Lingfu Zhang's Home Page

Lingfu Zhang

Princeton, NJ 08544-1000 USA
Email: lingfuz@math.princeton.edu

CV (Last update: Aug 26, 2019)

About
  • I'm a PhD student at the Princeton University Mathematics Department since fall 2017, and my advisor is Professor Allan Sly.
  • My primary interest is in probability theory, particularly in last passage percolation (LPP), interacting particle system, and Anderson localization. I'm also interested in problems related to percolation theory, random graphs, and exactly solvable models.

    Research
  • Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison (joint paper with Riddhipratim Basu and Shirshendu Ganguly), upcoming.
  • Stationary Distributions for the Voter Model in $d\geq 3$ are Bernoulli Shifts (joint paper with Allan Sly), arXiv:1908.09450
  • For the Voter Model on $\mathbb{Z}^d$, $d\geq 3$, we show that the (extremal) stationary distributions are Bernoulli shifts, and answer an open question asked by Steif and Tykesson. The proof is by explicit constructing the stationary distributions as factors of IID processes on $\mathbb{Z}^d$.
  • I gave a talk on this work at Duke University Probability Seminar on Oct 24, 2019. (slide).
  • Anderson-Bernoulli Localization on the 3D lattice and discrete unique continuation principle (joint paper with Linjun Li), arXiv:1906.04350
  • We consider the Anderson model with Bernoulli potential on $\mathbb{Z}^{3}$, and prove localization of eigenfunctions corresponding to eigenvalues near zero, the lower boundary of the spectrum. Our main contribution is the 3D discrete unique continuation, which says that any eigenfunction of harmonic operator with potential cannot be too small on a significant fractional portion of $\mathbb{Z}^{3}$.
  • Interlacing adjacent levels of $\beta$-Jacobi corners processes (joint paper with Vadim Gorin), Probability Theory and Related Fields 172, no. 3-4 (2018): 915-981. arXiv:1612.02321
  • We study the asymptotic of the global fluctuations for the difference between two adjacent level in the $\beta$--Jacobi corners process (multilevel and general $\beta$ extension of the classical Jacobi ensemble of random matrices). The limit is identified with the derivative of the $2d$ Gaussian Free Field. Our main tool is integral forms for the (Macdonald-type) difference operators originating from the Shuffle algebra.
  • I gave a talk on this work at MIT Integrable Probability Working Group on Nov 29, 2016. (slide).
  • Refinements of the 2-dimensional Strichartz estimate on the maximum wavepacket (joint paper with Hong Wang), arXiv:1611.10275
  • The Strichartz estimates for Schrödinger equations can be improved when the data is spread out in either physical or frequency space. In this paper we give refinements of the 2-dimensional homogeneous Strichartz estimate on the maximum size of a single wave packet. Different approaches are used in the proofs, including arithmetic approaches, polynomial partitioning, and the $l^2$ Decoupling Theorem, for different cases. We also give examples to show that the refinements we obtain cannot be further improved when $2 \leq p \leq 4$ and $p = 6$.

    Links
  • Princeton University Department of Mathematics
    Last updated on Aug 26, 2019