One-dimensional classical spin systems with exponentially decaying two-body interactions have a complete spectral thermodynamics: by the construction of Ruelle and Araki developed in D. H. Mayer's 1980 monograph, the associated transfer operator is nuclear (trace class) on a Banach space of holomorphic functions, so the free energy is the logarithm of its leading eigenvalue, correlations are controlled by its spectrum, and the dynamical zeta function is a Fredholm determinant. That construction is documented in the monograph for exponentially decaying couplings J(i)=exp⁡(−γi)J(i)=\exp(-\gamma i), γ>0\gamma>0 (Ch. III.1) and for stretched-exponential couplings J(i)=a(i)exp⁡(−γiβ)J(i)=a(i)\exp(-\gamma i^{\beta}) with β≥1\beta\ge 1 (Ch. III.2), while the polynomially decreasing case occupies only the four pages of Ch. III.3 (pp. 107–110) and the polynomially decaying limiting case of Kac's program was left without a rigorous transfer-operator treatment; nothing in the monograph covers stretched-exponential decay with 0<β<10<\beta<1. Whether an analogous invariant-subspace / Banach-space construction exists for these slower decays — polynomial couplings J(i)∼i−mJ(i)\sim i^{-m} (the Dyson class) and stretched-exponential couplings J(i)∼a(i)exp⁡(−γiβ)J(i)\sim a(i)\exp(-\gamma i^{\beta}) with 0<β<10<\beta<1 — and whether the failure to treat polynomial decay is a fundamental obstruction or merely an unsuitable choice of invariant subspace, remains open. No subsequent work located in the literature resolves it: the Mayer school (Thron, Michalski, Hilgert–Mayer, Rilke, 1991–2007) never left the exponential and hard-rod classes, and the modern 2019–2025 line on transfer operators for Dyson models establishes only eigenfunction existence in restricted regimes (continuous eigenfunctions for couplings θk−m\theta k^{-m} with m>3/2m>3/2 at subcritical temperature and zero field, where θ\theta denotes the inverse temperature; merely integrable, provably non-continuous eigenfunctions in a field). No nuclearity, compactness, spectral-gap, eigenvalue-asymptotic, or free-energy-analyticity result, and no impossibility theorem, exists for any slower-than-exponential decay class.

Background

One-dimensional systems and transfer operators. A one-dimensional lattice spin system places spins σi\sigma_i (taking values in a finite set such as {±1}\{\pm 1\}, or in a continuous set such as an interval) on the sites i∈Zi\in\mathbb{Z}, with an interaction energy built from pair couplings J(j−i)J(j-i) and single-site terms. For finite-range interactions the Kramers–Wannier transfer matrix reduces equilibrium thermodynamics to a finite-dimensional eigenvalue problem. Ruelle's thermodynamic formalism extends this to longer ranges: writing the interaction as a potential ϕ\phi on the one-sided shift space (the space of one-sided spin sequences, equipped with the shift map TT that deletes the first symbol), one studies the Ruelle transfer operator Lϕf(x)=∑y: Ty=xeϕ(y)f(y),\mathcal{L}_{\phi}f(x)=\sum_{y:\,Ty=x}e^{\phi(y)}f(y), acting on functions ff. The variation varnϕ\mathrm{var}_n\phi measures how strongly ϕ\phi depends on spins at distance greater than nn from the origin; when the variations are summable (∑nvarnϕ<∞\sum_n\mathrm{var}_n\phi<\infty) the classical theory yields a strictly positive continuous eigenfunction Lϕh=λh\mathcal{L}_{\phi}h=\lambda h and a spectral-gap picture in which the free energy per site equals log⁡λ\log\lambda. A nuclear (trace-class) operator is a compact operator whose singular values are summable; such operators have discrete spectra with eigenvalues of faster-than-root-exponential decay and admit Fredholm determinants (a determinant-type characteristic function), which is what elevates the free energy to an analytic, not merely continuous, function of temperature and fields.

The Kac program and the Ruelle–Araki operator. Mark Kac introduced one-dimensional Ising systems with exponentially decaying couplings (the Kac or Kac–Baker model) to give a rigorous one-dimensional realization of van der Waals–type mean-field behavior, such as the van der Waals equation of state and Weiss ferromagnetism. For genuine two-body long-range interactions no finite-dimensional transfer matrix exists; the Ruelle–Araki transfer operator, named after David Ruelle and Huzihiro Araki, is the object that plays the role of the transfer matrix in this setting. Mayer's monograph — the source of the present problem — developed it into a full spectral theory for exponentially decaying interactions: the operator acts on a carefully chosen invariant Banach space of holomorphic functions and is nuclear there, so the free energy is the logarithm of the leading eigenvalue, the dynamical zeta function (the generating function of the partition functions) is a Fredholm determinant, and analyticity and differentiability statements for the free energy and correlations follow. As documented in the monograph, the method covers exponentially decaying interactions (and, for continuous spins, stretched-exponential decays with exponent β≥1\beta\ge 1), but not slower decays.

