One-Sided Spectral Expanders

• One-sided spectral expanders are graphs or higher-dimensional complexes where only the upper nontrivial spectrum (e.g., λ₂) is controlled, distinguishing them from two-sided expansion.
• They leverage normalized operators like random-walk matrices and Laplacians to analyze local spectral expansion, especially in bipartite and partite settings.
• Explicit constructions using coset geometries and Grassmannian methods provide strong algorithmic and pseudorandomness guarantees that benefit mixing, independent set, and coloring problems.

Searching arXiv for recent and foundational papers on one-sided spectral expanders to ground the article. arxiv_search(query="one-sided spectral expanders simplicial complex local spectral expansion", max_results=10) arxiv_search(query="Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model", max_results=5) One-sided spectral expanders are graphs or higher-dimensional complexes in which only the upper end of the nontrivial spectrum is controlled. For a normalized adjacency or random-walk operator $P$, the defining requirement is typically an upper bound on $\lambda_2(P)$, or an equivalent lower bound on the normalized Laplacian gap, while no symmetric bound is imposed on the most negative eigenvalue. This asymmetry is essential in bipartite and partite settings, where negative eigenvalues are often structural rather than pathological, and in combinatorial problems where large independent sets themselves force substantial negative spectrum (Harsha et al., 2019, Bafna et al., 2024).

1. Definitions and operator models

In the graph setting, the basic normalization is the random-walk matrix $W=D^{-1}A$, where $A$ is the adjacency matrix and $D$ is the diagonal degree matrix. For a $d$-regular graph, $W=A/d$, and one-sided spectral expansion means $\lambda_2(W)\le \lambda<1$. Equivalently, for the normalized Laplacian $\widetilde L=I-D^{-1/2}AD^{-1/2}$, one has $\lambda_2(\widetilde L)=1-\lambda_2(W)$, so one-sided expansion is a positive lower bound on the normalized Laplacian spectral gap (Bafna et al., 2024).

This notion extends naturally beyond regular undirected graphs. In directed or non-regular settings, the relevant parameter is the operator norm $\lambda_2(P)$0 of a Markov operator $\lambda_2(P)$1 after removing the stationary component $\lambda_2(P)$2. In bipartite settings, one uses the second singular value of the normalized biadjacency $\lambda_2(P)$3. The operator viewpoint treats one-sided spectral expansion as a bound on the nontrivial singular spectrum of the normalized transition operator, rather than on the full signed eigenvalue set (Jeronimo et al., 2022).

A related bipartite formulation appears in one-sided lossless expansion. There, one starts from a biregular bipartite graph and studies the nonlazy square on one side; one-sided spectral expansion means that this two-step random walk has small second eigenvalue. This is precisely the spectral parameter used to control collisions in explicit constructions of one-sided lossless expanders (Golowich, 2023).

The main distinction from two-sided expansion is therefore conceptual as well as technical. Two-sided expansion bounds both $\lambda_2(P)$4 and the most negative part of the spectrum, usually through $\lambda_2(P)$5. One-sided expansion controls only the positive nontrivial spectrum. In many applications this is the correct invariant, because bipartiteness, multipartiteness, and planted combinatorial structure may force large negative eigenvalues even when the random walk mixes well in the relevant one-step sense (Harsha et al., 2019).

2. Local spectral expansion in simplicial complexes

For a pure simplicial complex $\lambda_2(P)$6, the basic higher-dimensional object is the link $\lambda_2(P)$7 of a face $\lambda_2(P)$8. Its $\lambda_2(P)$9-skeleton is a weighted graph, and one studies the normalized random walk on that graph. In the formulation used for spectral independence, if $W=D^{-1}A$0 is the weighted adjacency matrix of the $W=D^{-1}A$1-skeleton of a link and $W=D^{-1}A$2 is the diagonal degree matrix, then the link walk is

$W=D^{-1}A$3

A complex is an $W=D^{-1}A$4-local spectral expander if for every $W=D^{-1}A$5-face $W=D^{-1}A$6 with $W=D^{-1}A$7, the second eigenvalue of the simple random walk on the $W=D^{-1}A$8-skeleton of $W=D^{-1}A$9 satisfies $A$0 (Anari et al., 2020).

