A Note on Wetting Transition for Gradient Fields

Wetting of gradient fields: pathwise estimates

We consider the wetting transition in the framework of an effective interface model of gradient type, in dimension 2 and higher. We prove pathwise estimates showing that the interface is localized in the whole thermodynamically-defined partial wetting regime considered in earlier works. Moreover, we study how the interface delocalizes as the wetting transition is approached. Our main tool is reflection positivity in the form of the chessboard estimate.

Note on Bolthausen-Deuschel-Zeitouni's paper on the absence of a wetting transition for a pinned harmonic crystal in dimensions three and larger

The article [Bolthausen et al., 2000] provides a proof of the absence of a wetting transition for the discrete Gaussian free field conditioned to stay positive, and undergoing a weak delta-pinning at height 0. The proof is generalized to the case of a square pinning-potential replacing the delta-pinning, but it relies on a lower bound on the probability for the field to stay above the support of the potential, the proof of which appears to be incorrect. We provide a modified proof of the absence of a wetting transition in the square-potential case, which does not require the aforementioned low…

On the gradient dynamics associated with wetting models

We consider several critical wetting models. In the discrete case, these probability laws are known to converge, after an appropriate rescaling, to the law of a reflecting Brownian motion, or of the modulus of a Brownian bridge, according to the boundary conditions. In the continuous case, a corresponding convergence result is proven in this paper, which allows to approximate the law of a reflecting Brownian motion by the law of Brownian meander tilted by its local time near the origin. On the other hand, these laws can be seen as the reversible probability measure of some Markov processes, na…

(2+1)维 Solid-On-Solid 模型中的临界润湿

Critical wetting in the (2+1)D Solid-On-Solid model

本文研究了低温(2+1)D SOS界面在临界钉扎势下的行为，证明了在临界点λ=λ_w时表面发生离域且刚性高度变为⌊(1/6β)log n+1/3⌋，从而证实了Lacoin的猜想。

Lecture Notes on Gradient Flows and Optimal Transport

We present a short overview on the strongest variational formulation for gradient flows of geodesically $λ$-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009.

A note on diffusion limits for stochastic gradient descent

In the machine learning literature stochastic gradient descent has recently been widely discussed for its purported implicit regularization properties. Much of the theory, that attempts to clarify the role of noise in stochastic gradient algorithms, has widely approximated stochastic gradient descent by a stochastic differential equation with Gaussian noise. We provide a novel rigorous theoretical justification for this practice that showcases how the Gaussianity of the noise arises naturally.