Discrete Markov Spectrum

• Discrete Markov spectrum is the set of eigenvalues derived from Markov processes and quadratic forms, linking approximation properties with combinatorial structures.
• It is characterized by rigorous formulations using operator theory, snake graph constructions, and continued fraction matrix factorizations to yield explicit spectral values.
• The spectrum informs the analysis of Markov chains, metastability phenomena, and discrete Laplacians, offering practical insights for statistical estimation and data-driven modeling.

The discrete Markov spectrum, in its various formalizations, encapsulates the spectrum—typically the set of real or complex eigenvalues—associated with finite or countable-state Markov processes, as well as arising in number-theoretic, dynamical, and geometric contexts under “Markov-type” constraints. Across classical analysis, algebraic number theory, dynamical systems, probabilistic Markov chains, and spectral theory, the discrete Markov spectrum emerges as a central object determining extremal approximation properties, recurrence rates, metastable transition times, and algebraic-geometric quantities tied to discrete dynamical data. This article gives a rigorous, detailed account of the core mathematical definitions, underlying structures, and spectral-theoretic results governing the discrete Markov spectrum, with particular attention to the connections between operator theory, Diophantine approximation, combinatorial tree structures, metastable dynamics, and spectral gaps.

1. Classical Discrete Markov Spectrum: Number-Theoretic Origin

The classical discrete Markov spectrum is best understood in the context of indefinite binary quadratic forms. For a real quadratic form $f(x, y) = Ax^2 + Bxy + Cy^2$ of positive discriminant $\Delta(f) = B^2 - 4AC > 0$, the Markov minimum is

$$m(f) = \inf_{(x,y) \in \mathbb{Z}^2\setminus\{(0,0)\}} |f(x, y)|,$$

and its normalized value

$M(f) = \frac{m(f)}{\sqrt{\Delta(f)}}.$

The (classical) Markov spectrum is the set of all such $M(f)$ for varying $f$, and the discrete Markov spectrum is the special subset arising from the Markov equation: $x^2 + y^2 + z^2 = 3 x y z,$ whose positive integer solutions $(x, y, z)$, the Markov triples, generate the sequence of Markov numbers $n_i$. For each Markov number $n$, one sets

$r(n) = \frac{\sqrt{9 n^2 - 4}}{n},$

defining the discrete Markov spectrum $$\mathcal{M}_d = \{ r(n) \mid n \text{ a Markov number} \}$$. Markov's theorem establishes $\mathcal{M}_d$ as the segment of the Markov and Markov–Lagrange spectra below $3$, i.e., $\mathcal{M}_d = \mathcal{L}_{<3}$, where $$\mathcal{L}(\alpha) = \limsup_{q \to \infty} (1/(q \|q\alpha\|))$$ is the Lagrange spectrum and $\|x\|$ denotes distance to $\mathbb{Z}$ (Gyoda, 4 Dec 2025).

2. Combinatorial and Algebraic Structures: Trees and Snake Graphs

The set of Markov numbers has a rich combinatorial structure. Markov triples are related by "mutation" operations forming a binary tree: given $(a, b, c)$, one generates $(a, 3ab - c, b)$, with every positive solution reachable from the root $(1,1,1)$ by a sequence of mutations. This tree structure is encoded combinatorially via snake graphs: for each Markov number $n$, there exists a strongly admissible sequence $s(t) = (a_1, ..., a_k)$ yielded by labelling with a Farey rational slope $t$, and a corresponding snake graph $\mathcal{G}[a_1, ..., a_k]$. The number of perfect matchings $m(\mathcal{G}[s(t)])$ recovers the Markov number $n_t$.

Algebraically, these perfect matchings correspond to numerators and denominators of regular continued fractions, via continued-fraction matrix factorizations, and Cohn matrices attached to these sequences satisfy

$$C_t = \begin{pmatrix} 3 n_t - u_t & * \ n_t & u_t \end{pmatrix},$$

where the trace $3 n_t$ and determinant $1$ encode the Markov form structure. This yields an explicit, combinatorial route to generating all discrete Markov values and their continued-fraction expansions (Gyoda, 4 Dec 2025).

3. Generalized Discrete Markov Spectra

Recent developments consider generalizations of the Markov equation: $$x^2 + y^2 + z^2 + k_1 y z + k_2 z x + k_3 x y = (3 + k_1 + k_2 + k_3) x y z,$$ whose positive integer solutions are $(k_1, k_2, k_3)$-generalized Markov numbers. The corresponding tree structures, generalized snake graphs, and matrix factorizations extend the classical constructions. The generalized discrete Markov spectrum is then

$$\mathcal{M}_{k_1, k_2, k_3} = \Big\{ r_{k_1, k_2, k_3}(t) = \frac{\sqrt{((3 + k_1 + k_2 + k_3)n_t - k_{i_t})^2 - 4}}{n_t} : t \in \mathbb{Q}_{\geq 0} \cup \{\infty\} \Big\}.$$

It is proven that for all such $k_i$, $\mathcal{M}_{k_1, k_2, k_3} \subset \mathcal{L}$, i.e., the generalized discrete Markov spectra are contained in the Markov–Lagrange spectrum (Gyoda, 4 Dec 2025).

