This paper studies a surprising breakdown of monotonicity in the Ising energy field and gives a precise fix. The Ising energy field is the set of edges in a graph whose endpoints carry the same spin. One might expect that, when the interaction strengths (coupling constants) increase, this random set only grows in a stochastic sense. Past work shows that expectation is true for single edges but can fail for the whole set of edges. The authors introduce weakened forms of stochastic ordering that restore a useful sense of monotonicity and pin down exactly how much independent thinning is needed.

The two weakened notions are called p-weak domination and p-weak† domination. Both apply the same independent thinning to both measures before comparing them. Independent thinning means keeping each edge with some probability p, independently across edges (this is the familiar Bernoulli percolation). In p-weak domination one asks whether µ thinned at rate p is stochastically dominated by ν thinned at rate p. The p-weak† version imposes a symmetric requirement that also controls what happens to the closed edges, so it is stable under taking complements. For the Ising energy field the authors find exact formulas for the largest p that make these weakened dominations true.

A key technical ingredient is a new stochastic comparison for Fortuin–Kasteleyn (FK) percolation, also called the random-cluster model. FK percolation is a random set of edges weighted by the number of connected components and by independent edge probabilities. Theorem 1.1 gives inequalities that “mix’’ different edge-probability parameters and different cluster-weights q (in the regime q≥1). One useful special case is that, if you increase the per-edge probabilities pointwise from p to p̃, then the FK law at p is stochastically dominated by the FK law at p̃ intersected with an independent Bernoulli thinning of parameter p/p̃. This type of comparison is stronger than the usual monotonicity statements and it is what lets the authors compare the Ising energy fields using a single thinning parameter on both sides.