Abstract
AbstractWe investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.
Highlights
Partition functions play a central role in statistical physics
We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle
We prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal
Summary
Partition functions play a central role in statistical physics. The distribution of zeros of the partition functions is instrumental in describing phase changes in a variety of contexts. The following result from [PR20] shows that while the Cayley trees form a relatively small subset of the class of all graphs of bounded maximal degree, the zero free loci of these two classes are identical in the ferromagnetic case. Our main result shows that, contrary to the ferromagnetic case, the zero free locus for the Cayley trees is strictly larger than that of the class of all bounded degree graphs. We prove case (3) for the subclass in Gd+1 given by the spherically symmetric trees These trees have the advantage that dynamical methods can be used to describe the location of zeros, as indicated by the following lemma, whose proof will be given later . When all degrees kn are equal to d the tree Tn is said to be a (rooted) Cayley tree of degree d
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