Fine structure of the zeros of orthogonal polynomials: a progress report

Simon, Barry

doi:10.1090/conm/507/09963

“…To study the spacing of these zeros, we follow Lubinsky [81] (see also [82]) and consider the Christoffel-Darboux kernel associated to the orthogonal polynomials .26) This object has two interesting properties. First, if P (λ 1 ) = 0 then K L (λ 1 , λ 2 ) = 0 if P (λ 2 ) = 0 and…”

Section: No Random Matrix Statisticsmentioning

confidence: 99%

Alpha states demystified: Towards microscopic models of AdS$_2$ holography

Blommaert,

Iliesiu,

Kruthoff

2022

Preprint

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We continue our study of factorizing theories of dilaton gravity, characterized by a universal bilocal interaction. All such factorizing theories can be shown to have discrete spectra, distinguished only by their local dilaton potentials. We show how such theories can be used to construct all alpha-states in the Hilbert space of baby universes of ordinary JT gravity. Large classes of these theories with different local potentials are found to be non-perturbatively equivalent and have identical discrete spectra. This is a concrete example of how different bulk descriptions can give rise to the same boundary theory. Such equivalences manifest themselves as null states, which have to be quotiented out in order to construct a proper baby universe Hilbert space. Our results also allow us to revisit the mechanism discussed by Coleman, Giddings and Strominger and concretely link ensemble averaging to the appearance or disappearance of spacetime wormholes.We then investigate JT gravity deformed only by the universal bilocal interaction. In this theory, the only terms that do not cancel in a topological expansion are disks, which capture perturbative fluctuations around a two-dimensional black hole saddle. We find that this theory of black holes has an evenly spaced spectrum, instead of a quantum chaotic one. We present a dual quantum mechanical system with exactly the same discrete spectrum, and propose that this is an example of a new holographic duality between a two-dimensional theory of quantum gravity and a conventional quantum mechanics.

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“…To study the spacing of these zeros, we follow Lubinsky [81] (see also [82]) and consider the Christoffel-Darboux kernel associated to the orthogonal polynomials…”

Section: Jhep08(2022)071mentioning

confidence: 99%

Alpha states demystified — towards microscopic models of AdS2 holography

Blommaert

¹

,

Iliesiu

²

,

Kruthoff

³

2022

J. High Energ. Phys.

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We continue our study of factorizing theories of dilaton gravity, characterized by a universal bilocal interaction. All such factorizing theories can be shown to have discrete spectra, distinguished only by their local dilaton potentials. We show how such theories can be used to construct all alpha-states in the Hilbert space of baby universes of ordinary JT gravity. Large classes of these theories with different local potentials are found to be non-perturbatively equivalent and have identical discrete spectra. This is a concrete example of how different bulk descriptions can give rise to the same boundary theory. Such equivalences manifest themselves as null states, which have to be quotiented out in order to construct a proper baby universe Hilbert space. Our results also allow us to revisit the mechanism discussed by Coleman, Giddings and Strominger and concretely link ensemble averaging to the appearance or disappearance of spacetime wormholes.We then investigate JT gravity deformed only by the universal bilocal interaction. In this theory, the only terms that do not cancel in a topological expansion are disks, which capture perturbative fluctuations around a two-dimensional black hole saddle. We find that this theory of black holes has an evenly spaced spectrum, instead of a quantum chaotic one. We present a dual quantum mechanical system with exactly the same discrete spectrum, and propose that this is an example of a new holographic duality between a two-dimensional theory of quantum gravity and a conventional quantum mechanics.

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“…[15,16]), a much stronger statement holds: at any point of (−1, 0) ∪ (0, 1) they distribute very precisely in accordance with ω(x), complying with the so-called "clock behavior", see e.g. [19]. If, following [19], we enumerate the zeros x (n) j of P n as follows,…”

Section: Corollarymentioning

confidence: 99%

“…This shows that even the weak Lubinsky's "wiggle condition" (term coined by B. Simon, see e.g. [19,Theorem 3.6]) is not satisfied in a neighborhood of the jump of the weight. The kernel for x = y in ( 22) is written in the so-called integrable form.…”

Section: Corollarymentioning

confidence: 99%

Asymptotics of orthogonal polynomials for a weight with a jump on [-1,1]

Moreno,

Martinez-Finkelshtein,

Sousa

2009

Preprint

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We consider the orthogonal polynomials on [−1, 1] with respect to the weightwhere h is real analytic and strictly positive on [−1, 1], and Ξ c is a step-like function:. We obtain strong uniform asymptotics of the monic orthogonal polynomials in C, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as n → ∞. In particular, we prove for w c a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients.The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel-Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit clock behavior.For the asymptotic analysis we use the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problems characterizing the orthogonal polynomials. The local analysis at x = 0 is carried out in terms of confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.

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Fine structure of the zeros of orthogonal polynomials: a progress report

Abstract: Abstract. We present a complete theory of the asymptotics of the zeros of OPUC with Verblunsky coefficientsn ) where ∆ < 1 and |b | = b < 1.

Cited by 5 publications

References 41 publications

Alpha states demystified: Towards microscopic models of AdS$_2$ holography

Alpha states demystified: Towards microscopic models of AdS$_2$ holography

Alpha states demystified — towards microscopic models of AdS2 holography

Asymptotics of orthogonal polynomials for a weight with a jump on [-1,1]

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