Random parametrix layer (blueprint nodes D-Im / D-para, coefficient route) #
The random renormalized kernels RI_{m,κ} of Deng–Shen (arXiv:2607.10105)
§3.2: the deterministic closed-form integrand detIntegrand multiplied by
the Wick (chaos-projected) product of the noise at the single indices, then
integrated; the order-m parametrix piece P_m (paper (3.15)) and its
Fourier-mode coefficients (paper (3.23)); and the coefficient-level Neumann
route to H_ε = λ_ε⁻¹(G_ε − G) (DESIGN §5.2 refinement).
Junk-totalization (DESIGN §5.7) #
All tsum/Bochner integrals below default to 0 off summability /
integrability; downstream estimates carry the hypotheses excluding the junk
branches. In particular, the tsum in modeHcoeff itself has junk value
0 off the good event.
Wick factor at fixed points (node D-Im). For a pairing κ of the
m internal indices and an assembled tuple xt = (x, v, y), the value of
the Wick product :∏_{i ∈ S} ξ_ε(x_{i+1}): over the singles S of κ,
written as the explicit combinatorial chaos projection (Hermite/Wick
expansion): the sum over sub-pairings κ' of S of
(-1)^{#pairs κ'} · ∏_{pairs} η_ε(difference) · ∏_{new singles} ξ_ε.
The projection Proj_{|S|} is defined by this combinatorial formula, with no
Hilbert-space machinery; ChaosDecomposition proves its relation to the
L²(Ω) chaos projection.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Random renormalized integrand (node D-Im): the deterministic closed form (3.6) times the Wick factor of the singles.
Equations
- Anderson4D.randIntegrand M ρ ε κ xt ω = Anderson4D.detIntegrand ρ ε m κ xt * Anderson4D.wickAt M ρ ε κ xt ω
Instances For
The random renormalized kernel RI_{m,κ}(x,y) (paper Def 3.1 /
(3.6) with Wick factor): all m internal variables integrated against
paperMeasure^{⊗m}, with the coupling prefactor λ_ε^m. Junk-totalized
Bochner integral.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The order-m parametrix piece P_m(x,y) (paper (3.15)): the sum
of RI_{m,κ} over all partial pairings κ of the m internal indices.
(The full parametrix ∑_{m ≤ A} P_m is not needed for the formal
statements.)
Equations
- Anderson4D.parametrixP M ρ lam ε m x y ω = ∑ κ : Anderson4D.PartialPairing (Fin m), Anderson4D.randRI M ρ lam ε m κ x y ω
Instances For
Fourier-mode coefficients of P_m (paper (3.23)): the pairing of
P_m with the characters e_α ⊗ e_β in both external variables, against
paperMeasure in each. Junk-totalized iterated Bochner integrals.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The renormalized multiplication function λ_ε ξ_ε − C_ε (the potential
entering H_ε = G(λ_ε ξ_ε − C_ε)G + …, paper (3.1)–(3.11)).
Equations
- Anderson4D.multFun M ρ lam ε z ω = Anderson4D.lamEps lam ε * M.xiEps ρ ε ω z - Anderson4D.renormCEps ρ lam ε
Instances For
Neumann-series mode coefficient (the coefficient-level definition
from DESIGN §5.2). Its equivalence with the operator route is proved in
FredholmCoefficientBridge. The kernel of the n-th
Neumann term is a chain of n + 1 Green factors through n
multiplication points carrying multFun; it is paired with the characters
e_α ⊗ e_β. At n = 0 there are no multiplication factors and this is
the Ĝ-pairing. Junk-totalized iterated Bochner integrals.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Mode coefficient of H_ε = λ_ε⁻¹(G_ε − G) (coefficient route):
the n ≥ 1 tail of the Neumann series of G_ε, divided by λ_ε — the
n = 0 term is Ĝ, hence subtracted. The standard totalized tsum
has junk value 0 when the tail is not summable; the good-event estimates
exclude that branch.
Equations
- Anderson4D.modeHcoeff M ρ lam ε α β ω = (Anderson4D.lamEps lam ε)⁻¹ • ∑' (n : ℕ), Anderson4D.neumannCoeff M ρ lam ε (n + 1) α β ω