The 2022 theorem of Basor, Ehrhardt, Gharakhloo, Its and Li proves a strong Szegő-type limit theorem for bordered Toeplitz determinants DNB[ϕ;ψ]D_N^B[\phi;\psi] with Szegő-type symbols and borders ψ=q1ϕ+q2\psi=q_1\phi+q_2, and applies it to the Au-Yang–Perk bordered-Toeplitz representation of the next-to-diagonal correlation ⟨σ0,0σN−1,N⟩\langle\sigma_{0,0}\sigma_{N-1,N}\rangle of the anisotropic square-lattice Ising model in the low-temperature regime k>1k>1, establishing the long-range order (1−k−2)1/4(1-k^{-2})^{1/4} and the anisotropy-dependent first correction. The authors explicitly leave open the regimes outside this nondegenerate Szegő setting: (i) the critical regime k=1k=1 and the high-temperature regime k<1k<1, where the Ising symbol ϕ^\hat\phi develops Fisher–Hartwig singularities, deferred to a subsequent publication; and (ii) degenerate borders whose bordered constant F[ϕ;ψ]F[\phi;\psi] vanishes identically (e.g. ψ=ϕ z/(z−c)\psi=\phi\,z/(z-c) with ∣c∣>1|c|>1), where the leading term of the Szegő scaling disappears and the next nonzero contribution — obtained by the authors only numerically, with conjectured exact constants — must be determined and proved.

Background

A Toeplitz determinant is the determinant DN[ϕ]=det⁡(ϕj−k)0≤j,k≤N−1D_N[\phi]=\det(\phi_{j-k})_{0\le j,k\le N-1} of the matrix built from the Fourier coefficients ϕn=∫Tz−nϕ(z) dz/(2πiz)\phi_n=\int_{\mathbb T}z^{-n}\phi(z)\,dz/(2\pi iz) of a function ϕ\phi (the symbol) on the unit circle T\mathbb T. The strong Szegő limit theorem states that for symbols that are smooth, nonvanishing, and of zero winding number on T\mathbb T, DN[ϕ]∼G[ϕ]NE[ϕ]D_N[\phi]\sim G[\phi]^N E[\phi] as N→∞N\to\infty, where G[ϕ]=exp⁡[log⁡ϕ]0G[\phi]=\exp[\log\phi]_0 and E[ϕ]=exp⁡∑n≥1n[log⁡ϕ]n[log⁡ϕ]−nE[\phi]=\exp\sum_{n\ge1}n[\log\phi]_n[\log\phi]_{-n}. When the symbol has zeros, algebraic singularities, or jump discontinuities on T\mathbb T — so-called Fisher–Hartwig singularities, introduced by Fisher and Hartwig — the asymptotics change qualitatively: powers N∑βj2N^{\sum\beta_j^2}, gamma-function constants, and exponential factors appear. The Fisher–Hartwig asymptotics were proved in full generality for Toeplitz, Hankel, and Toeplitz+Hankel determinants by Deift, Its and Krasovsky, with uniform versions due to Fahs.

A bordered Toeplitz determinant replaces the last column of the Toeplitz matrix by Fourier coefficients of a second symbol ψ\psi: DNB[ϕ;ψ]D_N^B[\phi;\psi] is the determinant of the N×NN\times N matrix with entries ϕj−k\phi_{j-k} in the first N−1N-1 columns and ψj\psi_j in the last column. Basor, Ehrhardt, Gharakhloo, Its and Li (2022) proved the analogue of the strong Szegő theorem for these determinants: if ϕ\phi is of Szegő type (smooth, nonzero, zero winding number on T\mathbb T, analytically continuable across it) and ψ=q1ϕ+q2\psi=q_1\phi+q_2 with q1,q2q_1,q_2 rational functions with poles off T\mathbb T, then DNB[ϕ;ψ]=G[ϕ]NE[ϕ](F[ϕ;ψ]+O(e−cN))D_N^B[\phi;\psi]=G[\phi]^NE[\phi](F[\phi;\psi]+O(e^{-cN})), where the bordered constant F[ϕ;ψ]=[ϕ−−1ψ]0/[ϕ+]0F[\phi;\psi]=[\phi_-^{-1}\psi]_0/[\phi_+]_0 is explicit in terms of the Wiener–Hopf factorization ϕ=ϕ−ϕ+\phi=\phi_-\phi_+ (factors analytic outside and inside T\mathbb T, respectively).

