Edge-isoperimetric inequalities and influences

Isoperimetric Inequalities on Hexagonal Grids

We consider the edge- and vertex-isoperimetric probem on finite and infinite hexagonal grids: For a subset W of the hexagonal grid of given cardinality, we give a lower bound for the number of edges between W and its complement, and lower bounds for the number of vertices in the neighborhood of W and for the number of vertices in the boundary of W. For the infinite hexagonal grid the given bounds are tight.

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]。

Duality properties of strong isoperimetric inequalities on a planar graph and combinatorial curvatures

This paper is about hyperbolic properties on planar graphs. First, we study the relations among various kinds of strong isoperimetric inequalities on planar graphs and their duals. In particular, we show that a planar graph satisfies a strong isoperimetric inequality if and only if its dual has the same property, if the graph satisfies some minor regularity conditions and we choose an appropriate notion of strong isoperimetric inequalities. Second, we consider planar graphs where negative combinatorial curvatures dominate, and use the outcomes of the first part to strengthen the results of Hig…

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…

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.

A General Method to Determine Limiting Optimal Shapes for Edge-Isoperimetric Inequalities

For a general family of graphs on $\mathbb{Z}^n$, we translate the edge-isoperimetric problem into a continuous isoperimetric problem in $\mathbb{R}^n$. We then solve the continuous isoperimetric problem using the Brunn-Minkowski inequality and Minkowski's theorem on Mixed Volumes. This translation allows us to conclude, under a reasonable assumption about the discrete problem, that the shapes of the optimal sets in the discrete problem approach the shape of the optimal set in the continuous problem as the size of the set grows. The solution is the zonotope defined as the Minkowski sum of the …

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。

A note on fine graphs and homological isoperimetric inequalities

In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of attaching maps of $2$-cells and finitely many $2$-cells adjacent to any edge must have a fine $1$-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity, and show that a group $G$ is hyperbolic relative to a collection of subgroups $\mathcal P$ if and only if $G$ acts cocompactly with finite…

Isoperimetric Inequalities for Ramanujan Complexes and Topological Expanders

Expander graphs have been intensively studied in the last four decades. In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every $d$ there are unbounded degree simplicial complexes of dimension $d$ with these properties. However, a major open problem, formulated by Gromov, is whether bounded degree high dimensional expanders exist for $d \geq 2$. We present an explicit construction of bounded degree complexes of dimension $d=2$ which are topological…

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。