The exact Laplacian spectrum of the Berker–Ostlund diamond lattice

Abstract

We derive, in closed form, the complete spectrum of the combinatorial Laplacian L(L) on the level- L Berker–Ostlund diamond hierarchical lattice DL(b) with branching parameter b⩾2. Because the vertex degree grows exponentially under inflation, the renormalized eigenvalue equation is not a Laplacian at the coarser scale but a two-parameter matrix pencil αD−A, whose coefficients evolve under a quadratic renormalization map of the parameter plane. Iterating a Schur-complement block recursion yields an exact product formula for the characteristic polynomial χL(λ) over the singular polynomials Sℓ=2αℓ−βℓ and two boundary polynomials PL±, together with the exact multiplicity of every localized eigenvalue; the multiplicity contains a topological correction given by the cyclomatic number of the ancestral graph, and a dimension audit confirms deg⁡χL=VL. As a consequence, the empirical spectral measure is proved to converge, as L→∞, to an explicit purely atomic probability measure carried by the singular ladder, whose dominant atom sits at the molecular eigenvalue λ=2 with weight (b−1)/b. The relation between the accumulation set of the spectrum and the complex dynamics of the planar renormalization map is formulated as a set of precise conjectures.