Graph theory and statistical physics

SNIC 2017/1-146


SNAC Medium

Principal Investigator:

Per Håkan Lundow


Umeå universitet

Start Date:


End Date:


Primary Classification:

10104: Diskret matematik

Secondary Classification:

10304: Den kondenserade materiens fysik




The project concerns mainly the random cluster model and the spin glass model. The random cluster model is a natural generalisation of the percolation/Ising/Potts models without external field. It has a parameter $0<p<1$ and a model parameter $q>0$. Integer values of $q$ correspond to percolation ($q=1$), the Ising model (q=2) and the $q$-state Potts model ($q>2$). We are particularly interested in the case $0<q<1$ (where much less is known) and $q=1$. Percolation is expected (but not proved) to have critical dimension $d=6$ but for other $q$ it could be much higher. We intend to investigate this by looking at how different boundary conditions (free or periodic boundary) influence the finite-size scaling behaviour. We need data for different sizes, dimensions and boundary conditions and many $p$-values near $p_c$. The sheer size of the cubes will limit the possibilities for different sizes above $d=6$ so this will be highly experimental. We hope to write a program where the full cube is not stored but dynamically built as needed in computer memory. For $d=2$ it is known that the critical probability $p_c=\sqrt{q}/(1+\sqrt{q})$ when $q=>1$. Our numerical data collected on hebbe suggest that this also holds for $0<q<1$. We have also collected data on Schramm-Löwner Evolution (SLE) where we keep part of a 2D-lattice fixed and collect data on the length of a boundary curve surrounding the fixed part. The distribution of its length appears quite interesting which we would like to collect more data on. In the spin-glass model the interactions $J_{ij}$ between sites $i$ and $j$ are drawn from a statistical distribution, say Bernoulli, Gaussian, Laplace, uniform etc. Universality predicts that the choice of distribution does not affect the critical exponents. However, we have previously found different critical exponents for different distributions for $d=4$ and recently also for $d=5$ (PRE 95 (2017), 012112) meaning that universality does not hold. Very recently (to appear in PRE) our data for $d=2$ (following up on our PRE 93 (2016) 022119) suggested universality for continuous distribution but non-universality for discrete distributions. However, for $d=3$, our next objective, less is known. Due to equilibration problems it is very difficult to obtain good data. We thus need to use a major share of the applied time here, collecting data on different distributions where we can. However, we have some ideas for how to improve our programs to manage the equilibration problem which we also need to put to the test.