We give approximate counting and sampling algorithms for spin models on graphs of low threshold rank or high girth. Given any $n$-vertex graph $G$, let $k_η= k_η(G)$ count normalized-Laplacian eigenvalues strictly below $η$. For $\varepsilon > 0$, an $\varepsilon$-approximate counting algorithm outputs an estimate $\widehat Z$ of the partition function $Z$ satisfying $(1-\varepsilon)Z\le\widehat Z\le(1+\varepsilon)Z$ with probability at least $3/4$. An $\varepsilon$-approximate sampling algorithm outputs a configuration whose distribution is within total variation distance $\varepsilon$ of the target distribution. We give the following results: (a) Ferromagnetic Potts. Fix $q\ge2$ and $η\in(0,1]$. There exists $β_0(q,η)$ such that, for every inverse temperature $β\geβ_0$, the zero-field $q$-state Potts model admits approximate counting and sampling algorithms: (i) On arbitrary graphs in time $\exp(O(k_η))\operatorname{poly}(n,1/\varepsilon).$ (ii) Fix $Δ\ge 3$. On $Δ$-regular graphs of girth $g$ in time $\exp(O(n e^{-g})) \operatorname{poly}(n,1/\varepsilon).$ (b) Bipartite hard-core. Fix $Δ\ge3$ and $η\in(0,1]$. There exists $λ_0(Δ,η)$ such that, for every fugacity $λ\geλ_0$, the hard-core model admits approximate counting and sampling: (i) On $Δ$-regular vertex-transitive bipartite graphs in time $\exp(O (k_η))\operatorname{poly}(n,1/\varepsilon).$ (ii) On $Δ$-regular bipartite graphs of girth $g$ in $\exp(O(n e^{- g}))\operatorname{poly}(n,1/\varepsilon).$ All four results follow from verifying the hypotheses of a new framework we call C-3PO. At a high level, the C-3PO algorithm proceeds by enumerating small cuts and reconstructing the original measure through cluster expansions. Underlying the algorithm is an improvement of subspace enumeration methods via active regression.

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