Rigorous frozen-scale estimates for the Parisi–Zamponi matching front: the confined-spine adaptation of Schertzer–Tourniaire
Abstract
This note supplies the rigorous spectral foundation and the confined transport estimates behind Result III of the genealogy companion (the placement of the Parisi–Zamponi matching-front genealogy in the Kingman class, with precise object Kingman's coalescent). It carries out, for the confining forward (mean-density) operator L = (c/2)∂² + V∂ + c(1−2f) — the genealogical operator — the program that Schertzer–Tourniaire carry out for half-line branching Brownian motion. Although its transport drift is +V, L is self-adjoint in the weighted space L²(ρ), with ρ = exp((2/c)∫V), hence diagonalised by a symmetric Schrödinger operator H_L; this is what makes the estimates clean and the reproduction variance a grid-stable number. (The replicon operator A_c of the deposit, self-adjoint in L²(ρ⁻¹), is a different operator: it carries the velocity-selection exponent a = (1−c)/2, not the genealogy.) The foundation and the estimates — relaxation, heat-kernel, resolvent, escape, rate — are proved; the confining drift, in place of a killing boundary, makes each estimate cleaner than its Schertzer–Tourniaire counterpart, replacing their unbounded reproductive value and borderline integrability at the semi/fully-pushed transition by a Gaussian invariant measure and a sub-exponential reproductive value, integrable with room to spare. The assembly of these estimates into Kingman's coalescent, via the moment method of Schertzer–Tourniaire, is stated. The one structural point at which the confined front departs from Schertzer–Tourniaire — the conditioning scaffolding — is exactly what fixes the precise object, and is made explicit: the killing boundary forces a survival/Yaglom conditioning and selects their Brownian coalescent point process (a critical, constant-in-expectation size), whereas the comoving confinement here makes the population stationary with constant mass, which selects Kingman's coalescent — the two being coupled, along exactly this constant-size vs. critical axis, by Lambert–Schertzer.
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BibTeX
@misc{frisina2026rigorous,
title = {{Rigorous frozen-scale estimates for the Parisi–Zamponi matching front: the confined-spine adaptation of Schertzer–Tourniaire}},
author = {Giovanni Frisina},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = jun,
doi = {10.5281/zenodo.20946017},
url = {https://doi.org/10.5281/zenodo.20946017},
note = {CC BY},
}
RIS
TY - UNPB
TI - Rigorous frozen-scale estimates for the Parisi–Zamponi matching front: the confined-spine adaptation of Schertzer–Tourniaire
AU - Giovanni Frisina
PY - 2026
DA - 2026/06/13
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.20946017
UR - https://doi.org/10.5281/zenodo.20946017
AB - This note supplies the rigorous spectral foundation and the confined transport estimates behind Result III of the genealogy companion (the placement of the Parisi–Zamponi matching-front genealogy in the Kingman class, with precise object Kingman's coalescent). It carries out, for the confining forward (mean-density) operator L = (c/2)∂² + V∂ + c(1−2f) — the genealogical operator — the program that Schertzer–Tourniaire carry out for half-line branching Brownian motion. Although its transport drift is +V, L is self-adjoint in the weighted space L²(ρ), with ρ = exp((2/c)∫V), hence diagonalised by a symmetric Schrödinger operator H_L; this is what makes the estimates clean and the reproduction variance a grid-stable number. (The replicon operator A_c of the deposit, self-adjoint in L²(ρ⁻¹), is a different operator: it carries the velocity-selection exponent a = (1−c)/2, not the genealogy.) The foundation and the estimates — relaxation, heat-kernel, resolvent, escape, rate — are proved; the confining drift, in place of a killing boundary, makes each estimate cleaner than its Schertzer–Tourniaire counterpart, replacing their unbounded reproductive value and borderline integrability at the semi/fully-pushed transition by a Gaussian invariant measure and a sub-exponential reproductive value, integrable with room to spare. The assembly of these estimates into Kingman's coalescent, via the moment method of Schertzer–Tourniaire, is stated. The one structural point at which the confined front departs from Schertzer–Tourniaire — the conditioning scaffolding — is exactly what fixes the precise object, and is made explicit: the killing boundary forces a survival/Yaglom conditioning and selects their Brownian coalescent point process (a critical, constant-in-expectation size), whereas the comoving confinement here makes the population stationary with constant mass, which selects Kingman's coalescent — the two being coupled, along exactly this constant-size vs. critical axis, by Lambert–Schertzer.
ER -