Asymptotic correlations with corrections for the circular Jacobi <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e25" altimg="si6.svg"><mml:mi>β</mml:mi></mml:math>-ensemble

Forrester, Peter J.; Li, Shihao; Trinh, Allan K.

doi:10.1016/j.jat.2021.105633

Cited by 8 publications

(2 citation statements)

References 31 publications

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“…These steps, performed using computer algebra, verify that (2.30) solves (2.20), as required. To numerically evaluate the integral operators we use the Fredholm determinant Matlab toolbox by Folkmar Bornemann [11], and a Mathematica implementation by Allan Trinh, coauthor on some related works along the theme of finite size corrections to limit formulas in random matrix theory [26,27,28,23]. For the DE solutions u 0 (r), u 1 (r) needed for (2.19) we use a sequence of Taylor series expanded about various r points, beginning on the right (near +∞, to match the DE boundary conditions) and proceeding to the left.…”

Section: 2mentioning

confidence: 99%

Finite size corrections relating to distributions of the length of longest increasing subsequences

Forrester¹,

Mays²

2022

Preprint

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Considered are the large N , or large intensity, forms of the distribution of the length of the longest increasing subsequences for various models. Earlier work has established that after centring and scaling, the limit laws for these distributions relate to certain distribution functions at the hard edge known from random matrix theory. By analysing the hard to soft edge transition, we supplement and extend results of Baik and Jenkins for the Hammersley model and symmetrisations, which give that the leading correction is proportional to z −2/3 , where z 2 is the intensity of the Poisson rate, and provides a functional form as derivates of the limit law. Our methods give the functional form both in terms of Fredholm operator theoretic quantities, and in terms of Painlevé transcendents. For random permutations and their symmetrisations, numerical analysis of exact enumerations and simulations gives compelling evidence that the leading corrections are proportional to N −1/3 , and moreover provides an approximation to their graphical forms.

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Section: 2mentioning

confidence: 99%

Finite size corrections relating to distributions of the length of longest increasing subsequences

Forrester¹,

Mays²

2022

Preprint

Self Cite

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“…The existence of this limit, and moreover its explicit form as a β-dimensional integral, follows immediately from (2.5) [20], [21,Eq. (13.35)]; in fact starting with (2.5) it is possible to show that the rate of convergence to the limit is O(1/N 2 ) [26]. The significance of the existence of the limit, and moreover with corrections that are inverse powers of N, is that the differential operator characterising ρ (2),N (θ, 0) can similarly be expanded, with the leading order giving the differential operator for ρ…”

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confidence: 99%

Differential identities for the structure function of some random matrix ensembles

Forrester

2020

Preprint

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The structure function of a random matrix ensemble can be specified as the covariance of the linear statistics ∑ N j=1 e ik 1 λ j , ∑ N j=1 e −ik 2 λ j for Hermitian matrices, and the same with the eigenvalues λ j replaced by the eigenangles θ j for unitary matrices. As such it can be written in terms of the Fourier transform of the density-density correlation ρ (2) . For the circular β-ensemble of unitary matrices, and with β even, we characterise the bulk scaling limit of ρ (2) as the solution of a linear differential equation of order β + 1 -a duality relates ρ (2) with β replaced by 4/β to the same equation. Asymptotics obtained in the case β = 6 from this characterisation are combined with previously established results to determine the explicit form of the degree 10 palindromic polynomial in β/2 which determines the coefficient of |k| 11 in the small |k| expansion of the structure function for general β > 0. For the Gaussian unitary ensemble we give a reworking of a recent derivation and generalisation, due to Okuyama, of an identity relating the structure function to simpler quantities in the Laguerre unitary ensemble first derived in random matrix theory by Brézin and Hikami. This is used to determine various scaling limits, many of which relate to the dip-ramp-plateau effect emphasised in recent studies of many body quantum chaos, and allows too for rates of convergence to be established.

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Expanding the Fourier Transform of the Scaled Circular Jacobi $$\beta $$ Ensemble Density

Forrester,

Shen

2023

J Stat Phys

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The family of circular Jacobi $$\beta $$ β ensembles has a singularity of a type associated with Fisher and Hartwig in the theory of Toeplitz determinants. Our interest is in the Fourier transform of the corresponding $$N \rightarrow \infty $$ N → ∞ bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobi$$\beta $$ β ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for $$\beta = 2$$ β = 2 and $$\beta = 4$$ β = 4 , and the loop equation hierarchy. The polynomials in the variable $$u=2/\beta $$ u = 2 / β which occur in the expansion coefficents are found to have special properties analogous to those known for the structure function of the circular $$\beta $$ β ensemble, specifically in relation to the zeros lying on the unit circle $$|u|=1$$ | u | = 1 and interlacing. Comparison is also made with known results for the expanded Fourier transform of the density about a guest charge in the two-dimensional one-component plasma.

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Asymptotic correlations with corrections for the circular Jacobi β-ensemble

Cited by 8 publications

References 31 publications

Finite size corrections relating to distributions of the length of longest increasing subsequences

Finite size corrections relating to distributions of the length of longest increasing subsequences

Differential identities for the structure function of some random matrix ensembles

Expanding the Fourier Transform of the Scaled Circular Jacobi $$\beta $$ Ensemble Density

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