arXiv subjects
Publications and source records attributed to Li-Cheng Tsai.
We study the stochastic heat flow with constant initial data and analyze its spatial average on the scale of $\varepsilon\ll1$. We prove that the logarithm of the averaged process satisfies a pointwise central limit theorem: After being centered by $-\tfrac{1}{2}\log\log \varepsilon^{-1}$ and scaled down by $\sqrt{\log\log \varepsilon^{-1}}$, it converges in distribution to a standard Gaussian.
We establish that the family of polymer measures $M^{\theta}_{[s,t]}$ associated with the Stochastic Heat Flow (SHF), indexed by $\theta\in\mathbb{R}$, has a conditional Gaussian Multiplicative Chaos (GMC) structure. Namely, taking the random measure $M^{\theta}_{[s,t]}$ as the reference measure, we construct the path-space GMC with noise strength $a > 0$ and prove that the resulting random measure is equal in law to $M^{\theta+a}_{[s,t]}$. As two applications, we prove that the polymer measure and SHF tested against general nonnegative functions are almost surely strictly positive and that the SHF converges to $0$ as $\theta\to\infty$.
We prove that the stochastic heat flow Caravenna Sun Zygouras (2023), Tsai (2024) is a black noise in the sense of Tsirelson (2004). As a corollary, the 2d stochastic heat equation driven by a mollified spacetime white noise becomes asymptotically independent of that noise in the scaling limit.
We extend the construction of the semigroup of the two-dimensional delta-Bose gas in Gu, Quastel, and Tsai (2021) (based on Rajeev (1999) and Dimock and Rajeev (2004)) to a weighted $L^2$ space that allows exponentially growing functions. We further show that the semigroup of the mollified delta-Bose gas converges strongly to that of the delta-Bose gas.
The Stochastic Heat Flow (SHF) emerges as the scaling limit of directed polymers in random environments and the noise-mollified Stochastic Heat Equation (SHE), specifically at the critical dimension of two and near the critical temperature. The prior work Caravenna Sun Zygouras (2023) established the first construction of finite-dimensional distributions by demonstrating the universal (model-independent) convergence of discrete polymers. In this work, we present a new, independent approach to the SHF. We formulate the SHF as a continuous process and provide a set of axioms for its characterization. We establish both the uniqueness and existence of this process under our new formulation, with a key feature of these axioms being the matching of the first four moments.
We identify the explicit rate function for the upper-tail LDP of the parabolic Airy process and characterize the minimizing metrics, or equivalently the limit shape of the directed landscape, under the upper-tail conditioning. The LDP result answers Conjecture 10.1 in Das, Dauvergne, and Vir\'{a}g (2024, DDV), and the limit shape result gives the first proof of the limit shape in similar contexts beyond the case where the minimizer is piecewise linear. The starting point of our proof is the metric-level LDP for the directed landscape from DDV that reduces our work to solving a variational problem. Our proof demonstrates how the variational problem fits into the Jensen-Varadhan theory of hydrodynamic large deviations Jensen (2000), Varadhan (2004). The proof is PDE-based and uses geometric arguments, connecting the variational problem to weak solutions of Burgers' equation, and may be generalized to those models in the KPZ universality class that enjoy similar metric-level LDPs proven in DDV.
We consider the $n$-point, fixed-time large deviations of the KPZ equation with the narrow wedge initial condition. The scope consists of concave-configured, upper-tail deviations and a wide range of scaling regimes that allows time to be short, unit-order, and long. We prove the $n$-point large deviation principle and characterize, with proof, the corresponding spacetime limit shape. Our proof is based on the results -- from the companion paper Tsai (2023) -- on moments of the stochastic heat equation and utilizes ideas coming from a tree decomposition. Behind our proof lies the phenomenon where the major contribution of the noise concentrates around certain corridors in spacetime, and we explicitly describe the corridors.
We analyze the high moments of the Stochastic Heat Equation (SHE) via a transformation to the attractive Brownian Particles (BPs), which are Brownian motions interacting via pairwise attractive drift. In those scaling regimes where the particles tend to cluster, we prove a Large Deviation Principle (LDP) for the empirical measure of the attractive BPs. Under what we call the 1-to-$n$ initial-terminal condition, we characterize the unique minimizer of the rate function and relate the minimizer to the spacetime limit shapes of the Kardar--Parisi--Zhang (KPZ) equation in the upper tails. The results of this paper are used in the companion paper Lin and Tsai (2023) to prove an $n$-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape.
