Spectral Gap Conjecture Overview

• Spectral Gap Conjecture is a framework that relates the gap between the first two nontrivial eigenvalues of elliptic operators to geometric, combinatorial, and ergodic properties.
• It establishes sharp lower bounds for eigenvalue differences under various boundary conditions, with notable thresholds when extending from Dirichlet to Neumann and Robin parameters.
• Recent generalizations apply its principles to Schrödinger operators, random processes, and manifolds using techniques like log-concavity, one-dimensional comparisons, and representation theory.

The spectral gap conjecture refers to a series of foundational hypotheses and subsequent theorems relating the fundamental spectral gap (difference between the first two nontrivial eigenvalues) of elliptic operators—primarily Laplacians and Schrödinger operators—on domains, graphs, manifolds, and interacting particle systems, to explicit sharp lower bounds or representation-theoretic properties determined by domain geometry, combinatorics, or ergodic-theoretic parameters. Across diverse mathematical settings, “spectral gap” controls temporal relaxation, mixing rates, and geometric rigidity, so quantifying sharp lower bounds or identifying optimizing structures is a central analytical, probabilistic, and combinatorial challenge.

1. Spectral Gap Conjecture in Convex Domains: Dirichlet, Neumann, and Robin Laplacians

The classical spectral gap conjecture addresses the Dirichlet Laplacian $-\Delta$ on a bounded convex domain $\Omega\subset\mathbb{R}^n$ of diameter $D$, stating that the gap $\lambda_2-\lambda_1$ (difference between the first two Dirichlet eigenvalues) should be bounded below by the gap on an interval of the same diameter:

$$\lambda_2(\Omega) - \lambda_1(\Omega) \geq 3\pi^2/D^2.$$

This is now a theorem (“fundamental gap theorem” of Andrews–Clutterbuck (Andrews et al., 2010)), with equality in the limit as $\Omega$ degenerates to a thin interval. The proof establishes both a sharp eigenvalue comparison and a sharp modulus of log-concavity for the ground-state eigenfunction, via one-dimensional Sturm–Liouville theory and maximum-principle arguments.

Extensions to the Neumann ($\alpha=0$) and Robin boundary conditions:

$$\begin{cases} -\Delta u = \lambda u \text{ in } \Omega,\ \frac{\partial u}{\partial n} + \alpha u = 0 \text{ on } \partial\Omega \end{cases}$$

have been investigated, with sharp bounds for the spectral gap in both Neumann ($\pi^2/D^2$) and Dirichlet ($3\pi^2/D^2$) cases among convex domains of given diameter, and with a general Robin-gap conjecture proposed for nonnegative $\alpha$ (Kielty, 2021). For $\alpha\geq 0$, the optimal lower bound for the gap is achieved on the interval, and higher-dimensional “spreading” only increases the gap.

2. Degeneration and Failure for Negative Robin Parameters

Kielty (Kielty, 2021) demonstrates that the aforementioned Robin-gap conjecture fails for negative Robin parameters ($\alpha < 0$). For every fixed $\alpha<0$, one can construct convex domains (specifically, “double cones” $D_\theta$ of opening angle $\theta$ and fixed diameter) with Robin spectral gap decaying exponentially:

$$\text{Gap}(\alpha, D_\theta) = \lambda_2(\alpha, D_\theta) - \lambda_1(\alpha, D_\theta) \leq C \exp\{-4(1-\varepsilon)|\alpha|/\theta\}$$

as $\theta\to 0$. This establishes that, in contrast to the strictly positive lower bounds for $\alpha\geq 0$, the infimum of the Robin gap over all convex domains (even of fixed diameter) is zero for all $\alpha<0$; the conjecture delineates a sharp boundary at $\alpha=0$, with transition from gap-positive to gap-degenerate behavior. The proof employs explicit trial functions, reduction to one-dimensional Schrödinger-type problems on balls with weighted Laplacians, and analysis using confluent-hypergeometric functions.

3. Generalizations to Schrödinger Operators and Moduli of Convexity

The spectral gap comparison theorem (Andrews et al., 2010) extends the result beyond Laplacians to Schrödinger operators $-\Delta + V$ with convex potentials or more general moduli of convexity $\eta$:

$$V(x) + V(y) - 2V\left(\frac{x+y}{2}\right) \geq \eta(|x-y|), \quad \forall x, y \in \Omega.$$

One obtains:

$\lambda_2 - \lambda_1 \geq \mu_2 - \mu_1$

where $\mu_1 < \mu_2$ are the first two Dirichlet eigenvalues of the one-dimensional comparison operator on $[-D/2, D/2]$ with corresponding modulus. Analysis involves sharp log-concavity comparisons of eigenfunctions and modulus-of-continuity estimates for heat equations with drift, as well as probabilistic approaches using coupling by reflection (Gong et al., 2013).

