Quenched and Annealed measures

Question

I apologise in advance if the question is too "meta".

I'm trying to understand the difference of "quenched" and "annealed" in the following context:

Suppose you have some random quantity $\Lambda$, given by a law $\nu$. We have a set of measures, depending on this random quantity $\Lambda$, said "quenched" measures $P^\Lambda$, defined for every $\Lambda$.

We define an annealed measure $$P(\cdot):=\int_{\Xi}P^\Lambda(\cdot) d\nu(\Lambda),$$

so the "quenched" version is a measure in a particular fixed $\Lambda$, and the annealed is an average of the possible enviroments.

A "quenched" result would be something like

"If [statement] then $P^\Lambda([event])>[something]$ for $\nu$-almost every $\Lambda$"

An "annealed" statement would be

"If [statement] then $P([event])>[something]$"

My question is: How could a result be different for quenched and annealed measures if they are, in a sense, the same thing?

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In my particular situation I'm studying percolation on a random enviroment, where $\Lambda$ is a gives the random structure of the graph we're using. We're able to define $$p_c(\Lambda) = \sup\{p \in [0,1]: P^\Lambda_p(0 \leftrightarrow \infty)=0\}$$

and by ergodicity it's possible to show in that setting that $p_c$ is independent of $\Lambda$. The question is, is it possible to find results that are true in a quenched setting and false in the annealed one?

For instance, could we find exponential decay below $p_c$ in a quenched setting, whereas in the annealed version that's impossible?

Answer 1

Just had a meeting with my PhD advisor and he explained the issue. The answer is actually pretty simple (in fact I feel a bit silly now).

In my case where we were talking about exponential decay, a quenched result would be:

If $p<p_c$ then for $\nu-$almost every enviroment $\Lambda$ there's a constant $c(\Lambda)$ such that $$P_p^\Lambda(0 \leftrightarrow \partial B_n)\leq \exp(-c(\Lambda)n).$$

In that case the rate of my exponential decay depends on the enviroment itself. When trying to push this to the annealed version, my rates of decay could diminish fast enough, in a way that, in the annealed version after integrating on the enviroment, I no longer have an exponential decay.

Of course, if the rate was independent of the enviroment, or bounded by some global constant, the result in quenched models would imply it in the annealed version.

So, although we're modelling the same thing, there might be results valid in one setting and false in the other.