arXiv subjects

N. M. Ercolani

Publications and source records attributed to N. M. Ercolani.

6 recordsLinked to original sources

arXiv2019

Ginzburg-Landau equations on Riemann surfaces of higher genus

We study the Ginzburg-Landau equations on Riemann surfaces of arbitrary genus. In particular: - we construct explicitly the (local moduli space of gauge-equivalent) solutions in a neighbourhood of the constant curvature ones; - classify holomorphic structures on line bundles arising as solutions to the equations in terms of the degree, the Abel-Jacobi map, and symmetric products of the surface; - determine the form of the energy and identify when it is below the energy of the constant curvature (normal) solutions.

math.AP↗

arXiv2010

Caustics, counting maps and semi-classical asymptotics

This paper develops a deeper understanding of the structure and combinatorial significance of the partition function for Hermitian random matrices. The coefficients of the large N expansion of the logarithm of this partition function,also known as the genus expansion, (and its derivatives) are generating functions for a variety of graphical enumeration problems. The main results are to prove that these generating functions are in fact specific rational functions of a distinguished irrational (algebraic) function of the generating function parameters. This distinguished function is itself the generating function for the Catalan numbers (or generalized Catalan numbers, depending on the choice of parameter). It is also a solution of the inviscid Burgers equation for certain initial data. The shock formation, or caustic, of the Burgers characteristic solution is directly related to the poles of the rational forms of the generating functions. These results in turn provide new information about the asymptotics of recurrence coefficients for orthogonal polynomials with respect to exponential weights. One gains new insights into the relation between certain derivatives of the genus expansion and the asymptotic expansion of the first Painleve transcendent, related to the double-scaling limit. This work provides a precise expression of the Painleve asymptotic coefficients directly in terms of the coefficients of the partial fractions expansion of the rational form of the generating functions established here. Moreover, these insights point toward a more general program relating the first Painleve hierarchy and the higher order structure of the double-scaling limit to the specific rational structure of generating functions.

math-ph↗

arXiv2007

A variational theory for point defects in patterns

We derive a rigorous scaling law for minimizers in a natural version of the regularized Cross-Newell model for pattern formation far from threshold. These energy-minimizing solutions support defects having the same character as what is seen in experimental studies of the corresponding physical systems and in numerical simulations of the microscopic equations that describe these systems.

math.AP↗

arXiv2006

A Quick Derivation of the Loop Equations for Random Matrices

The "loop equations" of random matrix theory are a hierarchy of equations born of attempts to obtain explicit formulae for generating functions of map enumeration problems. These equations, originating in the physics of 2-dimensional quantum gravity, have lacked mathematical justification. The goal of this paper is to provide a complete and short proof, relying on a recently established complete asymptotic expansion for the random matrix theory partition function.

math-ph↗

arXiv2006

Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices

In this paper we derive a hierarchy of differential equations which uniquely determine the coefficients in the asymptotic expansion, for large $N$, of the logarithm of the partition function of $N \times N$ Hermitian random matrices. These coefficients are generating functions for graphical enumeration on Riemann surfaces. The case that we particularly consider is for an underlying measure that differs from the Gaussian weight by a single monomial term of degree $2ν$. The coupling parameter for this term plays the role of the independent dynamical variable in the differential equations. From these equations one may deduce functional analytic characterizations of the coefficients in the asymptotic expansion. Moreover, this ode system can be solved recursively to explicitly construct these coefficients as functions of the coupling parameter. This analysis of the fine structure of the asymptotic coefficients can be extended to multiple coupling parameters and we present a limited illustration of this for the case of two parameters.

math-ph↗

arXiv2002

Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration

We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generating functions for the enumeration of labelled maps according to genus and valence. Central to the analysis is a large N expansion for the mean density of eigenvalues, uniformly valid on the entire real axis.

math-ph↗