Graph theory and statistical physics

SNIC 2018/3-137


SNAC Medium

Principal Investigator:

Per Håkan Lundow


Umeå universitet

Start Date:


End Date:


Primary Classification:

10104: Discrete Mathematics

Secondary Classification:

10304: Condensed Matter Physics




The project concerns mainly the random cluster model, the spin glass model and the Ising model in high dimension. 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. 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 $d=5$ meaning that universality does not hold. For $d=3$, our next objective, less is known. Due to equilibration problems it is very difficult to obtain good data. However, we have recently realised that the face-centred cubic lattice is easier to equilibrate than the standard simple cubic lattice, probably due to the fact that the number of neighbours is higher. In combination with the more concentrated Laplace-distribution for the bonds this seems promising. For spin-glasses the order parameter is the spin-overlap between two independent spin systems (replicas) while using the same bonds. It seems however that equilibration runs somewhat faster when using more than two replicas and choose a random pair when measuring. Preliminary data look promising showing non-universality but we are only beginning to get relevant system sizes. We intend to revisit the Ising model at the critical dimension $d=4$. It has been assumed that the scaling behaves as the mean-field model with a logarithmic correction factor. Our study PRE 80,031104(2012) suggested otherwise but lattice sizes were still moderate. The time is ripe for a second look.