Asymptotic Spectrum Duality Explained

• Asymptotic spectrum duality is a unifying framework that links combinatorial, algebraic, and quantum invariants through optimization over convex sets of monotone, multiplicative, and additive functionals.
• The framework leverages semiring structures and Strassen-type theorems to connect maximization and minimization problems, establishing powerful minimax equalities for key operational bounds.
• It underpins analytic studies of limit objects and dual growth constants, offering continuous relaxations and metric refinements in graph theory, quantum information, and algebraic geometry.

Asymptotic spectrum duality is a unifying algebraic and analytic framework that characterizes the asymptotic behavior of combinatorial, algebraic, and quantum operational invariants. At its core, the duality relates maximization and minimization problems (such as Shannon capacity, LOCC-convertibility rates, or algebraic regularity constants) to optimization over convex sets of monotone, multiplicative, additive functionals—collectively called the asymptotic spectrum. The duality involves semiring structures, Strassen-type theorems, and, in recent developments, analytic metric and probabilistic refinements, unifying operational bounds and providing continuity and “distance” frameworks for their invariants.

1. Semiring Structures and Spectrum Definition

The foundation of asymptotic spectrum duality is the identification of the problem's objects (e.g., graphs, quantum states, graded ideals) with elements of a commutative semiring equipped with a relevant preorder. For instance, in the case of graphs, the semiring consists of isomorphism classes of finite graphs, with additive and multiplicative operations given by disjoint union ($G \sqcup H$) and strong product ($G \boxtimes H$), respectively. The preorder $G \leq H$ is defined by the existence of a cohomomorphism from $G$ to $H$ (or analogous morphisms in other settings) (Zuiddam, 2018, Boer et al., 25 Apr 2024).

Given such a structure $(S, +, \cdot, 0, 1, \leq)$, the asymptotic spectrum $\mathcal{X}$ is the set of non-negative, monotone, additive, and multiplicative semiring homomorphisms: $$\mathcal{X} = \{\varphi : S \to \mathbb{R}_{\ge 0} \mid \varphi(a+b) = \varphi(a) + \varphi(b), \, \varphi(a \cdot b) = \varphi(a)\varphi(b), \, \varphi(1) = 1,\, a \leq b \implies \varphi(a) \leq \varphi(b)\}.$$ Analogous constructions arise in quantum Shannon theory via projective or entanglement-assisted homomorphism preorders (Li et al., 2018), and in commutative algebra for invariants of graded families (DiPasquale et al., 2022).

2. Duality Theorems and Pointwise Characterizations

The principal theorems of asymptotic spectrum duality are Strassen-type minimax equalities: maximal (or minimal) asymptotic growth rates of key invariants are given as pointwise minima (or maxima) over the spectrum $\mathcal{X}$. Concretely, for the Shannon capacity $\Theta(G)$ of a finite graph $G$,

$$\Theta(G) = \min_{\varphi \in \mathcal{X}} \varphi(G),$$

where $\mathcal{X}$ encodes all upper bound methods monotone under $\leq$, additive under $\sqcup$, and multiplicative under $\boxtimes$. Similarly, the fractional clique-cover number $\bar{\chi}_f(G)$ emerges as a pointwise maximum (Boer et al., 25 Apr 2024, Zuiddam, 2018, Vrana, 2019).

This duality extends to quantum and entanglement-assisted graph invariants (e.g., quantum Shannon capacity), LOCC conversion rates via spectral points of the corresponding semiring (Li et al., 2018, Jensen et al., 2018), and algebraic growth rates (e.g., Waldschmidt or Seshadri constants) via sequence transforms (DiPasquale et al., 2022).

3. Spectrum Elements and Explicit Bounds

Within this framework, celebrated upper (and lower) bounds correspond to explicit elements of the spectrum. In graphs, notable spectral points include:

• Lovász theta function $\vartheta(G)$: semidefinite relaxation of the independence number.
• Fractional clique-cover number $\bar{\chi}_f(G)$: linear-programming relaxation.
• Fractional Haemers bound $H_f^F(G)$: field-dependent matrix rank method.
• Complement of fractional projective rank $\bar{\xi}_f(G)$: quantum and field-theoretic bounds.