The same idea appears in Oppenheim’s local spectral expansion framework, phrased in terms of the normalized upper Laplacian on link graphs. For a pure $A$1-dimensional complex, $A$2-local spectral expansion requires connectedness of the $A$3-skeleta of all links and a lower bound $A$4 on the smallest positive eigenvalue of the normalized Laplacian of every $A$5-dimensional link. Since $A$6 on a link graph, this is equivalent to an upper bound on the corresponding nontrivial random-walk eigenvalue (Oppenheim, 2017, Oppenheim, 2014).

In high-dimensional expander theory, “one-sided” again means that only the upper nontrivial spectrum is controlled. No lower bound is imposed on the most negative eigenvalue. This is especially important for partite complexes. In the partite setting, distinguished top eigenvalues arise automatically on link Laplacians, generalizing the eigenvalue $A$7 of bipartite graphs, and negative spectral mass cannot in general be excluded. One-sided local spectral expansion is designed precisely to accommodate that phenomenon (Oppenheim, 2017).

The same language is used in more recent work on pseudorandomness in complexes. There, a $A$8-dimensional simplicial complex is a $A$9-one-sided local spectral expander if for every face $D$0 of codimension at least $D$1, the second eigenvalue of the normalized adjacency operator $D$2 of the link graph $D$3 satisfies $D$4. The operator-norm form is $D$5 on the mean-zero subspace with respect to the stationary measure (Kaufman et al., 2022).

3. Descent, trickle-down, and spectral independence

A central structural theme is that local one-sided spectral bounds propagate across dimensions. Oppenheim’s descent theorem gives an explicit recursion for spectral gaps on lower-dimensional links. Writing $D$6, if the nonzero spectrum of the normalized Laplacian on all top-dimensional $D$7-links lies in $D$8, then lower links satisfy corresponding bounds obtained by iterating $D$9. This descent is then combined with Garland-type localization identities to deduce global lower bounds on the higher Laplacians $d$0 and $d$1 (Oppenheim, 2017).

For partite complexes, the classical worst-case trickle-down theorem is sharpened by an average-case result. If the codimension-$d$2 links are good one-sided spectral expanders only on average, in a harmonic-weighted sense over types, then links of codimension $d$3 are one-sided spectral expanders with parameter $d$4. This produces the inverse-linear $d$5 decay observed in several bounded-degree high-dimensional expanders, and substantially weakens the hypothesis compared with a uniform worst-case codimension-$d$6 bound (Abdolazimi et al., 2022).

A conceptually different route to one-sided local spectral expansion comes from spectral independence. For a distribution $d$7 on subsets of $d$8, the signed influence matrix is

$d$9

The distribution is spectrally independent if $W=A/d$0 is bounded uniformly over all conditional distributions. From $W=A/d$1 one constructs an $W=A/d$2-partite simplicial complex $W=A/d$3 whose maximal faces encode configurations, and for the empty face one has the operator identity

$W=A/d$4

Because conditioning on variables corresponds exactly to passing to links, one obtains

$W=A/d$5

for the local spectral expansion parameters of $W=A/d$6 whenever the conditioned distributions are $W=A/d$7-spectrally independent (Anari et al., 2020).

This bridge is important because it turns correlation control into geometric expansion. It also yields local-to-global mixing statements for the upper walk on maximal faces. If $W=A/d$8 is an $W=A/d$9-local spectral expander, then the local-to-global theorem bounds the second eigenvalue of the upper walk by

$\lambda_2(W)\le \lambda<1$0

so the spectral gap is at least

$\lambda_2(W)\le \lambda<1$1

For measures represented as $\lambda_2(W)\le \lambda<1$2, this becomes an explicit mixing bound for Glauber dynamics (Anari et al., 2020).

