Dominic Dotterrer, Matthew Kahle
We describe a natural topological generalization of edge expansion for graphs to regular CW complexes and prove that this property holds with high probability for certain random complexes.
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We introduce a new model of random $d$-dimensional simplicial complexes, for $d\geq 2$, whose $(d-1)$-cells have bounded degrees. We show that with high probability, complexes sampled according to this model are coboundary expanders. The construction relies on Keevash's recent result on designs [Ke14], and the proof of the expansion uses techniques developed by Evra and Kaufman in [EK15]. This gives a full solution to a question raised in [DK12], which was solved in the two-dimensional case by Lubotzky and Meshulam [LM13].
A Simple Sub-Polynomial Degree Coboundary Expander
本文提出了一种简单的组合构造，用于构建一个小多项式度复杂度的展开器，该展开器同时满足谱展开和边界展开的条件，并且在交换边界展开方面也有效，同时给出了具有良好一致测试的近线性大小的组合超图和一种简单的PCP构造。
Coboundary expansion for the union of determinantal hypertrees
本文研究随机行列式超树并集的高维扩张性质，证明足够多个独立 $d$ 维行列式超树的并集以高概率构成上边缘扩张子。
In this paper, we consider a variation on Cheeger numbers related to the coboundary expanders recently defined by Dotterer and Kahle. A Cheeger-type inequality is proved, which is similar to a result on graphs due to Fan Chung. This inequality is then used to study the relationship between coboundary expanders on simplicial complexes and their corresponding eigenvalues, complementing and extending results found by Gundert and Wagner. In particular, we find these coboundary expanders do not satisfy natural Buser or Cheeger inequalities.
We show a new framework for constructing good distance lattices from high dimensional expanders. For error-correcting codes, which have a similar flavor as lattices, there is a known framework that yields good codes from expanders. However, prior to our work, there has been no framework that yields good distance lattices directly from expanders. Interestingly, we need the notion of high dimensional expansion (and not only one dimensional expansion) for obtaining large distance lattices which are dense. Our construction is obtained by proving the existence of bounded degree high dimensional cos…
Coboundary expansion of coset complexes
本文证明了 Kaufman-Oppenheim 陪集复形具有非阿贝尔系数的上边界扩张，采用全局论证而非局部到全局方法，深化了高维扩张与理论计算机科学的联系。
The Cheeger Inequality and Coboundary Expansion: Beyond Constant Coefficients
本文研究了赋予广义系数群（层）的图的Cheeger型不等式，证明了图是良好谱展开子当且仅当其相对于常数层具有良好上边缘展开，并将结果应用于球面建筑给出了显式下界。
Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expanders. 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, according to these definitions, exist for d >= 2. We …
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…
Coboundary and cosystolic expansion without dependence on dimension or degree
本文给出了几族高维膨胀子的余systolic膨胀和上边缘膨胀的新界限，其不依赖于维数、次数或系数群，改进了Gromov拓扑重叠常数等结果，并基于颜色限制、谱证明和构造极短锥等新技术实现。
Let X be a 2-dimensional simplicial complex. The degree of an edge e is the number of 2-faces of X containing e. The complex X is an ε-expander if the coboundary d_1(ϕ) of every Z_2-valued 1-cochain ϕ\in C^1(X;Z_2) satisfies |support(d_1(ϕ))| \geq ε|\supp(ϕ+d_0(ψ))| for some 0-cochain ψ. Using a new model of random 2-complexes we show the existence of an infinite family of 2-dimensional ε-expanders with maximum edge degree d, for some fixed ε>0 and d.
Following Gromov, the coboundary expansion of building-like complexes is studied. In particular, it is shown that for any $n \geq 1$, there exists a constant $ε(n)>0$ such that for any $0 \leq k <n$ the $k$-th coboundary expansion constant of any $n$-dimensional spherical building is at least $ε(n)$.