The driving application is the two-dimensional Ising model on the square lattice with horizontal and vertical couplings Jh,JvJ_h,J_v. With Sh,v=sinh⁡(2Jh,v/kBT)S_{h,v}=\sinh(2J_{h,v}/k_BT) and k=ShSvk=S_hS_v, the model is critical (T=TcT=T_c) when k=1k=1; k>1k>1 is the low-temperature (ordered) phase and k<1k<1 the high-temperature (disordered) phase. The diagonal correlation is the pure Toeplitz determinant DN[ϕ^]D_N[\hat\phi] with ϕ^(z)=(1−k−1z−1)/(1−k−1z)\hat\phi(z)=\sqrt{(1-k^{-1}z^{-1})/(1-k^{-1}z)}, which is of Szegő type exactly when k>1k>1; at k=1k=1 and k<1k<1 the symbol develops Fisher–Hartwig singularities on T\mathbb T, the source of the classical critical power-law decay and of the exponentially damped high-temperature asymptotics of the diagonal correlation. Au-Yang and Perk showed that the next-to-diagonal correlation ⟨σ0,0σN−1,N⟩\langle\sigma_{0,0}\sigma_{N-1,N}\rangle (spins separated by N−1N-1 horizontal and NN vertical steps) equals the bordered Toeplitz determinant DNB[ϕ^;ψ^]D_N^B[\hat\phi;\hat\psi] with ψ^(z)=(Cvzϕ^(z)+Ch)/(Sv(z−c∗))\hat\psi(z)=(C_vz\hat\phi(z)+C_h)/(S_v(z-c_*)), Ch,v=cosh⁡(2Jh,v/kBT)C_{h,v}=\cosh(2J_{h,v}/k_BT), c∗=−Sh/Svc_*=-S_h/S_v. Applying their theorem to this pair, Basor et al. proved that in the low-temperature regime the next-to-diagonal long-range order (the N→∞N\to\infty limit of the correlation, i.e. the square of the spontaneous magnetization (1−k−2)1/8(1-k^{-2})^{1/8}) equals (1−k−2)1/4(1-k^{-2})^{1/4}, the same as along the diagonal and horizontal directions, and computed the first subleading correction ⟨σ0,0σN−1,N⟩=(1−k−2)1/4(1+12π(1−k−2)(Cv−2+(k2−1)−1)N−2k−2N(1+O(N−1)))\langle\sigma_{0,0}\sigma_{N-1,N}\rangle=(1-k^{-2})^{1/4}\big(1+\frac{1}{2\pi(1-k^{-2})}(C_v^{-2}+(k^2-1)^{-1})N^{-2}k^{-2N}(1+O(N^{-1}))\big), whose coefficient encodes the anisotropy Jh≠JvJ_h\ne J_v.

Two gaps in this theory are explicitly left open by the authors. First, the theorem requires ϕ^\hat\phi of Szegő type, i.e. k>1k>1; the critical (k=1k=1) and high-temperature (k<1k<1) regimes, where ϕ^\hat\phi carries Fisher–Hartwig singularities, are deferred to a future publication. Second, the bordered constant can vanish identically — for example for borders ψ=ϕ z/(z−c)\psi=\phi\,z/(z-c) with ∣c∣>1|c|>1 — in which case the leading term of the Szegő-type asymptotics disappears and the next nonzero contribution must be computed; the authors analyze this case numerically (their Section 4.2), fitting an expansion whose constants they conjecture to be exactly g1=3ln⁡k−1+ln⁡c−1g_1=3\ln k^{-1}+\ln c^{-1} and gL=−5/2g_L=-5/2, but explicitly refrain from providing rigorous details.

Problem Statement

Extend the asymptotic theory of bordered Toeplitz determinants beyond the nondegenerate Szegő-type regime completed by Basor, Ehrhardt, Gharakhloo, Its and Li, in the two directions they explicitly left open. The two components are instances of a single question — what is the asymptotic behavior of DNB[ϕ;ψ]D_N^B[\phi;\psi] when the hypotheses of the bordered strong Szegő theorem fail, either because the symbol ϕ\phi is Fisher–Hartwig singular or because the bordered constant F[ϕ;ψ]F[\phi;\psi] vanishes — and completion is a set of rigorously proved asymptotic formulas, with explicit constants and error bounds, covering the following regimes.

(A) Fisher–Hartwig regimes (singular symbol). Prove the N→∞N\to\infty asymptotics of the bordered Toeplitz determinant DNB[ϕ^;ψ^]D_N^B[\hat\phi;\hat\psi] for the Ising symbol pair at the critical temperature k=1k=1 and in the high-temperature regime k<1k<1, and thereby of the next-to-diagonal correlation ⟨σ0,0σN−1,N⟩\langle\sigma_{0,0}\sigma_{N-1,N}\rangle: the leading behavior (critical power-law decay for k=1k=1; exponential decay with power-law corrections for k<1k<1), the explicit constants, and at least the first subleading correction including its dependence on the anisotropy Jh≠JvJ_h\ne J_v, with rigorous error terms.

(B) Degenerate borders (vanishing bordered constant). Determine and rigorously justify the asymptotics of DNB[ϕ;ψ]D_N^B[\phi;\psi] when the bordered constant vanishes, F[ϕ;ψ]=0F[\phi;\psi]=0, so that the leading term of the Szegő scaling disappears: compute the next nonzero contribution with error bounds. The general result is required; it must include the case ψ=ϕ z/(z−c)\psi=\phi\,z/(z-c) with ∣c∣>1|c|>1 for the Ising symbol ϕ=ϕ^\phi=\hat\phi, k>1k>1, for which the source's Section 4.2 numerics suggest a specific expansion with constants conjectured to be exactly g1=3ln⁡k−1+ln⁡c−1g_1=3\ln k^{-1}+\ln c^{-1} and gL=−5/2g_L=-5/2: proving (or correcting) this expansion is part of the problem.

The deliverables in (A) and (B) are scientifically linked and jointly serve one overarching objective: a complete strong-asymptotic description of the bordered Toeplitz determinants representing next-to-diagonal Ising correlations, valid in all temperature regimes and in the degenerate-border cases.

References

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