Comparing the Lee-Yang gap with the mass gap in the sub-critical planar Ising model

Technical Summary: Comparing the Lee–Yang Gap with the Mass Gap in the Sub-critical Planar Ising Model

Problem Statement
The paper investigates the quantitative relationship between two fundamental spectral quantities in the sub-critical (disordered) phase of the planar Ising model: the mass gap ($m_\beta$) and the Lee–Yang gap ($L_\beta$).

• The mass gap ($m_\beta$) characterizes the exponential decay rate of the two-point correlation function (the inverse correlation length) in the absence of a magnetic field.
• The Lee–Yang gap ($L_\beta$) is defined as the distance from the origin to the closest zero of the partition function in the complex magnetic field plane (specifically, the magnitude of the closest purely imaginary zero).

While the Combes–Thomas estimate for Schrödinger operators relates the distance to the spectrum with the spatial decay rate of the Green's function, a rigorous quantitative analogue for the Ising model was lacking. Previous work established qualitative connections (exponential decay implies a zero-free region), but the precise scaling relation between $m_\beta$ and $L_\beta$ near the critical point $\beta_c$ remained an open problem. The authors aim to prove that these two gaps are comparable, specifically establishing that $m_\beta$ scales as $L_\beta^{8/15}$ (or conversely, $L_\beta \sim m_\beta^{15/8}$) as $\beta \uparrow \beta_c$.

Methodology
The proof is divided into establishing a lower bound and an upper bound for the mass gap in terms of the Lee–Yang gap.

1. Lower Bound ($m_\beta \gtrsim L_\beta^{8/15}$):

   • The authors utilize the Lee–Yang product formula (Hadamard factorization) to relate the derivatives of the pressure (log-partition function) to the Lee–Yang zeros.
   • They employ standard estimates for the planar Ising model, specifically the asymptotic behavior of the susceptibility ($\chi_\beta \sim m_\beta^{-7/4}$) and the critical magnetization ($E_{\beta_c, h}[\sigma_0] \sim h^{1/15}$).
   • By combining the variance of the magnetization (linked to susceptibility) with the monotonicity of the magnetization in the field, they derive the inequality $L_\beta^{14/15} \lesssim m_\beta^{7/4}$, which yields the desired power law.

2. Upper Bound ($m_\beta \lesssim L_\beta^{8/15}$):

   • This direction is more technically involved and relies on a finite block expansion combined with random-current moment bounds and a specific correlation inequality.
   • Random-Current Moment Bounds: The authors establish Gaussian bounds on the magnetization of finite blocks using correlation inequalities (specifically the Aizenman inequality and Griffiths-Kelly-Sherman inequalities).
   • Correlation Inequality: They utilize a result by Ding, Song, and Sun (DSS) which bounds the covariance of spins in arbitrary real fields by their zero-field correlation. This allows them to compare conditional distributions of spins in a block given different boundary conditions (specifically, arbitrary configurations vs. all-plus configurations).
   • Finite Block Expansion: Using the DSS inequality, they construct a "zero-free criterion." They define a block $T_\ell$ and a surrounding shell $S_\ell$. If the zero-field correlation between these contracted sets ($q_\beta(\ell)$) is sufficiently small, the partition function is non-zero in a specific complex neighborhood.
   • Planar Crossing Bounds: To verify that the correlation $q_\beta(\ell)$ decays exponentially at the scale of the correlation length ($\ell \sim 1/m_\beta$), the authors adapt methods from percolation theory (Duminil-Copin and Manolescu; Köhler-Schindler and Tassion). They use RSW (Russo-Seymour-Welsh) type crossing estimates for the FK-Ising model to show that separated wired sets have exponentially small connection probabilities in the sub-critical phase.
   • Combining the zero-free criterion with the susceptibility bound ($\chi_\beta \sim m_\beta^{-7/4}$) and the block size $\ell \sim 1/m_\beta$, they derive the lower bound on $L_\beta$ in terms of $m_\beta$.

Key Contributions and Results

• Main Theorem (Theorem 1.1): The authors prove that for the planar Ising model with free boundary conditions on square boxes, there exist constants $c, C > 0$ such that for $\beta$ sufficiently close to $\beta_c$:
  $$c L_\beta^{8/15} \leq m_\beta \leq C L_\beta^{8/15}$$
  This confirms the scaling prediction derived from continuum field theory (Fonseca and Zamolodchikov), where the Lee–Yang edge occurs at a fixed value of $h_{cont}/|m_{cont}|^{15/8}$.
• Uniform Factorial Bounds: The paper establishes uniform bounds on the field derivatives of the pressure (log-partition function). Specifically, for any $k \geq 1$:
  $$\frac{1}{|\Lambda_L|} \left| \partial_h^{2k} \log Z_{\beta, L}(h) \bigg|_{h=0} \right| \leq (2k)! C^{2k} m_\beta^{2 - 15k/4}$$
  This result follows directly from the Lee–Yang product and the established gap bounds.
• Random-Field Ising Model (RFIM) Conjecture: The authors discuss the RFIM, distinguishing between spatial decay and the complex-field gap. They conjecture that for the RFIM, while the mass gap remains positive, the Lee–Yang gap may vanish almost surely ($L_\beta^\eta = 0$) due to the existence of large regions with nearly vanishing random fields that mimic the ordered phase.
• Obstruction to Shorter Proofs: In the Appendix, the authors analyze a potential shorter proof based on the monotonicity of Ursell functions asserted in a recent paper by CJN. They provide a counterexample to a specific auxiliary inequality used in that work, explaining why the block expansion and planar crossing arguments are necessary for the current proof.

Significance
The paper provides the first rigorous quantitative comparison between the Lee–Yang gap and the mass gap in the sub-critical planar Ising model. While the qualitative link (exponential decay $\iff$ zero-free region) was known, this work establishes the precise critical exponent ($15/8$) connecting the two.
The authors note that this comparison is motivated by the Combes–Thomas estimate in Schrödinger operators, suggesting a deep structural analogy between the spectral gap of quantum operators and the Lee–Yang zeros of statistical mechanical models. The result validates the continuum scaling prediction for the Yang–Lee edge in the lattice model. The work also highlights the necessity of planar crossing estimates and random-current techniques in establishing these bounds, as simpler arguments based on Ursell function monotonicity appear to face obstructions.