In the two-dimensional Ising model on a finite cylinder, Dobrushin-type boundary conditions force domain walls whose incremental free energies were recently computed exactly as single Fourier-type integrals and analysed by steepest descent (Squarcini, Nowakowski, Abraham and Maciołek, 2025), yielding the angle-dependent surface tension, two point tensions, and a 'rattling' hierarchy of exponentially small terms describing repeated reflections of the interface between the cylinder edges. Two parts of that asymptotic theory are explicitly incomplete. First, the contour deformation assumes the integrand has no isolated poles between the real axis and the steepest-descent path; with nonzero surface fields a 'wetting pole' does occur, and its residue contribution to the finite-size free energy — with its coarse-grained interpretation — is not treated. Second, the saddle-point exponent is expanded only to quadratic order, so higher-order Laplace corrections, which can dominate lower-order rattling terms, are missing; no uniform expansion consistently ordering both exists. The open problem is to complete the theory: derive the surface-field extensions of the interface-structure integrals, classify their singularities, incorporate the wetting-pole residue contributions, and construct a complete uniform asymptotic expansion with controlled error bounds over the physical parameter regime, including the limiting interface angles at which the point tensions diverge.

Background

The planar Ising model with nearest-neighbour couplings J1,J2>0J_1,J_2>0 is the archetypal exactly solved statistical system: Onsager's 1944 solution and its fermionic reformulations (Kaufman's spinor analysis; the Schultz–Mattis–Lieb many-fermion reduction) underlie all later exact interfacial results. Wrap the model into a cylinder of circumference MM and height NN. Dobrushin-type boundary conditions — boundary spins fixed to +1+1 except on segments where they are fixed to −1-1 — force domain walls (interfaces) into the system, and the object of interest is the incremental (excess) free energy −ln⁡(Z+−/Z++)-\ln(Z_{+-}/Z_{++}), where Z+−Z_{+-} and Z++Z_{++} are the partition functions with and without reversed boundary segments. Two interfacial structures are basic: an interface running from one cylinder edge to the other with endpoint separation ss, with partition-function ratio ζA(N,s)\zeta_{\mathrm A}(N,s), and a 'droplet' pinned to one edge, with ratio ζB(N,s)\zeta_{\mathrm B}(N,s).

By the generalized Wick theorem within Dobrushin's program — the fermionic (Pfaffian) reduction of the transfer matrices — both ratios reduce, in the limit M→∞M\to\infty, to single integrals, e.g.

ζA(N,s)=∫−ππdω2π eisωcosh⁡[Nγ(ω)]+sinh⁡[Nγ(ω)]cos⁡δ∗(ω),\zeta_{\mathrm A}(N,s)=\int_{-\pi}^{\pi}\frac{\mathrm{d}\omega}{2\pi}\,\frac{\mathrm{e}^{\mathrm{i}s\omega}}{\cosh[N\gamma(\omega)]+\sinh[N\gamma(\omega)]\cos\delta^{*}(\omega)},

where cosh⁡γ(ω)=c1⋆c2−s1⋆s2cos⁡ω\cosh\gamma(\omega)=c_1^{\star}c_2-s_1^{\star}s_2\cos\omega with ci=cosh⁡2Kic_i=\cosh 2K_i, si=sinh⁡2Kis_i=\sinh 2K_i, Ki=Ji/(kBT)K_i=J_i/(k_{\mathrm B}T), and ⋆\star denoting the Kramers–Wannier dual coupling defined by tanh⁡K⋆=e−2K\tanh K^{\star}=\mathrm{e}^{-2K}; δ∗(ω)\delta^{*}(\omega) is the Bogoliubov–Valatin rotation angle — the mixing angle that diagonalizes the quadratic form of the underlying fermions. More generally one meets the class I(M,N)=∫−ππdω2π g(ω) e−Nγ(ω)+iMωI(M,N)=\int_{-\pi}^{\pi}\frac{\mathrm{d}\omega}{2\pi}\,g(\omega)\,\mathrm{e}^{-N\gamma(\omega)+\mathrm{i}M\omega}, ubiquitous in exact Ising asymptotics.

