Lingfu Zhang's Home Page

Lingfu Zhang
(张灵夫)

Linde Hall 358, Caltech, Pasadena, CA
Email: lingfuz at caltech dot edu

About me


  • I am an Assistant Professor in Mathematics at Caltech, since the summer of 2024. Previously I was a Miller Fellow at UC Berkeley Department of Statistics, hosted by Professor Shirshendu Ganguly. I did my PhD at the Princeton University Mathematics Department, advised by Professor Allan Sly.
  • My research is in the area of probability, and within this area, I'm interested in various problems, connected with mathematical physics, computer science, combinatorics, and statistics. Topics I've studied include the Last Passage Percolation and related exactly solvable models in the KPZ universality class, the Anderson model of localization, and problems on Markov chains, such as cutoffs and factors of IID/local sampling algorithms.
  • My research is supported by the National Science Foundation through the standard grant DMS-2246664 (PI) in the Probability program (2023-2026).


    • Education


    Ph.D. in Mathematics,   Princeton University   (2017 - 2022).

    B.S. in Mathematics and B.S. in Computer Science (Course 18 and 6-3),   MIT   (2014 - 2017).

    Research

    My papers can also be found on arXiv here. See also my Google Scholar page.

    Accepted for publication

    Pearcey universality at cusps of polygonal lozenge tiling
    (with Jiaoyang Huang and Fan Yang)
    Comm. Pure Appl. Math. 77 (9), 3708-3784 (September 2024).
    arXiv Journal Slides Abstract
    We study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases proved in Aggarwal-Huang and Aggarwal-Gorin, these are believed to be the three types of edge statistics that can arise in a generic polygon. Our proof is via a local coupling of the random tiling with non-intersecting Bernoulli random walks (NBRW). To leverage this coupling, we establish an optimal concentration estimate for the tiling height function around the cusp. As another step and also a result of potential independent interest, we show that the local statistics of NBRW around a cusp converge to the Pearcey process when the initial configuration consists of two parts with proper density growth, via careful asymptotic analysis of the determinantal formula.


    Infinite order phase transition in the slow bond TASEP
    (with Sourav Sarkar and Allan Sly)
    Comm. Pure Appl. Math. 77 (6), 3107-3140 (June 2024).
    arXiv Journal Slides talk by Allan Sly Abstract
    In the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from \(1\) to \(1-\varepsilon\). Janowsky and Lebowitz posed the question of whether such very small perturbations could affect the macroscopic current. Different groups of physicists, using a range of heuristics and numerical simulations reached opposing conclusions on whether the critical value of \(\varepsilon\) is 0. This was ultimately resolved rigorously by Basu-Sidoravicius-Sly which established that \(\varepsilon_c=0\).
    Here we study the effect of the current as \(\varepsilon\) tends to \(0\) and in doing so explain why it was so challenging to predict on the basis of numerical simulations. In particularly we show that the current has an infinite order phase transition at \(0\), with the effect of the perturbation tending to \(0\) faster than any polynomial. Our proof focuses on the Last Passage Percolation formulation of TASEP where a slow bond corresponds to reinforcing the diagonal. We give a multiscale analysis to show that when \(\varepsilon\) is small the effect of reinforcement remains small compared to the difference between optimal and near optimal geodesics. Since geodesics can be perturbed on many different scales, we inductively bound the tails of the effect of reinforcement by controlling the number of near optimal geodesics and giving new tail estimates for the local time of (near) geodesics along the diagonal.


    Cutoff profile of the Metropolis biased card shuffling
    Ann. Probab. 52(2): 713-736 (March 2024).
    arXiv Journal Slides talk Abstract
    We consider the Metropolis biased card shuffling (also called the multi-species ASEP on a finite interval or the random Metropolis scan). Its convergence to stationary was believed to exhibit a total-variation cutoff, and that was proved a few years ago by Labbé and Lacoin. In this paper, we prove that (for \(N\) cards) the cutoff window is in the order of \(N^{1/3}\), and the cutoff profile is given by the GOE Tracy-Widom distribution function. This confirms a conjecture by Bufetov and Nejjar. Our approach is different from Labbé-Lacoin, by comparing the card shuffling with the multi-species ASEP on \(\mathbb{Z}\), and using Hecke algebra and recent ASEP shift-invariance and convergence results. Our result can also be viewed as a generalization of the Oriented Swap Process finishing time convergence of Bufetov-Gorin-Romik, which is the TASEP version (of our result).


