The classical random walk isomorphism theorems relate the local times of a continuous-time random walk to the square of a Gaussian free field. The Gaussian free field is a spin system (or sigma model) that takes values in Euclidean space; in this work, we generalise the classical isomorphism theorems to spin systems taking values in hyperbolic and spherical geometries. The corresponding random walks are no longer Markovian: they are the vertex-reinforced and vertex-diminished jump processes. We also investigate supersymmetric versions of these formulas, which give exact random walk representations.

The proofs are based on exploiting the continuous symmetries of the corresponding spin systems. The classical isomorphism theorems use the translation symmetry of Euclidean space, while in hyperbolic and spherical geometries the relevant symmetries are Lorentz boosts and rotations, respectively. These very short proofs are new even in the Euclidean case.

To illustrate the utility of these new isomorphism theorems, we present several applications. These include simple proofs of exponential decay for spin system correlations, exact formulas for the resolvents of the joint processes of random walks together with their local times, and a new derivation of the Sabot–Tarrès magic formula for the limiting local time of the vertex-reinforced jump process.

The second ingredient is a new Mermin–Wagner theorem for hyperbolic sigma models. This result is of intrinsic interest for the sigma models, and together with the aforementioned isomorphism theorems, implies our main theorem on the VRJP, namely, that it is recurrent in two dimensions for any translation invariant finite-range initial jump rates.

We also use supersymmetric hyperbolic sigma models to study the arboreal gas. This is a model of unrooted random forests on a graph, where the probability of a forest $F$ with $|F|$ edges is multiplicatively weighted by a parameter $\beta |F|>0$. In simple terms, it can be defined to be Bernoulli bond percolation with parameter $p=\beta 1+\beta$ conditioned to be acyclic, or as the $q\to 0$ limit with $p=\beta q$ of the random cluster model.

It is known that on the complete graph $KN$ with $\beta =\alpha /N$ there is a phase transition similar to that of the Erdős--Rényi random graph: a giant tree percolates for $\alpha >1$ and all trees have bounded size for . In contrast to this, by exploiting an exact relationship with the hyperbolic sigma model, we show that the forest constraint is significant in two dimensions: trees do not percolate on $\backslash Z2$ for any finite $\beta >0$. This result is again a consequence of our hyperbolic Mermin–Wagner theorem, and is used in conjunction with a version of the principle of dimensional reduction. To further illustrate our methods, we also give a spin-theoretic proof of the phase transition on the complete graph.