Renjie Feng

 

I am a Postdoctoral Research Fellow in Mathematics and AI at the School of Mathmatics and Statistics and Sydney Mathematical Research Institute at the University of Sydney, supervised by Prof. Geordie Williamson.

My academic profiles at ResearchGate, and Google Scholar.

Address: Sydney Mathematical Research Institute L4.47
University of Sydney NSW 2006
Australia

Email: renjie.feng@sydney.edu.au

Research Interests

My research focuses on probability theory and its applications in data science, machine learning, deep learning, AI, statistics, computer science, information theory, applied mathematics, and quantum information theory.

Publications

  1. Smallest gaps between zeros of stationary Gaussian processes (with D. Yao and F. Götze)
    Journal of Functional Analysis, Volume 287, Issue 4, 15 August 2024, 110493.
  2. Critical radius and supermum of random waves over Riemannian manifolds (with D.Yao and R.Adler)
    submitted for review, 2024.
  3. Smallest distances between zeros of Gaussian analytic functions (with D. Yao)
    submitted, under review at Advances in Mathematics , 2024.
  4. Small gaps of GSE (with J. Li and D. Yao)
    arXiv:2409.03324, submitted.
  5. Determinantal point processes on spheres: multivariate linear statistics (with F. Götze and D. Yao)
    submitted for review, 2023.
  6. U-statistics of infinite Ginibre ensemble and Wiener chaos (with D. Yao)
    submitted for review, 2022.
  7. Principal minors of GOE (with G. Tian, D. Wei, and D. Yao)
    arXiv:2205.05732, under review at Annals of Applied Probability.
  8. Large gaps of CUE and GUE (with D. Wei)
    under reviw at Annals of Probability.
  9. The Berry-Esseen theorem for circular β-ensemble (with G. Tian and D. Wei)
    Annals of Applied Probability, 33(6B): 5050-5070 (2023).
  10. Small gaps of circular β-ensemble (with D. Wei)
    Annals of Probability, 49(2): 997-1032 (2021).
  11. Small gaps of GOE (with G. Tian and D. Wei)
    GAFA, 29 (2019), no. 6, 1794-1827.
  12. Spectrum of SYK model (with G. Tian and D. Wei)
    Peking Mathematical Journal, 2, 41-70 (2019).
  13. Spectrum of SYK model II: Central limit theorem (with G. Tian and D. Wei)
    Random Matrices: Theory and Applications, 10(04), 2150037 (2021).
  14. Spectrum of SYK model III: Large deviations and concentration of measures (with G. Tian and D. Wei)
    Random Matrices: Theory and Applications, 09(02), 2050001 (2020).
  15. Zeros of repeated derivatives of random polynomials (with D. Yao)
    Analysis & PDE, 12(6), 1489-1512, 2019.
  16. Critical radius and supremum of random spherical harmonics (with R. Adler)
    Annals of Probability, 47(2), 1162-1184 (2019).
  17. Correlations between zeros and critical points of random analytic functions
    Transactions of the AMS, 371(8), 5247-5265 (2019).
  18. Critical radius and supremum of random spherical harmonics II (with X. Xu and R. Adler)
    Electronic Communications in Probability, 23, paper no. 50, 11 pp. (2018).
  19. Conditional expectations of random holomorphic fields on Riemann surfaces
    International Mathematics Research Notices, 2017(14), 4406-4434.
  20. Critical values of Gaussian SU(2) random polynomials (with Z. Wang)
    Proceedings of the American Mathematical Society, Vol. 144, No. 2 (February 2016), pp. 487-502.
  21. Critical values of random analytic functions on complex manifolds (with S. Zelditch)
    Indiana University Mathematics Journal, 63(3), 651-686 (2014).
  22. Addendum to "Critical Values of Random Analytic Functions on Complex Manifolds" (with S. Zelditch)
    Indiana University Mathematics Journal, Vol. 66, No.1 (2017),pp.23-29.
  23. Median and mean of the Supremum of L2 normalized random holomorphic fields (with S. Zelditch)
    Journal of Functional Analysis, 266(8), 5085-5107 (2014).
  24. Random Riesz energies on compact Kähler manifolds (with S. Zelditch)
    Transactions of the AMS, 365(10), 5579-5604 (2013).
  25. Large deviations for zeros of P(φ)_2 random polynomials (with S. Zelditch)
    Journal of Statistical Physics, 143, 619-635 (2011).
  26. Bergman metrics and geodesics in the space of Kähler metrics on principally polarized abelian varieties
    Journal of the Institute of Mathematics of Jussieu, 11(1), 1-25 (2012).
  27. The global existence and convergence of the Calabi flow on ℂn/ ℤn+iℤn (with H. Huang)

    Journal of Functional Analysis, 263(4), 1129-1146 (2012).

  28. Szasz analytic functions and noncompact Kähler toric manifolds
    Journal of Geometric Analysis, 22 (1), 107-131 (2012).
  29. Periodic solutions of Abreu's equation (with G. Szekelyhidi)
    Mathematical Research Letters, 18(6), 1271-1279 (2011).
  30. Extreme gap problems in random matrix theory
    Surveys in Geometric Analysis Volume 3, 2019 (2024):19.
  31. Zeros and Critical Points of Random Fields
    PhD thesis, Northwestern University, 2012.

PPT for U-statistics of DPPs

In this PPT, I will present my recent work with D. Yao on the U-statistics of the determinantal point processes (DPPs). I studied this problem with D.Yao since 2022 on the U-statistics of infinite Ginibre ensemble, which seems a much easier model to work with, but it's not. Our original motivation is to study its random topology inspired by the research on that of Poisson point process conducted by research group of R. Adler - one of my coauthors, where we found the study of its U-statistics is a must. Our first contribution in 2022 is to derive a graph representation for the cumulants of U-statistics of any DPPs, which generalizes Soshnikov's formula on the linear statistics of DPPs. Based on this new graph formula, we derived the first 2 orders of U-statistics for the infinite Ginibre ensemble. The 1st order is the Gaussian limit; when it's degenerate, the 2nd order is a mixture of chi-squared distribution and Gaussian.

In 2023, we studied another model with F. Götze, based on the graph representation again, we discovered that the complete Wiener chaos exists for the spherical case associated with the spectral projection kernel with respect to the Laplace operator. This model seems quite complicated but the main result is much better. When considering the higher order degeneration of the U-statistics, the leading order terms of the cumulants are given by the so-called complete paring graphs in exactly the same pattern as the i.i.d. case. To the best of our knowledge, this is the first result demonstrating the existence of complete Wiener chaos for DPPs.

U-statistics of determinantal point processes and Wiener chaos.

Seminar and Conference Talks since 2020