Non-Gaussian Surface Pinned by a Weak Potential

Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

We investigate level-set percolation of the discrete Gaussian free field on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the level-set of the Gaussian free field below a level $α$ disconnects the discrete blow-up of a compact set $A$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the Gaussian free field deviate from a specific multiple of the harmonic potential of $A$, when disconnection occurs. These bounds, combined with the findings of the recent article [12],…

Disorder and wetting transition: the pinned harmonic crystal in dimension three or larger

We consider the Lattice Gaussian free field in $d+1$ dimensions, $d=3$ or larger, on a large box (linear size $N$) with boundary conditions zero. On this field two potentials are acting: one, that models the presence of a wall, penalizes the field when it enters the lower half space and one, the «pinning potential» , that rewards visits to the proximity of the wall. The wall can be soft, i.e. the field has a finite penalty to enter the lower half plane, or hard when the penalty is infinite. In general the pinning potential is disordered and it gives on average a reward h in $\mathbb{R}$ (a neg…

Weak universality of dynamical $Φ^4_3$: non-Gaussian noise

We consider a class of continuous phase coexistence models in three spatial dimensions. The fluctuations are driven by symmetric stationary random fields with sufficient integrability and mixing conditions, but not necessarily Gaussian. We show that, in the weakly nonlinear regime, if the external potential is a symmetric polynomial and a certain average of it exhibits pitchfork bifurcation, then these models all rescale to $Φ^4_3$ near their critical point.

Continuous interfaces with disorder: Even strong pinning is too weak in 2 dimensions

We consider statistical mechanics models of continuous height effective interfaces in the presence of a delta-pinning at height zero. There is a detailed mathematical understanding of the depinning transition in 2 dimensions without disorder. Then the variance of the interface height w.r.t. the Gibbs measure stays bounded uniformly in the volume for any positive pinning force and diverges like the logarithm of the pinning force when it tends to zero. How does the presence of a quenched disorder term in the Hamiltonian modify this transition? We show that an arbitarily weak random field term is…

Hölder continuity and bounds for fundamental solutions to non-divergence form parabolic equations

We consider the non-degenerate second-order parabolic partial differential equations of non-divergence form with bounded measurable coefficients (not necessary continuous). Under some assumptions it is known that the fundamental solution to the equations exists uniquely, has the Gaussian bounds and is locally Hölder continuous. In the present paper, we concern the Gaussian bounds and the lower bound of the index of the Hölder continuity with respect to the initial point. We use the pinned diffusion processes for the probabilistic representation of the fundamental solutions and apply the coupli…

Return probability and recurrence for the random walk driven by two-dimensional Gaussian free field

Given any $γ>0$ and for $η=\{η_v\}_{v\in \mathbb Z^2}$ denoting a sample of the two-dimensional discrete Gaussian free field on $\mathbb Z^2$ pinned at the origin, we consider the random walk on~$\mathbb Z^2$ among random conductances where the conductance of edge $(u, v)$ is given by $\mathrm{e}^{γ(η_u + η_v)}$. We show that, for almost every~$η$, this random walk is recurrent and that, with probability tending to~1 as $T\to \infty$, the return probability at time~$2T$ decays as $T^{-1+o(1)}$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from…

电缆系统上高斯自由场的渗流：反例

Percolation for the Gaussian free field on the cable system: counterexamples

该研究针对一般瞬态图上高斯自由场渗流的临界参数条件，构造反例证明现有两个弱条件非必要，揭示有质量图的典型临界值特征并建立相关场与随机交错的同构关系。

Weak convergence towards two independent Gaussian processes from a unique Poisson process

We consider two independent Gaussian processes that admit a representation in terms of a stochastic integral of a deterministic kernel with respect to a standard Wiener process. In this paper we construct two families of processes, from a unique Poisson process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As an application of this result we obtain families of processes that converge in law towards fractional Brownian motion and sub-fractional Brownian motion.

Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes

For a Gaussian process $X$ and smooth function $f$, we consider a Stratonovich integral of $f(X)$, defined as the weak limit, if it exists, of a sequence of Riemann sums. We give covariance conditions on $X$ such that the sequence converges in law. This gives a change-of-variable formula in law with a correction term which is an Itô integral of $f"'$ with respect to a Gaussian martingale independent of $X$. The proof uses Malliavin calculus and a central limit theorem from [10]. This formula was known for fBm with $H=1/6$ [9]. We extend this to a larger class of Gaussian processes.

Gaussian过程的弱近似：从再生成过程出发

Weak approximation for Gaussian processes from renewal processes

本文研究了从再生成过程构建高斯过程近似的方法，证明了某些进程可用于近似分数布朗运动等，并提供了确保收敛的充分条件。

Limit Laws for Maxima of Contracted Stationary Gaussian Sequences

The principal results of this contribution are the weak and strong limits of maxima of contracted stationary Gaussian random sequences. Due to the random contraction we introduce a modified Berman condition which is sufficient for the weak convergence of the maxima of the scaled sample. Under a stronger assumption the weak convergence is strengthened to almost convergence.

关于两用户弱高斯干扰信道中信令传输的高斯码本最优性

On the Optimality of Gaussian Code-books for Signaling over a Two-Users Weak Gaussian Interference Channel

本文证明两用户弱高斯干扰信道的容量区域可通过单字母高斯码本达到，并利用至多两个时隙的时分复用实现容量区域的上凹包络，且Han-Kobayashi可达速率区域采用单字母高斯码本即可达到最优边界。

粗略与高斯均值回归随机波动模型的弱误差近似

Weak error approximation for rough and Gaussian mean-reverting stochastic volatility models

本文研究了粗略与高斯均值回归随机波动模型的弱误差近似，通过积分核欧拉方案得到离散化粗略奥本海姆-乌伦贝克过程的弱收敛速率，并在多项式测试函数下获得相同收敛速率。