2015/10/21 by Ewain Gwynne, Xin Sun, Gwynne, Ewain +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.1510.06346
67 pages, 7 figures
arxiv created 2015/10/21 · openalex publication_date 2015/10/21 · arxiv updated 2015/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove scaling limit results for the finite-volume version of the inventory accumulation model of Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. In particular, we prove that the random walk associated with the finite-volume version of this model converges in the scaling limit to a correlated Brownian motion Z conditioned to stay in the first quadrant for two units of time and satisfy Z(2) = 0. We also show that the times which describe complementary connected components of FK loops in the discrete model converge to the π/2-cone times of Z. Combined with recent results of Duplantier, Miller, and Sheffield, our results imply that many interesting functionals of the FK loops on a finite-volume FK planar map (e.g. their boundary lengths and areas) converge in the scaling limit to the corresponding "quantum" functionals of the CLEκ loops on a 4/√κ-Liouville quantum gravity sphere for κ∈ (4,8). Our results are finite-volume analogues of the scaling limit theorems for the infinite-volume version of the inventory accumulation model proven by Sheffield (2011) and Gwynne, Mao, and Sun (2015).