It isn't that unusual to use 8mm rope in caving on vertical (at least in Europe) especially in deeper caves with more rope to carry down and of course, back out again.
In the US cavers tend to rig pitches from a single anchor and take care that there are no sharp bits of rock the rope could come in contact with and use rope which is more abrasion-resistant. In Europe we will place intermediate anchors (a "rebelay") on longer pitches which helps prevent rope rub and speeds up abseil and ascent as the pitch is split into sections. Also this means that thinner rope can be used as there should be no chance of abrasion because of rubbing on rock.
Ropes are rigged such that the chances of the rope rubbing anywhere on rock is reduced to nil (or as near as possible) because when prusiking back up ropes, any rope rub will damage the rope and with thinner diameter ropes, the safety factor with respect to damage is much reduced.
I often visit a local old lead mine with a 100 metre shaft which is generally at an angle 45 degrees or steeper with several large steps or platforms cut into the rock at different points. Even though the angle is nowhere near vertical, the shaft is not straight as it wasn't bored by a tunnelling machine but formed partly be natural activity and partly by miners. This means that a single length or rope would rub at various points, both on the lower side of the shaft and also on the upper side.
Cavers do not use Figure of Eight or similar descenders due to the fact that they twist the rope and can be dropped when attaching or detaching to or from the rope. Cavers use either racks (for example) or "bobbin" type descenders (for example) as these do not twist the rope and remain attached to your harness even when loading or unloading the rope.
Although you intend to descend a slope, if it is of any real length other than a few metres, I would treat it as a normal abseil (or rappel) and would be happy to use 8mm rope (bearing in mind my points above). However, below this diameter I would be worried about the ability to absorb any shock in a fall or any possibility of damage from abrasion. Even on a slope, a fall can occur. For example an intermediate belay could give way.