Manuscript Psy02-B@2 · Psy02-B-v2.tex · theorem · cfd06e2e

Theorem: Exact uniform concentration criterion

Track A3: concentration and temporal fluctuations › Separate scalar benchmark · lines 635–664 · proof included or linked

Psy02-B@2#thm:concentration

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For the spectral laws in the displaced states,

W_{2} (μ_{ϵ, t}, δ_{o_{t}})^{2} = ϵ Tr (ρ_{ϵ, t} Φ (v_{t})^{2}) .

Uniform convergence on $0 \leq t \leq T$ is equivalent to

t \leq T sup ϵ Tr (ρ_{ϵ, t} Φ (v_{t})^{2}) \to 0.

A sufficient bound is

ϵ Tr (ρ_{ϵ, t} Φ (v_{t})^{2}) \leq ϵ (4 n_{ϵ, t} + 2) \norm v_{t}^{2} .

Proof. The displacement identity gives the target-centered second moment. Distance to a point mass has that exact squared Wasserstein value. The number estimate follows from

\norm Φ (v) ψ^{2} \leq 2 \norm v^{2} (\norm N^{1/2} ψ^{2} + \norm (N + 1)^{1/2} ψ^{2}) .

Sum over a normal-state decomposition.

Displayed formulas in this passage

unlabeled · line 638
$W_{2} (μ_{ϵ, t}, δ_{o_{t}})^{2} = ϵ Tr (ρ_{ϵ, t} Φ (v_{t})^{2}) .$
unlabeled · line 643
$t \leq T sup ϵ Tr (ρ_{ϵ, t} Φ (v_{t})^{2}) \to 0.$
unlabeled · line 647
$ϵ Tr (ρ_{ϵ, t} Φ (v_{t})^{2}) \leq ϵ (4 n_{ϵ, t} + 2) \norm v_{t}^{2} .$
unlabeled · line 657
$\norm Φ (v) ψ^{2} \leq 2 \norm v^{2} (\norm N^{1/2} ψ^{2} + \norm (N + 1)^{1/2} ψ^{2}) .$

Theorem: Exact uniform concentration criterion

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