The fine structure constant - can it genuinely be a random variable? The question

Does it make sense, and are there physical reasons to think about the
fine structure constant as a (very concentrated) probability
distribution rather than a single real number?

The same question applies to other physical constants  like the mass of a proton or the plank constant if you wish.
more
The question is not if these constant are not constants but varies with time or from one place of the universe to another (that people sometime discuss and was asked here several times);  the question is if these constants are really constants, but they are described not by a single real number but by a probability distribution.
We can also ask if the fine structure constant is a random variable, does it have interesting physical consequences? Could it have consequences for computing quantities where we use it for computation, such as QED computations (say, for energy levels of the Hydrogen atom). Perhaps the effect on such calculation will be more interesting than simple averaging over this random variable.
Sort-of-related
A related mathematical fact is that in critical lattice systems of statistical mechanics, quantities (like the size of the connected cluster containing the origin) often have distributions which are "smeared" - the variance behave like square root of the expectation. (So the fluctuations do not vanish as $n$ goes to infinity.)  I am not aware of such quantities that are directly related to fundamental physical constants though.
background
This (speculative) question arose from a discussion on my blog as a sharp alternative to the following (speculative, as well) possibility: Quantum computers will allow the computation of more and more digits of the fine structure constant, just like we can compute with a digital computer more and more digits of e and π.
Related question: QCD and QED with unlimited computational power - how precise are they going to be? ; ON TCS-stackexchenge Quantum-algorithms-for-qed-computations-related-to-the-fine-structure-constants;
 A: There seems to be no empirical motivation to consider this possibility. But if we approach it as a pure "what if" anyway, it opens a big can of worms. 
(1) How do we even implement the idea in a principled way? We might take a given calculation in which the fine-structure constant is employed, and substitute a sampling of Gil's "very concentrated probability distribution" for each use of the constant (thus making the outcome of the calculation a random variable too). But there are usually several ways of doing a physical calculation. Do we need to worry that different methods, when "randomized" in this way, will yield different random variables for the final answer?
(2) In applied quantum field theory, one deals with quantities that run with energy scale, as described by renormalization group flow. The experimental data is used to make this framework concrete, by telling us (e.g.) a coupling constant's effective value at a particular energy. Then the RG equations (beta functions, etc) tell us what the effective value would be at other scales, should we need that information.
If we are trying to implement the notion of "physical constant as random variable" in a way that fits the deepest physical principles we know, this framework is probably the one that we want to adhere to. 
The best idea I have, would be for the "constant" to actually be the VEV (vacuum expectation value) of a new field. For example, in the standard model, the yukawa couplings are just numbers, but there are beyond-standard-model theories in which they are VEVs of new fields (called flavons in this context). 
Perhaps one can do the same for the two electroweak couplings which combine to give us the observed electromagnetic coupling, and then look for effects on the constancy of the W and Z masses, and other sensitive electroweak observables.
(3) Finally... in string theory, the string coupling is a VEV, the VEV of the dilaton field. So there may be, in theory, an (unobservably small?) effect exactly such as Gil proposes. But it would be best to hear about this from someone who actually knows string theory.