The 2025 analysis (Squarcini, Nowakowski, Abraham and Maciołek) extracts the large-M,NM,N behaviour of such integrals by the steepest-descent method: with M=[Ntan⁡ϑ]M=[N\tan\vartheta] ([ ⋅ ][\,\cdot\,] denoting the integer part), the exponent −NL(ω)-NL(\omega), L(ω)=γ(ω)−iωtan⁡ϑL(\omega)=\gamma(\omega)-\mathrm{i}\omega\tan\vartheta, has a unique saddle point on the imaginary axis, ωs=ivs(ϑ)\omega_{\mathrm s}=\mathrm{i}v_{\mathrm s}(\vartheta), and the steepest-descent path (SDP — the contour through the saddle on which the exponent is real and decays fastest away from it) is determined exactly. Expanding the exponent to quadratic order at the saddle yields the angle-dependent surface tension τ(ϑ)\tau(\vartheta) — the exact quantity introduced by Onsager for the lattice axes and computed at general orientation by Abraham and Upton — plus an entropic N1/2N^{1/2} fluctuation factor. Expanding gg in a geometric series in the exponentially small quantity e−2Nγ(ω)\mathrm{e}^{-2N\gamma(\omega)} produces a hierarchy of terms whose saddle points sit at angles ϑn\vartheta_n obeying tan⁡ϑn=tan⁡ϑ/(2n+1)\tan\vartheta_n=\tan\vartheta/(2n+1) for the crossing interface and tan⁡ϑn=tan⁡ϑ/(2n)\tan\vartheta_n=\tan\vartheta/(2n) for the droplet: these are the 'rattling' contributions, in which the interface reflects nn times off the cylinder walls, even and odd reflection numbers being topologically distinct. The amplitudes are expressed through two point tensions — τp\tau_{\mathrm p}, the free-energy cost of the point where the interface is pinned to the boundary, and τfcp\tau_{\mathrm{fcp}}, the analogous cost for fluctuating contact points — with τp→∞\tau_{\mathrm p}\to\infty as ϑ→π/2\vartheta\to\pi/2, τfcp→∞\tau_{\mathrm{fcp}}\to\infty as ϑ→0\vartheta\to 0, and the constraint that when one diverges the other vanishes.

Two parts of this theory are explicitly incomplete, and both gaps constitute the present problem. (i) Pole-free assumption: the whole contour deformation assumes g(ω)g(\omega) has no isolated poles in the region swept between the original real axis and the SDP. With nonzero boundary fields (surface fields — magnetic fields acting only on edge spins), the setting of wetting, in which the interface unbinds from a wall, such a pole does occur: the 'wetting pole', known from the exact corner-filling solutions of Abraham and Maciołek and from exact results for Ising strips with boundary fields. Analogous bound-state poles occur in integrable field theories (Squarcini and Tinti 2023) and in the Baxter (eight-vertex) model (Abraham and Owczarek 1990) — the exactly solved generalization of the six-vertex (ice-type) model in which all eight arrow configurations around a lattice vertex are allowed. When the deformation crosses such a pole its residue must be added, producing a new exponential contribution whose coarse-grained meaning in the present setting is unknown. (ii) Non-uniform expansion: at each saddle the exponent is truncated at quadratic order; the neglected higher-order Laplace corrections produce subleading terms that, in the source's own words, 'typically dominate over the lower order terms in the series', so the published expansion is not a controlled asymptotic expansion — no uniform scheme consistently ordering the geometric-series terms and the beyond-quadratic saddle corrections exists. The limiting-angle divergences of the point tensions, and the vanishing of the axis surface tensions τ(0)=γ(0)\tau(0)=\gamma(0) and τ(π/2)=γ^(0)\tau(\pi/2)=\hat\gamma(0) — where γ^\hat\gamma is defined by cosh⁡γ^(ω)=c1c2⋆−s1s2⋆cos⁡ω\cosh\hat\gamma(\omega)=c_1c_2^{\star}-s_1s_2^{\star}\cos\omega, the same expression with the Kramers–Wannier dual applied to the other coupling — as T→TcT\to T_{\mathrm c}, mark the boundaries of the regime over which such a uniform expansion is sought.

