The Scaling Limit of the Volume of Loop–O(n) Quadrangulations

We study the volume of rigid loop-<i>O</i>(<i>n</i>) quadrangulations with a boundary of length 2<i>p</i> in the non-generic critical regime, for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>]</mml:mo></mml:mrow> </mml:math> . We prove that, as the half-perimeter <i>p</i> goes to infinity, the volume scales in distribution to an explicit random variable. This limiting random variable is described in terms of the multiplicative cascades of Chen et al. (Ann Inst Henri Poincaré D 7(4):535-584, 2020), or alternatively (in the dilute case) as the law of the area of a unit-boundary <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>γ</mml:mi></mml:math> -quantum disc, as determined by Ang and Gwynne (Ann Inst Henri Poincaré D 57(1): 1-53, 2021), for suitable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>γ</mml:mi></mml:math> . Our arguments go through a classification of the map into several regions, where we rule out the contribution of bad regions to be left with a tractable portion of the map. One key observable for this classification is a Markov chain which explores the nested loops around a size-biased vertex pick in the map, making explicit the spinal structure of the discrete multiplicative cascade. We stress that our techniques enable us to include the boundary case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> , that we define rigorously, and where the nested cascade structure is that of a critical branching random walk. In that case the scaling limit is given by the limit of the derivative martingale and is inverse-exponentially distributed, which answers a conjecture of Aïdékon and Da Silva (Probab Theory Relat Fields 183(1):125-166, 2022).