Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths

Kuijlaars, Arno B. J.; Román, Pablo

doi:10.1016/j.jat.2010.06.003

Cited by 15 publications

(20 citation statements)

References 26 publications

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“…In that situation, all paths, after proper scaling, initially stay away from the hard edge at x = 0, but at a certain critical time t * the lowest paths hit the hard edge and are stuck to it from then on. The limiting mean density of the paths at time t is characterized by a non-real solution of an algebraic equation of order three; see also [38,Appendix] and [42] for an interpretation in terms of a vector equilibrium problem with two measures. If t = t * , the local correlations obey the usual scaling limit from the random matrix theory, leading to the sine, Airy, and Bessel kernels [38].…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

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

Delvaux

Kuijlaars

Román

et al. 2012

JAMA

View full text Add to dashboard Cite

show abstract

“…Such functions appear as symbols of banded Toeplitz matrices [6], and we need certain results [7,13] that were derived in that context. Although we will not use Toeplitz matrices in this paper, we still refer to s as the symbol.…”

Section: Results From the Literaturementioning

confidence: 99%

“…Next, we state a result of Kuijlaars and Román [13] on polynomials satisfying certain recurrence relations. It will be the key ingredient of the proof of Theorem 2.2 given in Section 7.…”

Section: Results From the Literaturementioning

confidence: 99%

“…We are going to analyze these recurrence coefficients as n → ∞ and obtain the vector equilibrium problem from this analysis. This approach is similar in spirit to the analysis of [13] for the recurrence coefficients of multiple orthogonal polynomials in a model of non-intersecting squared Bessel paths.…”

Section: Aim Of This Papermentioning

confidence: 99%

“…Then the proof of Theorem 2.3 follows the approach outlined in [13,Section 7]. So, to obtain an external field V 1 acting on ν 1 and an upper constraint σ acting on ν 2 we will need that Γ 1 (ξ) is an increasing and Γ 2 (ξ) a decreasing set as a function of ξ.…”

Section: Analysis Of the Symbol S 1 In The One-cut Casementioning

confidence: 99%

See 2 more Smart Citations

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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Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

Delvaux

2012

Mathematische Nachrichten

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We consider banded block Toeplitz matrices Tn with n block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of Tn for n → ∞ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix.For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of Tn and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if Tn is a Hessenberg matrix.

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Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths

Cited by 15 publications

References 26 publications

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

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

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

Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

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