An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture

Bourgain's slicing problem and KLS isoperimetry up to polylog

We prove that Bourgain's hyperplane conjecture and the Kannan-Lovász-Simonovits (KLS) isoperimetric conjecture hold true up to a factor that is polylogarithmic in the dimension.

对数界：等周不等式与凸集的切片

Logarithmic bounds for isoperimetry and slices of convex sets

本文证明 Bourgain 切片猜想与 KLS 等周猜想在 $\mathbb{R}^n$ 中至多相差 $\sqrt{\log n}$ 因子，关键新工具是改进的对数凹 Lichnerowicz 不等式。

凸性、自由拉普拉斯算子的谱及其几何推论：自由KLS界

Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound

该研究针对满足$CD(c,\infty)$条件的自由拉普拉斯算子，证明其预解式紧性等算子代数性质，得到自由KLS界等几何推论，还探讨了与标准半圆模型的谱比较问题。

随机正则图等周数的下界

Lower bounds for the isoperimetric numbers of random regular graphs

本文针对随机正则图，研究$u=1/2$情形下$d\ge3$时顶点等周数的渐近几乎必然下界、$d\to\infty$时的顶点等周数渐近下界，以及边等周数的对应下界并分析其大$d$表现。

Intrinsic isoperimetry of the giant component of supercritical bond percolation in dimension two

We study the isoperimetric subgraphs of the giant component $\textbf{C}_n$ of supercritical bond percolation on the square lattice. These are subgraphs of $\textbf{C}_n$ having minimal edge boundary to volume ratio. In contrast to the work of Biskup, Louidor, Procaccia and Rosenthal, the edge boundary is taken only within $\textbf{C}_n$ instead of the full infinite cluster. The isoperimetric subgraphs are shown to converge almost surely, after rescaling, to the collection of optimizers of a continuum isoperimetric problem emerging naturally from the model. We also show that the Cheeger constan…

The isoperimetric constant of the random graph process

The isoperimetric constant of a graph $G$ on $n$ vertices, $i(G)$, is the minimum of $\frac{|\partial S|}{|S|}$, taken over all nonempty subsets $S\subset V(G)$ of size at most $n/2$, where $\partial S$ denotes the set of edges with precisely one end in $S$. A random graph process on $n$ vertices, $\widetilde{G}(t)$, is a sequence of $\binom{n}{2}$ graphs, where $\widetilde{G}(0)$ is the edgeless graph on $n$ vertices, and $\widetilde{G}(t)$ is the result of adding an edge to $\widetilde{G}(t-1)$, uniformly distributed over all the missing edges. We show that in almost every graph process $i(\…