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 $p'<p$? (Clearly it works for $\delta$ sufficiently large and $p'=p$, but I would like to say that one can find such a $p'=p'(\delta)<p$ 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.


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