A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field

Abstract

We make a few elementary observations that relate directly the items mentioned in the title. In particular, we note that when one superimposes the random current model related to the Ising model with an independent Bernoulli percolation model with well-chosen weights, one obtains exactly the FK-percolation (or random cluster model) associated with the Ising model. We also point out that this relation can be interpreted via loop-soups, combining the description of the sign of a Gaussian Free Field on a discrete graph knowing its square (and the relation of this question with the FK-Ising model) with the loop-soup interpretation of the random current model.

5 pages

References in corpus (2)

• Convergence of the two-dimensional random walk loop soup clusters to CLE
• Decomposition of Brownian loop-soup clusters

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• Lecture notes on the Gaussian Free Field
• The planar Ising model and total positivity
• Covariant Symanzik identities
• Inverting the coupling of the signed Gausssian free field with a loop soup
• Conformal invariance of double random currents II: tightness and properties in the discrete
• An equivalence between gauge-twisted and topologically conditioned scalar Gaussian free fields
• Cluster explorations of the loop soup on a metric graph related to the Gaussian free field
• On monotonicity and couplings of random currents and the loop-$\mathrm{O}(1)$-model
• Vector-valued Gaussian free field conditioned to avoid a ball: Entropic repulsion of the norm and Freezing of spins