Discontinuity in the Asymptotic Behavior of Planar Orthogonal Polynomials Under a Perturbation of the Gaussian Weight

Lee, Seung-Yeop; Yang, Meng

doi:10.1007/s00220-017-2888-8

Cited by 27 publications

(44 citation statements)

References 22 publications

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“…From our point of view, the reason to restrict to

| z | < 1

is that this is a more interesting case than

| z | > 1

: one should expect from that for each

z \in C

for which

| z | > 1

prefixlog false| trueprefixdet false(G_{N} - z false) false| - double-struckE prefixlog false| trueprefixdet false(G_{N} - z false) false|

converges in law to a real‐valued Gaussian random variable — there should be no

N^{γ^{2} / 8}

appearing in this case. We expect that this could be proven using a similar approach as the one we take here (using the results of with

| z | > 1

), but we do not explore this further. Note that another reason to distinguish between

| z | < 1

and

| z | > 1

is that in our normalization, the unit disk is the support of the equilibrium measure for the Ginibre ensemble, so it is the set where the eigenvalues should accumulate in the large

N

limit.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In this section, we recall some basic facts about the complex Ginibre ensemble, such as the distribution of the eigenvalues, how expectations of suitable functions of eigenvalues of Ginibre random matrices can be expressed in terms of determinants of complex moment matrices, as well as how such questions relate to orthogonal polynomials. We also recall results from , which show that the orthogonal polynomials associated to the expectation

double-struckE false| trueprefixdet false(G_{N} - z false) |^{γ}

also satisfy suitable orthogonality conditions on certain contours in the complex plane. Then in Section 3, we apply these results to transform the analysis of

double-struckE false| trueprefixdet false(G_{N} - z false) |^{γ}

into a question of the asymptotic analysis of a suitable RHP.…”

Section: The Ginibre Ensemble and Orthogonal Polynomialsmentioning

confidence: 88%

“…We also point out that if

normalΣ

were the unit circle, would look like a basic orthogonality condition for polynomials on the unit circle. Thus (as in ), it is fruitful to define a dual family of polynomials which are orthogonal to the polynomials

p_{j}

with respect to the pairing coming from . We now recall how these dual orthogonal polynomials are constructed and how their leading order coefficient is related to

χ_{j}

.…”

Section: The Ginibre Ensemble and Orthogonal Polynomialsmentioning

confidence: 99%

“…Again, we offer a sketch of a proof in Appendix , as such a result is not formulated precisely in this form in . We now turn to the RHP and the differential identity related to

D_{N - 1} false(F false)

.…”

Section: The Ginibre Ensemble and Orthogonal Polynomialsmentioning

confidence: 99%

“…In this section, we recall from the asymptotic solution of the RHP for

Y_{N}

. In fact, we will consider a minor generalization of their situation where we study the asymptotics of

Y_{N + k}

, where

k

is a fixed integer — for our differential identity, we only need

k = 0

and

k = 1

.…”

Section: Solving the Riemann–hilbert Problem For Yn Asymptoticallymentioning

confidence: 99%

See 4 more Smart Citations

On the moments of the characteristic polynomial of a Ginibre random matrix

Webb

Wong

2018

Proc. London Math. Soc.

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In this article, we study the large N asymptotics of complex moments of the absolute value of the characteristic polynomial of an N×N complex Ginibre random matrix with the characteristic polynomial evaluated at a point in the unit disk. More precisely, we calculate the large N asymptotics of double-struckEfalse|trueprefixdetfalse(GN−zfalse)|γ, where GN is an N×N matrix whose entries are i.i.d. and distributed as N−1/2Z, Z being a standard complex Gaussian, Re (γ)>−2, and |z|<1. This expectation is proportional to the determinant of a complex moment matrix with a symbol which is supported in the whole complex plane and has a Fisher–Hartwig type of singularity: 0truedet(∫Cwiw¯jfalse|w−zfalse|γe−Nfalse|wfalse|2d2wfalse)i,j=0N−1. We study the asymptotics of this determinant using recent results due to Lee and Yang concerning the asymptotics of orthogonal polynomials with respect to the weight |w−z|γe−N|w|2d2w along with differential identities familiar from the study of asymptotics of Toeplitz and Hankel determinants with Fisher–Hartwig singularities. To our knowledge, even in the case of one singularity, the asymptotics of the determinant of such a moment matrix whose symbol has support in a two‐dimensional set and a Fisher–Hartwig singularity have been previously unknown.

