arXiv subjects

Li-Cheng Tsai

Publications and source records attributed to Li-Cheng Tsai.

32 records · Page 2Linked to original sources

arXiv2019

Another look into the Wong-Zakai Theorem for Stochastic Heat Equation

Consider the heat equation driven by a smooth, Gaussian random potential: \begin{align*} \partial_t u_{\varepsilon}=\tfrac12Δu_{\varepsilon}+u_{\varepsilon}(ξ_{\varepsilon}-c_{\varepsilon}), \ \ t>0, x\in\mathbb{R}, \end{align*} where $ξ_{\varepsilon}$ converges to a spacetime white noise, and $c_{\varepsilon} $ is a diverging constant chosen properly. For any $ n\geq 1 $, we prove that $ u_{\varepsilon} $ converges in $ L^n $ to the solution of the stochastic heat equation. Our proof is probabilistic, hence provides another perspective of the general result of Hairer and Pardoux \cite{Hairer15a}, for the special case of the stochastic heat equation. We also discuss the transition from homogenization to stochasticity.

math.PR↗

arXiv2019

Criticality of a randomly-driven front

Consider an advancing `front' $ R(t) \in \mathbb{Z}_{\geq 0} $ and particles performing independent continuous time random walks on $ (R(t),\infty)\cap\mathbb{Z} $. Starting at $R(0)=0$, whenever a particle attempts to jump into $R(t)$ the latter instantaneously moves $k \ge 1$ steps to the right, absorbing all particles along its path. We take $ k $ to be the minimal random integer such that exactly $ k $ particles are absorbed by the move of $ R $, and view the particle system as a discrete version of the Stefan problem \begin{align*} &\partial_t u_*(t,ξ) = \tfrac12 \partial^2_ξ u_*(t,ξ), \quad ξ>r(t), &u_*(t,r(t))=0, &\tfrac{d~}{dt}r(t) = \tfrac12 \partial_ξu_*(t,r(t)), &t\mapsto r(t) \text{ non-decreasing }, \quad r(0):=0. \end{align*} For a constant initial particles density $u_*(0,ξ)=ρ{\bf 1}_{\{ξ>0\}}$, at $ρ<1$ the particle system and the PDE exhibit the same diffusive behavior at large time, whereasat $ρ\ge 1$ the PDE explodes instantaneously. Focusing on the critical density $ ρ=1 $, we analyze the large time behavior of the front $ R(t) $ for the particle system, and obtain both the scaling exponent of $R(t)$ and an explicit description of its random scaling limit. Our result unveils a rarely seen phenomenon where the macroscopic scaling exponent is sensitive to the amount of initial local fluctuations. Further, the scaling limit demonstrates an interesting oscillation between instantaneous super- and sub-critical phases. Our method is based on a novel monotonicity as well as PDE-type estimates.

math.PR↗

arXiv2019

Exceedingly Large Deviations of the Totally Asymmetric Exclusion Process

Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the integer lattice $ \mathbb{Z} $. We study the functional Large Deviations of the integrated current $ \mathsf{h}(t,x) $ under the hyperbolic scaling of space and time by $ N $, i.e., $ \mathsf{h}_N(t,ξ) := \frac{1}{N}\mathsf{h}(Nt,Nξ) $. As hinted by the asymmetry in the upper- and lower-tail large deviations of the exponential Last Passage Percolation, the TASEP exhibits two types of deviations. One type of deviations occur with probability $ \exp(-O(N)) $, referred to as speed-$ N $; while the other with probability $ \exp(-O(N^{2})) $, referred to as speed-$ N^2 $. In this work we study the speed-$ N^2 $ functional LDP of the TASEP, and establishes (non-matching) large deviation upper and lower bounds.

math.PR↗

arXiv2018

KPZ equation limit of higher-spin exclusion processes

We prove that under a particular weak scaling, the 4-parameter interacting particle system introduced by Corwin and Petrov converges to the KPZ equation. This expands the relatively small number of systems for which weak universality of the KPZ equation has been demonstrated.

math.PR↗

arXiv2018

Exact lower tail large deviations of the KPZ equation

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 $ Φ_-(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.

math.PR↗

arXiv2018

Coulomb-gas electrostatics controls large fluctuations of the KPZ equation

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 which 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 straight-forward 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, 063203 (2017)] via a WKB approximation analysis of a non-local, non-linear integro-differential equation generalizing Painlevé II which Amir et al. [Comm. Pure Appl. Math. 64, 466 (2011)] related to the KPZ one-point distribution.

