expansions of F beyond the planar limit as well as the double-scaling limit are also discussed.
A hodograph transformation for a wide family of multidimensional nonlinear partial differential e... more A hodograph transformation for a wide family of multidimensional nonlinear partial differential equations is presented. It is used to derive solutions of the heavenly equation (dispersionless Toda equation) as well as a family of explicit ultra-hyperbolic selfdual vacuum spaces admiting only one Killing vector which is not selfdual, we also give the corresponding explicit Einstein--Weyl structures.
We examine the phase structure and the critical processes of the spectral curves that arise in th... more 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.
A∂-formalism for studying dispersionless integrable hierarchies is applied to the dKP hierarchy. ... more A∂-formalism for studying dispersionless integrable hierarchies is applied to the dKP hierarchy. Connections with the theory of quasiconformal mappings on the plane are described and some clases of explicit solutions of the dKP hierarchy are presented.
Journal of Physics A: Mathematical and Theoretical, 2014
Theoretical and Mathematical Physics, 2011
Complete description of the singular sectors of the 1-layer Benney system (classical long wave eq... more Complete description of the singular sectors of the 1-layer Benney system (classical long wave equation) and dToda system is presented. Associated Euler-Poisson-Darboux equations E(1/2,1/2) and E(-1/2,-1/2) are the main tool in the analysis. A complete list of solutions of the 1-layer Benney system depending on two parameters and belonging to the singular sector is given. Relation between Euler-Poisson-Darboux equations E(ε, ε) with opposite sign of ε is discussed.
Journal of High Energy Physics, 2013
We study the superpotentials, quantum parameter space and phase transitions that arise in the stu... more 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.
Physics Letters B, 2005
A class of exact solutions of the dispersionless Toda hierarchy constrained by a string equation ... more A class of exact solutions of the dispersionless Toda hierarchy constrained by a string equation is obtained. These solutions represent deformations of analytic curves with a finite number of nonzero harmonic moments. The corresponding τ-functions are determined and the emergence of cusps is studied.
Physics Letters B, 1998
We show how Ramond free neutral Fermi fields lead to a τ-function theory of BKP type which descri... more We show how Ramond free neutral Fermi fields lead to a τ-function theory of BKP type which describes iso-orthogonal deformations of systems of ortogonal curvilinear coordinates. We also provide a vertex operator representation for the classical Ribaucour transformation.
Physics Letters B, 2006
A scheme for solving quasiclassical string equations is developed to prove that genus-zero Whitha... more A scheme for solving quasiclassical string equations is developed to prove that genus-zero Whitham hierarchies describe the deformations of planar domains determined by rational conformal maps. This property is applied in normal matrix models to show that deformations of simplyconnected supports of eigenvalues under changes of coupling constants are governed by genus-zero Whitham hierarchies.
Physics Letters A, 2011
The singular sector of zero genus case for the Hermitian random matrix model in the large N limit... more The singular sector of zero genus case for the Hermitian random matrix model in the large N limit is analyzed. It is proved that the singular sector of the hodograph solutions for the underlying dispersionless Toda hierarchy and the singular sector of the 1-layer Benney (classical long wave equation) hierarchy are deeply connected. This property is due to the fact that the hodograph equations for both hierarchies describe the critical points of solutions of Euler-Poisson-Darboux equations E(a, a), with a = −1/2 for the dToda hierarchy and a = 1/2 for the 1-layer Benney hierarchy.
Physics Letters A, 1999
The bilinear equations of the N-component KP and BKP hierarchies and a corresponding extended Miw... more The bilinear equations of the N-component KP and BKP hierarchies and a corresponding extended Miwa transformation allow us to generate quadrilateral and circular lattices from conjugate and orthogonal nets, respectively. The main geometrical objects are expressed in terms of Baker functions.
Nuclear Physics B, 2011
We present a method to compute the genus expansion of the free energy of Hermitian matrix models ... more 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.
Mathematical Methods in the Applied Sciences, 2010
In this work we use Riemann-Hilbert problems for multiple orthogonal polynomials in order to deri... more In this work we use Riemann-Hilbert problems for multiple orthogonal polynomials in order to derive string equations associated to Lax-Orlov pairs operators. These string equations provide us with a useful tool to analyze the large n-limit of the related hierarchies. The results are finally applied to the study of the associated random matrix models (Gaussian Hermitian matrix models with an external source) and non-intersecting Brownian motions starting from a fix point.
Journal of Physics A: Mathematical and Theoretical, 2007
Journal of Physics A: Mathematical and Theoretical, 2009
Journal of Physics A: Mathematical and General, 2003
Journal of Physics A: Mathematical and General, 2001
The dispersionless limit of the scalar nonlocal ∂-problem is derived. It is given by a special cl... more The dispersionless limit of the scalar nonlocal ∂-problem is derived. It is given by a special class of nonlinear first-order equations. A quasi-classical version of the ∂-dressing method is presented. It is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of Beltrami tupe equations. The universal Whitham hierarchy and, in particular, the dispersionless KP hierarchy turn out to be rings of symmetries for the quasi-classical ∂problem.
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Papers by LUIS MARTINEZ ALONSO