These invariants are spectrum members due to their monotonicity, additivity, and multiplicativity, and collectively explain the equivalence of techniques across information and graph theory. In quantum or non-commutative settings, spectrum elements diverge: some classical spectral points fail to survive quantum monotonicity constraints, but Lovász theta and fractional Haemers bounds over $\mathbb{R}$ or $\mathbb{C}$ do (Li et al., 2018).

In commutative algebra, growth constants like the Waldschmidt constant and Seshadri constant occupy dual points in the asymptotic spectrum, and a dual transformation interchanges their roles, reciprocating their asymptotic values (DiPasquale et al., 2022).

4. Metric and Probabilistic Refinements

Recent advances generalize the duality from extremal (pointwise) bounds to analytic and probabilistic frameworks. The asymptotic spectrum distance is defined as

$$d(G, H) = \sup_{\varphi \in \mathcal{X}} |\varphi(G) - \varphi(H)|,$$

yielding a pseudometric (metric on spectrum-quotiented objects) that quantifies the analytic distance between objects in terms of all known spectrum bounds (Boer et al., 25 Apr 2024). Convergence in this metric ensures continuity of invariants like Shannon capacity, enabling graph limit constructions and analytic approximations.

Probabilistic refinements assign to each spectrum element $f$ and graph $G$ a function $f(G, P)$ on probability distributions $P$ over $V(G)$, representing exponential growth rates on typical type classes and encoding a concave entropy-like structure: $F(G, P) := \log f(G, P) = H_{C_f(G)}(P),$ where $C_f(G)$ is a canonically associated convex corner (Vrana, 2019). This parameterizes the spectrum by convex sets and bridges combinatorial invariants with entropy, convex duality, and information-theoretic methodologies.

5. Applications: Limit Objects and Sequence Dualities

Asymptotic spectrum duality underpins the analytic paper of object limits and dual invariants. In the theory of graph limits, sequences of fraction graphs $(E_{p_n/q_n})$ converge in spectrum distance to infinite circle graphs $E_r^o$ or $E_r^c$, producing genuine infinite-graph points unattainable by finite graphs. This realizes new analytic, infinite-dimensional structures in extremal graph theory and explains continuity phenomena for graph invariants such as the Lovász theta and Haemers bounds (Boer et al., 25 Apr 2024).

In commutative algebra, duality interchanges subadditive and superadditive sequences (e.g., symbolic power degrees and regularities), with the sequence transform

$a^*_n = \sup\{ m : a_m \leq n \}$

mapping growth constants $\alpha \mapsto 1/\alpha$, encapsulating reciprocity between asymptotic regularity and Seshadri constants (DiPasquale et al., 2022).

A table highlighting analogies across settings:

Domain	Semiring Elements	Spectrum Invariant	Spectral Duality
Graph Theory	Graphs	$\vartheta$, $\bar{\chi}_f$, $H^F_f$	$\Theta(G) = \min_{\varphi} \varphi(G)$
Quantum Info/LOCC	Quantum states	Rényi Monotones, etc.	LOCC rates via $\inf_{\varphi}$ formulas
Algebraic Geometry	Graded Ideals	Waldschmidt, Seshadri, Reg	Reciprocation, $\alpha \leftrightarrow 1/\alpha$

6. Generalizations and Outlook

Asymptotic spectrum duality, via analytic, algebraic, and operational interpretations, has unified and generalized bounds, continuity frameworks, and duality principles across combinatorics, quantum information, and algebraic geometry. The analytic metric and entropy-rich convex-corner perspectives pave the way for continuous relaxations, limit object analysis, and new infinite-dimensional techniques. Finding further examples—identifying semirings and dual paradigms in new domains—remains a prominent research direction (Boer et al., 25 Apr 2024, DiPasquale et al., 2022, Vrana, 2019).