bibkey: nacu2003monotone authors: Şerban Nacu year: 2003 title: Glauber Dynamics On The Cycle Is Monotone doi: null url: https://arxiv.org/abs/math/0305056v1 claim: A monotone finite Ising kernel has an increasing second-eigenvalue eigenfunction; the linear gap eigenfunction in the source additionally uses zero-field ferromagnetic cycle dynamics. strata_touched: [] license: citation-only triage: anchor

Increasing eigenfunctions and the hard-core application boundary

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Lemma 3, attributed to Y. Peres, obtains an increasing eigenfunction for the second eigenvalue by applying powers of a monotone kernel to increasing functions and taking spectral limits. Lemma 4 uses the Ising FKG inequality to identify the eigenvalue of a strictly increasing eigenfunction. Those are distinct arguments with different required hypotheses.

Lemma 5 and Corollary 6 specialize to ferromagnetic Ising on a cycle, with zero external field, positive couplings and uniform vertex selection. The source proves that linear spin functions form an invariant subspace and obtains a linear gap eigenfunction. It also treats a straight line through a zero-boundary-coupling limit. This is an Ising result, not a hard-core path theorem.

For the positive weighted hard-core path, particle-hole inversion on one bipartite class gives a monotone legal configuration poset. To use the projection method there, one must establish monotone conditional updates and show that increasing functions span this legal carrier. Principal-upset indicators provide the finite-poset spanning argument; the continuous-time semigroup then avoids any negative transition eigenvalue issue.

The hard-core site update contains a neighboring quadratic occupation product at interior sites. Nacu’s linear invariance cannot be imported into that model. An increasing gap eigenfunction can instead be paired with the strictly ordered staggered occupation sum through the actual positive-conductance Dirichlet form. The resulting overlap bound and its classical/quantum correlation application are separate deductions from the hard-core law.

Nonzero slow overlap does not identify a linear signal’s immediate Rayleigh quotient with the gap, nor imply a uniform finite-time readout, bounded polynomial degree or physical control protocol.