The Brydges–Kennedy–Abdesselam–Rivasseau forest interpolation formula #
Root facade: importing this module brings in the whole development.
For a finite vertex set V and a function ρ : (Edge V → ℝ) → ℝ smooth
(C^∞) on the edge-coupling space (BKARContDiff), the BKAR forest
interpolation formula states
ρ(1,…,1) = ∑_{forests F} ∫_{[0,1]^{E(F)}} ∂_{E(F)} ρ (x^F(u)) du,
summing over all acyclic edge subsets of the complete graph on V (the
empty forest contributing ρ(0,…,0)), where ∂_{E(F)} is the mixed
partial derivative in the edge variables of F and x^F(u) is the
path-minimum interpolation: the coordinate of x^F(u) at an edge {i, j}
is the minimum of u along the unique forest path joining i to j, and
0 when i and j lie in different components of F.
The flagship formal statement is
BKAR.bkar_formula_forestIndex_cube_contributions in BKAR.Formula, with
sector-form and empty-forest-split variants alongside, and a threshold /
layer-cake API for the interpolation points in the BKAR.Threshold files.
The formalization assumes ρ is C^∞ (BKARContDiff = ContDiff ℝ ⊤)
where the classical statement needs only C^{|V|-1}; this is a deliberate
strengthening of the hypothesis.
References #
- D. Brydges, T. Kennedy, Mayer expansions and the Hamilton–Jacobi equation, J. Statist. Phys. 48 (1987) 19–49.
- A. Abdesselam, V. Rivasseau, Trees, forests and jungles: a botanical garden for cluster expansions, in Constructive Physics (Palaiseau 1994), Lecture Notes in Physics 446, Springer, 1995. arXiv:hep-th/9409094.