The semiclassical limit of unit-area quantum disks
Abstract
Fix W∈(0,2) and let γ↓0. We prove that a weight-W quantum disk conditioned to have unit quantum area converges, in the maximum-embedded strip, to the deterministic field ϕW(t+iθ)=log(W/(4π))−2logcosh(Wt/2) and its constant-curvature area measure. The proof establishes a joint large-deviation principle for the radial Brownian input, the lateral Gaussian free field, and the total Gaussian multiplicative chaos mass. Diverging-order moment bounds justify both adjoining the noncontinuous area functional and retaining its unbounded positive power in the unit-area expectation. An exact radial Girsanov shift centers the candidate, and reflection together with the sharp Onofri inequality proves its unique optimality. This yields concentration without a quadratic expansion of the running maximum or an order-one partition-function asymptotic. We also identify the field rate with the renormalized Liouville action in the maximum embedding.
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BibTeX
@misc{fan2026semiclassical,
title = {{The semiclassical limit of unit-area quantum disks}},
author = {Yuchen Fan and Zhenfeng Tu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02436},
eprint = {2610.02436},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02436},
note = {CC0},
}
RIS
TY - UNPB
TI - The semiclassical limit of unit-area quantum disks
AU - Yuchen Fan
AU - Zhenfeng Tu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02436
UR - https://arxiv.org/abs/2610.02436
AB - Fix W∈(0,2) and let γ↓0. We prove that a weight-W quantum disk conditioned to have unit quantum area converges, in the maximum-embedded strip, to the deterministic field ϕ_W(t+iθ)=log(W/(4π))−2logcosh(Wt/2) and its constant-curvature area measure. The proof establishes a joint large-deviation principle for the radial Brownian input, the lateral Gaussian free field, and the total Gaussian multiplicative chaos mass. Diverging-order moment bounds justify both adjoining the noncontinuous area functional and retaining its unbounded positive power in the unit-area expectation. An exact radial Girsanov shift centers the candidate, and reflection together with the sharp Onofri inequality proves its unique optimality. This yields concentration without a quadratic expansion of the running maximum or an order-one partition-function asymptotic. We also identify the field rate with the renormalized Liouville action in the maximum embedding.
ER -