Li-Cheng Tsai

Bio: Li-Cheng Tsai is an academic researcher from Rutgers University. The author has contributed to research in topics: Mathematics & Rate function. The author has an hindex of 14, co-authored 40 publications receiving 625 citations. Previous affiliations of Li-Cheng Tsai include Columbia University & Stanford University.

Weakly Asymmetric Non-Simple Exclusion Process and the Kardar–Parisi–Zhang Equation

Abstract: We analyze a class of non-simple exclusion processes and the corresponding growth models by generalizing the discrete Cole–Hopf transformation of Gartner (Stoch Process Appl, 27:233–260, 1987). We identify the main non-linearity and eliminate it by imposing a gradient type condition. For hopping range at most 3, using the generalized transformation, we prove the convergence of the exclusion process toward the Kardar–Parisi–Zhang (kpz) equation. This is the first universality result under the weak asymmetry concerning interacting particle systems. While this class of exclusion processes are not explicitly solvable, under the weak asymmetry we obtain the exact one-point limiting distribution for the step initial condition by using the previous result of Amir et al. (Commun Pure Appl Math, 64(4): 466–537, 2011) and our convergence result.

KPZ equation limit of higher-spin exclusion processes

Abstract: We prove that under a particular weak scaling, the 4-parameter interacting particle system introduced by Corwin and Petrov [Comm. Math. Phys. 343 (2016) 651–700] converges to the Kardar–Parisi–Zhang (KPZ) equation. This expands the relatively small number of systems for which weak universality of the KPZ equation has been demonstrated.

Coulomb-Gas Electrostatics Controls Large Fluctuations of the Kardar-Parisi-Zhang Equation.

Abstract: We establish a large deviation principle for the Kardar-Parisi-Zhang (KPZ) equation, providing precise control over the left tail of the height distribution for narrow wedge initial condition. Our analysis exploits an exact connection between the KPZ one-point distribution and the Airy point process-an infinite particle Coulomb gas that arises at the spectral edge in random matrix theory. We develop the large deviation principle for the Airy point process and use it to compute, in a straightforward and assumption-free manner, the KPZ large deviation rate function in terms of an electrostatic problem (whose solution we evaluate). This method also applies to the half-space KPZ equation, showing that its rate function is half of the full-space rate function. In addition to these long-time estimates, we provide rigorous proof of finite-time tail bounds on the KPZ distribution, which demonstrate a crossover between exponential decay with exponent 3 (in the shallow left tail) to exponent 5/2 (in the deep left tail). The full-space KPZ rate function agrees with the one computed in Sasorov et al. [J. Stat. Mech. (2017) 063203JSMTC61742-546810.1088/1742-5468/aa73f8] via a WKB approximation analysis of a nonlocal, nonlinear integrodifferential equation generalizing Painleve II which Amir et al. [Commun. Pure Appl. Math. 64, 466 (2011)CPMAMV0010-364010.1002/cpa.20347] related to the KPZ one-point distribution.

ASEP(q,j) converges to the KPZ equation

Abstract: We show that a generalized Asymmetric Exclusion Process called ASEP(q,j) introduced by Carinci, Giardina, Redig and Sasamoto converges to the Cole-Hopf solution to the KPZ equation under weak asymmetry scaling.

Exact lower tail large deviations of the KPZ equation

Abstract: Consider the Hopf--Cole solution $ h(t,x) $ of the KPZ equation with narrow wedge initial condition. Regarding $ t\to\infty $ as a scaling parameter, we provide the first rigorous proof of the Large Deviation Principle (LDP) for the lower tail of $ h(2t,0)+\frac{t}{12} $, with speed $ t^2 $ and an explicit rate function $ \Phi_-(z) $. This result confirms existing physic predictions [Sasorov, Meerson, Prolhac 17], [Corwin, Ghosal, Krajenbrink, Le Doussal, Tsai 18], and [Krajenbrink, Le Doussal, Prolhac 18]. Our analysis utilizes the formula from [Borodin, Gorin 16] to convert LDP of the KPZ equation to calculating an exponential moment of the Airy point process. To estimate this exponential moment, we invoke the stochastic Airy operator, and use the Riccati transform, comparison techniques, and certain variational characterizations of the relevant functional.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Quantum Inverse Scattering Method and Correlation Functions

Abstract: One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Shape Fluctuations and Random Matrices

Abstract: We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

The KPZ fixed point

Abstract: An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process (TASEP) with arbitrary right-finite initial condition. The method is by solving the biorthogonal ensemble/non-intersecting path representation found by [Sas05; BFPS07]. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data. In the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula, in terms of analogous kernels based on Brownian motion, for the transition probabilities of the scaling invariant Markov process at the centre of the KPZ universality class. The formula readily reproduces known special self-similar solutions such as the Airy$_1$ and Airy$_2$ processes. The process takes values in real valued functions which look locally like Brownian motion, and is Holder $1/3-$ in time. Both the KPZ fixed point and TASEP are shown to be stochastic integrable systems in the sense that the time evolution of their transition probabilities can be linearized through a new Brownian scattering transform and its discrete analogue.