Variational Principles and Rearrangement Inequalities for asymmetric Operators on Periodic Lattices

What the math gives to ML

The paper gives a constructive Perron-type variational description of the principal eigenvalue for irreducible asymmetric nearest-neighbor operators, expressed through positive profiles and periodic logarithmic correctors. The transferable asset is that a directed positive operator's growth rate can be bounded or optimized using only local edge ratios, avoiding full eigendecomposition and exposing a learnable certificate of instability. This suggests regularizing positive recurrent or state-space neural transitions by minimizing a smooth upper bound on their principal growth rate, while jointly optimizing the corrector as a low-cost auxiliary variable.