Phase Transitions and Equilibrium Measures in Random Matrix Models
@article{MartnezFinkelshtein2013PhaseTA, title={Phase Transitions and Equilibrium Measures in Random Matrix Models}, author={Andrei Mart{\'i}nez-Finkelshtein and Ram'on Orive and Evguenii Rakhmanov}, journal={Communications in Mathematical Physics}, year={2013}, volume={333}, pages={1109-1173} }
The paper is devoted to a study of phase transitions in the Hermitian random matrix models with a polynomial potential. In an alternative equivalent language, we study families of equilibrium measures on the real line in a polynomial external field. The total mass of the measure is considered as the main parameter, which may be interpreted also either as temperature or time. Our main tools are differentiation formulas with respect to the parameters of the problem, and a representation of the…
10 Citations
Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlev\'e II equation
- Mathematics
- 2022
We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative…
On external fields created by fixed charges
- EconomicsJournal of Mathematical Analysis and Applications
- 2018
Quantum Mechanics in symmetry language
- Physics
- 2013
We consider symmetry as a foundational concept in quantum mechanics and rewrite quantum mechanics and measurement axioms in this description. We argue that issues related to measurements and physical…
On point-mass Riesz external fields on the real axis
- EconomicsJournal of Mathematical Analysis and Applications
- 2020
Asymptotics for the Partition Function in Two-Cut Random Matrix Models
- Mathematics
- 2014
AbstractWe obtain large N asymptotics for the random matrix partition function
$$Z_N(V)=\int_{\mathbb{R}^N} \prod_{i <…
Gravitational lensing by eigenvalue distributions of random matrix models
- Physics
- 2018
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…
An equilibrium problem on the sphere with two equal charges
- Physics
- 2019
We study the equilibrium measure on the two dimensional sphere in the presence of an external field generated by two equal point charges. The support of the equilibrium measure is known as the…
Equilibrium problems in weakly admissible external fields created by pointwise charges
- MathematicsJ. Approx. Theory
- 2019
Zero distribution for Angelesco Hermite–Padé polynomials
- MathematicsRussian Mathematical Surveys
- 2018
This paper considers the zero distribution of Hermite–Padé polynomials of the first kind associated with a vector function whose components are functions with a finite number of branch points in the…
References
SHOWING 1-10 OF 122 REFERENCES
Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields
- Mathematics
- 2000
The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example, in random matrix theory: The limiting mean density of eigenvalues is precisely…
Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies
- Mathematics
- 2010
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…
Breakdown of universality in multi-cut matrix models
- Mathematics
- 2000
We solve the puzzle of the disagreement between orthogonal polynomials methods and mean-field calculations for random N×N matrices with a disconnected eigenvalue support. We show that the difference…
First Colonization of a Spectral Outpost in Random Matrix Theory
- Mathematics
- 2007
We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the…
New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field
- Mathematics
- 1998
In this paper we use techniques from the theory of ODEs and also from inverse scattering theory to obtain a variety of results on the regularity and support properties of the equilibrium measure for…
Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration
- Mathematics
- 2002
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…
Partition function for multi-cut matrix models
- Mathematics
- 2006
We consider the partition function Z N of a random matrix model with polynomial potential V(ξ) = t 1 ξ + t 2 ξ 2 + ··· + t 2d ξ 2d . It is known that the second logarithmic derivative of Z N with…
Orthogonal polynomial ensembles in probability theory
- Mathematics
- 2005
We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an {it orthogonal polynomial ensemble}. The most prominent…
Eigenvalue density in Hermitian matrix models by the Lax pair method
- Mathematics
- 2009
In this paper, a new method is discussed to derive the eigenvalue density in a Hermitian matrix model with a general potential. The density is considered on one interval or multiple disjoint…