High Dimensional Expanders (HDX)

• High Dimensional Expanders (HDX) are simplicial complexes that generalize expander graphs to higher dimensions by leveraging robust topological and spectral properties.
• They enable diverse applications such as enhanced coding theory, property and agreement testing, and quantum LDPC code constructions through explicit combinatorial methods.
• Their structure, characterized by coboundary and cosystolic expansion, underpins breakthroughs in combinatorics and drives advances in both theoretical and applied research.

A high-dimensional expander (HDX) generalizes the expander graph paradigm to simplicial complexes of dimension $d>1$, leveraging both topological and spectral properties in higher dimensions. Over the last decade, these objects have emerged as fundamental combinatorial and analytic structures bridging pure mathematics, theoretical computer science, and quantum information. HDXs are defined via the expansion of links of all faces—generalizing spectral and combinatorial expansion from graphs to simplicial complexes—and their explicit construction has driven advances in coding theory, agreement testing, property testing, and small-set expansion phenomena.

1. Foundations: Simplicial Complexes, Cohomology, and Laplacians

Let $X$ be a pure $d$-dimensional finite simplicial complex with vertex set $X(0)=\{v_1,\ldots,v_n\}$. Its $i$-simplices, $X(i)$, are the subsets of $X(0)$ of size $i+1$ contained in a $d$-simplex. The combinatorial structure of $X$ is encoded via chain and cochain groups ($C_i(X, \mathbb F)$, $C^i(X, \mathbb F)$) with associated boundary ($\partial_{i+1}$) and coboundary ($\delta_i$) operators. Cohomology groups $$H^i(X, \mathbb F) = \ker(\delta_i)/\mathrm{im}(\delta_{i-1})$$ measure the failure of exactness at each level.

Spectral properties are probed by the Hodge Laplacians: $$\Delta_i = \delta_i^* \delta_i + \delta_{i-1}\delta_{i-1}^*,$$ and the $i$th spectral gap $\lambda_i(X)$ is defined as the minimal nonzero eigenvalue of $\Delta_i$. The Hodge decomposition partitions $$C^i(X, \mathbb R) = \mathrm{im}(\delta_{i-1}) \oplus \ker(\Delta_i) \oplus \mathrm{im}(\delta_i^*)$$, with $\ker(\Delta_i) \cong H^i(X, \mathbb R)$ (Lubotzky, 2017).

2. Spectral and Combinatorial Expansion: Key Notions

2.1 Spectral Expansion

For each $(d-2)$-face $\tau\in X(d-2)$, the link $\operatorname{lk}_X(\tau)$ is the complex of faces containing $\tau$. The 1-skeleton of the link is required to be a $\lambda$-spectral expander, i.e., its (normalized) adjacency operator has nontrivial eigenvalues bounded by $\lambda$. This local-spectral expansion propagates new forms of high-dimensional mixing and agreement (Lubotzky, 2017, Bafna et al., 2020).

2.2 Coboundary and Cosystolic Expansion

Coboundary expansion (over $\mathbb F_2$) is defined by the coboundary-expansion constant: $$h_i(X) = \min_{f\notin B^i} \frac{\Vert \delta_i f \Vert}{\Vert [f] \Vert},$$ where $[f]=f+B^i$ and $\Vert f \Vert$ is the normalized weight. $X$ is an $\varepsilon$-coboundary expander if $h_i(X)\geq \varepsilon$ for $i=0,\dots,d-1$.

Cosystolic expansion strengthens this via lower bounds on

$$v_i(X) = \min_{f\in Z^i \setminus B^i} \Vert \delta_i f \Vert, \quad \pi_i(X) = \min_{f\in Z^i \setminus B^i} \Vert f \Vert$$

yielding the $\varepsilon$-cosystolic expander property, which, by Gromov, implies topological overlap (Lubotzky, 2017).

2.3 Cheeger-Type and Garland Inequalities

Generalizations of Cheeger inequalities relate spectral gaps to combinatorial isoperimetric parameters of higher-dimension cochains. Garland's method establishes that if each $(d-2)$-face link is a good expander graph (with spectral gap $\mu$), then the top-dimensional Laplacian satisfies $\lambda_{d-1}(X)\ge 1+(d-1)(1-\sqrt{1-\mu})^2$, and similar estimates propagate to lower skeleta (Lubotzky, 2017).

3. Constructions and Explicit Families

HDXs with strong local-spectral expansion have been constructed using several frameworks:

3.1 Bruhat–Tits Buildings and Ramanujan Complexes

Quotients of affine Bruhat–Tits buildings by suitable lattices in simple algebraic groups over local fields yield finite complexes of bounded degree. Garland’s method and Property (T) imply uniform spectral expansion in all skeletons. If spherical representations in $L^2$ are tempered, the resulting $X$ is a Ramanujan complex, generalizing optimal spectral bounds for graphs (Lubotzky, 2017, O'Donnell et al., 2022).