We consider the variational problem associated with the Freidlin--Wentzell Large Deviation Principle of the Stochastic Heat Equation (SHE). The logarithm of the minimizer of the variational problem gives the most probable shape of the solution of the Kardar--Parisi--Zhang equation conditioned on achieving certain unlikely values. Taking the SHE with the delta initial condition and conditioning the value of its solution at the origin at a later time, under suitable scaling, we prove that the logarithm of the minimizer converges to an explicit function as we tune the value of the conditioning to $ 0 $. Our result confirms the physics prediction Kolokolov and Korshunov (2009), Meerson, Katzav, and Vilenkin (2016), Kamenev, Meerson, and Sasorov (2016).
In this article we review the ideas in Tsai (2018) toward proving the one-point, lower-tail large deviation principle for the Kardar--Parisi--Zhang equation.
We consider the variational problem associated with the Freidlin--Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation (SHE). For a general class of initial-terminal conditions, we show that a minimizer of this variational problem exists, and any minimizer solves a system of imaginary-time Nonlinear Schr\"{o}dinger equations. This system is integrable. Utilizing the integrability, we prove that the formulas from the physics work Krajenbrink and Le Doussal (2021) hold for every minimizer of the variational problem. As an application, we consider the Freidlin--Wentzell LDP for the SHE with the delta initial condition. Under a technical assumption on the poles of the reflection coefficients, we prove the explicit expression for the one-point rate function that was predicted in the physics works Le Doussal, Majumdar, Rosso, and Schehr (2016) and Krajenbrink and Le Doussal (2021). Under the same assumption, we give detailed pointwise estimates of the most probable shape in the upper-tail limit.
In this paper, we consider the KPZ equation under the weak noise scaling. That is, we introduce a small parameter $\sqrt{\varepsilon}$ in front of the noise and let $\varepsilon \to 0$. We prove that the one-point large deviation rate function has a $\frac{3}{2}$ power law in the deep upper tail. Furthermore, by forcing the value of the KPZ equation at a point to be very large, we prove a limit shape of the KPZ equation as $\varepsilon \to 0$. This confirms the physics prediction in Kolokolov and Korshunov (2007), Kolokolov and Korshunov (2009), Meerson, Katzav, and Vilenkin (2016), Kamenev, Meerson, and Sasorov (2016), and Le Doussal, Majumdar, Rosso, and Schehr (2016).
We consider the large deviations from the hydrodynamic limit of the Totally Asymmetric Simple Exclusion Process (TASEP). This problem was studied in Jensen (2000) and Varadhan (2004) and was shown to be related to entropy production in the inviscid Burgers equation. Here we prove the full large deviation principle. Our method relies on the explicit formula of Matetski, Quastel, and Remenik (2016) for the transition probabilities of the TASEP.
We establish the Freidlin--Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation with multiplicative noise in one spatial dimension. That is, we introduce a small parameter $ \sqrt{\varepsilon} $ to the noise, and establish an LDP for the trajectory of the solution. Such a Freidlin--Wentzell LDP gives the short-time, one-point LDP for the KPZ equation in terms of a variational problem. Analyzing this variational problem under the narrow wedge initial data, we prove a quadratic law for the near-center tail and a $ \frac52 $ law for the deep lower tail. These power laws confirm existing physics predictions Kolokolov and Korshunov (2007), Kolokolov and Korshunov (2009), Meerson, Katzav, and Vilenkin (2016), Le Doussal, Majumdar, Rosso, and Schehr (2016), and Kamenev, Meerson, and Sasorov (2016).
Consider the solution $\mathcal{Z}(t,x)$ of the one-dimensional stochastic heat equation, with a multiplicative spacetime white noise, and with the delta initial data $\mathcal{Z}(0,x) = \delta(x)$. For any real $p>0$, we obtained detailed estimates of the $p$-th moment of $e^{t/12}\mathcal{Z}(2t,0)$, as $t\to\infty$, and from these estimates establish the one-point upper-tail large deviation principle of the Kardar-Parisi-Zhang equation. The deviations have speed $t$ and rate function $\Phi_+(y)=\frac{4}{3}y^{3/2}$. Our result confirms the existing physics predictions [Le Doussal, Majumdar, Schehr 16] and also [Kamenev, Meerson, Sasorov 16].
We study the stochastic heat equation in two spatial dimensions with a multiplicative white noise, as the limit of the equation driven by a noise that is mollified in space and white in time. As the mollification radius $ \varepsilon\to 0 $, we tune the coupling constant near the critical point, and show that the single time correlation functions converge to a limit written in terms of an explicit non-trivial semigroup. Our approach consists of two steps. First we show the convergence of the resolvent of the (tuned) two-dimensional delta Bose gas, by adapting the framework of Dimock and Rajeev (2004) to our setup of spatial mollification. Then we match this to the Laplace transform of our semigroup.
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.
In this article we study the stochastic six vertex model under the scaling proposed by Borodin and Gorin (2018), where the weights of corner-shape vertices are tuned to zero, and prove Conjecture 6.1 therein: that the height fluctuation converges in finite dimensional distributions to the solution of stochastic telegraph equation.