On the equilibrium problem for vector potentials

Gonchar, A A; Rakhmanov, Evguenii Andreevich

doi:10.1070/rm1985v040n04abeh003638

Cited by 91 publications

(72 citation statements)

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“…Vector equilibrium problems with mutual interaction can be traced back to [27,28], see also [46]. The equilibrium problem with constraints appeared before mainly in the asymptotic analysis of discrete orthogonal polynomials, see e.g.…”

Section: A Vector Equilibrium Problemmentioning

confidence: 99%

Non-intersecting squared Bessel paths with one positive starting and ending point

Delvaux

Kuijlaars

Román

et al. 2012

JAMA

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We consider a model of n non-intersecting squared Bessel processes with one starting point a > 0 at time t = 0 and one ending point b > 0 at time t = T . After proper scaling, the paths fill out a region in the tx-plane. Depending on the value of the product ab the region may come to the hard edge at 0, or not. We formulate a vector equilibrium problem for this model, which is defined for three measures, with upper constraints on the first and third measures and an external field on the second measure. It is shown that the limiting mean distribution of the paths at time t is given by the second component of the vector that minimizes this vector equilibrium problem. The proof is based on a steepest descent analysis for a 4 × 4 matrix valued Riemann-Hilbert problem which characterizes the correlation kernel of the paths at time t. We also discuss the precise locations of the phase transitions.

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Section: A Vector Equilibrium Problemmentioning

confidence: 99%

Non-intersecting squared Bessel paths with one positive starting and ending point

Delvaux

Kuijlaars

Román

et al. 2012

JAMA

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“…Given h 2 N we can consider the problem of minimization of the weighted energy I j ðmÞ in the class MðhÞ: In fact, this is a particular instance of the vector-valued equilibrium problem for the vector potentials: the restriction of the solution m to a particular subinterval ½a iÀ1 ; a i solves the equilibrium problem in the presence of the external field jointly generated by j and the potential of the remaining part of m: Thus, the following lemma is a direct consequence of the well-known results (see [8,20, Theorem VIII. Moreover, m h is characterized by the following property: for i ¼ 1; .…”

Section: Vector Equilibrium Problem and Zero Distributionmentioning

confidence: 86%

“…Namely, we describe the limit of the sequence of normalized counting measures nðE n Þ=N under assumptions (6), (8) in terms of the solution of a certain extremal problem for vector logarithmic potentials. The main results are stated in Section 2, their proofs are presented in Section 3, and particular cases are discussed in Section 4.…”

Section: Heine-stieltjes and Van Vleck Polynomialsmentioning

confidence: 99%

Asymptotic Properties of Heine–Stieltjes and Van Vleck Polynomials

Martínez–Finkelshtein¹,

Saff²

2002

Journal of Approximation Theory

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We study the asymptotic behavior of the zeros of polynomial solutions of a class of generalized Lam! e e differential equations, when their coefficients satisfy certain asymptotic conditions. The limit distribution is described by an equilibrium measure in the presence of an external field, generated by charges at the singular points of the equation. Moreover, a case of non-positive charges is considered, which leads to an equilibrium with a non-convex external field. # 2002 Elsevier Science (USA)

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“…The vector case without external field has been treated in [6] and [12]. Our proof for the general case follows the scheme proposed in [12].…”

Section: Functions Of Second Type and Orthogonalitymentioning

confidence: 97%

Rate of Convergence of Generalized Hermite–Padé Approximants of Nikishin Systems

Prieto

Lagomasino

2004

Constr Approx

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Abstract. We study the rate of convergence of interpolating simultaneous rational approximations with partially prescribed poles to so called Nikishin systems of functions. To this end, a vector equilibrium problem in the presence of a vector external field is solved which is used to describe the asymptotic behavior of the corresponding second type functions which appear. Generalized Hermite-Padé approximantsLet ∆ 1 be a bounded interval of the real line R. By M(∆ 1 ) we denote the set of all Borel measures with constant sign (positive or negative) whose support supp(·) is contained in ∆ 1 and contains infinitely many points. Let s = (s 1 , · · · s m ) be a vector of measures belonging to M(∆ 1 ). The Markov function corresponding to the measure s i ∈ M(∆ 1 ) is given byCertainly, s i is a holomorphic function in C \ ∆ 1 . We restrict our attention to a special class of vector measures introduced by E. M. Nikishin in [11] and adopt the notation introduced in [8]. Let σ 1 and σ 2 be two measures supported on R with constant sign and let ∆ 1 , ∆ 2 denote the convex hull of supp(σ 1 ) and supp(σ 2 ) respectively; that is,When it is convenient we use the differential notation of a measure. Then σ 1 , σ 2 is a measure with constant sign and supported on supp(σ 1 ) ⊂ ∆ 1 .Definition 1. Given a system of closed bounded intervals ∆ 1 , . . . , ∆ m satisfying ∆ j−1 ∩ ∆ j = ∅, j = 2, . . . , m, and finite Borel measures σ 1 , . . . , σ m with constant sign and Co(supp(σ j )) = ∆ j , we define inductivelyWe say that S = (

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On the equilibrium problem for vector potentials

Cited by 91 publications

References 1 publication

Non-intersecting squared Bessel paths with one positive starting and ending point

Non-intersecting squared Bessel paths with one positive starting and ending point

Asymptotic Properties of Heine–Stieltjes and Van Vleck Polynomials

Rate of Convergence of Generalized Hermite–Padé Approximants of Nikishin Systems

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