I can explain why there's a tendency for the cubes to clump, but I'll have to guess a little about exactly how it happens.
In thermal equilibrium, things tend toward a state of lowest 'free
energy'. Ultimately, that's because such a state is can be constructed
with a larger number of quantum states of the the system and its
environment, and nature seems to shake things up so that any allowed
quantum state is equally likely.
For ice, the molecules at the surface have higher free energy than
those in the middle. largely, that's because they are missing some of
the bonds that the middle molecules have. So left to itself for a very
long time, ice tends to form lumps with low surface area. In the case
of ice cubes resting on each other, there is also an increase of free
energy at the points where the cubes are in contact and the pressure
squeezes the ice. Melting those contact points so the cubes fit
together better without high-pressure spots reduces the free energy.
Pressure tends to melt ice, because liquid water has a lower volume.
Apparently a thin layer of molecules at the surface can melt,
migrating out away from the contact points. Farther out, where the
pressure is lower and the the water can freeze between the cubes with
very little surface area, it refreezes.
In addition, if there is enough moisture in the atmosphere and
the cubes are cold enough, some water can freze right out of the
atmosphere, filling in the gaps between the cubes.
What I'm not at all sure of is the relative importance of the
pressure effect and the simple reduction of surface area effect. Also,
I'm not sure how about the relative imporatnce of the vapor and of the
surface liquid in moving molecules around.
You can check the hypothesis that moisture from the air is
condensing on your ice cubes and then freezing by looking for fuzzy
frost on your ice cubes. Another idea of what could be going on is that
your ice cubes are melting a bit and then re-freezing if the
temperature in your ice bucket goes through cycles of warming and
cooling. This will certainly speed up the process of clumping ice cubes
(published on 10/22/2007)