Renjie Feng

 

Postdoctoral research fellow in the School of Mathmatics and Statistics and Sydney Mathematical Research Institute at the University of Sydney, supervised by Prof. Geordie Williamson. I got Ph.D. from Northwestern University under the supervision of Steve Zelditch.

My academic profile at the Faculty of Science and School of Mathematics and Statistics of the University of Sydney.

My personal homepage at Google Sites.

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

Email: renjie.feng@sydney.edu.au

Research Interests

I am a member of the Pure Mathematics research group, within the subgroup Geometry, Topology and Analysis.

My reseach covers various areas in analysis, including probability theory, geometry and their intersection, especially random matrices and random geometry.

Publications

The links for all papers are available at my Google Scholar.

Random matrices
  1. Small gaps for GSE (with J. Li and D. Yao)
    near completion, available upon request.
  2. Determinantal point processes on spheres: multivariate linear statistics (with F. Götze and D. Yao)
    arXiv:2301.09216, submitted to Annals of Probability.
  3. Principal minors of GOE (with G. Tian, D. Wei, and D. Yao)
    arXiv:2205.05732, submitted to Transactions of the AMS.
  4. Large gaps of CUE and GUE (with D. Wei)
    Submitted to Annals of Probability.
  5. Normality of circular β-ensemble (with G. Tian and D. Wei)
    Annals of Applied Probability, 33(6B): 5050-5070 (2023).
  6. Small gaps of circular β-ensemble (with D. Wei)
    Annals of Probability, 49(2): 997-1032 (2021).
  7. Small gaps of GOE (with G. Tian and D. Wei)
    GAFA, 29 (2019), no. 6, 1794-1827.
  8. Spectrum of SYK model (with G. Tian and D. Wei)
    Peking Mathematical Journal, 2, 41-70 (2019).
  9. Spectrum of SYK model II: Central limit theorem (with G. Tian and D. Wei)
    Random Matrices: Theory and Applications, 10(04), 2150037 (2021).
  10. 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).
    11. Random geometry
  11. Smallest distances between zeros of Gaussian analytic functions (with D. Yao)
    Submitted to Annals of Probability, 2023.
  12. Smallest gaps between zeros of stationary Gaussian processes (with D. Yao and F. Götze)
    Under minor revision at Journal of Functional Analysis, 2023.
  13. Critical radius and supremum of random waves on Riemannian manifolds (with D. Yao and R. Adler)
    in preparation.
  14. Zeros of repeated derivatives of random polynomials (with D. Yao)
    Analysis & PDE, 12(6), 1489-1512, 2019.
  15. Critical radius and supremum of random spherical harmonics (with R. Adler)
    Annals of Probability, 47(2), 1162-1184 (2019).
  16. Correlations between zeros and critical points of random analytic functions
    Transactions of the AMS, 371(8), 5247-5265 (2019).
  17. 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).
  18. Conditional expectations of random holomorphic fields on Riemann surfaces
    International Mathematics Research Notices, 2017(14), 4406-4434.
  19. 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.
  20. Critical values of random analytic functions on complex manifolds (with S. Zelditch)
    Indiana University Mathematics Journal, 63(3), 651-686 (2014).
  21. Median and mean of the Supremum of L2 normalized random holomorphic fields (with S. Zelditch)
    Journal of Functional Analysis, 266(8), 5085-5107 (2014).
  22. Random Riesz energies on compact Kähler manifolds (with S. Zelditch)
    Transactions of the AMS, 365(10), 5579-5604 (2013).
  23. Large deviations for zeros of P(φ)_2 random polynomials (with S. Zelditch)
    Journal of Statistical Physics, 143, 619-635 (2011).
    24. Complex geometry and PDEs
  24. 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).
  25. 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).
  26. Szasz analytic functions and noncompact Kähler toric manifolds
    Journal of Geometric Analysis, 22 (1), 107-131 (2012).
  27. Periodic solutions of Abreu's equation (with G. Szekelyhidi)
    Mathematical Research Letters, 18(6), 1271-1279 (2011).

