Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups

高维 m-依赖数据的双重归纳 CLT

Dual Induction CLT for High-dimensional m-dependent Data

本文研究高维 m-依赖随机向量和的 CLT 逼近，提出连接反集中与 Berry–Esseen 界的双重归纳方法，在弱矩条件下得到尖锐且近最优的样本复杂度。

Logarithmic Sobolev inequalities for finite spin systems and applications

We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard-core model with fugacity. This leads to two separate branches of applications. The first branch is given by mixing time estimates of the Glauber dynamics. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, this also yields exponential decay of t…

Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding on the average fibers

In this paper we show how to apply classical probabilistic tools for partial sums $\sum_{j=0}^{n-1}φ\circτ^j$ generated by a skew product $τ$, built over a sufficiently well mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $φ$, we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate deviations principle, several exponential concentration inequalities and Rosenthal type moment estimates for skew products with $α, ϕ$ or $ψ$ mixing base maps and expanding on the aver…

随机磁场伊辛模型中的涨落

Fluctuations in random field Ising models

本文在高温区域建立了线性统计量的中心极限定理，并给出了贝叶斯-埃辛界限。研究应用至具有离散和连续自旋的随机磁场伊辛模型，并推导了淬火和退火的CLT，包括Erdős-Rényi等图结构的伊辛模型和Hopfield自旋玻璃模型。

相位边界自旋系统的近似FKG不等式，及其在指数随机图模型中心极限定理中的应用

Approximate FKG inequalities for phase-bound spin systems, with applications to central limit theorems for exponential random graphs

本文研究了相位边界自旋系统中近似的FKG不等式，证明了在相变共存区域中，每个相内部确实满足近似的FKG不等式，并利用此结果完成了各个相内的中心极限定理证明，回答了Bianchi等人提出的问题。

Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup

It has been recently understood (arXiv:1212.2885, arXiv:1310.4764, arXiv:1410.0605) that for a general class of percolation models on $\mathbb{Z}^d$ satisfying suitable decoupling inequalities, which includes i.a.\ Bernoulli percolation, random interlacements and level sets of the Gaussian free field, large scale geometry of the unique infinite cluster in strongly percolative regime is qualitatively the same; in particular, the random walk on the infinite cluster satisfies the quenched invariance principle, Gaussian heat-kernel bounds and local CLT. In this paper we consider the random walk lo…

分支随机游走的Chernoff界

Chernoff bounds for branching random walks

本文提出了一种更一般的分支随机游走框架，并由此推导出BRW的Chernoff型界，同时讨论了随机游走与BRW之间的关系。

Transition asymptotics for reaction-diffusion in random media

We describe a universal transition mechanism characterizing the passage to an annealed behavior and to a regime where the fluctuations about this behavior are Gaussian, for the long time asymptotics of the empirical average of the expected value of the number of random walks which branch and annihilate on ${\mathbb Z}^d$, with stationary random rates. The random walks are independent, continuous time rate $2dκ$, simple, symmetric, with $κ\ge 0$. A random walk at $x\in{\mathbb Z}^d$, binary branches at rate $v_+(x)$, and annihilates at rate $v_-(x)$. The random environment $w$ has coordinates $…