Bruhat-Tits Trees — Hecke Operators as Random Walks

The Tree T_p

For a prime p, the Bruhat-Tits tree is an infinite (p+1)-regular tree. Its vertices represent lattice classes — ℤ_p-lattices in ℚ_p², up to scaling. Two vertices connect when one lattice contains the other with index p.

The Hecke operator T_p acts on functions on this tree: (T_pf)(v) = Σ f(w) summed over all neighbors w of v. This is the adjacency operator — or equivalently, one step of a random walk.

Eigenforms of T_p are harmonic functions on the tree. The spherical function h_s(v) = ρ^d(v,o) is an eigenfunction with eigenvalue λ = p^s + p^1-s, where d(v,o) is the distance to the root.