Mass functions and asymptotic behavior of caloric functions on affine buildings

Abstract We study the large-time asymptotic behavior of solutions to the discrete-time heat equation, i.e., caloric functions, on affine buildings, including those without transitive group actions. For each $$p \in [1, \infty ]$$ p ∈ [ 1 , ∞ ] , we introduce a notion of a p -mass function and prove that caloric functions with initial data belonging to certain weighted- $$\ell ^1$$ ℓ 1 spaces or to the radial $$\ell ^1$$ ℓ 1 class, asymptotically decouple as the product of this mass function and the heat kernel. These results extend classical analogues from Euclidean spaces and symmetric spaces of non-compact type to the non-Archimedean setting, and remain valid even for exotic buildings beyond the Bruhat–Tits framework. We characterize the spatial concentration of heat kernels in p -norms and describe the geometry of associated critical regions. Our results highlight substantial differences in the asymptotic regimes depending on the value of p , and clarify the interplay between volume growth and heat diffusion.