Statistical mechanics of graphity models

Graphity models are characterized by configuration spaces in which states correspond to graphs and Hamiltonians that depend on local properties of graphs such as the degrees of vertices and numbers of short cycles. As statistical systems, graphity models can be studied analytically by estimating their partition functions or numerically by Monte Carlo simulations. Results presented here are based on both of these approaches and give new information about the high- and low-temperature behavior of the models and the transitions between them. In particular, it is shown that matter degrees of freedom must play an important role in order for the low-temperature regime to be described by graphs resembling interesting extended geometries.

Figure 1

Interaction moves on graphs.

Figure 2

Lowest-energy graphs in the basic model with $N=36$ and with (a) $r=-2.5$ and (b) $r=2.5$. The results are obtained through simulation at $\beta =0.2$.

Figure 3

Average energy per node as function of $N$ in two series of simulations. Data points in the rising series are obtained through simulation with $v_0=$, $r=-2.5$, and fixed $\beta =0.17$. Data points in the flat series correspond to $\beta$ scaling logarithmically with $N$. Error bars represent the standard error on the mean. The lines shown merely connect the data points and do not represent any fit.

Figure 4

Average energy per node for systems with $r=-2.5$ and $N$ equal to (left to right) 60, 120, 180, and 240. Error bars on the points (some too small to be seen) represent the standard error on the mean. The lines do not represent computed data and are only shown to aid the eye distinguish between the four series.

Figure 5

Normalized specific heat for systems with $r=-2.5$ and $N$ equal to (left to right) 60, 120, 180, and 240. Error bars are computed using the bootstrap resampling method. The lines do not represent computed data.

Figure 6

Graphs with lowest criterion $C$ for $N=36$ and $r=+2.5$. Results are obtained with $\beta =0.2$ and the variance potential set to (a) $\mu =0.8$, (b) $\mu =3$, and (c) $\mu =4$.

Figure 7

Two graphs representing excitations of the $r=-2.5$ and high $\mu$ model.

Figure 8

Low-criterion quartic graphs. (a) is obtained with $r=-1.5$, $\beta =0.6$, and $\mu =32$. (b) is produced with $r=-2.5$, $\beta =0.17$, and $\mu =3.8$.