Length Distributions in Loop Soups

Statistical lattice ensembles of loops in three or more dimensions typically have phases in which the longest loops fill a finite fraction of the system. In such phases it is natural to ask about the distribution of loop lengths. We show how to calculate moments of these distributions using $C{P}^{n-1}$ or $R{P}^{n-1}$ and $\mathrm{O}(n)$ $\sigma$ models together with replica techniques. The resulting joint length distribution for macroscopic loops is Poisson-Dirichlet with a parameter $\theta$ fixed by the loop fugacity and by symmetries of the ensemble. We also discuss features of the length distribution for shorter loops, and use numerical simulations to test and illustrate our conclusions.

Figure 1

$\mathcal{L}{P}_{\mathrm{link}}(l)$ versus $l/\mathcal{L}$, on a double logarithmic scale for the indicated $\theta$ values. Inset: comparison of data for $L=40$ and $L=100$. Dashed line has slope $-3/2$. Fluctuations at small $l$ arise from loops of length equal to a few lattice spacings.

Figure 2

Behavior of loop length distribution in PD regime: $\mathcal{L}{P}_{\mathrm{link}}(l)/\theta$ versus $l/f\mathcal{L}$, showing dependence on PD parameter $\theta$. Points: simulation data; curves from Eq. (11). Inset: the measured ratio $\mathcal{L}{P}_{\mathrm{link}}(l)/[1-l/(f\mathcal{L})]$ is close to its theoretical value $\theta$ except for $l\approx f\mathcal{L}$, where statistical errors are large.

Figure 3

Comparison of simulation data (points) with theoretical values (lines) for ratios of moments of loop lengths. In the main panel we plot the one-loop ratio $R$ and in the inset the two-loop ratios $Q_2$, $Q_3$, and $Q_4$. Universality is tested by the comparison of results for different lattices ($K$, $L$ and unoriented), defined in Refs. 22, 24, 25.