| Riassunto: | A spin system is a framework in which the vertices of a graph are assigned spins
from a finite set. The interactions between neighbouring spins give rise to weights, so a
spin assignment can also be viewed as a weighted graph homomorphism. The problem
of approximating the partition function (the aggregate weight of spin assignments) or of
sampling from the resulting probability distribution is typically intractable for general
graphs.
In this work, we consider arbitrary spin systems on bipartite expander ∆-regular
graphs, including the canonical class of bipartite random ∆-regular graphs. We develop
fast approximate sampling and counting algorithms for general spin systems whenever the
degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees
that the spin system is in the so-called low-temperature regime. Our approach generalises
the techniques of Jenssen et al. and Chen et al. by showing that typical configurations
on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the
partition function in O˜(n
2
) time, where n is the size of the graph.
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