The probability of electron-hole pair recombination as a function of physical proximity When we shine line of an appropriate wavelength at a metal, e.g. gold, such that there is sufficient energy to promote an electron from the valence band to the conduction band, we'll generate with some probability a set of electron-hole pairs.  We can then, to my understanding, rough approximate the electrons and holes as Brownian particles with distinct three-dimensional diffusion coefficients for movement in the metal (conditional on lattice defects / etc. being randomly or uniformly distributed).
Now, when a diffusing electron comes with some critical Euclidean distance $r$ of a hole, the "particles" should recombine with some probability and annihilate one-another.  
My question is... how can we characterize this process in principle in some common metals or semiconductors like silicon?  Is there a known distance-dependent relationship / probability distribution for recombination?  Is this relationship conserved over different metals / metalloids?  To be clear, I'm not talking about some overall probability distribution for recombination as a function of time - I'm strictly asking about how the physical distance between the electron and hole effects the annihilation probability. 
Here's a cartoon drawing from Wikipedia of an electron and a hole diffusing to the cathode and anode, respectively, of a silicon-based photovoltaic chip: http://en.wikipedia.org/wiki/File:Silicon_Solar_cell_structure_and_mechanism.svg.  Of course, here there's a built-in potential to induce the appropriate directed Brownian motion of the electron and hole, but this should of no consequence for my above question.
 A: The electrons and holes are quasiparticles which can be hardly represented by a free-particle model due to their exchange-correlation coupling. For instance, at the extremely low densities, when low-energy electron and hole arrive at very short distance to each other, they do not necessary recombine, they probably form exciton, a stable bound state of the electron-hole pair. After some time, exciton can recombine. At high densities, the exciton lifetime is very small due to Coulomb interactions with other particles and it recombines faster. This time can be computed or be estimated from experimental absorption spectra from broadening of the  excitonic lines. At highly dense electron-hole plasma, the recombination probability is computed using simple Fermi's Golden Rule. Here you have to know recombination mechanism in order to compute a matrix element: luminescence, phonon-assisted, through traps. Each of these processes has own probability. In all these cases, dependence on the density is determined via the Fermi-Dirac distribution of electrons and holes in k-space that enters Fermi's Golden Rule.
