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.

In this paper we propose and study a class of simple, nonparametric, yet interpretable measures of association between two random variables X and Y taking values in general topological spaces. These nonparametric measures -defined using the theory of reproducing kernel Hilbert spaces -capture the strength of dependence between X and Y and have the property that they are 0 if and only if the variables are independent and 1 if and only if one variable is a measurable function of the other. Further, these population measures can be consistently estimated using the general framework of geometric graphs which include k-nearest neighbor graphs and minimum spanning trees. Moreover, a sub-class of these estimators are also shown to adapt to the intrinsic dimensionality of the underlying distribution. Some of these empirical measures can also be computed in near linear time. Under the hypothesis of independence between X and Y , these empirical measures (properly normalized) have a standard normal limiting distribution. Thus, these measures can also be readily used to test the hypothesis of mutual independence between X and Y . In fact, as far as we are aware, these are the only procedures that possess all the above mentioned desirable properties. Furthermore, when restricting to Euclidean spaces, we can make these sample measures of association finite-sample distribution-free, under the hypothesis of independence, by using multivariate ranks defined via the theory of optimal transport. The correlation coefficient proposed in Dette et al. [31], Chatterjee [22] and Azadkia and Chatterjee [7] can be seen as a special case of this general class of measures.

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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.

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