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“…From our point of view, the reason to restrict to

| z | < 1

is that this is a more interesting case than

| z | > 1

: one should expect from that for each

z \in C

for which

| z | > 1

prefixlog false| trueprefixdet false(G_{N} - z false) false| - double-struckE prefixlog false| trueprefixdet false(G_{N} - z false) false|

converges in law to a real‐valued Gaussian random variable — there should be no

N^{γ^{2} / 8}

appearing in this case. We expect that this could be proven using a similar approach as the one we take here (using the results of with

| z | > 1

), but we do not explore this further. Note that another reason to distinguish between

| z | < 1

and

| z | > 1

is that in our normalization, the unit disk is the support of the equilibrium measure for the Ginibre ensemble, so it is the set where the eigenvalues should accumulate in the large

N

limit.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

double-struckE false| trueprefixdet false(G_{N} - z false) |^{γ}

also satisfy suitable orthogonality conditions on certain contours in the complex plane. Then in Section 3, we apply these results to transform the analysis of

double-struckE false| trueprefixdet false(G_{N} - z false) |^{γ}

into a question of the asymptotic analysis of a suitable RHP.…”

Section: The Ginibre Ensemble and Orthogonal Polynomialsmentioning

confidence: 88%

“…We also point out that if

normalΣ

p_{j}

with respect to the pairing coming from . We now recall how these dual orthogonal polynomials are constructed and how their leading order coefficient is related to

χ_{j}

.…”

Section: The Ginibre Ensemble and Orthogonal Polynomialsmentioning

confidence: 99%

“…Again, we offer a sketch of a proof in Appendix , as such a result is not formulated precisely in this form in . We now turn to the RHP and the differential identity related to

D_{N - 1} false(F false)

.…”

Section: The Ginibre Ensemble and Orthogonal Polynomialsmentioning

confidence: 99%

“…In this section, we recall from the asymptotic solution of the RHP for

Y_{N}

. In fact, we will consider a minor generalization of their situation where we study the asymptotics of

Y_{N + k}

, where

k

is a fixed integer — for our differential identity, we only need

k = 0

and

k = 1

.…”

Section: Solving the Riemann–hilbert Problem For Yn Asymptoticallymentioning

confidence: 99%

See 3 more Smart Citations

On the moments of the characteristic polynomial of a Ginibre random matrix

Webb

Wong

2018

Proc. London Math. Soc.

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Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges

Lee

Yang

2023

Comm Pure Appl Math

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We consider the orthogonal polynomial pn(z) with respect to the planar measure supported on the whole complex plane where dA is the Lebesgue measure of the plane, N is a positive constant, {c1, …, cν} are nonzero real numbers greater than −1 and are distinct points inside the unit disk. In the scaling limit when n/N = 1 and n → ∞ we obtain the strong asymptotics of the polynomial pn(z). We show that the support of the roots converges to what we call the “multiple Szegő curve,” a certain connected curve having ν + 1 components in its complement. We apply the nonlinear steepest descent method [9,10] on the matrix Riemann‐Hilbert problem of size (ν + 1) × (ν + 1) posed in [22]. © 2023 Wiley Periodicals, LLC.

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Large gap asymptotics on annuli in the random normal matrix model

Charlier

2023

Math. Ann.

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We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large n asymptotics of the form $$\begin{aligned} \exp \Bigg ( C_{1} n^{2} + C_{2} n \log n + C_{3} n + C_{4} \sqrt{n} + C_{5}\log n + C_{6} + {\mathcal {F}}_{n} + \mathcal {O}\Big ( n^{-\frac{1}{12}}\Big )\Bigg ), \end{aligned}$$ exp ( C 1 n 2 + C 2 n log n + C 3 n + C 4 n + C 5 log n + C 6 + F n + O ( n - 1 12 ) ) , where n is the number of points of the process. We determine the constants $$C_{1},\ldots ,C_{6}$$ C 1 , … , C 6 explicitly, as well as the oscillatory term $${\mathcal {F}}_{n}$$ F n which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only $$C_{1},\ldots ,C_{4}$$ C 1 , … , C 4 were previously known, (ii) when the hole region is an unbounded annulus, only $$C_{1},C_{2},C_{3}$$ C 1 , C 2 , C 3 were previously known, and (iii) when the hole region is a regular annulus in the bulk, only $$C_{1}$$ C 1 was previously known. For general values of our parameters, even $$C_{1}$$ C 1 is new. A main discovery of this work is that $${\mathcal {F}}_{n}$$ F n is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.

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Discontinuity in the Asymptotic Behavior of Planar Orthogonal Polynomials Under a Perturbation of the Gaussian Weight

Cited by 27 publications

References 22 publications

On the moments of the characteristic polynomial of a Ginibre random matrix

On the moments of the characteristic polynomial of a Ginibre random matrix

Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges

Large gap asymptotics on annuli in the random normal matrix model

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