The Fourier spectrum of critical percolation

Quantitative noise sensitivity and exceptional times for percolation

One goal of this paper is to prove that dynamical critical site percolation on the planar triangular lattice has exceptional times at which percolation occurs. In doing so, new quantitative noise sensitivity results for percolation are obtained. The latter is based on a novel method for controlling the "level k" Fourier coefficients via the construction of a randomized algorithm which looks at random bits, outputs the value of a particular function but looks at any fixed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff dimension of the set of percolating t…

On the spectrum of lamplighter groups and percolation clusters

Let $G$ be a finitely generated group and $X$ its Cayley graph with respect to a finite, symmetric generating set $S$. Furthermore, let $H$ be a finite group and $H \wr G$ the lamplighter group (wreath product) over $G$ with group of "lamps" $H$. We show that the spectral measure (Plancherel measure) of any symmetric "switch--walk--switch" random walk on $H \wr G$ coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on the cluster of the group identity for Bernoulli site percolation on $X$ with parameter $p = 1/|H|$. The return …

第一通过渗透中的高阶导数

Higher-order derivatives of first-passage percolation with respect to the environment

研究第一通过渗透中边权为i.i.d.随机变量的导数，推导通过时间方差表达式并分析其结构与性质。

零温度 Edwards-Anderson 模型的谱性质

Spectral properties of the zero temperature Edwards-Anderson model

本文研究零温度 Edwards-Anderson 模型的谱性质，通过引入依赖“势垒”构造的逾渗型论证，获得了 Fourier 谱样本大小的新概率下界。

傅里叶谱与和集类型问题

The Fourier spectrum and sumset type problems

引入并研究连续参数化的傅里叶谱，建立基本理论并通过和集类型问题展示其应用。

被动标量湍流的Obukhov-Corrsin谱通过异常正则化

The Obukhov--Corrsin spectrum of passive scalar turbulence through anomalous regularization

该研究通过异常正则化证明了被动标量湍流的Obukhov-Corrsin谱在特定条件下的成立，利用了傅里叶空间的能量等式和加权晶格Poincaré不等式。

On the Distribution of the Fourier Spectrum of Halfspaces

Bourgain showed that any noise stable Boolean function $f$ can be well-approximated by a junta. In this note we give an exponential sharpening of the parameters of Bourgain's result under the additional assumption that $f$ is a halfspace.

平面渗流中谱集与关键集的熵与奇异性

Entropy and singularity of spectral and pivotal sets in planar percolation

本文证明平面临界渗流中谱集与关键集的香农熵与总影响量可比，并揭示二者渐近互异，重叠衰减指数为11/12，最小对称差距离与总影响量可比。

A generalised inductive approach to the lace expansion

The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den Hollander and the authors. We give conditions under which the solution to the recursion relation behaves as a Gaussian, both in Fourier space and in terms of a local central limit theorem. These conditions are shown elsewhere to hold for sufficiently spread-out models of networks of self-avoiding walks in dimensions $d>4$, and for critical oriented percol…

返回时间定理、自相关性与具有空傅里叶-博尔谱的序列

The Return Times Theorem, Auto-Correlation and Sequences with an Empty Fourier-Bohr Spectrum

本文基于Bourgain等人返回时间定理的证明，探讨了自相关性的作用，并通过自相关性对满足点wise定理的序列进行更内在的刻画，同时关联到满足均值定理的序列，这些序列恰好是具有空傅里叶-博尔谱的序列。

An interlacing technique for spectra of random walks and its application to finite percolation clusters

A comparison technique for finite random walks on finite graphs is introduced, using the well-known interlacing method. It yields improved return probability bounds. A key feature is the incorporation of parts of the spectrum of the transition matrix other than just the principal eigenvalue. As an application, an upper bound of the expected return probability of a random walk with symmetric transition probabilities is found. In this case, the state space is a random partial graph of a regular graph of bounded geometry and transitive automorphism group. The law of the random edge-set is assumed…

降至渗流阈值的Bethe-Hessian矩阵

The Bethe-Hessian down to the Percolation Threshold

该研究将Bethe-Hessian矩阵负特征值数量与植入模型特征值对应关系的结论，从d≥2推广至最优条件d>1，通过2-核测试向量构造与马尔可夫随机场各向同性基的方法实现稀疏情形下的波动控制。