Repulsion of an evolving surface on walls with random heights

Dynamic entropic repulsion for interacting interfaces

The dynamic entropic repulsion for the Ginzburg-Landau $\nablaϕ$ interface model was discussed in [Deuschel-N. 2007] and the asymptotics of the height of the interface was identified. This paper studies a similar problem for two interfaces on the wall which are interacting with one another by the exclusion rule. Each leading order of the asymptotics of height is $\sqrt{\log t}$ as $t\to\infty$ for the system on $\integer^d,\,d\ge3$, $\log t$ for the system on $\integer^2$. The coefficient of the leading term for each interface is also identified.

Entropic repulsion in $|\nabla ϕ|^p$ surfaces: a large deviation bound for all $p\geq 1$

We consider the $(2+1)$-dimensional generalized solid-on-solid (SOS) model, that is the random discrete surface with a gradient potential of the form $|\nablaϕ|^{p}$, where $p\in [1,+\infty]$. We show that at low temperature, for a square region $Λ$ with side $L$, both under the infinite volume measure and under the measure with zero boundary conditions around $Λ$, the probability that the surface is nonnegative in $Λ$ behaves like $\exp(-4βτ_{p,β} L H_p(L) )$, where $β$ is the inverse temperature, $τ_{p,β}$ is the surface tension at zero tilt, or step free energy, and $H_p(L)$ is the entropic…

Dynamics of $(2+1)$-dimensional SOS surfaces above a wall: Slow mixing induced by entropic repulsion

We study the Glauber dynamics for the $(2+1)\mathrm{D}$ Solid-On-Solid model above a hard wall and below a far away ceiling, on an $L\times L$ box of $\mathbb{Z}^2$ with zero boundary conditions, at large inverse-temperature $β$. It was shown by Bricmont, El Mellouki and Fröhlich [J. Stat. Phys. 42 (1986) 743-798] that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height $H\asymp(1/β)\log L$. As an essential step in understanding the effect of entropic repulsion on the Glauber dynamics we determine the equilibrium height $H$ to within an additi…

被约束保持在地板上方的三维 Ising 界面的熵排斥

Entropic repulsion of 3D Ising interfaces conditioned to stay above a floor

本文研究三维 Ising 界面在硬地板约束下的熵排斥，确定临界地板高度 h_n^*=(c_*+o(1))log n，并验证其与 SOS 近似的标度预测一致。

The serial harness interacting with a wall

The serial harnesses introduced by Hammersley describe the motion of a hypersurface of dimension d embedded in a space of dimension $d+1$. The height assigned to each site i of Z^d is updated by taking a weighted average of the heights of some of the neighbors of i plus a ``noise'' (a centered random variable). The surface interacts by exclusion with a ``wall'' located at level zero: the updated heights are not allowed to go below zero. We show that for any distribution of the noise variables and in all dimensions, the surface delocalizes. This phenomenon is related to the so called ``entropic…

Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs

Consider the free field on a fractal graph based on a high-dimensional Sierpinski carpet (e.g. the Menger sponge), that is, a centered Gaussian field whose covariance is the Green's function for simple random walk on the graph. Moreover assume that a "hard wall" is placed at height zero so that the field stays positive everywhere. We prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph, thereby extending a result of Bolthausen, Deuschel, and Zeitouni for the free field on $\mathbb{Z}^d$, $d \geq 3$, to the …