arrive at a complete quantitative understanding of phase transitions in terms of the geometrical properties of the loop gas under consideration.[@ABH; @NguyenSudbo; @loops; @Hoveetal; @pre01; @geoPotts; @ht; @anomalous] The resulting geometric description of critical phenomena ties together various strains of theoretical physics: statistical mechanics, quantum field theory, polymer physics, and percolation theory–all to be discussed below.

The plan of these notes is as follows. In Sec. \[sec:lft\], the particle content of the linear O($N$) symmetric quantum field theory, defined on a hypercubic lattice, is studied in the spirit of Feynman’s sum-over-paths approach, starting from the noninteracting theory in Secs. \[sec:noninteracting\] and \[sec:loopgas\]. Interactions are perturbatively included in Sec. \[sec:phi4\], where also the spacetime interpretation of Feynman diagrams is pursued. In Sec. \[sec:spinmodel\], the strong-coupling limit, where the field theory reduces to the nonlinear O($N$) spin model, is considered. The high-temperature representation of the nonlinear model is shown to provide a purely geometric representation, directly connected to Feynman’s spacetime approach for the linear model. As a case study, the high-temperature representation of the Ising model, corresponding to $N=1$, on a square lattice is detailed in Sec. \[sec:IM\].

In Sec. \[sec:CP\], the critical properties of the O($N$) universality class are examined from 