4. Spectral Gap Conjectures in Random Walks, Interchange, and Exclusion Processes

Aldous’ spectral gap conjecture for Markov chains and combinatorial processes—proved by Caputo, Liggett, Richthammer—asserts that the spectral gap of the interchange process on a finite graph equals that of the underlying single-particle random walk. Extensions include:

• Generalized exclusion processes (GEP): $\lambda_1^{\text{GEP}} = k \lambda_1^{\text{RW}}$, where $k$ is the site capacity (Kanegae et al., 2023).
• Zero-range processes: the spectral gap is bounded below by that of a single-particle chain multiplied by an explicit rate factor (Hermon et al., 2018).
• Symmetric inclusion process and Brownian energy process: universal comparison theorems relating their spectral gaps to that of underlying random walks, with equality in log-concave stationary measure regimes (Kim et al., 2023).
• Markov chains on symmetric groups (Aldous property): second-largest eigenvalue of Cayley graphs generated by transpositions and certain conjugacy classes is attained by the standard representation (Li et al., 2022, Parzanchevski et al., 2018).

The key technical ingredients include operator inequalities (notably the “octopus inequality”) and representation-theoretic decompositions in symmetric-group group algebras (Alon et al., 2023, Cesi, 2013).

5. Gap Formulas in Invariant Geometries, Manifolds, and Other Models

Spectral gap conjectures and their analogues appear in broader settings:

• Gap bounds on conformally flat and negatively curved manifolds under convexity/horoconvexity constraints (Khan et al., 24 Apr 2024): explicit dimension- and diameter-controlled lower bounds obtained via log-concavity methods, coupling of diffusions, and comparison to one-dimensional models. Notably, convexity alone does not suffice in hyperbolic geometry—horoconvexity is necessary.
• Resonance gaps in random Schottky surfaces: the spectral gap for resonances of the Laplacian on random covers is shown to match the optimal half-plane predicted by the Jakobson–Naud conjecture, with probabilistic confirmation in the large-degree regime (Calderón et al., 31 Jul 2024).
• Algebraic graphs: minimal spectral gap conjectures for regular graphs are established for cubic and quartic (k=3,4) cases (Aldous–Fill conjecture), with explicit graph constructions and path-like block decomposition yielding $\mu_2 \sim (1+o(1)) \frac{2k\pi^2}{3n^2}$ for large $n$ (Abdi et al., 2019, Abdi et al., 2020).

6. Representation-Theoretic Principles and Absorption Phenomena

Advances in representation theory have led to general principles regarding the persistence of spectral gap under tensor products and other operations. The spectral gap absorption principle (Gorfine, 10 Apr 2025) states:

If a unitary representation $\pi$ of a semisimple algebraic group $G$ over a local field has spectral gap, then so does every tensor product $\pi\otimes\rho$ (for arbitrary $\rho$). This is formalized via filtration of the unitary dual $\widehat{G}$ by $L^p$-integrability of matrix coefficients; spectral gap is equivalent to membership in some $\widehat{G}_p$ for $p<\infty$. These filtration-ideal arguments resolve conjectures of Bader–Sauer and Bekka–Valette regarding the non-density of the restriction map and the general absorption of spectral gap.

7. Limitations, Failures, and Open Problems

• The extension of the Robin gap conjecture to negative parameters fails entirely: infimal spectral gaps over convex domains drop to zero (Kielty, 2021). This marks a fundamental structural threshold in boundary behavior.
• In the setting of normal Cayley graphs, the Aldous property is not universal for arbitrary invariant sets: additional structure or restrictions (conjugacy class type, number of fixed points) are required for second eigenvalue domination (Li et al., 2022, Parzanchevski et al., 2018).
• For spectral gap conjectures in exclusion-type interacting systems, generalization beyond cases with explicit product invariance or log-concavity remains challenging.
• For regular graphs with higher degrees ($k\geq 5$), full structural characterizations of minimizers and matching gap formulas are open, with indications of complexity escalation in block decomposition.

This body of work establishes the spectral gap conjecture as a nexus in analysis, probability, combinatorics, representation theory, and mathematical physics. The sharp identification of gap bounds, the explicit construction of extremal objects (domains, graphs, algebraic elements), and understanding of failures and transitions underpin both foundational theory and applications in mixing, relaxation, and rigidity.