Why don't black holes quickly eat up the galaxy/universe? It's my understanding that the radius of the event horizon of a black hole is proportional to it's mass.
Which means the surface area of a black hole is proportional to it's mass squared.
Lets just assume that black holes eat stuff that by chance &quot;runs into them&quot;.
Using this (perhaps naive) assumption, this would mean black holes gain mass at a rate proportional to their surface area, which is the same as their radius squared, which is their mass squared.
So then we get something like:
$\frac{dm}{dt} \propto m^2$
$m = 1 / (C - t)$
Which means that all black holes should grow to an infinite size in a finite length of time.
Indeed they will eventually get to a point where even small amounts of mass will grow the volume of the black hole so much (because volume increase is proportional to the current mass squared) and even if the density of the universe is very low this will eventually mean any growth will set off runaway growth because each atom the black hole swallows increases it's volume such that on average that new volume contains a new atom.
Is it just that the constant factors here are very high, or are one of my assumption quite incorrect? I say quite incorrect because even if they're slightly incorrect (the growth rate is proportional to mass not mass squared) many of these issues still occur on long enough timescales.
Your rough analysis is basically correct; everything may well end up in black holes eventually. But the timescale for that is very long indeed.
The cross-section for direct interaction as you describe is vanishingly small. The diameter of a ten solar mass black hole is 60 km. The rate at which it scoops up interstellar material from a cylinder of diameter 60 km is totally negligible. e.g. A speed of 10 km/s with respect to interstellar gas and a typical gas density of $10^6$ H atoms per cubic metre, would result in an accretion rate of about 1.5 kg/year.
The chances of direct star to star collisions in a galaxy is exceedingly rare, so the chances of a wandering star crashing directly into a black hole are even smaller.
With supermassive black holes, the issue with your analysis is that they sit in the centres of galaxies and stars do not encounter them by chance. The stars and gas in a galaxy have angular momentum that must be conserved. This takes them on orbits which do not go near the black hole unless perturbed in some way. These supermassive black holes feed while they can, but when all the easily available matter has been consumed, they become dormant. This is one of the reasons that the peak of quasar activity was about 10-12 billion years ago.