3.2 Coset Geometries and Chevalley Group Constructions

Coset complexes $C(G;H_0,\dots,H_d)$ for finite groups (notably Chevalley groups) and families of subgroups yield partite $d$-complexes with explicit control on the expansion properties via representation theory of the underlying group. The link expansion parameter can be computed in terms of group-theoretic data, with bounds $\lambda_2\leq \sqrt{2/p}$ for large $p$, decaying with field size (O'Donnell et al., 2022, Peralta et al., 2024).

3.3 Covering and Lifting Constructions

Recent approaches include forming covers or lifts of initial HDXs, either via group-theoretic cocycles or combinatorial rules, e.g., Bilu–Linial-type local lifts preserve certain face degrees and local expansion up to explicit error terms (Dikstein, 2022, Yaacov et al., 2024). Iterated sparsification of Grassmannian posets generates HDXs of subpolynomial degree (Dikstein et al., 2024, Golowich, 2023).

3.4 Product, Tensor, and Combinatorial Models

Tensoring complete complexes with random or expander graphs, reweighted and appropriately adjusted, yields combinatorial HDXs with provable spectral gaps matching $\Omega(1/k^2)$ for $k$-dimensional walks, with information-theoretic upper bounds at $O(1/k)$ (Golowich, 2021, Liu et al., 2019, Dikstein et al., 2024).

4. Geometric, Topological, and Probabilistic Expansion

4.1 Geometric and Topological Overlap

An HDX exhibits $\varepsilon$-geometric overlap if in every affine map $f:X\to\mathbb R^d$, a constant fraction of $d$-faces map over a single point. Coboundary and cosystolic expansion imply, via Gromov’s theorem, the existence of a positive topological overlap constant, foundational for geometric embeddings and applications (Lubotzky, 2017, Dikstein et al., 2024).

4.2 Concentration and Hypercontractivity

Recent advances establish optimal measure concentration and reverse hypercontractivity for HDX: for $X$ $k$-dimensional HDX and $f:X(i)\to [0,1]$, it holds

$$\Pr_{s\in X(k)}\left[\,|\mathbb{E}_{t\subseteq s}[f(t)] - \mu| \geq \varepsilon \,\right] \leq \exp(-\varepsilon^2 k/i)$$

with the “trickling-down threshold” marking a sharp threshold below which such concentration is universal (Dikstein et al., 2024). Hypercontractivity and global symmetrization results for HDX extend Bourgain’s and Bonami’s theorems to non-product domains (Bafna et al., 2021, Hopkins, 2024).

5. Algorithmic and Coding-Theoretic Applications

5.1 Locally Testable and List-Decodable Codes

Coboundary expansion and the global expansion of HDX imply the existence of locally testable codes (LTCs) with strong parameters (amplified local testability), removing the need for local-to-global patching on codes via link expansion (Kaufman et al., 2021). HDX are now central in the construction of explicit, high-rate, locally list-decodable codes with polylogarithmic query and decoding complexity, leveraging belief propagation and strongly explicit routing on HDX skeletons (Dikstein et al., 30 Jan 2026, Dikstein et al., 2024).

5.2 Property Testing, Agreement Testing, and PCPs

HDX structure underpins combinatorial property testers for cohomological properties and agreement testing, foundational for low-error PCP constructions and probabilistically checkable proofs with optimal soundness (Lubotzky, 2017, Dikstein et al., 2024). The agreement property and robust local spectral mixing translate to strong derandomized testing frameworks.

5.3 Quantum LDPC and Lattices

HDX enable the construction of homological quantum low-density parity-check (LDPC) codes, which require families of cosystolic expanders for their distance and rate properties (Lubotzky, 2017). Furthermore, explicit cosystolic expansion over $\mathbb Z$ enables the direct construction of lattices with simultaneous large normalized distance and density (Kaufman et al., 2018).

6. Open Problems and Future Directions

• Characterize when arithmetic (e.g., Bruhat–Tits, Chevalley, or Kac–Moody) quotients are uniformly cosystolic expanders from spectral/representation-theoretic criteria (Lubotzky, 2017, Peralta et al., 2024).
• Develop random or combinatorial constructions of bounded-degree HDX with optimal or near-optimal expansion (Dikstein et al., 2024).
• Establish sharp high-dimensional Cheeger inequalities relating combinatorial and spectral expansion for all cohomological dimensions.
• Extend zeta-function formalisms (e.g., Ihara) and higher-dimension Riemann Hypotheses to the HDX regime.
• Construct new locally testable and PCP-friendly codes directly exploiting explicit, bounded-degree HDX beyond currently available group and design-theoretic examples.
• Explore further ramifications in topological combinatorics (embedding and overlap), metric geometry (coarse embeddings), and rigidity via cohomology vanishing (Lubotzky, 2017).

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The rapidly evolving landscape of HDXs features new construction paradigms, structural theorems (e.g., eigenstripping, level decomposition, $q$-norm expansion), improved small-set expansion, and a spectrum of applications across coding theory, computational hardness, and topological analysis (Bafna et al., 2020, Kaufman et al., 11 Dec 2025, Hopkins, 2024). The area remains characterized by tight interactions between algebra, topology, and analysis, with a rich set of open conjectures driving ongoing research.