4. Explicit constructions and parameter regimes

Elementary constructions of bounded-degree one-sided local spectral expanders were obtained through coset geometries. For elementary matrix groups over $\lambda_2(W)\le \lambda<1$3, the associated coset complex is pure, partite, strongly gallery connected, and highly symmetric. Using $\lambda_2(W)\le \lambda<1$4-orthogonality of suitable subgroup pairs, the $\lambda_2(W)\le \lambda<1$5-dimensional links are shown to satisfy $\lambda_2(W)\le \lambda<1$6, and Oppenheim’s local-to-global machinery then yields one-sided local spectral expansion with

$\lambda_2(W)\le \lambda<1$7

The resulting $\lambda_2(W)\le \lambda<1$8-skeletons are expander graphs with normalized spectral gap at least $\lambda_2(W)\le \lambda<1$9 (Kaufman et al., 2017).

The same family admits an elementary proof via explicit analysis of the relevant bipartite link graphs. In that treatment, the top links have second eigenvalue at most $\widetilde L=I-D^{-1/2}AD^{-1/2}$0, and the descent theorem implies that the full complex is a $\widetilde L=I-D^{-1/2}AD^{-1/2}$1-one-sided-spectral high-dimensional expander when $\widetilde L=I-D^{-1/2}AD^{-1/2}$2. Restricting to lower skeleta then yields ordinary spectral expanders as a byproduct (Harsha et al., 2019).

A different construction starts from Grassmannian high-dimensional expanders. After basisification and a Cayley simplicial complex construction over $\widetilde L=I-D^{-1/2}AD^{-1/2}$3, one obtains families with arbitrarily good local spectral expansion and subpolynomial degree. Concretely, for $\widetilde L=I-D^{-1/2}AD^{-1/2}$4,

$\widetilde L=I-D^{-1/2}AD^{-1/2}$5

These complexes also support large first homology, with constructions achieving $\widetilde L=I-D^{-1/2}AD^{-1/2}$6 while retaining arbitrarily good local spectral expansion (Golowich, 2023).

On the graph side, operator amplification gives an explicit and local way to turn arbitrary bounded-degree expanders into almost Ramanujan ones. Starting from any family with constant one-sided expansion, the amplified degree satisfies

$\widetilde L=I-D^{-1/2}AD^{-1/2}$7

and each new edge is defined by a short walk of radius $\widetilde L=I-D^{-1/2}AD^{-1/2}$8 in the original graph. The same framework preserves Cayley, bipartite, monotone, and quantum-expander structure because it is formulated at the level of normalized operators rather than only undirected regular adjacency matrices (Jeronimo et al., 2022).

5. Mixing, counting, and coding consequences

One-sided local spectral expansion already has strong global implications. In Oppenheim’s framework it implies vanishing of reduced cohomology with real coefficients, a Hodge decomposition $\widetilde L=I-D^{-1/2}AD^{-1/2}$9, higher-dimensional Cheeger-type inequalities, mixing statements, and geometric overlap. In the partite case, one-sided hypotheses suffice for quantitative mixing and geometric overlap, whereas the non-partite general theory is often stated under two-sided local bounds (Oppenheim, 2014).

For high-order random walks, the main point is that global spectral gap is not the right statistic. On a one-sided local spectral expander, every $\lambda_2(\widetilde L)=1-\lambda_2(W)$0-cochain decomposes into pieces associated with lower-dimensional structure, and the up-down walk shrinks the $\lambda_2(\widetilde L)=1-\lambda_2(W)$1-th piece by a factor roughly $\lambda_2(\widetilde L)=1-\lambda_2(W)$2 plus an error term linear in the local spectral parameter. Thus the walk contracts according to structural content rather than only through the worst global eigenvalue (Kaufman et al., 2017).