    Shift-Invariance of the Colored TASEP and Finishing Times of the Oriented Swap Process
    Adv. Math., Volume 415, 15 February 2023.
    arXiv Journal Slides Abstract
    We prove a new shift-invariance property of the colored TASEP. From the shift-invariance of the colored six-vertex model (proved in Borodin-Gorin-Wheeler or Galashin), one can get a shift-invariance property of the colored TASEP at one time, and our result generalizes this to multiple times. Our proof takes the one time shift-invariance as an input, and uses analyticity of the probability functions and induction arguments. We apply our shift-invariance to prove a distributional identity between the finishing times of the oriented swap process and the point-to-line passage times in exponential last-passage percolation, which is conjectured by Bisi-Cunden-Gibbons-Romik and Bufetov-Gorin-Romik, and is also equivalent to a purely combinatorial identity related to the Edelman-Greene correspondence. With known results from last-passage percolation, we also get new asymptotic results on the colored TASEP and the finishing times of the oriented swap process.


    Convergence of the environment seen from geodesics in exponential last passage percolation
    (with James B. Martin and Allan Sly)
    J. Eur. Math. Soc. accepted.
    arXiv Slides online talk talk by James Martin Abstract
    A well-known question in the planar first-passage percolation model concerns the convergence of the empirical distribution along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on \(\mathbb{Z}^2\) with i.i.d. exponential weights, and provide explicit formulae for the limiting distributions, which depend on the asymptotic direction. For example, for geodesics in the direction of the diagonal, the limiting weight distribution has density \((1/4+x/2+x^2/8)e^{-x}\), and so is a mixture of Gamma(1,1), Gamma(2,1) and Gamma(3,1) distributions with weights 1/4, 1/2, and 1/4 respectively. More generally, we study the local environment as seen from vertices along the geodesics (including information about the shape of the path and about the weights on and off the path in a local neighborhood). We consider finite geodesics from (0,0) to \(n\boldsymbol{\rho}\) for some vector \(\boldsymbol{\rho}\) in the first quadrant, in the limit as \(n\to\infty\), as well as the semi-infinite geodesic in direction \(\boldsymbol{\rho}\). We show almost sure convergence of the empirical distributions along the geodesic, as well as convergence of the distribution around a typical point, and we give an explicit description of the limiting distribution.
    We make extensive use of a correspondence with TASEP as seen from a single second-class particle for which we prove new results concerning ergodicity and convergence to equilibrium. Our analysis relies on geometric arguments involving estimates for the last-passage time, available from the integrable probability literature.


    Disjoint optimizers and the directed landscape
    (with Duncan Dauvergne)
    Mem. Amer. Math. Soc., Vol. 303. No. 1524. (November 19, 2024).
    arXiv Journal related talk by Duncan Dauvergne (second half) Abstract
    We study maximal length collections of disjoint paths, or `disjoint optimizers', in the directed landscape. We show that disjoint optimizers always exist, and that their lengths can be used to construct an extended directed landscape. The extended directed landscape can be built from an independent collection of extended Airy sheets, which we define via last passage percolation across the Airy line ensemble. We show that the extended directed landscape and disjoint optimizers are scaling limits of the corresponding objects in Brownian last passage percolation. As two consequences of this work, we show that one direction of the Robinson-Schensted-Knuth bijection passes to the KPZ limit, and we find a criterion for geodesic disjointness in the directed landscape that uses only a single Airy line ensemble.


    Stationary distributions for the Voter Model in \(d\geq 3\) are factors of IID
    (with Allan Sly)
    Ann. Probab. 50 (4), 1589-1609, (July 2022).
    arXiv Journal Slides Abstract
    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 gives explicit constructions of the stationary distributions as factors of IID processes on \(\mathbb{{Z}}^d\).


    Anderson-Bernoulli localization on the 3D lattice and discrete unique continuation principle
    (with Linjun Li)
    Duke Math. J. 171(2): 327-415 (1 February 2022).
    arXiv Journal Slides talk short online talk Abstract
    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 \(\mathbb{{Z}}^{3}\).