We discuss two combinatorical ways of generalizing the definition of expander graphs and Ramanujan graphs, to quotients of buildings of higher dimension. The two possible definitions are equivalent for affine buildings, giving the notion of an Lp-expander complex. We calculate explicit spectral gaps on many combinatorical operators, on any Lp-expander complex. We associate with any complex a natural "zeta function", generalizing the Ihara-Hashimoto zeta function of a finite graph. We generalize a well known theorem of Hashimoto, showing that a complex is Ramanujan if and only if the zeta funct…
Expander graphs in general, and Ramanujan graphs in particular, have been of great interest in the last three decades with many applications in computer science, combinatorics and even pure mathematics. In these notes we describe various efforts made in recent years to generalize these notions from graphs to higher dimensional simplicial complexes.
New cosystolic high-dimensional expanders from KMS groups
本文利用Kac-Moody-Steinberg群的有限商群构造KMS复形，首次给出任意维数下余上同调展开图的陪集复形构造，并证明其链结为余边界展开图，从而获得无限族有界度余上同调展开图。
Coboundary and cosystolic expansion are notions of expansion that generalize the Cheeger constant or edge expansion of a graph to higher dimensions. The classical Cheeger inequality implies that for graphs edge expansion is equivalent to spectral expansion. In higher dimensions this is not the case: a simplicial complex can be spectrally expanding but not have high dimensional edge-expansion. The phenomenon of high dimensional edge expansion in higher dimensions is much more involved than spectral expansion, and is far from being understood. In particular, prior to this work, the only known bo…
In this work we introduce a new notion of expansion in higher dimensions that is stronger than the well studied cosystolic expansion notion, and is termed {\em Collective-cosystolic expansion}. We show that tensoring two cosystolic expanders yields a new cosystolic expander, assuming one of the complexes in the product, is not only cosystolic expander, but rather a collective cosystolic expander. We then show that the well known bounded degree cosystolic expanders, the Ramanujan complexes are, in fact, collective cosystolic expanders. This enables us to construct new bounded degree cosystolic …
In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer affirmatively an open question raised by Gromov.
We show that the high dimensional expansion property as defined by Gromov, Linial and Meshulam, for simplicial complexes is a form of testability. Namely, a simplicial complex is a high dimensional expander iff a suitable property is testable. Using this connection, we derive several testability results.
We construct an explicit family of 3XOR instances which is hard for $O(\sqrt{\log n})$ levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems can be constructed explicitly in deterministic polynomial time. Our construction is based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov. Our construction offers an i…
Swap cosystolic expansion
该研究提出单纯复形的新扩张性质交换上收缩扩张，证明球面建筑与LSV拉马努金复形的相关扩张下界，结合姊妹研究表明这类复形可支撑低接受度一致性测试。
From Grassmannian to Simplicial High-Dimensional Expanders
本文提出一种基于格拉斯曼HDX的凯莱单纯复形构造，实现次多项式度数与任意好局部谱扩张，并证明其1-同调群维数可达Ω(log²N)，超越以往构造。
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in material science, and in other settings. Stochastic topology may also be considered as a null hypothesis for topological data analysis. In this chapter we overview combinatorial aspects of stochastic topology. We focus on the topological and geometric properties of random simplicial complexes. We introduce a few of the fundamental models in Section 23.1. We…
In this paper, we further explore the local-to-global approach for expansion of simplicial complexes that we call local spectral expansion. Specifically, we prove that local expansion in the links imply the global expansion phenomena of mixing and geometric overlapping. Our mixing results also give tighter bounds on the error terms compared to previously known results.