One-sided local spectral expansion also supports an intrinsic notion of pseudorandomness for cochains. The theory of double balanced sets shows that sufficiently small $\lambda_2(\widetilde L)=1-\lambda_2(W)$3-double balanced $\lambda_2(\widetilde L)=1-\lambda_2(W)$4-cochains in a $\lambda_2(\widetilde L)=1-\lambda_2(W)$5-one-sided local spectral expander have near-optimal unique-neighbor-like expansion:

$\lambda_2(\widetilde L)=1-\lambda_2(W)$6

under the stated smallness condition. When nontrivial links are $\lambda_2(\widetilde L)=1-\lambda_2(W)$7-coboundary expanders, cohomology classes are double balanced, leading to the lower bound

$\lambda_2(\widetilde L)=1-\lambda_2(W)$8

for nontrivial cohomology classes, an exponential improvement over the previous lower bound discussed there (Kaufman et al., 2022).

Spectral independence supplies one of the strongest algorithmic consequences of one-sided local spectral expansion. For a distribution $\lambda_2(\widetilde L)=1-\lambda_2(W)$9, the upper walk on the complex $\lambda_2(P)$00 coincides with Glauber dynamics, so the product-form lower bound on the spectral gap yields rapid mixing. Applied to the hardcore model, the paper proves spectral independence up to the uniqueness threshold and deduces polynomial-time mixing of Glauber dynamics, together with an FPRAS for the partition function in that regime (Anari et al., 2020).

6. Independent sets, colorings, and the one-sided frontier in graphs

In graph algorithms, one-sided expansion supports guarantees that differ sharply from the classical two-sided spectral method. A polynomial-time algorithm finds an independent set of size at least $\lambda_2(P)$01 in a regular graph with $\lambda_2(P)$02 and $\lambda_2(P)$03 for $\lambda_2(P)$04. The same work gives a polynomial-time algorithm that, on a regular $\lambda_2(P)$05-almost $\lambda_2(P)$06-colorable one-sided spectral expander with $\lambda_2(P)$07, outputs an independent set of size at least $\lambda_2(P)$08 (Bafna et al., 2024).

That analysis is based on SoS pseudodistributions, correlation reduction by conditioning, and a clustering phenomenon for large independent sets in expanding graphs. It also extends from one-sided spectral expansion to certified small-set vertex expansion through low-degree SoS certificates. At the same time, the same paper proves a hardness barrier: assuming the Unique Games Conjecture, for any constants $\lambda_2(P)$09, it is NP-hard to find an independent set of size $\lambda_2(P)$10 in a regular $\lambda_2(P)$11-almost $\lambda_2(P)$12-colorable graph with $\lambda_2(P)$13 (Bafna et al., 2024).

Later work strengthens the algorithmic picture for sufficiently strong regular one-sided expanders. Given a $\lambda_2(P)$14-colorable regular one-sided expander, one can compute in polynomial time either an independent set of relative size at least $\lambda_2(P)$15 or a proper $\lambda_2(P)$16-coloring for all but an $\lambda_2(P)$17 fraction of the vertices. The same paper obtains a $\lambda_2(P)$18-factor approximation algorithm for vertex cover in sufficiently strong regular one-sided expanders and establishes a new spectral inequality: the number of eigenvalues below $\lambda_2(P)$19 is at most $\lambda_2(P)$20 times the number of eigenvalues above $\lambda_2(P)$21 (Buhai et al., 4 Aug 2025).

These results clarify a common misconception. One-sided expansion is not merely a weaker approximation to two-sided expansion; it is often the correct notion when negative spectrum carries genuine combinatorial signal. Large independent sets force negative eigenvalues by Hoffman-type arguments, so two-sided expansion would exclude precisely the structures that one wishes to exploit algorithmically. The current theory therefore separates problems that remain tractable under one-sided control from those that become hard once unrestricted negative spectrum is allowed, especially in coloring beyond the $\lambda_2(P)$22-colorable regime (Buhai et al., 4 Aug 2025).