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 Painlevé II which Amir et al. [Commun. Pure Appl. Math. 64, 466 (2011)CPMAMV0010-364010.1002/cpa.20347] related to the KPZ one-point distribution.

We study a stochastic PDE limit of the height function of the dynamic asymmetric simple exclusion process (dynamic ASEP). A degeneration of the stochastic Interaction Round-a-Face (IRF) model of [Bor17], dynamic ASEP has a jump parameter q ∈ (0, 1) and a dynamical parameter α > 0. It degenerates to the standard ASEP height function when α goes to 0 or ∞. We consider very weakly asymmetric scaling, i.e., for ε tending to zero we set q = e −ε and look at fluctuations, space and time in the scales ε −1 , ε −2 and ε −4 . We show that under such scaling the height function of the dynamic ASEP converges to the solution of the space-time Ornstein-Uhlenbeck (OU) process. We also introduce the dynamic ASEP on a ring with generalized rate functions. Under the very weakly asymmetric scaling, we show that the dynamic ASEP (with generalized jump rates) on a ring also converges to the solution of the space-time OU process on [0, 1] with periodic boundary conditions.

We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite (Z + -indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on 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 the Gaussian field corresponding to the solution of the Additive Stochastic Heat Equation (ashe) on R + with Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a 1 4 -Fractional Brownian Motion ( 1 4 -fbm). In particular, we prove a conjecture of Pal and Pitman [17] about the asymptotic Gaussian fluctuation of the ranked particles.Date: November 9, 2018. 2010 Mathematics Subject Classification. Primary 60K35; Secondary 60H15, 82C22.

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