2022/02/18 by Abraham Levitan, Levitan, Abraham · 1 citation
Materials Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Material Dynamics and Properties #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.2202.09436
5 pages, 4 figures. Originally prepared for 8.334, statistical physics 2 at MIT
arxiv created 2022/02/18 · openalex publication_date 2022/02/18 · arxiv updated 2022/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Percolation refers to an interesting class of problems related to the properties of disordered systems, usually formulated in terms of objects randomly placed on an underlying lattice or continuum. Despite the simplicity of the setup, most percolative systems undergo a phase transition from a disconnected state with many disjoint clusters to a state where a finite fraction of the lattice sites are connected to a single cluster. As in the case of thermodynamic phase transitions, power law dependencies generically near the critical percolation threshold. The origin of these dependencies can be understood through the lens of scaling and renormalization, and indeed many quantitative results can be acquired using these tools. In this paper we study the percolation problem on a hierarchical lattice, where exact results for the critical exponents can be obtained from a decimation procedure. We calculate analytic results for the full set of geometric critical exponents and confirm their consistency with simulation. Finally, we set up an interesting renormalization group for the conductivity of the system and use it to computationally extract the conductivity exponent t.