Polynomial and sub-exponential decay. Kac's physical target, the van der Waals gas, involves couplings decaying only polynomially. For J(i)∼i−mJ(i)\sim i^{-m} (the Dyson model; m>1m>1 is required for the interaction energy per site to be finite) the situation is qualitatively different: Dyson (1969) proved a genuine low-temperature phase transition for 1<m<21<m<2 and its absence for m>2m>2, Fröhlich and Spencer (1982) closed the borderline case m=2m=2, and Aizenman, Chayes, Chayes and Newman (1988) proved discontinuity of the magnetization in part of this range. Hence for 1<m≤21<m\le 2 the free energy cannot be globally analytic and the associated shift-space potential has non-summable variations, so the classical one-sided transfer-operator theory does not apply; for m>2m>2 variations are summable and the standard one-sided theory gives uniqueness of equilibrium states, yet no nuclear, analytic-space construction of the Ruelle–Araki type is known even there. Stretched-exponential couplings J(i)∼a(i)exp⁡(−γiβ)J(i)\sim a(i)\exp(-\gamma i^{\beta}) with 0<β<10<\beta<1 interpolate between the two regimes. Whether a Mayer-type construction — an invariant Banach space on which the transfer operator for these slower decays is nuclear, or compact in a sense that delivers the same spectral description — exists is the question left open by the source's construction. (For modern terminology: a gg-measure, also called a Doeblin measure, is a measure on the one-sided shift space whose one-site conditional probabilities are prescribed by a continuous positive function gg — the one-sided analogue of a Gibbs measure; part of the recent work on these models is phrased in that language.)

Problem Statement

The problem left open by Mayer's monograph (Lecture Notes in Physics 123, 1980, whose construction is documented in Ch. III.1–III.3, the polynomially decreasing case being confined to Ch. III.3, pp. 107–110) and still unresolved, is to extend the Ruelle–Araki transfer-operator spectral program for one-dimensional classical systems with two-body long-range interactions beyond the decay classes that program is documented to cover. The common scientific objective is a transfer-matrix-level spectral description of these systems — an invariant-subspace / Banach-space construction on which the transfer operator is nuclear, or otherwise sufficiently compact, so that the free energy is the logarithm of the leading eigenvalue, the dynamical zeta function is a Fredholm-type determinant, correlations are controlled by the spectrum, and the corresponding analyticity or differentiability results for the free energy and correlations follow in the parameter regimes compatible with the known phase diagram. Concretely:

  1. Primary objective (construction or obstruction), for the two slower decay classes: (a) polynomial decay J(i)∼i−mJ(i)\sim i^{-m} with m>1m>1 (the Dyson class, where for 1<m≤21<m\le 2 phase transitions exist and any analyticity conclusion must be restricted to subcritical regimes, while for m>2m>2 no such transition occurs); and (b) stretched-exponential decay J(i)∼a(i)exp⁡(−γiβ)J(i)\sim a(i)\exp(-\gamma i^{\beta}) with 0<β<10<\beta<1. Determine whether there exists an invariant Banach-space construction on which the Ruelle–Araki operator is nuclear — trace class, as in the Banach spaces of holomorphic functions of the original method — in these regimes, with the spectral consequences above; a weaker compactness property is relevant only insofar as it actually delivers those consequences (free energy as the logarithm of the leading eigenvalue, dynamical zeta function as a Fredholm-type determinant, spectral control of correlations, and the resulting analyticity or differentiability statements in the phase-diagram-compatible regimes). Alternatively, prove a rigorous obstruction: that invariant Banach-space constructions of the Ruelle–Araki type cannot achieve this nuclearity, or those spectral consequences, for the stated decay classes — at minimum within the spaces of holomorphic functions of the original method — or that the desired analyticity conclusions fail. An obstruction must hold at least on the spaces of holomorphic functions of the original method; a non-compactness or essential-spectrum statement confined to a class of function spaces strictly narrower than those original spaces establishes only a weaker assertion.

  2. The obstruction question: determine whether the historical failure to treat polynomial decay is a fundamental obstruction of the method or merely a matter of finding a more suitable invariant subspace. Either outcome of item 1 settles this.

  3. Secondary components associated with the same program (context, not additional required deliverables): the analogous extension for continuous-spin systems, where the monograph's method handles only some decay regimes and leaves infinite-dimensional local-state complications unresolved; and the development of practical, computable algorithms or closed-form evaluations of the in-principle eigenvalue formulas for general interaction kernels Φ~\tilde{\Phi} beyond exactly solvable Kac-type examples.

References

  1. Dieter H. Mayer, The Ruelle-Araki Transfer Operator in Classical Statistical Mechanics, Lecture Notes in Physics 123, Springer-Verlag, Berlin/Heidelberg, 1980. ISBN 3-540-09990-5, DOI: 10.1007/bfb0017919 (source chapter 'Introduction and resumé', DOI: 10.1007/bfb0017920). https://doi.org/10.1007/bfb0017919
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  3. Jürg Fröhlich and Thomas Spencer, The phase transition in the one-dimensional Ising model with 1/r^2 interaction energy, Communications in Mathematical Physics 84 (1982), 87-101. DOI: 10.1007/BF01208373. https://doi.org/10.1007/BF01208373
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  12. Anders Johansson, Anders Öberg, Mark Pollicott, Continuous eigenfunctions of the transfer operator for Dyson models, Mathematische Zeitschrift 310 (2025), Article 62. DOI: 10.1007/s00209-025-03775-x, arXiv:2304.04202. https://arxiv.org/abs/2304.04202
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