Rigorous proof of initial-FCS extensivity for dimer and matrix-product states

Prove the boundary-splitting, linked-cluster, and extensivity results needed to rigorously establish the thermodynamic full-counting statistics of finite-range conserved charges in dimer product states, and extend the proof where possible to suitable integrable matrix product states.

Background

The initial BMFT measure requires an extensive large-deviation principle for charge fluctuations. The appendix argues that periodic dimer product states exhibit a dichotomy: a charge is either fixed exactly, producing zero fluctuations, or has an extensive variance and a nontrivial extensive FCS.

Although the structural argument identifies the expected exceptional and extensive cases, the authors do not provide a complete mathematical proof. They specifically identify boundary splitting, linked-cluster control, and the interchange of thermodynamic limits as the missing ingredients, with possible extensions beyond dimer states to suitable matrix product states.

References

We do not attempt a complete proof here. A detailed mathematical treatment of boundary splitting and FCS extensivity for dimer product states, together with an investigation of extensions to suitable classes of matrix product states, will be presented in a separate publication.

— Large-scale dynamics of integrable quenches  (2609.09139 - Horvath et al., 8 Sep 2026) in Appendix A, Section “Extensivity of initial fluctuations”