We prove a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to as mixing, or quasi-randomness). Recently, an analogue of this Lemma was proved for simplicial complexes of arbitrary dimension, provided that the skeleton of the complex is complete. More precisely, it was shown that a concentrated spectrum of the simplicial Hodge Laplacian implies a similar type of expansion as in graphs. In this paper we remove the assumpt…
Local to global machinery plays an important role in the study of simplicial complexes, since the seminal work of Garland [G] to our days. In this work we develop a local to global machinery for general posets. We show that the high dimensional expansion notions and many recent expansion results have a generalization to posets. Examples are fast convergence of high dimensional random walks generalizing [KO,AL], an equivalence with a global random walk definition, generalizing [DDFH] and a trickling down theorem, generalizing [O]. In particular, we show that some posets, such as the Grassmannia…
Non-Abelian expansion of congruence KMS complexes
本文证明经典类型 Chevalley 群同余子群下的 Grave de Peralta 与 Valentiner–Branth KMS 复形具有非 Abel 系数上边缘展开，为该现象新增四类构造。
An Improved Trickle-Down Theorem for Partite Complexes
本研究提出分部复形滴流定理的强化版本，证明余维2面链环的平均谱扩张性质可推导至更高余维面的链环，并将其应用于有界度高维扩张子的谱扩张界分析。
Topological expanders, coarse geometry and thick embeddings of complexes
本文引入一族次线性函数量化有界度单纯复形的拓扑扩张，证明其单调性并计算欧氏空间等实例的上下界，进而证明高重 horocyclic 树乘积到若干粗几何空间不存在粗嵌入。
High dimensional expanders and coset geometries
本文利用陪集几何为交换含幺环上有限生成代数的初等矩阵群构造对称、自足且有界度的新高维扩张子，并得到接近 Ramanujan 界的扩张图及对称 LDPC 码应用。
Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree
本文构造了显式的无限族二维Cayley复形，其度数多项式且特征值可控，并推广到加权高维情形，利用代数曲线有理点求值实现。
The total-variation cutoff phenomenon has been conjectured to hold for simple random walk on all transitive expanders. However, very little is actually known regarding this conjecture, and cutoff on sparse graphs in general. In this paper we establish total-variation cutoff for simple random walk on Ramanujan complexes of type $\widetilde{A}_{d}$ $(d\geq1)$. As a result, we obtain explicit generators for the finite classical groups $\mathrm{PGL}_{n}(\mathbb{F}_{q})$ for which the associated Cayley graphs exhibit total-variation cutoff.
High-dimensional expanders generalize the notion of expander graphs to higher-dimensional simplicial complexes. In contrast to expander graphs, only a handful of high-dimensional expander constructions have been proposed, and no elementary combinatorial construction with near-optimal expansion is known. In this paper, we introduce an improved combinatorial high-dimensional expander construction, by modifying a previous construction of Liu, Mohanty, and Yang (ITCS 2020), which is based on a high-dimensional variant of a tensor product. Our construction achieves a spectral gap of $Ω(\frac{1}{k^2…
Topological expansion for posets and the homological $k$-connectivity of random $q$-complexes
本文针对超越单纯复形的高维展开问题，开发了新的拓扑技术，证明完全q-复形的相关性质，建立随机q-复形连通性的尖锐阈值，还解决了一项同调猜想。
We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$ vertices. We show that for $p=Ω(\log n/n)$, the eigenvalues of these matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of $(k-2)$-dimensional faces. Garland's result concerns the Laplacian; we develop an analogo…
In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes an…
Random walks on a graph reflect many of its topological and spectral properties, such as connectedness, bipartiteness and spectral gap magnitude. In the first part of this paper we define a stochastic process on simplicial complexes of arbitrary dimension, which reflects in an analogue way the existence of higher dimensional homology, and the magnitude of the high-dimensional spectral gap originating in the works of Eckmann and Garland. The second part of the paper is devoted to infinite complexes. We present a generalization of Kesten's result on the spectrum of regular trees, and of the conn…
Explicit Lossless Vertex Expanders
本文首次显式构造了常数度无损顶点扩展器，并推广到双正则二分图，进而实现具有线性时间解码的新量子LDPC码族。
We show that the natural Glauber dynamics mixes rapidly and generates a random proper edge-coloring of a graph with maximum degree $Δ$ whenever the number of colors is at least $q\geq (\frac{10}{3} + ε)Δ$, where $ε>0$ is arbitrary and the maximum degree satisfies $Δ\geq C$ for a constant $C = C(ε)$ depending only on $ε$. For edge-colorings, this improves upon prior work \cite{Vig99, CDMPP19} which show rapid mixing when $q\geq (\frac{11}{3}-ε_0 ) Δ$, where $ε_0 \approx 10^{-5}$ is a small fixed constant. At the heart of our proof, we establish a matrix trickle-down theorem, generalizing Oppenh…
