Coarse-grained spectral bounds for multicyclic networks

Establish whether similarly small sets of kinetic observables can sharpen thermodynamic spectral bounds for multicyclic Markov networks, where overlapping cycles share transitions and cycle-wide rate products are not independent, without requiring full knowledge of the generator.

Background

The results are proved for unicyclic Markov processes, where forward and backward cycle-rate products provide a natural additional kinetic scale. The authors identify multicyclic networks as a particularly important extension because overlapping cycles share transitions, making cycle-wide rate products interdependent. The unresolved issue is whether a comparably small collection of kinetic observables can yield generator-dependent improvements to thermodynamic spectral bounds in that more complicated setting.

References

A particularly natural next step is to extend this viewpoint to multicyclic networks, where overlapping cycles share transitions and cycle-wide rate products are no longer independent. The central problem is then whether similarly small sets of kinetic observables can sharpen thermodynamic spectral bounds without requiring full knowledge of the generator.

— Coarse-grained kinetic scale tightens thermodynamic spectral bounds of Markov cycles  (2608.22934 - Xu et al., 24 Aug 2026) in Section Discussion