2020/07/13 by James R. Lee, Lee, James R. · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2007.06548
openalex publication_date 2020/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the validity of the "Einstein relations" in the general setting of unimodular random networks. These are equalities relating scaling exponents: dw = df + ζ and ds = 2 df/dw, where dw is the walk dimension, df is the fractal dimension, ds is the spectral dimension, and ζ is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that if df and ζ ≥ 0 exist, then dw and ds exist, and the aforementioned equalities hold. Moreover, our primary new estimate is the relation dw ≥ df + ζ, which is established for all ζ ∈ ℝ. For the uniform infinite planar triangulation (UIPT), this yields the consequence dw=4 using df=4 (Angel 2003) and ζ=0 (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2017 and Ding-Gwynne 2020). The conclusion dw=4 had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is that dw = df for the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, since df > 2 (Ding and Gwynne 2020). For the random walk on ℤ2 driven by conductances from an exponentiated Gaussian free field with exponent γ> 0, one has df = df(γ) and ζ=0 (Biskup, Ding, and Goswami 2020). This yields ds=2 and dw = df, confirming two predictions of those authors.