  In O($N$) non-linear $\sigma$-models on the lattice, the Wolff cluster
algorithm is based on rewriting the functional integral in terms of mutually
independent clusters. Through improved estimators, the clusters are directly
related to physical observables. In the $(N-1)$-d O($N$) model (with an
appropriately constrained action) the clusters carry an integer or half-integer
topological charge. Clusters with topological charge $\pm 1/2$ are denoted as
merons. Similarly, in the 2-d O(2) model the clusters carry pairs of
semi-vortices and semi-anti-vortices (with vorticity $\pm 1/2$) at their
boundary. Using improved estimators, meron- and semi-vortex-clusters provide
analytic insight into the topological features of the dynamics. We show that
the histograms of the cluster-size distributions scale in the continuum limit,
with a fractal dimension $D$, which suggests that the clusters are physical
objects. We demonstrate this property analytically for merons and non-merons in
the 1-d O(2) model (where $D=1$), and numerically for the 2-d O(2), 2-d O(3),
and 3-d O(4) model, for which we observe fractal dimensions $D < d$. In the
vicinity of a critical point, a scaling law relates $D$ to a combination of
critical exponents. In the 2-d O(3) model, meron- and multi-meron-clusters are
responsible for a logarithmic ultraviolet divergence of the topological
susceptibility.
