2018/07/31 by Jian Ding, Ewain Gwynne · 3 citations
Mathematics · Physics and Astronomy · #Combinatorics #Exponent #Fractal #Limiting #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Monotonic function #Physics #Quantum mechanics #Renormalization group #Stochastic processes and statistical mechanics #Subadditivity #Theoretical and Computational Physics #Universality (dynamical systems) #Upper and lower bounds #math-ph #math.MP #math.PR
paper · pdf · doi:10.1007/s00220-019-03487-4
56 pages, 7 figues; final version, to appear in CMP
arxiv created 2019/04/22 · openalex publication_date 2019/06/26 · arxiv updated 2020/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We prove that for each \(γ ∈ (0,2)\) , there is an exponent \(dγ > 2\) , the “fractal dimension of \(γ \) -Liouville quantum gravity (LQG)”, which describes the ball volume growth exponent for certain random planar maps in the \(γ \) -LQG universality class, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of \(γ \) -LQG distances such as Liouville graph distance and Liouville first passage percolation. We also show that \(dγ \) is a continuous, strictly increasing function of \(γ \) and prove upper and lower bounds for \(dγ \) which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for \(γ =√(2)\) (which corresponds to spanning-tree weighted planar maps) our bounds give \(3.4641 ≤ d√(2) ≤ 3.63299\) and in the limiting case we get \(4.77485 ≤ lim γ → 2- dγ ≤ 4.89898\) .