    Mean Field Behavior during the Big Bang for Coalescing Random Walk
    (with Jonathan Hermon, Shuangping Li, and Dong Yao)
    Ann. Probab. 50 (5), 1813-1884, (September 2022).
    arXiv Journal Slides by Dong Yao Abstract
    In this paper we consider the coalescing random walk model on general graphs \(G=(V,E)\). We set up a unified framework to study the leading order of decay rate of \(P_t\), the expectation of the fraction of occupied sites at time $t$, particularly for the `Big Bang' regime where \(t \ll t_{coal}:=\mathbb{E}[\inf\{s:\text{There is only one particle at time }s \}]\). Our results show that \(P_t\) satisfies certain `mean field behavior', if the graphs satisfy certain `transience-like' conditions.
    We apply this framework to two families of graphs: (1) graphs given by configuration model with degree \(\ge 3\), and (2) finite and infinite vertex-transitive graphs. In the first case, we show that for \(1 \ll t \ll |V|\), \(P_t\) decays in the order of \(t^{-1}\), and \((tP_t)^{-1}\) is approximately the probability that two particles starting from the root of the corresponding unimodular Galton-Watson tree never collide after one of them leaves the root, which is also roughly \(|V|/(2t_{meet})\), where \(t_{meet}\) is the mean meeting time of two independent walkers. By taking the local weak limit, for the corresponding unimodular Galton-Watson tree we prove convergence of \(tP_t\) as \(t\to\infty\). For the second family of graphs, if we take a growing sequence of finite vertex-transitive graphs \(G_n=(V_n, E_n)\), such that \(t_{meet}=O(|V_n|)\), and the inverse of the spectral gap \(t_{rel}\) is \(o(|V_n|)\), we show for \(t_{rel} \ll t \ll t_{coal}\), \((tP_t)^{-1}\) is approximately the probability that two random walks never meet before time \(t\), and also \(|V|/(2t_{meet})\). In addition, we define a certain natural `uniform transience' condition, and show that in the transitive setup it implies the above for all \(1 \ll t\ll t_{coal}\). Such estimates of \(tP_t\) are also obtained for all infinite transient transitive unimodular graphs, in particular, all transient transitive amenable graphs.


    The Ising model on trees and factor of IID
    (with Danny Nam and Allan Sly)
    Comm. Math. Phys. 389, 1009–1046 (2022).
    arXiv Journal Slides online talk (with Allan Sly) Abstract
    We study the ferromagnetic Ising model on the infinite \(d\)-regular tree under the free boundary condition. This model is known to be a factor of IID in the uniqueness regime, when the inverse temperature \(\beta\ge 0\) satisfies \(\tanh \beta \le (d-1)^{-1} \). However, in the reconstruction regime (\(\tanh \beta > (d-1)^{-\frac{1}{2}}\) ), it is not a factor of IID. We construct a factor of IID for the Ising model beyond the uniqueness regime via a strong solution to an infinite dimensional stochastic differential equation which partially answers a question of Lyons. The solution \(\{X_t(v) \}\) of the SDE is distributed as \(X_t(v) = t\tau_v + B_t(v)\), where \(\{\tau_v \}\) is an Ising sample and \(\{B_t(v) \}\) are independent Brownian motions indexed by the vertices in the tree. Our construction holds whenever \(\tanh \beta \le c(d-1)^{-\frac{1}{2}}\), where \(c>0\) is an absolute constant.


    A phase transition for repeated averages
    (with Sourav Chatterjee, Persi Diaconis, and Allan Sly)
    Ann. Probab. 50(1): 1-17 (January 2022).
    arXiv Journal Slides short online talk Abstract
    Let \(x_1,\ldots,x_n\) be a fixed sequence of real numbers. At each stage, pick two indices \(I\) and \(J\) uniformly at random and replace \(x_I\), \(x_J\) by \((x_I+x_J)/2\), \((x_I+x_J)/2\). Clearly all the coordinates converge to \((x_1+\cdots+x_n)/n\). We determine the rate of convergence, establishing a sharp cutoff transition answering a question of Jean Bourgain.


    Temporal correlation in last passage percolation with flat initial condition via Brownian comparison
    (with Riddhipratim Basu and Shirshendu Ganguly)
    Comm. Math. Phys. 383, 1805–1888 (2021).
    arXiv Journal Slides online talk related talk by Riddhipratim Basu (first half) Abstract
    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 (SIGMA 2016), 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 exponent for the droplet initial condition recently rigorously established in Ferrari and Occelli (2018), Basu and Ganguly (2018), and requires novel arguments. Key ingredients include the understanding of geodesic geometry and recent advances in quantitative comparison of geodesic weight profiles to Brownian motion using the Brownian Gibbs property. The proof methods are expected to be applicable for a wider class of initial data.


