# Strict stochastic domination of “thinned out” random cluster model

Fix some $$q\geq 1$$ and denote by $$X_p$$ a random variable sampled from the law of the random cluster model with parameters $$p,q$$ on some graph $$G$$ and with, say, free boundary conditions.

Define the "thinned out" $$X_p^\delta$$ as follows: Sample (an independent copy of) $$X_p$$ and then close each open edge independently with probability $$\delta>0$$. Equivalently one may define $$X_p^\delta$$ as the pointwise product of $$X_p$$ and an independent Bernoulli percolation process with parameter $$1-\delta$$.

Can one prove that $$X_p^\delta$$ is stochastically dominated by some $$X_{p'}$$ with $$p'? (Clearly it works for $$\delta$$ sufficiently large and $$p'=p$$, but I would like to say that one can find such a $$p'=p'(\delta) for every $$\delta>0$$.)

My attempt: We have the finite energy property, i.e. no matter what values we condition on outside of some fixed edge $$e$$, the probability of that edge being open is in $$\{p,p/(p+q(1-p))\}$$; in particular it is bounded away from $$1$$ (and $$0$$). So by Lemma 1.1 we can stochastically bound the law of $$X_p$$ from above (and below) by some Bernoulli percolation measure. Clearly $$X_p^\delta$$ (as product of $$X_p$$ and some independent Bernoulli process) satisfies the same stochastic inequality with an even smaller bound, but I'm not sure if I can translate this back into stochastic domination in terms of a random cluster model. I guess I would need some (independent) process $$Y$$ which is stochastically bounded from below by some Bernoulli percolation process such that $$YX_p$$ (pointwise product) has the law of $$X_{p'}$$, but I don't know how to construct such a $$Y$$.

Theorem 1.6 (MON) seems helpful.