交错网格上的等周不等式

更简单的等周不等式

Isoperimetric Inequalities Made Simpler

本文提出一种更简单的方法证明超立方体上的等周不等式，给出Talagrand经典结果及Eldan-Gross猜想的初等证明，并加强Friedgut junta定理，将敏感度p-矩条件从p=1推广到p∈(1/2,1]。

Edge Isoperimetric Inequalities for Powers of the Hypercube

For positive integers $n$ and $r$, we let $Q_n^r$ denote the $r$th power of the $n$-dimensional discrete hypercube graph, i.e., the graph with vertex-set $\{0,1\}^n$, where two 0-1 vectors are joined if they are Hamming distance at most $r$ apart. We study edge isoperimetric inequalities for this graph. Harper, Bernstein, Lindsey and Hart proved a best-possible edge isoperimetric inequality for this graph in the case $r=1$. For each $r \geq 2$, we obtain an edge isoperimetric inequality for $Q_n^r$; our inequality is tight up to a constant factor depending only upon $r$. Our techniques also yi…

Edge-isoperimetric inequalities for the symmetric product of graphs

The $k$-th symmetric product of a graph $G$ with vertex set $V$ with edge set $E$ is a graph with vertices as $k$-sets of $V$, where two $k$-sets are connected by an edge if and only if their symmetric difference is an edge in $E$. Using the isoperimetric properties of the vertex-induced subgraphs of $G$ and Sobolev inequalities on graphs, we obtain various edge-isoperimetric inequalities pertaining to the symmetric product of certain families of finite and infinite graphs.

Z^k上一类图的点等周不等式

Vertex Isoperimetric Inequalities for a Family of Graphs on Z^k

本文考虑一类顶点集为Z^k的图族，其中两个顶点之间的l \infty -距离为1时，它们由一条边连通.证明了这类图的最优点等周不等式。也就是说，给定一个正整数n，我们找到一个大小为n的集合A \subset Z^k，使得与A中某个顶点共享一条边的顶点数最小。这些最小边界集是嵌套的，证明使用了压缩技术。我们还显示了一种方法，计算某些子集在这个家庭的图的顶点边界。这种计算和等周不等式允许我们间接地找到使计算边界的函数最小化的集合。

Site percolation and isoperimetric inequalities for plane graphs

We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least $7$ have site percolation threshold bounded away from $1/2$, which was conjectured by Benjamini and Schramm, and make progress on a…

路径强积上的离散等周不等式

Discrete isoperimetric inequalities on the strong products of paths

本文研究两条路径强积图中的离散等周问题，刻画了使非空顶点集的顶点边界大小最小化的集合情形。

高亏格随机三角剖分中的距离与等周不等式

Distances and isoperimetric inequalities in random triangulations of high genus

本研究证明亏格与规模n成正比的均匀随机三角剖分直径高概率下为log n量级、多数点对距离差至多为加性常数，核心工具是具备独立研究价值的等周不等式。

关于精细图与同调等周不等式的注记

A note on fine graphs and homological isoperimetric inequalities

Groves和Manning在相对双曲性的同调刻画的框架下，提出了一个问题：具有线性同调等周不等式的单连通$2$ -复$X$，$2$ -胞腔和多个$2$-胞腔的附连映射的长度的界是否一定具有精细的$1$ -骨架.我们对这个问题提供了肯定的答案。我们重新审视了相对双曲性的同调表征，并表明，一个组$G$是双曲相对于一个集合的子群$\mathcal P$当且仅当$G$的行为cocompanies有限边稳定的连接$2$维胞腔复形的线性同调等周不等式和$\mathcal P$是一个集合的代表共轭类的顶点稳定。

Ramanujan复形与拓扑扩张的等周不等式

Isoperimetric Inequalities for Ramanujan Complexes and Topological Expanders

在过去的四十年里，扩展图得到了广泛的研究。近年来出现了一个膨胀子的高维理论，并研究了几个变体。其中最突出的是上边界扩展和拓扑扩展。已知对每一个$d$都存在维数为$d$的具有这些性质的无界度单纯复形.然而，一个主要的开放问题，制定Gromov，是是否有界的程度高维扩展存在$d \geq 2$。我们提出了一个明确的结构维数$d=2$，这是拓扑扩张的有界度复形，从而回答了Gromov的问题是肯定的。在Serre关于同余子群性质的一个猜想的条件下，这些子族的无穷子族也给出了一族有界度的上边界扩张子。主要的技术工具是Ramanujan复形的新的等周不等式。我们证明了这些复合物的$F_2$收缩不变量的线性尺寸界，这似乎是第一个线性$F_2$收缩界。由这些等周不等式导出了展开式的结果。

Hamming立方体上的锐利等周不等式II：临界指数

Sharp isoperimetric inequalities on the Hamming cube II: The critical exponent

本文在临界指数β=1/2处证明了Hamming立方体的锐利等周不等式，解决了Kahn和Park的猜想，并给出了布尔值函数的锐利L¹泊松不等式。