    Optimal exponent for coalescence of finite geodesics in exponential last passage percolation
    Electron. Commun. Probab. 25 (2020), paper no. 74, 14 pp.
    arXiv Journal Abstract
    In this note, we study the model of 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. For the coalescing point of these paths, we show that the probability for it being \(kR\) far away from the origin is in the order of \(R^{-2/3}\). This is motivated by a recent work of Basu, Sarkar, and Sly, where the same estimate was obtained for semi-infinite geodesics, and the optimal exponent for the finite case was left open.


    Interlacing adjacent levels of \(\beta\)-Jacobi corners processes
    (with Vadim Gorin)
    Probab. Theory Relat. Fields 172, 915–981 (2018).
    arXiv Journal Slides Abstract
    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
    (with Hong Wang), 2016.
    In: Contemporary Mathematics Volume 792, 2024.
    arXiv eBook Abstract
    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 \(p = 6\).

    Preprints

    Characterization of the directed landscape from the KPZ fixed point
    (with Duncan Dauvergne), 2024.
    arXiv Abstract
    We show that the directed landscape is the unique coupling of the KPZ fixed point from all initial conditions at all times satisfying three natural properties: independent increments, monotonicity, and shift commutativity. Equivalently, we show that the directed landscape is the unique directed metric on \(\mathbb{R}^2\) with independent increments and KPZ fixed point marginals. This gives a framework for proving convergence to the directed landscape given convergence to the KPZ fixed point. We apply this framework to prove convergence to the directed landscape for a range of models, some without exact solvability: asymmetric exclusion processes with potentially non-nearest neighbour interactions, exotic couplings of ASEP, the random walk and Brownian web distance, and directed polymer models. All of our convergence theorems are new except for colored ASEP and the KPZ equation, where we provide alternative proofs.


    A convergence framework for Airy\(_\beta\) line ensemble via pole evolution
    (with Jiaoyang Huang), 2024.
    arXiv Slides Abstract
    The Airy\(_\beta\) line ensemble is an infinite sequence of random curves. It is a natural extension of the Tracy-Widom\(_\beta\) distributions, and is expected to be the universal edge scaling limit of a range of models in random matrix theory and statistical mechanics. In this work, we provide a framework of proving convergence to the Airy\(_\beta\) line ensemble, via a characterization through the pole evolution of meromorphic functions satisfying certain stochastic differential equations. Our framework is then applied to prove the universality of the Airy\(_\beta\) line ensemble as the edge limit of various continuous time processes, including Dyson Brownian motions with general \(\beta\) and potentials, Laguerre processes and Jacobi processes.


    Airy\(_\beta\) line ensemble and its Laplace transform
    (with Vadim Gorin and Jiaming Xu), 2024.
    arXiv Slides Abstract
    The Airy\(_\beta\) line ensemble is a random collection of continuous curves, which should serve as a universal edge scaling limit in problems related to eigenvalues of random matrices and models of 2d statistical mechanics. This line ensemble unifies many existing universal objects including Tracy-Widom distributions, eigenvalues of the Stochastic Airy Operator, Airy\(_2\) process from the KPZ theory. Here \(\beta>0\) is a real parameter governing the strength of the repulsion between the curves.
    We introduce and characterize the Airy\(_\beta\) line ensemble in terms of the Laplace transform, by producing integral formulas for its joint multi-time moments. We prove two asymptotic theorems for each \(\beta>0\): the trajectories of the largest eigenvalues in the Dyson Brownian Motion converge to the Airy\(_\beta\) line ensemble; the extreme particles in the G\(\beta\)E corners process converge to the same limit.
    The proofs are based on the convergence of random walk expansions for the multi-time moments of prelimit objects towards their Brownian counterparts. The expansions are produced through Dunkl differential-difference operators acting on multivariate Bessel generating functions.


