Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions

Dimitrov, Evgeni; Knizel, Alisa

doi:10.1016/j.jfa.2018.12.008

Cited by 14 publications

(39 citation statements)

References 62 publications

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“…The first work of this kind was due to Johansson [29] where Gaussian fluctuations for β-ensembles with analytic potentials are shown using loop equations. This approach was further extended in the articles of Kriecherbauer-Shcherbina [30], Borot-Guionnet [10,11], Shcherbina [40], Borodin-Gorin-Guionnet [8], and Dimitrov-Knizel [18]. Dumitriu-Paquette [20] established global fluctuations for the general β-Jacobi ensemble through the tridiagonalization representation of this ensemble [19].…”

Section: Introductionmentioning

confidence: 99%

Fluctuations of $β$-Jacobi Product Processes

Ahn¹

2019

Preprint

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We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index β = 1, 2, 4 respectively) where time corresponds to the number of terms in the product. More generally, we consider the β-Jacobi product process obtained by extrapolating to arbitrary β > 0. When the time scaling is preserved, we show that the global fluctuations are jointly Gaussian with explicit covariances. For time growing linearly with matrix size, we show convergence of moments after suitable rescaling. When β = 2, our results imply that the right edge converges to a process which interpolates between the Airy point process and a deterministic configuration. This process connects a time-parametrized family of point processes appearing in the works of Akemann-Burda-Kieburg and Liu-Wang-Wang across time. In the arbitrary β > 0 case, our results show tightness of the particles near the right edge. The limiting moment formulas correspond to expressions for the Laplace transform of a conjectural β-generalization of the interpolating process.

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Section: Introductionmentioning

confidence: 99%

Fluctuations of $β$-Jacobi Product Processes

Ahn¹

2019

Preprint

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“…A closely related loop equation formalism has been applied recently in the study of a q-boxed plane partition model relating to an example of (1.1) on an exponential measure [17]. This is part of our motivation, in addition to its relevance to moments, to consider whether q-orthogonal polynomial ensembles satisfy such an integration by part formula, and whether it can be applied to formulate a q-difference equation for the Laplace transform of the density.…”

Section: Linear Discrete Casementioning

confidence: 99%

$q$-Pearson pair and moments in $q$-deformed ensembles

Forrester¹,

Li²,

Shen³

et al. 2021

Preprint

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The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the q-lattice setting is considered. We take up the task of initiating a systematic study of the corresponding moments of the density from two complementary viewpoints. The first requires knowledge of the ensemble average with respect to a general Schur polynomial, from which the spectral moments follow as a corollary. In the case of little q-Laguerre weight, a particular 3 φ 2 basic hypergeometric polynomial is used to express density moments. The second approach is to study the q-Laplace transform of the un-normalised measure. Using integrability properties associated with the q-Pearson equation for the q-classical weights, a fourth order qdifference equation is obtained, generalising a result of Ledoux in the continuous classical cases.

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“…Our expression for the particle-particle interaction ((1.7) and (5.2)) agrees with [8, (82)] for configurations X on the right semi-axis R >0 . Dimitrov and Knizel [19] studied general (q, t)-deformed discrete beta ensembles in which the particle-particle interaction corresponds to a higher level of the hypergeometric hierarchy. However, the problem settings and the results of [8] and [19] are very different from what is being done in the present paper.…”

Section: 4mentioning

confidence: 99%

Macdonald-level extension of beta ensembles and large-N limit transition

Olshanski

2020

Preprint

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We introduce and study a family of (q, t)-deformed discrete N -particle beta ensembles, where q and t are the parameters of Macdonald polynomials. The main result is the existence of a large-N limit transition leading to random point processes with infinitely many particles. Contents( • ; q, t, s) 3.2. Koornwinder polynomials 3.3. Multivariable big q-Jacobi polynomials ϕ λ|N ( • ; q, t; a, b, c, d) 4. One-particle hypergeometric ensembles 5. N-particle hypergeometric ensembles Key words and phrases. Beta-ensembles; large-N limit transition; Macdonald polynomials; multivariable interpolation polynomials; q-Selberg integral; big q-Jacobi polynomials; Koornwinder polynomials. 5.1. Particle configurations 23 5.2. Degenerate series of hypergeometric measures 23 5.3. Some estimates 26 5.4. Principal series of hypergeometric measures 29 6. Main result: large-N limit transition 29 6.1. Preliminaries 29 6.2. The measures M q,t;α,β,γ,δ N 30 6.3. The coherency relation 31 6.4. Main result 33 7. Concluding remarks 34 7.1. Big q-Jacobi symmetric functions 34 7.2. Degeneration to discrete beta-ensembles 34 7.3. Degeneration to continuous beta-ensembles 36 References 37

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Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions

Cited by 14 publications

References 62 publications

Fluctuations of $β$-Jacobi Product Processes

Fluctuations of $β$-Jacobi Product Processes

$q$-Pearson pair and moments in $q$-deformed ensembles

Macdonald-level extension of beta ensembles and large-N limit transition

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