2022/02/01 by Benedikt Stufler, Stufler, Benedikt
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2202.00592
openalex publication_date 2022/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the random simple cubic planar graph Cn with an even number n of vertices admits a novel uniform infinite cubic planar graph (UICPG) as quenched local limit. We describe how the limit may be constructed by a series of random blow-up operations applied to the dual map of the type~III Uniform Infinite Planar Triangulation established by Angel and Schramm (Comm. Math. Phys., 2003). Our main technical lemma is a contiguity relation between Cn and a model where the networks inserted at the links of the largest 3-connected component of Cn are replaced by independent copies of a specific Boltzmann network. We prove that the number of vertices of the largest 3-connected component concentrates at κn for κ≈ 0.85085, with Airy-type fluctuations of order n2/3. The second-largest component is shown to have significantly smaller size Op(n2/3).