The Scaling-Limit Genealogy of the Parisi–Zamponi Matching Front: A Conditional Kingman Coalescent and the Schertzer–Tourniaire Adaptation to Confining Drift
Abstract
The genealogy of the Parisi–Zamponi matching front was identified, at the frozen (comoving, equilibrated) scale, with Kingman's coalescent: a self-adjoint genealogical operator with a strictly positive spectral gap, an exponentially ergodic spine, a finite reproduction variance with finite moments of all orders, and binary branching with a constant-size stationary population (Zenodo 10.5281/zenodo.20572800 and the spectral-foundation companion). This note is the successor that lifts that frozen-scale statement toward the full jamming dynamics through the scaling limit, and isolates with precision the single deep input the lift requires. We do four things. (I) A self-contained account of the frozen-scale genealogy on the forward operator L, recomputing at the selected exponent the pushed top eigenvalue λ = −0.528 and gap Δ_L = 0.468, the bulk-localized spine measure Φ₀², the finite reproduction variance γ = 1.061 with bounded moment ratios, the k-spine binary-merger ladder C(k,2)·γ with exact rate ratios 3, 2, 5/3, 3/2, 7/5 and vanishing triple driver, and the timescale separation NΔ_L/γ → ∞. (II) We make explicit the structural contrast with the Schertzer–Tourniaire theory of fully pushed fronts: there a killing boundary forces a survival/Yaglom conditioning that selects the Brownian coalescent point process at the critical, constant-in-expectation size; here the linear confinement pins a stationary front of constant mass with no killing boundary, and the stationary normalization selects Kingman's coalescent. The mechanism that fixes which coalescent is therefore precisely the absence of a killing boundary. (III) We state the conditional scaling-limit theorem: under the construction of the scaling limit of the matching-profile selection — the input that the operator-mechanism deposit, Parisi–Zamponi and Charbonneau–Kurchan–Parisi–Urbani–Zamponi all take as external — the genealogy of the full dynamics is Kingman's coalescent, the implication being made rigorous by the spatial reduction (a three-spine computation closes the simultaneous-merger channel) and the finite-variance/finite-moment estimates that transfer from Schertzer–Tourniaire with margin in the confined setting. (IV) We formulate that external input, (P2), sharply, and record what transfers and what does not. In this, the note is the genealogical counterpart of the operator-mechanism deposit: it gives the conditional closure of the genealogy (if the scaling limit is constructed, then Kingman) exactly as that paper gives the conditional closure of the profile selection (if 0 ≤ f ≤ 1, then a+b=1) — the genealogy side and the identity side of the same matching front, each rigorous modulo the same single external input. The operator structure on which the construction stands — the reciprocally weighted self-adjoint pair and its exact spectral relations — is the subject of the companion "Beyond the Shadow", whose shared weight ρ generates the spine measure used here. All numerical statements are reproduced from the verified profile by the consolidated driver, reproduced in full in the appendix.
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BibTeX
@misc{frisina2026scalinglimit,
title = {{The Scaling-Limit Genealogy of the Parisi–Zamponi Matching Front: A Conditional Kingman Coalescent and the Schertzer–Tourniaire Adaptation to Confining Drift}},
author = {Giovanni Frisina},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = jun,
doi = {10.5281/zenodo.20946278},
url = {https://doi.org/10.5281/zenodo.20946278},
note = {CC BY},
}
RIS
TY - UNPB
TI - The Scaling-Limit Genealogy of the Parisi–Zamponi Matching Front: A Conditional Kingman Coalescent and the Schertzer–Tourniaire Adaptation to Confining Drift
AU - Giovanni Frisina
PY - 2026
DA - 2026/06/17
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.20946278
UR - https://doi.org/10.5281/zenodo.20946278
AB - The genealogy of the Parisi–Zamponi matching front was identified, at the frozen (comoving, equilibrated) scale, with Kingman's coalescent: a self-adjoint genealogical operator with a strictly positive spectral gap, an exponentially ergodic spine, a finite reproduction variance with finite moments of all orders, and binary branching with a constant-size stationary population (Zenodo 10.5281/zenodo.20572800 and the spectral-foundation companion). This note is the successor that lifts that frozen-scale statement toward the full jamming dynamics through the scaling limit, and isolates with precision the single deep input the lift requires. We do four things. (I) A self-contained account of the frozen-scale genealogy on the forward operator L, recomputing at the selected exponent the pushed top eigenvalue λ = −0.528 and gap Δ_L = 0.468, the bulk-localized spine measure Φ₀², the finite reproduction variance γ = 1.061 with bounded moment ratios, the k-spine binary-merger ladder C(k,2)·γ with exact rate ratios 3, 2, 5/3, 3/2, 7/5 and vanishing triple driver, and the timescale separation NΔ_L/γ → ∞. (II) We make explicit the structural contrast with the Schertzer–Tourniaire theory of fully pushed fronts: there a killing boundary forces a survival/Yaglom conditioning that selects the Brownian coalescent point process at the critical, constant-in-expectation size; here the linear confinement pins a stationary front of constant mass with no killing boundary, and the stationary normalization selects Kingman's coalescent. The mechanism that fixes which coalescent is therefore precisely the absence of a killing boundary. (III) We state the conditional scaling-limit theorem: under the construction of the scaling limit of the matching-profile selection — the input that the operator-mechanism deposit, Parisi–Zamponi and Charbonneau–Kurchan–Parisi–Urbani–Zamponi all take as external — the genealogy of the full dynamics is Kingman's coalescent, the implication being made rigorous by the spatial reduction (a three-spine computation closes the simultaneous-merger channel) and the finite-variance/finite-moment estimates that transfer from Schertzer–Tourniaire with margin in the confined setting. (IV) We formulate that external input, (P2), sharply, and record what transfers and what does not. In this, the note is the genealogical counterpart of the operator-mechanism deposit: it gives the conditional closure of the genealogy (if the scaling limit is constructed, then Kingman) exactly as that paper gives the conditional closure of the profile selection (if 0 ≤ f ≤ 1, then a+b=1) — the genealogy side and the identity side of the same matching front, each rigorous modulo the same single external input. The operator structure on which the construction stands — the reciprocally weighted self-adjoint pair and its exact spectral relations — is the subject of the companion "Beyond the Shadow", whose shared weight ρ generates the spine measure used here. All numerical statements are reproduced from the verified profile by the consolidated driver, reproduced in full in the appendix.
ER -