An Efficient Monte Carlo Method to Make a Geometric Graph with a Fixed Connectivity
Abstract
We present a Markov chain Monte Carlo (MCMC) method to make a geometric graph that satisfies the following two conditions: (i) The degree of each vertex is fixed to a positive integer k. (ii) The probability that two vertices located on a ddimensional hypercubic lattice are connected by an edge is proportional to d_{ij}^{  α }, where d_{ij} is the distance between the two vertices and α is a positive exponent. We introduce a reverse update method and a listbased update method for the MCMC method. The graph is updated efficiently by the MCMC method since the two update methods work complementarily. We also investigate a ferromagnetic Ising model defined on the geometric graph as a test case. As a result, we have confirmed that the nature of ferromagnetic transition significantly depends on the exponent α.
 Publication:

Journal of the Physical Society of Japan
 Pub Date:
 October 2020
 DOI:
 10.7566/JPSJ.89.104006
 arXiv:
 arXiv:2004.00920
 Bibcode:
 2020JPSJ...89j4006S
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Disordered Systems and Neural Networks
 EPrint:
 26 pages, 21 figures