Recent Seminar and Conference Talks

  1. A quick introduction to random matrices and extreme gap problems
    The Australian National University, June, 2024 (upcoming).
  2. Determinantal point processes on spheres: Multivariate linear statistics
    Stochastic processes seminar, The University of Melbourne, March, 2024.
  3. Extreme spacings for classical random matrices
    Random matrix theory group seminar, The University of Melbourne, March, 2024.
  4. A graphical formula for cumulants of multivariate linear statistics of determinantal point processes
    Probability and Mathematical Statistics, 67th Annual Meeting of the Australian Mathematical Society, Dec, 2023.
  5. Determinantal point processes on spheres: Multivariate linear statistics
    Applied Maths Seminar, The University of Sydney, Oct, 2023.
  6. A quick introduction to random matrices and extreme gap problems
    SMRI seminar, The University of Sydney, Sep, 2023.
  7. Determinantal point processes on the sphere
    Research Seminar on Probability and Geometry, Ruhr-Universität Bochum, Germany, May, 2023.
  8. Determinantal point processes on spheres: Multivariate linear statistics
    Seminar Bielefeld-Melbourne-Seoul Random Matrices (online), March, 2023.
  9. Determinantal point processes on spheres: Multivariate linear statistics
    Institute of Applied Mathematics, Chinese Academy of Sciences, Feb, 2023.
  10. Extreme gap problems and the SYK model
    Lille days on Point Processes and stochastic Geometry, Université de Lille, France, Oct, 2022.
  11. Extreme gap problems for classical random matrices
    MPI-Oberseminar, Max Planck Institute for Mathematics, Germany, Aug, 2022.
  12. Extreme gaps for classical random matrices and the SYK model
    Seminar Mathematical Physics, Bielefeld University, Germany, March, 2022.
  13. Extreme spacings of classical random matrices and the SYK model
    Oberseminar Stochastics, University of Bonn, Germany, Jan, 2022.
  14. Random matrices and our recent results
    MATH seminar, Bilkent University (online), March, 2021.
  15. Some recent results in random matrices
    Institute of Applied Mathematics, Chinese Academy of Sciences, Oct, 2020.

Key Research Findings

    I studied complex geometry and PDEs during my Ph.D. After my graduation, I have been working on random matrices, random geometry, and their extensions. Here is a summary of the key findings from my selected publications.

  1. The extreme gap problem for eigenvalues of random matrices is classical and fundamental. The Poisson nature of the smallest gaps of determinantal point processes, e.g., CUE and GUE, is understood very well, but there was no previous knowledge beyond the determinantal structure. In [7], we finally overcome the determinantal structure, and derived the smallest gaps for GOE which has a Pfaffian structure. The method is based on the observation that the pair of the smallest gaps will stick together, causing the system to become a two-component log-gas in the limit. All the essential computations turn out to be some type of Selberg Integral. The method can be applied to CβE for integer β [6].
  2. However, for GSE, this method no longer works since the Selberg Integral for the two-component log-gas of GSE is intractable. In [1], we developed some very brutal-force computations to tackle it, and this systematic method can be used to study the smallest gaps of any Pfaffian point processes with an explicit kernel matrix.
  3. The largest gaps between eigenvalues are significantly more challenging compared to the smallest gaps. To prove the Poisson limit of the largest gaps, one has to prove the splitting of the gap probability for relative large scale intervals or domains, which is in fact a type of large deviation result. In [4], we proved the Poisson limit for the largest gaps of CUE and the interior of GUE based on their determinantal structures. The problem regarding the largest gaps is quite open for other classical ensembles, e.g., COE and CSE.
  4. Very recent, in [11] we studied the smallest geodesic distances between zeros of Gaussian random holomorphic sections over Riemann surfaces. The main result is that the smallest geodesic distances tend to a Poisson point process with a universal rate. The result applies specifically to the classical SU(2) random polynomials. Again, the problem regarding the largest geodesic distances is open.
  5. The Sachdev-Ye-Kitaev model (SYK) is very topical recently in high energy physics, which is to simply model a black hole. It is also a beautiful quantum spin glass model and a random matrix model. In our 3 series papers [8,9,10], we tried to establish some fundamental results from mathematical point of view regarding the distribution of eigenvalues, including the global density, the central limit theorem and the large deviation principle. But more problems are left unsolved, especially the behaviors of the largest eigenvalues, e.g., the growth order, the global distribution and its rescaling limit.
  6. Last year, within random matrix theory, in [2] we studied the U-statistics (rather than linear statistics) for determinantal point processes. We studied a model case on spheres, and we have uncovered a surprising connection with Wiener chaos. To prove this, we first derived a graphical representation for the cumulant of the U-statistics, which extends the famous Soshnikov's formula to multivariate cases. Our computations can be applied to any determinantal point processes, and Wiener chaos are expected.
  7. In random geometry, in [15] we derived the supremum of the random waves on spheres by Weyl's tube formula. The essential part is to show that there is a lower bound for the critical radius of the embedding of the spheres into higher dimensional space via the spherical harmonics. In the paper under construction [13], we further proved that such results hold for any Riemannian manifolds, and the critical radius has a universal limit which has its own interest in Riemannian geometry.