超临界零程过程中不变测度的热力学极限

Continuum Percolation for Gaussian zeroes and Ginibre eigenvalues

We study continuum percolation on certain negatively dependent point processes on \R^2. Specifically, we study the Ginibre ensemble and the planar Gaussian zero process, which are the two main natural models of translation invariant point processes on the plane exhibiting local repulsion. For the Ginibre ensemble, we establish the uniqueness of infinite cluster in the supercritical phase. For the Gaussian zero process, we establish that a non-trivial critical radius exists, and we prove the uniqueness of infinite cluster in the supercritical regime.

Local limit theorems for a directed random walk on the backbone of a supercritical oriented percolation cluster

We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions $d+1$ with $d \ge 3$ being the spatial dimension. For this random walk we prove an annealed local central limit theorem and a quenched local limit theorem. The latter shows that the quenched transition probabilities of the random walk converge to the annealed transition probabilities reweighted by a function of the medium centred at the target site. This function is the density of the unique measure which is invariant for the point of view of the particle, is absolutely continuous…

超临界远程渗流在多项式增长图上的研究：截断单臂指数

Supercritical long-range percolation on graphs of polynomial growth: the truncated one-arm exponent

研究超临界远距离渗流在多项式增长图上的行为，推导有限簇半径和体积的边界，并给出无限分量等周维数的下界。

Ergodicity of supercritical SDEs driven by $α$-stable processes and heavy-tailed sampling

Let $α\in(0,2)$ and $d\in\mathbb{N}$. Consider the following stochastic differential equation (SDE) driven by $α$-stable process in $\mathbb{R}^d$: $$ dX_t=b(X_t)dt+σ(X_{t-})d L^α_t, \quad X_0=x\in\mathbb{R}^d, $$ where $b:\mathbb{R}^d\to\mathbb{R}^d$ and $σ:\mathbb{R}^d\to\mathbb{R}^d\otimes\mathbb{R}^d$ are locally $γ$-Hölder continuous with $γ\in(0\vee(1-α)^+,1]$, $L^α_t$ is a $d$-dimensional rotationally invariant $α$-stable process. Under some dissipative and non-degenerate assumptions on $b,σ$, we show the $V$-uniformly exponential ergodicity for the semigroup $P_t$ associated with $(X_t…

具有双稳态热力学极限的随机稳定全局耦合映射

Stochastically stable globally coupled maps with bistable thermodynamic limit

我们研究系统的全球耦合区间映射，其中相同的个人地图有两个扩展，分数线性，到分支，并通过一个参数引入耦合-共同的所有个人地图-这取决于在一个分析的方式上的平均场的系统。我们展示：1）在我们所考虑的耦合参数范围内，有限大小耦合系统总是有唯一的严格正的解析不变概率密度，并且所有的有限大小耦合系统的相关性都是指数衰减的。2)对于相同的参数范围内，自洽Perron-Frobenius算子，捕获相应的无限大系统的基本方面（出现上述的限制时，系统的大小趋于无穷大），经历了一个超临界的干草叉分叉从一个唯一的稳定平衡共存的两个稳定和一个不稳定的平衡。

Thermodynamic formalism and large deviations for multiplication-invariant potentials on lattice spin systems

We introduce the multiplicative Ising model and prove basic properties of its thermodynamic formalism such as existence of pressure and entropies. We generalize to one-dimensional "layer-unique" Gibbs measures for which the same results can be obtained. For more general models associated to a $d$-dimensional multiplicative invariant potential, we prove a large deviation theorem in the uniqueness regime for averages of multiplicative shifts of general local functions. This thermodynamic formalism is motivated by the statistical properties of multiple ergodic averages.

动态环境中的高尔顿 - 沃森过程

Galton-Watson processes in dynamical environments

研究动态环境中高尔顿 - 沃森过程，通过定义模型探讨其三种行为，在一致超临界时研究灭绝概率，临界时研究坏环境集，给出部分情况下坏环境集的豪斯多夫维数。