Law of large numbers and central limit theorems through Jack generating functions

Huang, Jiaoyang

doi:10.1016/j.aim.2020.107545

Cited by 7 publications

(5 citation statements)

References 43 publications

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“…The pull-back map Ω and other expressions in the statement are independent of θ. This is in strong agreement with the results in [BG15] for the β-Jacobi corners process and in [Hua21] for certain Jack processes.…”

Section: Introduction and Main Resultssupporting

confidence: 91%

“…The main point here is that if we do this alternative scaling, then in the formulas for the covariances C θ θ,µ and C θ θ,µ , the parameter θ enters simply as a linear prefactor, as opposed to where in C θ,µ the θ appears inside the integral in a nontrivial way. This linearity of the covariances in θ −1 is in strong agreement with the results in [26] for the β-Jacobi corners process and in [28] for Jack processes.…”

Section: Central Limit Theoremsupporting

confidence: 89%

“…As mentioned earlier, when θ ̸ = 1 one cannot use techniques based on determinantal point processes, nor the Schur generating functions in [BG18,BG19]. One possible approach is based on the Jack generating functions in [Hua21], which have been used to prove GFF asymptotics in certain Markov processes based on Jack symmetric functions. The existing work in that direction is in a dynamical setting of N walkers that evolve without crossing as opposed to corners processes.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Asymptotics of discrete β-corners processes via two-level discrete loop equations

Dimitrov

Knizel

2022

Prob. Math. Phys.

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We introduce and study a class of discrete particle ensembles that naturally arise in connection with classical random matrix ensembles, log-gases and Jack polynomials. Under technical assumptions on a general analytic potential we prove that the global fluctuations of these ensembles are asymptotically Gaussian with a universal covariance that remarkably differs from its counterpart in random matrix theory. Our main tools are certain novel algebraic identities that we have discovered. They play a role of discrete multilevel analogues of loop equations. 1. Introduction 247 2. Nekrasov's equations 257 3. Particular setup 264 4. Application of Nekrasov's equations 271 5. Central limit theorem 283 6. Multilevel extensions and examples 294 7. Continuous limit 305 Appendix A. 321 Appendix B.331 References 340 * Note that we have reversed the order of the indices so that ℓ 1 is now largest and ℓ N is the smallest -this convention is more consistent with the symmetric function origin of (1-3).

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Section: Introduction and Main Resultssupporting

confidence: 91%

Section: Central Limit Theoremsupporting

confidence: 89%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Asymptotics of discrete β-corners processes via two-level discrete loop equations

Dimitrov

Knizel

2022

Prob. Math. Phys.

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“…According to the law of large numbers (Huang, 2021), the population mean μi* is approximately equal to the sample mean μi. Therefore, μi* can be used to represent the mean value of the area loss when the i -th member is corroded.…”

Section: Area-loss Limit Of Members Based On Reliability Theory and N...mentioning

confidence: 99%

Area-loss effect analysis of pin-jointed structures

Deng

Samy

Yuan

2023

Advances in Structural Engineering

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Corrosion will cause cross-sectional area loss of steel members, which will lead to redistribution of internal force and affect the performance of structures. In this paper, a generalized formula for the internal force variation of pin-jointed structures due to variation of external force, cross-sectional area, and initial length is derived. The area-loss sensitivity coefficient ( ASC) and the area-loss evenness coefficient ( AEC) based on statistical measures are then defined. By calculating these two coefficients, the influence of single member’s area-loss on the behavior of the structure as well as the evenness of internal force variation are evaluated. Besides, a method to determine the area-loss limit of pin-jointed structures is proposed based on the structural reliability theory and mathematical nonlinear programming. According to the above process, the effect of members’ area loss on structural properties is analyzed without complicated numerical approaches such as FEM, and then the safety of corroded structures is assessed preliminarily. The proposed method is verified by two numerical examples, and provides a convenient tool for pre-manufacturing, maintenance during service life, and structural health monitoring of pin-jointed structures.

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“…By taking q → 1, Macdonald symmetric polynomials recover the Jack symmetric polynomials with parameter θ. The global fluctuations of these measures related to Jack symmetric polynomials have been previously studied in [34,54,69] by different methods. We postpone the proof of Claim 3.11 till Appendix A. Theorem 1.1 gives the dynamical loop equation for transition probabilities generalizing the ascending Macdonald process (3.31).…”

Section: Loop Equations For the Ascending Processmentioning

confidence: 99%

Dynamical Loop Equation

Gorin¹,

Huang²

2022

Preprint

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We introduce dynamical versions of loop (or Dyson-Schwinger) equations for large families of two-dimensional interacting particle systems, including Dyson Brownian motion, Nonintersecting Bernoulli/Poisson random walks, β-corners processes, uniform and Jack-deformed measures on Gelfand-Tsetlin patterns, Macdonald processes, and (q, κ)-distributions on lozenge tilings. Under technical assumptions, we show that the dynamical loop equations lead to Gaussian field type fluctuations.As an application, we compute the limit shape for (q, κ)-distributions on lozenge tilings and prove that their height fluctuations converge to the Gaussian Free Field in an appropriate complex structure.

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Law of large numbers and central limit theorems through Jack generating functions

Cited by 7 publications

References 43 publications

Asymptotics of discrete β-corners processes via two-level discrete loop equations

Asymptotics of discrete β-corners processes via two-level discrete loop equations

Area-loss effect analysis of pin-jointed structures

Dynamical Loop Equation

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