Lingfu Zhang's Home Page

Lingfu Zhang

Fine hall 410, Princeton, NJ 08544 USA
Email: lingfuz at princeton dot math dot edu

CV (Last update: Aug 26, 2019)

About me



  • I'm a PhD student at the Princeton University Mathematics Department, and my advisor is Professor Allan Sly.
  • My primary interest is in probability theory, particularly in first and last passage percolation (either exactly solvable or not), interacting particle system, and Anderson localization problems. I’m also interested in some other problems related to statistical mechanics on lattice (such as Ising/Potts model, percolation theory, reinforced random walk, hard-core model).

    Education



    Ph.D. in Mathematics, Princeton University   (Sep. 2017 - present).

    B.S. in Mathematics, Massachusetts Institute of Technology   (Sep. 2014 - Jun. 2017).

    Research


    Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison (joint paper with Riddhipratim Basu and Shirshendu Ganguly), draft available upon request.
    We consider directed last passage percolation on \(\mathbb{{Z}}^2\) with exponential passage times on the vertices. A topic of great interest is the coupling structure of the weights of geodesics between points as they are varied in space and time with various initial conditions with a particular case of importance being the flat initial data which corresponds to line-to-point last passage times. Settling a conjecture by Ferrari and Spohn., we show that for the passage times from the line \(x+y=0\) to the points \((r,r)\) and \((n,n)\), denoted \(X_{r}\) and \(X_{n}\) respectively, as \(n\to \infty\) and \(\frac{r}{n}\) is small but bounded away from zero, the covariance satisfies \(\mbox{Cov}(X_{r},X_{n})=\Theta((\frac{r}{n})^{4/3+o(1)} n^{2/3})\), thereby establishing \(4/3\) as the temporal covariance exponent. This differs from the corresponding exponents for the droplet and other curved initial conditions recently rigorously established and requires significantly novel arguments. In particular, our proof relies on recent advances in quantitative comparison of geodesic weight profiles to Brownian motion and is expected to go through for a wider class of initial data.


    Optimal Estimates for Coalescence of Finite Geodesics in Exponential Last Passage Percolation, draft available upon request.
    For the directed last passage percolation on \(\mathbb{{Z}}^2\), with i.i.d. exponential weight, we consider the maximum paths from vertices \(\left(0, \left\lfloor k^{2/3} \right\rfloor\right)\) and \(\left(\left\lfloor k^{2/3} \right\rfloor, 0\right)\) to \((n, n)\), respectively. We study the distance to the coalescing point of these two paths, and show that with probability in the order of \(R^{-2/3}\), it is at least \(kR\) far away. This is inspired by a recent work by Basu, Sarkar, and Sly, where the estimate was obtained for semi-infinite geodesics, and the finite case was left open.


    Empirical Distribution Along Geodesics in Exponential Last Passage Percolation (joint paper with Allan Sly), in preparation.


    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 \(\mathbb{{Z}}^d\).


    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.


    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 \(p = 6\).

    Talks


  • Anderson-Bernoulli localization near the edge on the 3D lattice (slide). Stanford University (Stanford, CA) Applied Math Seminar, Dec 4, 2019.
  • Constructing extremal stationary distributions for the Voter Model in \(d\geq 3\) as factors of IID (slide). Duke University (Durham, NC) Probability Seminar, Oct 24, 2019.
  • ICTS (Bangalore, India): Universality in random structures: Interfaces, Matrices, Sandpiles, Jan 25, 2019.
  • Interlacing adjacent levels of \(\beta\)-Jacobi corners processes (slide). Massachusetts Institute of Technology (Cambridge, MA) Integrable Probability Working Group, Nov 29, 2016.
  • Massachusetts Institute of Technology (Cambridge, MA) Summer Program for Undergraduate Research Conference, Aug 5, 2016.



    Last updated on Nov 20, 2019