The question of finding expander graphs with strong vertex expansion properties such as unique neighbor expansion and lossless expansion is central to computer science. A barrier to constructing these is that strong notions of expansion could not be proven via the spectral expansion paradigm. A very symmetric and structured family of optimal spectral expanders (i.e., Ramanujan graphs) was constructed using number theory by Lubotzky, Phillips and Sarnak, and was subsequently generalized by others. We call such graphs Number Theoretic Ramanujan Graphs. These graphs are not only spectrally optima…
Random walks on bounded degree expander graphs have numerous applications, both in theoretical and practical computational problems. A key property of these walks is that they converge rapidly to their stationary distribution. In this work we {\em define high order random walks}: These are generalizations of random walks on graphs to high dimensional simplicial complexes, which are the high dimensional analogues of graphs. A simplicial complex of dimension $d$ has vertices, edges, triangles, pyramids, up to $d$-dimensional cells. For any $0 \leq i < d$, a high order random walk on dimension $i…
Spectral and Homological Bounds on k-Component Edge Connectivity
本文提出结合谱图理论与同调理论的框架，通过高维单纯复形推导 k-分量边连通性的紧界，以多维刻画网络稳健性并验证其在基础设施网络及随机图、扩张图中的适用性。
Random walks on regular bounded degree expander graphs have numerous applications. A key property of these walks is that they converge rapidly to the uniform distribution on the vertices. The recent study of expansion of high dimensional simplicial complexes, which are the high dimensional analogues of graphs, calls for the natural generalization of random walks to higher dimensions. In particular, a high order random walk on a $2$-dimensional simplicial complex moves at random between neighboring edges of the complex, where two edges are considered neighbors if they share a common triangle. W…
Improved Small Set Expansion in High Dimensional Expanders
本文改进了高维展开器中小集展开的结果，通过结合不同方法实现了对集合大小的指数级提升。
We study the stability of covers of simplicial complexes. Given a map $f:Y\to X$ that satisfies almost all of the local conditions of being a cover, is it close to being a genuine cover of $X$? Complexes $X$ for which this holds are called cover-stable. We show that this is equivalent to $X$ being a cosystolic expander with respect to non-abelian coefficients. This gives a new combinatorial-topological interpretation to cosystolic expansion which is a well studied notion of high dimensional expansion. As an example, we show that the $2$-dimensional spherical building $A_{3}(\mathbb{F}_q)$ is c…
Cheeger inequalities on simplicial complexes
本文提出了一种方法，通过结合 simplicial 拓扑学、符号图、Gromov 填充半径等，解决了 simplicial 复合体上 Cheeger 型不等式的问题，并推广到 $p$-Laplacians。
Toward a KKL Theorem for any HDX
本文提出一种局部到全局方法，证明任何链接满足KKL定理的单纯复形整体也满足KKL型定理，并给出组合高维扩展器上非扩展函数的首次刻画。
Consider a random geometric 2-dimensional simplicial complex $X$ sampled as follows: first, sample $n$ vectors $\boldsymbol{u_1},\ldots,\boldsymbol{u_n}$ uniformly at random on $\mathbb{S}^{d-1}$; then, for each triple $i,j,k \in [n]$, add $\{i,j,k\}$ and all of its subsets to $X$ if and only if $\langle{\boldsymbol{u_i},\boldsymbol{u_j}}\rangle \ge τ, \langle{\boldsymbol{u_i},\boldsymbol{u_k}}\rangle \ge τ$, and $\langle \boldsymbol{u_j}, \boldsymbol{u_k}\rangle \ge τ$. We prove that for every $\varepsilon > 0$, there exists a choice of $d = Θ(\log n)$ and $τ= τ(\varepsilon,d)$ so that with h…
The coboundary expansion generalizes the classical graph expansion to the case of the general simplicial complexes, and allows the definition of the higher-dimensional Cheeger constants $h_k(X)$ for an arbitrary simplicial complex $X$, and any $k\geq 0$. In this paper we investigate the value of $h_1(Δ^{[n]})$ - the first Cheeger constant of a simplex with $n$ vertices. It is known, due to the pioneering work of Meshulam and Wallach, that \[\lceil n/3\rceil\geq h_1(Δ^{[n]})\geq n/3, \textrm{ for all } n,\] and that the equality $h_1(Δ^{[n]})=n/3$ is achieved when $n$ is divisible by $3$. Here …
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