    Non-Markovianity of \(2K−B\) and a degeneration
    (with Yang Chu), 2024. In submission.
    arXiv Abstract
    We study the process of 2K−B, where B is a standard one-dimensional Brownian motion and K is its concave majorant. In light of Pitman's 2M−B theorem, it was recently conjectured by Ouaki and Pitman that 2K−B has the law of the BES(5) process. The two processes share properties such as Brownian scaling, time inversion and quadratic variation, and the same one point distribution and infinitesimal generator, among many other evidences; and it remains to prove that 2K−B is Markovian. However, we show that this conjecture is false. To better understand the similarity between these two processes, we study a degeneration of 2K−B. We show it is a mixture of BES(3), and get other properties including multiple points distribution, infinitesimal generator, and path decomposition at future infimum. We also further investigate the Markovian structure and the filtrations of 2K−B,B and K.


    Brownian bridge limit of polymers under upper-tail large deviation
    (with Shirshendu Ganguly and Milind Hegde), 2023. In submission.
    arXiv Slides by Milind Hegde talk by Milind Hegde Abstract
    For models in the KPZ universality class, such as the zero temperature model of planar last passage-percolation (LPP) and the positive temperature model of directed polymers, its upper tail behavior has been a topic of recent interest, with particular focus on the associated path measures (i.e., geodesics or polymers). For Exponential LPP, diffusive fluctuation had been established in Basu-Ganguly. In the directed landscape, the continuum limit of LPP, the limiting Gaussianity at one point, as well as of related finite-dimensional distributions of the KPZ fixed point, were established, using exact formulas in Liu and Wang-Liu. It was further conjectured in these works that the limit of the corresponding geodesic should be a Brownian bridge. We prove it in both zero and positive temperatures; for the latter, neither the one-point limit nor the scale of fluctuations was previously known. Instead of relying on formulas (which are still missing in the positive temperature literature), our arguments are geometric and probabilistic, using the results on the shape of the weight and free energy profiles under the upper tail from Ganguly-Hegde as a starting point. Another key ingredient involves novel coalescence estimates, developed using the recently discovered shift-invariance Borodin-Gorin-Wheeler in these models. Finally, our proof also yields insight into the structure of the polymer measure under the upper tail conditioning, establishing a quenched localization exponent around a random backbone.


    Discrete geodesic local time converges under KPZ scaling
    (with Shirshendu Ganguly), 2022. In submission.
    arXiv Abstract
    The directed landscape constructed in [DOV18] produces a directed, planar, random geometry, and is believed to be the universal scaling limit of two-dimensional first and last passage percolation models in the Kardar-Parisi-Zhang (KPZ) universality class. Geodesics in this random geometry form an important class of random continuous curves exhibiting fluctuation theory quite different from that of Brownian motion. In this vein, counterpart to Brownian local time, BLT (a self-similar measure supported on the set of zeros of Brownian motion), a local time for geodesics, GLT, was recently constructed and used to study fractal properties of the directed landscape in [GZ22]. It is a classical fact and can be proven using the Markovian property of Brownian motion that the uniform discrete measure on the set of zeros of the simple random walk converges to BLT. In this paper, we prove the ``KPZ analog'' of this by showing that the local times for discrete geodesics in pre-limiting integrable last passage percolation models converge to GLT under suitable scaling guided by KPZ exponents.
    In absence of any Markovianity, our arguments rely on the recently proven convergence of geodesics in pre-limiting models to that in the directed landscape [DV21]. However, this input concerns macroscopic properties and is too coarse to capture the microscopic information required for local time analysis. To relate the macroscopic and microscopic behavior, a key ingredient is an a priori smoothness estimate of the local time in the discrete model, relying on geometric ideas such as the coalescence of geodesics as well as their stability under perturbations of boundary data.


