30 research outputs found
Orthogonal polynomials in the normal matrix model with a cubic potential
We consider the normal matrix model with a cubic potential. The model is
ill-defined, and in order to reguralize it, Elbau and Felder introduced a model
with a cut-off and corresponding system of orthogonal polynomials with respect
to a varying exponential weight on the cut-off region on the complex plane. In
the present paper we show how to define orthogonal polynomials on a specially
chosen system of infinite contours on the complex plane, without any cut-off,
which satisfy the same recurrence algebraic identity that is asymptotically
valid for the orthogonal polynomials of Elbau and Felder. The main goal of this
paper is to develop the Riemann-Hilbert (RH) approach to the orthogonal
polynomials under consideration and to obtain their asymptotic behavior on the
complex plane as the degree of the polynomial goes to infinity. As the
first step in the RH approach, we introduce an auxiliary vector equilibrium
problem for a pair of measures on the complex plane. We then
formulate a matrix valued RH problem for the orthogonal polynomials
in hand, and we apply the nonlinear steepest descent method of Deift-Zhou to
the asymptotic analysis of the RH problem. The central steps in our study are a
sequence of transformations of the RH problem, based on the equilibrium vector
measure , and the construction of a global parametrix. The main
result of this paper is a derivation of the large asymptotics of the
orthogonal polynomials on the whole complex plane. We prove that the
distribution of zeros of the orthogonal polynomials converges to the measure
, the first component of the equilibrium measure. We also obtain
analytical results for the measure relating it to the distribution of
eigenvalues in the normal matrix model which is uniform in a domain bounded by
a simple closed curve.Comment: 57 pages, 8 figure
Two-Point Correlation Functions and Universality for the Zeros of Systems of SO(n+1)-invariant Gaussian Random Polynomials
We study the two-point correlation functions for the zeroes of systems of
-invariant Gaussian random polynomials on and systems
of -invariant Gaussian analytic functions. Our result
reflects the same "repelling," "neutral," and "attracting" short-distance
asymptotic behavior, depending on the dimension, as was discovered in the
complex case by Bleher, Shiffman, and Zelditch. For systems of the -invariant Gaussian analytic functions we also obtain a
fast decay of correlations at long distances.
We then prove that the correlation function for the -invariant Gaussian analytic functions is "universal,"
describing the scaling limit of the correlation function for the restriction of
systems of the -invariant Gaussian random polynomials to any
-dimensional submanifold . This provides a
real counterpart to the universality results that were proved in the complex
case by Bleher, Shiffman, and Zelditch. (Our techniques also apply to the
complex case, proving a special case of the universality results of Bleher,
Shiffman, and Zelditch.)Comment: 28 pages, 1 figure. To appear in International Mathematics Research
Notices (IMRN
The Mother Body Phase Transition in the Normal Matrix Model
The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth.
In this present paper we consider the normal matrix model with cubic plus linear potential. In order to regularize the model, we follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain Ω that we determine explicitly by finding the rational parametrization of its boundary.
We also study in detail the mother body problem associated to Ω. It turns out that the mother body measure ÎŒâ displays a novel phase transition that we call the mother body phase transition: although âΩ evolves analytically, the mother body measure undergoes a âone-cut to three-cutâ phase transition.
To construct the mother body measure, we define a quadratic differential Ï on the associated spectral curve, and embed ÎŒâ into its critical graph. Using deformation techniques for quadratic differentials, we are able to get precise information on ÎŒâ. In particular, this allows us to determine the phase diagram for the mother body phase transition explicitly.
Following previous works of Bleher & Kuijlaars and Kuijlaars & LĂłpez, we consider multiple orthogonal polynomials associated with the normal matrix model. Applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain strong asymptotic formulas for these polynomials. Due to the presence of the linear term in the potential, there are no rotational symmetries in the model. This makes the construction of the associated g-functions significantly more involved, and the critical graph of Ï becomes the key technical tool in this analysis as well
Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit
We consider the double scaling limit in the random matrix ensemble with an
external source \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM defined on Hermitian matrices, where is a diagonal matrix with two eigenvalues of equal multiplicities. The value is critical since the eigenvalues
of accumulate as on two intervals for and on one
interval for . These two cases were treated in Parts I and II, where
we showed that the local eigenvalue correlations have the universal limiting
behavior known from unitary random matrix ensembles. For the critical case
new limiting behavior occurs which is described in terms of Pearcey
integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish
this result by applying the Deift/Zhou steepest descent method to a -matrix valued Riemann-Hilbert problem which involves the construction of a
local parametrix out of Pearcey integrals. We resolve the main technical issue
of matching the local Pearcey parametrix with a global outside parametrix by
modifying an underlying Riemann surface.Comment: 36 pages, 9 figure
Exact solution of the six-vertex model with domain wall boundary condition. Critical line between ferroelectric and disordered phases
This is a continuation of the papers [4] of Bleher and Fokin and [5] of
Bleher and Liechty, in which the large asymptotics is obtained for the
partition function of the six-vertex model with domain wall boundary
conditions in the disordered and ferroelectric phases, respectively. In the
present paper we obtain the large asymptotics of on the critical line
between these two phases.Comment: 22 pages, 6 figures, to appear in the Journal of Statistical Physic