A mathematical preprint reports conditional lower bounds on how quickly the ground state of a Pauli-Fierz model can fade across space, but it also identifies a problem in the direct route used for the full model. The stochastic integral at the heart of that route depends on the path taken, and the range of a key control parameter narrows as the time horizon grows. As a result, the estimates cannot be made uniform in time.
The analytical target is the normalized ground state, written as phi_g, and specifically its spatially resolved norm in Fock space. The resulting statements are lower bounds on that norm under the model assumptions, not measurements of a separate sample.
To obtain the bounds, the authors use a Feynman-Kac formula, positivity of a relevant ground-state component, and estimates for exponential moments of a double stochastic integral. In the full model, that integral retains its dependence on the entire path.
The full model's time problem
In the full model, the exponential-moment estimate is conditional in two linked ways. The parameter controlling the moment must remain below the reciprocal of a model constant multiplied by the time horizon T. Within that range, the expected exponential of the formal double stochastic integral is bounded by a factor that depends on the parameter times an exponential in T.
That dependence on T is the paper's central technical obstacle. As T increases, the allowed parameter range shrinks, while the integral retains its dependence on the whole path. The direct Agmon-distance strategy therefore does not provide the uniform-in-T control needed for the desired full-model lower bound. This limits the stated approach, not necessarily every possible method.
Even so, the analysis obtains a conditional pointwise lower bound for the full model when the relevant potential difference belongs to the required class, the geometric path conditions hold, and the exponents satisfy a further restriction involving T. For potentials trapped between matching polynomial lower and upper bounds, the full-model norm has paired exponential lower and upper bounds. In both directions, the leading spatial power is the distance raised to n+1, for a positive tolerance smaller than b, with b below one-half.
The paper also gives a separate full-model proposition in terms of Agmon distance, a path-based quantity used to measure decay. In that argument, the exponent is chosen using a parameter greater than one, the full-model constant and the time horizon. The proposition supplies a lower bound for sufficiently distant points and every admissible path from the origin to the point, with additional correction terms that depend on the tolerance parameter.
A simpler path in the dipole model
The dipole approximation behaves differently in the relevant estimate. Its double stochastic integral has a deterministic integrand, which makes the exponential-moment analysis substantially simpler and removes the dependence of the admissible parameter range on T.
Here the exponential moment grows at most exponentially with the time horizon, while the allowed parameter range stays fixed as T changes. That difference lets the paper state a conditional Agmon-distance lower bound: for sufficiently large distance and every admissible path, the ground-state norm is at least a constant times an exponential controlled by the path's Agmon length, with any positive tolerance added to the exponent.
For a potential that grows no faster than a constant times the distance raised to 2n, the dipole result gives a lower bound whose exponent has leading power n+1. Its coefficient depends explicitly on the potential-growth constant, n, the theorem's exponent and the tolerance, and the statement holds beyond a corresponding distance threshold.
The ground-state positivity result provides part of the foundation for these arguments: in the full model, the vacuum component has a continuous version and is strictly positive at each spatial position.
What the estimates mean
The supplied front matter identifies the work as arXiv:2608.25719v2, dated 13 September 2026. The full-model moment result is a conditional analytical bound rather than an empirical estimate, and the dipole result depends on the stated cutoff and coupling assumptions. The conclusions therefore apply only within the potential and path conditions used in the proofs.
The authors state that the dipole Agmon theorem may not be optimal. The time-dependent obstruction also leaves open whether a uniform-in-T estimate can be obtained for the path-dependent full-model integral. For now, the paper's conclusions remain conditional mathematical decay statements tied to the assumptions of each model.
Paper data and sources
Original title: Lower bounds on the spatial decay of the ground state of the Pauli-Fierz model
Authors: Fumio Hiroshima, Yuki Tsujimoto
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text