The semiclassical limit of unit-area quantum disks

Source abstract

Fix $W\in(0,2)$ and let $γ\downarrow0$. We prove that a weight-$W$ quantum disk conditioned to have unit quantum area converges, in the maximum-embedded strip, to the deterministic field $φ_W(t+iθ)=\log(W/(4π))-2\log\cosh(Wt/2)$ 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.