Unique recurrence properties of Markov numbers within such trees, as articulated by the generalized uniqueness conjecture, are resolved negatively: the same value may arise from distinct branches for nontrivial $k_i$ (Karpenkov et al., 2018). Moreover, structural results relate sets such as $\mathcal{M}_{2,2,2}$ and $\mathcal{M}_{0,0,0}$ via explicit scaling: $\mathcal{M}_{2,2,2} = 3\,\mathcal{M}_{0,0,0}$.

4. Spectral Properties in Discrete Markov Chains

The notion of a discrete Markov spectrum is also central in the spectral analysis of finite-state Markov chains. For a transition matrix $P\in\mathbb{R}^{N\times N}$, the spectrum $\sigma(P)$ is the set of eigenvalues, which by the Perron–Frobenius theorem all lie within the unit disk, and $1$ is always a simple eigenvalue corresponding to the stationary distribution. The spectrum is additionally constrained to the Karpelevič region $\Theta_N$, a subset of the unit disk determined by Farey fractions; its boundary consists of analytic arcs associated with extremal stochastic matrices, and each nontrivial eigenvalue encodes transient behavior, mixing rates, or periodicities in the chain (Chruściński et al., 10 Apr 2025).

For continuous-time Markov generators, after rescaling by a maximal rate $\nu_c$, the spectra of the normalized generator fall within a "modified" Karpelevič region. Stochastic matrices at the boundary of $\Theta_N$ (for low $N$) have explicit realizing forms, and the combinatorics of eigenvalue locations are described in terms of Farey intervals and analytic arcs (Chruściński et al., 10 Apr 2025).

5. Metastable and Structured Markov Chains: Spectral Asymptotics

Specializing to finite grids approximating SDE dynamics in multiwell potentials (e.g., stochastic differential equations driven by Lévy flights), spectral analysis reveals a characteristic separation of spectral scales. For the generator $Q^\epsilon$ approximating small noise dynamics in an $n$-well potential,

• There exist exactly $n$ eigenvalues $\lambda^\epsilon_i = O(\epsilon^\alpha)$, where $\alpha$ is the stability index of the Lévy noise, converging to the spectrum of a reduced $n \times n$ generator $Q$ as $\epsilon\to 0$.
• A pronounced spectral gap separates these eigenvalues from the rest, corresponding to metastability: sojourn times in wells are of order $1/\lambda^\epsilon_2$, and the leading dynamics are governed by interwell transitions.
• The eigenvectors are nearly constant on subsets associated with individual wells, with the principal eigenvector corresponding to the equilibrium distribution (Burghoff et al., 2015).

This structure allows one to reduce the long-term metastable dynamics to effective jump processes between wells, quantitatively predicted by the top of the discrete Markov spectrum.

6. Discrete Markov Spectra in Complexes and Operator Theory

Discrete Markov spectra generalize naturally to higher-order combinatorial and geometric structures. On an abstract finite simplicial complex $K$ of dimension $N$, the spectrum of the discrete Hodge Laplacian $\Delta_k$ arises as a form of "discrete Markov spectrum," determining the rates of convergence of up-walk and down-walk Markov processes on oriented $k$-simplices. The zero-eigenspace encodes $k$-dimensional homology (via the $k$-th Betti number), and the spectral gap $\lambda_k$ controls the mixing rate of the associated chain. Notably, the irreducibility of the chain is guaranteed for top-dimensional down-walks if and only if the underlying complex is orientable, a result that extends the spectral-geometric correspondence from graphs to higher-dimensional complexes (Eidi et al., 2023).

7. Broader Spectral Methods and Statistical Estimation

Spectral decompositions underpin statistical and data-driven analyses of Markov models, including state aggregation, compression, and the recovery of lumpable partitions. For a finite-state Markov chain, spectral features and singular vectors can be estimated efficiently from data, with theoretical guarantees governed by the Markov rank, mixing times, and spectral gaps. Low-rank Markov models allow statistically optimal recovery of leading spectral subspaces and feature spaces, and spectral clustering methods group state space blocks via eigenvector constancy. The ability to estimate discrete Markov spectra with provable accuracy from finite observations connects the abstract operator-theoretic spectrum to practical data-driven modeling (Zhang et al., 2018).

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In summary, the discrete Markov spectrum unites number-theoretic, combinatorial, operator-theoretic, and statistical frameworks via a common spectral analytic structure. Its manifestations—classically as the spectrum of minima obtained from the Markov equation and its generalizations; probabilistically as the eigenstructure of Markov transition operators and their continuous-time generators; and geometrically in the Laplacians of complexes—link diverse mathematical disciplines through sharp spectral invariants, recurrence properties, and algebraic-combinatorial identities. Recent advances integrate combinatorial constructions (snake graphs), generalizations to broader Diophantine equations, and operator-theoretic spectral regions to offer a comprehensive theory with both deep structural results and direct statistical modeling implications (Gyoda, 4 Dec 2025, Chruściński et al., 10 Apr 2025, Eidi et al., 2023, Burghoff et al., 2015, Karpenkov et al., 2018, Zhang et al., 2018).