    Fractal geometry of the space-time difference profile in the directed landscape via construction of geodesic local times
    (with Shirshendu Ganguly), 2022. In submission.
    arXiv Slides Abstract
    The Directed Landscape, a random directed metric on the plane (where the first and the second coordinates are termed spatial and temporal respectively), was constructed in the breakthrough work of Dauvergne, Ortmann, and Virág, and has since been shown to be the scaling limit of various integrable models of Last Passage percolation, a central member of the Kardar-Parisi-Zhang universality class. It exhibits several scale invariance properties making it a natural source of rich fractal behavior. Such a study was initiated in Basu-Ganguly-Hammond, where the difference profile i.e., the difference of passage times from two fixed points (say \((\pm 1,0)\)), was considered. Owing to geodesic geometry, it turns out that this difference process is almost surely locally constant. The set of non-constancy is connected to disjointness of geodesics and inherits remarkable fractal properties. In particular, it has been established that when only the spatial coordinate is varied, the set of non-constancy of the difference profile has Hausdorff dimension \(1/2\), and bears a rather strong resemblance to the zero set of Brownian motion. The arguments crucially rely on a monotonicity property, which is absent when the temporal structure of the process is probed, necessitating the development of new methods.
    In this paper, we put forth several new ideas, and show that the set of non-constancy of the 2D difference profile and the 1D temporal process (when the spatial coordinate is fixed and the temporal coordinate is varied) have Hausdorff dimensions \(5/3\) and \(2/3\) respectively. A particularly crucial ingredient in our analysis is the novel construction of a local time process for the geodesic akin to Brownian local time, supported on the ''zero set'' of the geodesic. Further, we show that the latter has Hausdorff dimension \(1/3\) in contrast to the zero set of Brownian motion which has dimension \(1/2\).




    Honors and Awards

  • 2024 Bernoulli Society New Researcher Award.
  • 2023 ICCM Best Thesis Award (formerly New World Mathematics Awards) Doctor Thesis Award, Gold Prize.
  • 2022 Miller Fellow, UC Berkeley.
  • 2017 Centennial Fellowship, Princeton University.
  • 2016 The Hartley Rogers Jr. Prize, MIT.
  • 2015 Putnam Fellow, the 75th William Lowell Putnam Math Competition.
  • 2013 Gold Medal, the 54th International Mathematical Olympiad (IMO).


    • Teaching and Grading

  • MAT 104 Calculus II, Spring 2022, Assistant Instructor.
  • MAT 215 Single Variable Analysis with an Introduction to Proofs, Fall 2021, Preceptor.
  • MAT 385 Probability Theory, Spring 2021, Assistant Instructor.
  • MAT 104 Calculus II, Fall 2020, Preceptor.
  • MAT 385 Probability Theory, Spring 2020, Assistant Instructor.
  • MAT 216 Accelerated Honors Analysis I, Fall 2019, Assistant Instructor.
  • MAT 375 Introduction to Graph Theory, Spring 2019, Assistant Instructor.
  • MAT 204 Advanced Linear Algebra with Applications, Fall 2018, Assistant Instructor.


    • I'm contributing to and instructing at the Berkeley Math Circle.

