Infinite loops in random loop models on bounded-degree graphs occur strictly later than infinite clusters in the associated Bernoulli percolation.
Critical temperature of Heisenberg models on regular trees, via random loops
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abstract
We estimate the critical temperature of a family of quantum spin systems on regular trees of large degree. The systems include the spin-$\frac12$ XXZ model and the spin-1 nematic model. Our formula is conjectured to be valid for large-dimensional cubic lattices. Our method of proof uses a probabilistic representation in terms of random loops.
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Critical Parameters for Loop and Bernoulli Percolation
Infinite loops in random loop models on bounded-degree graphs occur strictly later than infinite clusters in the associated Bernoulli percolation.