cond-mat.stat-mech↗

arXiv2018

Stochastic Telegraph Equation Limit for the Stochastic Six Vertex Model

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.

math.PR↗

arXiv2018

Stationary Distributions of the Atlas Model

In this article we study the Atlas model, which constitutes of Brownian particles on $ \mathbb{R} $, independent except that the Atlas (i.e., lowest ranked) particle $ X_{(1)}(t) $ receive drift $ γdt $, $ γ\in\mathbb{R} $. For any fixed shape parameter $ a>2γ_- $, we show that, up to a shift $ \frac{a}{2}t $, the entire particle system has an invariant distribution $ ν_a $, written in terms an explicit Radon-Nikodym derivative with respect to the Poisson point process of density $ a e^{aξ} dξ$. We further show that $ ν_a $ indeed has the product-of-exponential gap distribution $ π_a $ derived in Sarantsev and Tsai (2016). As a simple application, we establish a bound on the fluctuation of the Atlas particle $ X_{(1)}(t) $ uniformly in $ t $, with the gaps initiated from $ π_a $ and $ X_{(1)}(0)=0 $.

math.PR↗

arXiv2017

Optimal Surviving Strategy for Drifted Brownian Motions with Absorption

We study the 'Up the River' problem formulated by Aldous (2002), where a unit drift is distributed among a finite collection of Brownian particles on $ \mathbb{R}_+ $, which are annihilated once they reach the origin. Starting $ K $ particles at $ x=1 $, we prove a conjecture of Aldous (2002) that the 'push-the-laggard' strategy of distributing the drift asymptotically (as $ K\to\infty $) maximizes the total number of surviving particles, with approximately $ \frac{4}{\sqrtπ} K^{1/2} $ surviving particles. We further establish the hydrodynamic limit of the particle density, in terms of a two-phase PDE with a moving boundary, by utilizing certain integral identities and coupling techniques.

math.PR↗

arXiv2017

Stationary Gap Distributions for Infinite Systems of Competing Brownian Particles

Consider the infinite Atlas model: a semi-infinite collection of particles driven by independent standard Brownian motions with zero drifts, except for the bottom-ranked particle which receives unit drift. We derive a continuum one-parameter family of product-of-exponentials stationary gap distributions, with exponentially growing density at infinity. This result shows that there are infinitely many stationary gap distributions for the Atlas model, and hence resolves a conjecture of Pal and Pitman (2008) in the negative. This result is further generalized for infinite systems of competing Brownian particles with generic rank-based drifts.

math.PR↗

arXiv2016

ASEP(q,j) converges to the KPZ equation

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.

math.PR↗

arXiv2015

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

We analyze a class of non-simple exclusion processes and the corresponding growth models by generalizing Gaertners Cole-Hopf transformation. 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 concerning interacting particle systems in the context of KPZ universality class. While this class of exclusion processes are not explicitly solvable, we obtain the exact one-point limiting distribution for the step initial condition by using the previous result of Amir et al. (2011) and our convergence result.

math.PR↗

arXiv2015

Infinite Dimensional Stochastic Differential Equations for Dyson's Model

In this paper we show the strong existence and the pathwise uniqueness of an infinite-dimensional Stochastic Differential Equation (SDE) corresponding to the bulk limit of Dyson's Brownian Motion (DBM), for all $β\geq 1$. Our construction applies to an explicit and general class of initial conditions, including the lattice configuration $\{x_i\}=\mathbb{Z}$ and the sine process. We further show the convergence of the finite to infinite-dimensional SDE. This convergence concludes the determinantal formula of Katori and Tanemura (2010) for the solution of this SDE at $β=2$.

math.PR↗

arXiv2015

Equilibrium Fluctuation of the Atlas Model

We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite ($ \mathbb{Z}_+ $-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on $ \mathbb{R}_+ $. In this context, we show that the joint law of ranked particles, after being centered and scaled by $t^{-1/4}$, converges as $t \to \infty$ to the Gaussian field corresponding to the solution of the additive stochastic heat equation on $\mathbb{R}_+$ with Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a $ \frac{1}{4} $-fractional Brownian motion. In particular, we prove a conjecture of Pal and Pitman (2008) about the asymptotic Gaussian fluctuation of the ranked particles.

math.PR↗