Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

Duits, Maurice; Kuijlaars, Arno B. J.

doi:10.1002/cpa.20269

Cited by 50 publications

(143 citation statements)

References 78 publications

(191 reference statements)

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“…It easily follows from these rules and the residue assumptions on the branch points ω j x * that the following relations hold in the case d odd: 12) and in the case d even, the relations (9.6)-(9.7) still hold, and we now have be a basis of the vector space of all meromorphic functions that have a pole of order at most d…”

Section: Fourth Transformation T → Smentioning

confidence: 97%

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The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Kuijlaars

López-García

2015

Nonlinearity

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We investigate the asymptotic behavior of a family of multiple orthogonal polynomials that is naturally linked with the normal matrix model with a monomial potential of arbitrary degree d + 1. The polynomials that we investigate are multiple orthogonal with respect to a system of d analytic weights defined on a symmetric (d + 1)-star centered at the origin. In the first part we analyze in detail a vector equilibrium problem involving a system of d interacting measures (µ1, . . . , µ d ) supported on star-like sets in the plane. We show that in the subcritical regime, the first component µ * 1 of the solution to this problem is the asymptotic zero distribution of the multiple orthogonal polynomials. It also characterizes the domain where the eigenvalues in the normal matrix model accumulate, in the sense that the Schwarz function associated with the boundary of this domain can be expressed explicitly in terms of µ * 1 . The second part of the paper is devoted to the asymptotic analysis of the multiple orthogonal polynomials. The asymptotic results are obtained again in the subcritical regime, and they follow from the Deift/Zhou steepest descent analysis of a Riemann-Hilbert problem of size (d + 1) × (d + 1). The vector equilibrium problem and the Riemann-Hilbert problem that we investigate are generalizations of those studied recently by Bleher-Kuijlaars in the case d = 2.

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Section: Fourth Transformation T → Smentioning

confidence: 97%

“…In this paper we study monomial V of higher degree, that is, 12) where d ≥ 3. Our initial expectation was that the approach of [7] for the case d = 2 would carry through, with proper but non-essential modifications.…”

Section: Aim Of the Papermentioning

confidence: 99%

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Kuijlaars

López-García

2015

Nonlinearity

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“…In [8] two of us studied the limiting eigenvalue behavior of the matrix M 1 in the two-matrix model (1.1) for the case of an even polynomial V with positive leading coefficient and…”

Section: A Vector Equilibrium Problemmentioning

confidence: 99%

“…The vector equilibrium problem from [8] asks to minimize the energy functional Here, dz is the complex line element on iR.…”

Section: A Vector Equilibrium Problemmentioning

confidence: 99%

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A vector equilibrium problem for the two-matrix model in the quartic/quadratic case

2011

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We consider the two sequences of biorthogonal polynomials (p k,n )and (q k,n ) ∞ k=0 related to the Hermitian two-matrix model with potentials V (x) = x 2 /2 and W (y) = y 4 /4+ty 2 . From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p n,n as n → ∞. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t = 0 reduces to the vector equilibrium problem that was given recently by two of us. A novel feature is that for t < 0 an external field is active on the third measure which introduces a new type of critical behavior for a certain negative value of t.We also prove a general result about the interlacing of zeros of biorthogonal polynomials.

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Random matrix model with external source and a constrained vector equilibrium problem

Bleher

Delvaux

Kuijlaars

2010

Comm Pure Appl Math

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We consider the random matrix model with external source in the case where the potential V .x/ is an even polynomial and the external source has two eigenvalueṡ a of equal multiplicity. We show that the limiting mean eigenvalue distribution of this model can be characterized as the first component of a pair of measures . 1 ; 2 / that solve a constrained vector equilibrium problem. The proof is based on the steepest-descent analysis of the associated Riemann-Hilbert problem for multiple orthogonal polynomials.We illustrate our results in detail for the case of a quartic double-well potential V .x/ D 1 4 x 4 t 2 x 2 . We are able to determine the precise location of the phase transitions in the ta-plane, where either the constraint becomes active or the two intervals in the support come together (or both).

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Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

Cited by 50 publications

References 78 publications

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

A vector equilibrium problem for the two-matrix model in the quartic/quadratic case

Random matrix model with external source and a constrained vector equilibrium problem

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