    Invited Seminars/Conferences

  • Tsinghua Sanya International Mathematics Forum (Sanya, China): Jan, 2025.
  • University of Edinburgh (Edinburgh, UK) North British Probability Seminar, Nov 26, 2024.
  • Oberwolfach Research Institute for Mathematics (Oberwolfach, Germany): Mini-Workshop Mixing Times in the Kardar-Parisi-Zhang Universality Class, Nov, 2024.
  • Bernoulli-IMS 11th World Congress in Probability and Statistics (Bochum, Germany), Invited Session, Aug 13, 2024.
  • Chinese Academy of Sciences (Beijing, China) Hua Loo-Keng Lecture Series, Jul, 2024.
  • Institute for Pure & Applied Mathematics (Los Angeles, CA): Geometry, Statistical Mechanics, and Integrability culminating workshop, Jun, 2024.
  • [Online] Lehigh University/U of Minnesota LU-UMN joint probability seminar, Mar 22, 2024.
  • Fields Institute (Toronto, Canada): KPZ meets KPZ workshop, Mar, 2024.
  • Stanford University (Stanford, CA) Probability Seminar, Jan 29, 2024.
  • University of Pennsylvania (Philadelphia, PA) Statistics Seminar, Jan 24, 2024.
  • Banff International Research Station (Alberta, Canada): Randomness and Quasiperiodicity in Mathematical Physics, Jan, 2024.
  • University of Chicago (Chicago, IL) CAM/Statistics Joint Colloquium, Jan 8, 2024.
  • [Online] Brown University APMA Seminar, Dec 18, 2023.
  • UC Berkeley (Berkeley, CA) Math Guest Lecture Series, Dec 15, 2023.
  • Carnegie Mellon University (Pittsburgh, PA) Math Colloquium, Dec 12, 2023.
  • Stanford University (Stanford, CA) Mathematics Colloquium, Nov 30, 2023.
  • Caltech (Pasadena, CA) Mathematics Colloquium, Nov 28, 2023.
  • Stanford University (Stanford, CA) Probability Seminar, Nov 6, 2023.
  • CalTech/UCLA/USC (Los Angeles, CA) LA Probability Forum, Nov 2, 2023.
  • National University of Singapore (Singapore): Workshop on Random Systems, Oct 2023.
  • Simons Center for Geometry and Physics (Stony Brook, NY): The Asymmetric Simple Exclusion Process, Oct, 2023.
  • MIT (Cambridge, MA) Probability Seminar, Sep 18, 2023.
  • MIT (Cambridge, MA) Integrable Probability Working Group, Sep 14, 2023.
  • 43rd Conference on Stochastic Processes and their Applications (Lisbon, Portugal), Invited Session, Jul, 2023.
  • Chinese Academy of Sciences (Beijing, China) Seminar, Jun 26, 2023.
  • Peking University (Beijing, China) Summer School on Probability, lecture series, Jun 12-16, 2023.
  • [Online] The second International Conference for Chinese Young Probability Scholars, Apr 30, 2023.
  • University of Pennsylvania (Philadelphia, PA) Probability Seminar, Apr 12, 2023.
  • Princeton University (Princeton, NJ) Probability Seminar, Apr 6, 2023.
  • Columbia University (New York, NY) Probability and the City Seminar, Mar 31, 2023.
  • Stony Brook University (Stony Brook, NY) Probability and Combinatorics seminar, Mar 27, 2023.
  • University of Washington (Seattle, WA) Probability Seminar, Feb 6, 2023.
  • UC Davis (Davis, CA) Probability Seminar, Jan 25, 2023.
  • UC Berkeley (Berkeley, CA) Harmonic Analysis and Differential Equations Seminar, Oct 25, 2022.
  • UC Berkeley (Berkeley, CA) Probability Seminar, Oct 12, 2022.
  • University of Chicago (Chicago, IL) Calderón-Zygmund Analysis Seminar, May 2, 2022.
  • University of Chicago (Chicago, IL) Probability Seminar, Apr 29, 2022.
  • Princeton University (Princeton, NJ) PACM Graduate Student Seminar, Apr 22, 2022.
  • [Online] Purdue University Probability Seminar, Apr 20, 2022.
  • University of Pennsylvania (Philadelphia, PA) Penn/Temple Probability Seminar, Feb 22, 2022.
  • [Online] Chinese Webinar on Analysis & PDE, Feb 12, 2022.
  • [Online] University of Washington Probability Seminar, Feb 7, 2022.
  • [Online] Stanford University Probability Seminar, Jan 10, 2022.
  • MIT (Cambridge, MA) Probability Seminar, Nov 29, 2021.
  • [Online] University of Utah Stochastics Seminar, Nov 19, 2021.
  • [Online] KTH Random Matrix Seminar, Nov 16, 2021.
  • UW Madison Probability Seminar, Nov 11, 2021.
  • [Online] Columbia/NYU Courant Probability and the City Seminar, Oct 29, 2021.
  • [Online] Columbia/Princeton Probability Day, May 7, 2021.
  • [Online] UC San Diego Group Actions Seminar, May 4, 2021.
  • [Online] UC Berkeley Probability Seminar, Apr 21, 2021.
  • [Online] ETH/Geneva/Cambridge Percolation Today Webinar, Feb 16, 2021.
  • [Online] Junior Integrable Probability Seminar, Dec 17, 2020.
  • [Online] Integrable probability mini-workshop at Online Open Probability School, Jun 12, 2020.
  • [Online] University of Kansas Probability and Statistics Seminar, Apr 29, 2020.
  • Stanford University (Stanford, CA) Applied Math Seminar, Dec 4, 2019.
  • Duke University (Durham, NC) Probability Seminar, Oct 24, 2019.
  • ICTS, Tata Institute of Fundamental Research (Bangalore, India): Universality in random structures: Interfaces, Matrices, Sandpiles, Jan 25, 2019.
  • MIT (Cambridge, MA) Integrable Probability Working Group, Nov 29, 2016.


    • I co-organized the Berkeley Probability Seminar in Spring 2023.

    According to Math Sci Net, my Erdős number is 2 (thanks to a paper with Persi Diaconis).