Problem Statement

Complete the steepest-descent asymptotic theory of the two-dimensional Ising interface-structure partition functions — the class of integrals

I(M,N)=∫−ππdω2π g(ω) e−Nγ(ω)+iMω,cosh⁡γ(ω)=c1⋆c2−s1⋆s2cos⁡ω,I(M,N)=\int_{-\pi}^{\pi}\frac{\mathrm{d}\omega}{2\pi}\,g(\omega)\,\mathrm{e}^{-N\gamma(\omega)+\mathrm{i}M\omega},\qquad\cosh\gamma(\omega)=c_1^{\star}c_2-s_1^{\star}s_2\cos\omega,

where ci=cosh⁡2Kic_i=\cosh 2K_i and si=sinh⁡2Kis_i=\sinh 2K_i with Ki=Ji/(kBT)>0K_i=J_i/(k_{\mathrm B}T)>0 the dimensionless nearest-neighbour couplings of the planar Ising model, ⋆\star denoting the Kramers–Wannier dual coupling defined by tanh⁡K⋆=e−2K\tanh K^\star=\mathrm{e}^{-2K}, and g(ω)g(\omega) the kernel of a Dobrushin-type interfacial structure (in particular the ratios ζA(N,s)\zeta_{\mathrm A}(N,s) and ζB(N,s)\zeta_{\mathrm B}(N,s) of Squarcini, Nowakowski, Abraham and Maciołek 2025) — along the two directions the source work identifies as unresolved.

  1. Pole and residue contributions. Extend the exact contour-deformation/steepest-descent analysis to integrands whose g(ω)g(\omega) possesses isolated poles in the region swept between the real axis and the steepest-descent path (SDP — the contour through the saddle point on which the exponent decreases fastest away from it). Concretely: (a) extend the exact calculation of the interface incremental free energies to nonzero boundary (surface) fields, obtaining the corresponding single-integral representations; (b) classify the singularities of the resulting kernels — in particular determine the location of the wetting pole relative to the saddle point and the SDP across the wetting phase diagram of the surface field; (c) include the residue contributions in the large-M,NM,N asymptotics and give their physical, coarse-grained interpretation (as surface-tension-, point-tension-, or genuinely new wetting-type contributions) on the same footing as the existing rattling decomposition.

  2. Uniform asymptotic expansion. Construct a complete asymptotic expansion of the incremental free energies in which the geometric ('rattling') series in e−2Nγ(ω)\mathrm{e}^{-2N\gamma(\omega)} and the higher-order (beyond-quadratic) Laplace corrections around each saddle point are simultaneously retained and consistently ordered, with an explicit bound on the truncation error, valid uniformly over the physical parameter regime (T<TcT<T_{\mathrm c} fixed, 0<ϑ<π/20<\vartheta<\pi/2, large pinning separations ss), and characterizing the approach to the limiting angles ϑ→0\vartheta\to0 and ϑ→π/2\vartheta\to\pi/2 at which τfcp\tau_{\mathrm{fcp}} and τp\tau_{\mathrm p} diverge.

A complete answer states exactly which exponentially small NN-dependent contributions exist for these integrals, computes them explicitly (including residue terms when poles intervene), and derives the ordering and error control.

References

  1. Alessio Squarcini, Piotr Nowakowski, Douglas B. Abraham, Anna Maciołek, 'Partition function for several Ising model interface structures', Eur. Phys. J. B 98, 186 (2025); arXiv:2506.17170; DOI 10.1140/epjb/s10051-025-01030-0; https://arxiv.org/abs/2506.17170
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  10. Alessio Squarcini, Antonio Tinti, 'Interfacially adsorbed bubbles determine the shape of droplets', SciPost Phys. 15, 164 (2023); DOI 10.21468/SciPostPhys.15.4.164; https://doi.org/10.21468/SciPostPhys.15.4.164
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