Localization_and_Rigidity_in_Discrete_Relational_Variational_Functionals

We study a class of discrete variational functionals defined on a countable set with a symmetric interaction kernel and internal degrees of freedom valued in a compact Lie group. The modelis formulated purely in terms of relational data, without assuming any underlying geometricstructure.We establish existence of global minimizers using the direct method in the calculus of variations. A structural result shows that admissible pair interactions are rigid at quadratic order: under natural symmetry and regularity assumptions, the second variation is necessarily ofDirichlet type. In addition, conjugation-invariant loop contributions admit a universal quadraticexpansion near the identity, determined by the trace in the chosen representation.We analyze the stability of uniform configurations via a quadratic form associated witha linear operator combining local and interaction terms. A stability threshold is identified,beyond which the uniform state becomes unstable. In this regime, we construct non-uniformconfigurations with strictly lower energy and show that they are exponentially localized undersuitable assumptions on the interaction kernel.Finally, we prove rigidity in the orientation sector: the second variation with respect tointernal variables is nonnegative, and its kernel consists precisely of global symmetry modes.All results are obtained in the discrete setting and do not rely on any continuum approximation.