Magnetization Distribution on Fractals and Percolation Lattices

TL;DR

This paper investigates the magnetization distribution of the Ising model on fractals and percolation lattices, revealing non-trivial behavior below a certain temperature and estimating the ramification of percolation clusters.

Contribution

It provides a novel analysis of magnetization distributions on fractals and percolation clusters, linking fixed points and crossover temperatures to structural properties.

Findings

Magnetization distribution is non-trivial below a specific temperature.

Crossover temperature scales inversely with the number of iterations.

Estimated ramification of percolation clusters at threshold is approximately 2.3.

Abstract

We study the magnetization distribution of the Ising model on two regular fractals (a hierarchical lattice, the regular simplex) and percolation clusters at the percolation threshold in a two dimensional imbedding space. In all these cases, the only fixed point is $T=0$. In the case of the two regular fractals, we show that the magnetization distribution is non trivial below $T^{*} \simeq A^{*}/n$, with $n$ the number of iterations, and $A^{*}$ related to the order of ramification. The cross-over temperature $T^{*}$ is to be compared with the glass cross-over temperature $T_g \simeq A_g/n$. An estimation of the ratio $T^{*}/T_g$ yields an estimation of the order of ramification of bidimensional percolation clusters at the threshold ($C = 2.3 \pm 0.2$).