the perspective of particle trajectories or high-temperature graphs. In Sec. \[sec:fractal\], their fractal structure is investigated, while in Sec. \[sec:proliferation\], the proliferation of these lines is shown to drive the O($N$) phase transition. In Sec. \[sec:ces\], the fractal structure of the worldlines or high-temperature graphs is related to the critical behavior. The general relations are first applied to self-avoiding random walks, corresponding to the $N\to0$ limit, in two dimensions (Sec. \[sec:saw\]), and to arbitrary $-2 \leq
N \leq 2$ (Sec. \[sec:on\]) also in two dimensions, where many exact results are available for comparison.

In Sec. \[sec:mc\], the high-temperature representation of the Ising model is investigated by means of Monte Carlo simulations. The Metropolis update used to generate the high-temperature graphs is introduced in Sec. \[sec:plaq\], while the numerical results are summarized in Sec. \[sec:num\].

In Sec. \[sec:appl\], the conclusions for the O($N$) universality class are extended to charged systems (Sec. \[sec:higgs\]) and to Bose-Einstein condensation (Sec. \[sec:bec\]) to demonstrate the generality of the spacetime approach to phase transitions.

Finally, in Sec. \[sec:dual\], the worldlines or high-temperature graphs are reinterpreted as line defects in two (Sec. \[sec:peierls\]), three (Sec. \[sec:vortex\]), and four (Sec. \[sec:monopole\]) dimensions, respectively, and the so-called dual field 