Graph theory and statistical physics
Dnr:

SNIC 2019/3-152

Type:

SNIC Medium Compute

Principal Investigator:

Per Håkan Lundow

Affiliation:

Umeå universitet

Start Date:

2019-04-01

End Date:

2020-04-01

Primary Classification:

10104: Discrete Mathematics

Secondary Classification:

10304: Condensed Matter Physics

Webpage:

## Abstract

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. We have earlier studied the influence of the boundary for the Ising model in 5 dimensions. A Monte Carlo study (Nucl Phys B 889 (2014) 249) then gave that free boundary conditionsleads to susceptibility growing as O(L^2) as opposed to O(L^{5/2}) for periodic boundary conditions. We now intend to see how intermediate boundary conditions affects the exponent. For example, how many boundary edges should we remove to get O(L^{9/4})? 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 rather moderate $L\le 80$. We have already improved dramatically on this but we are striving for $L=256$ which requires an extra computational effort.