受硬壁约束的树上高斯自由场 I：界

Statistics of fermions in a $d$-dimensional box near a hard wall

We study $N$ noninteracting fermions in a domain bounded by a hard wall potential in $d \geq 1$ dimensions. We show that for large $N$, the correlations at the edge of the Fermi gas (near the wall) at zero temperature are described by a universal kernel, different from the universal edge kernel valid for smooth potentials. We compute this $d$ dimensional hard edge kernel exactly for a spherical domain and argue, using a generalized method of images, that it holds close to any sufficiently smooth boundary. As an application we compute the quantum statistics of the position of the fermion closes…

Soft and hard wall in a stochastic reaction diffusion equation

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a "soft" repulsion from the boundary. We finally show how a "hard" repulsion can be obtained by an extra di…

随机正规矩阵：硬壁附近的特征值相关性

Random normal matrices: eigenvalue correlations near a hard wall

本文研究了处于被推离相的平面Coulomb系统在硬壁附近的配对相关函数，给出了相关核在微观和宏观区域的渐近行为，并发现了“半硬边缘”过渡界面。

Enhanced interface repulsion from quenched hard-wall randomness

We consider the harmonic crystal on the d-dimensional lattice, d larger or equal to 3, that is the centered Gaussian field $ϕ$ with covariance given by the Green function of the simple random walk on $Z^d$. Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition the field to be larger than an IID field $σ$ (which is also independent of $ϕ$), for every x in a large region $D_N=ND\cap \Z^d$, with N a positive integer and $D \subset\R^d$. We are mostly motivated by results for given typical realizations of the $σ$ (quenched set-up), since the c…

一维奔跑翻滚粒子的吸收边界条件与硬墙边界条件之间的关联

Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle

本文研究一维奔跑翻滚粒子在任意外力下吸收边界与硬墙边界的对偶性，建立逸出概率与硬墙位置累积分布的关系，并通过稳态显式计算、Fokker-Planck 推导及数值模拟验证。

d≥3维整数格点ℤᵈ上随机环境中离散高斯自由场的硬壁排斥

Hard wall repulsion for the discrete Gaussian free field in random environment on $\mathbb{Z}^d$, $d\geq 3$

本文研究d≥3维随机环境中离散高斯自由场的硬壁事件，建立其大偏差渐近与条件场的分布性质，克服平移不变性缺失问题，证明了熵排斥现象。

Nonintersecting Brownian bridges between reflecting or absorbing walls

We study a model of nonintersecting Brownian bridges on an interval with either absorbing or reflecting walls at the boundaries, focusing on the point in space-time at which the particles meet the wall. These processes are determinantal, and in different scaling limits when the particles approach the reflecting (resp. absorbing) walls we obtain hard-edge limiting kernels which are the even (resp. odd) parts of the Pearcey and tacnode kernels. We also show that in the single time case, our hard-edge tacnode kernels are equivalent to the ones studied by Delvaux [16], defined in terms of a $4\tim…

Harmonically confined long-ranged interacting gas in the presence of a hard wall

In this paper, we compute exactly the average density of a harmonically confined Riesz gas of $N$ particles for large $N$ in the presence of a hard wall. In this Riesz gas, the particles repel each other via a pairwise interaction that behaves as $|x_i - x_j|^{-k}$ for $k>-2$, with $x_i$ denoting the position of the $i^{\rm th}$ particle. This density can be classified into three different regimes of $k$. For $k \geq 1$, where the interactions are effectively short-ranged, the appropriately scaled density has a finite support over $[-l_k(w),w]$ where $w$ is the scaled position of the wall. Whi…