Why is the bottom of the Mariana Trench not frozen? I have found following information on Wikipedia:

The temperature is $1$ to $4\,°\mathrm C$ and the pressure is about $0.156$ GPa.

The Graph tells us that it's liquid.
However, I don't quite understand why the temperature is not below $0\,°\mathrm C$. Light is unable to reach these depths and thus unable to transfer heat.
My reasoning is that eventually particles with higher kinetic energy will have risen due to convection, lowering the temperature below $0\,°\mathrm C$, resulting in freezing the water.

 A: You are basically asking why water does not freeze at the bottom of the trench. It is basically because of two reasons:

*

*it is very important to understand the difference between fresh and salt water. We are talking about salt water here and the freezing point (phase change) is below $0^\circ \,\rm{C}$, at around $-2^\circ \,\rm{C}$. This takes your question as to why water at the bottom of the trench cannot reach $-2^\circ \,\rm{C}$.


*you are asking about the bottom of the trench, where the water is in direct contact with the ground. To reach freezing (phase change), the ground would need to have the same (or even colder) $-2^\circ \,\rm{C}$. In reality, because of geothermal energy, the ground is way warmer then that.
A: Your intuition is that warm water rises, so eventually the coldest water molecules end up at the bottom and freeze. There are two problems with this: 
(1) individual water molecules do not really feel much of buoyancy if they move fast (=are hot). The buoyancy of warm fluids and gases is because warm molecules bounce more vigorously against each other, producing an expansion of the medium that lowers its density and this makes it tend to move upwards if it is surrounded by colder, more dense medium. So there will not be any strong sorting of molecules, just of water parcels.
(2) Water conducts heat, and a cold water parcel surrounded by warmer water or rock will tend to heat up. This is the main reason the deepest ocean is not arbitrarily cold. There is some heat coming from the upper warmer layers even if they do not mix, and there is geothermal heat (around 60 mW/m$^2$) from below. 
There are further issues. There are slow currents (the thermohaline circulation) driven by the polar regions dripping cold, salty water into the deep, and upwellings in some regions where the deep water gets mixed with warm surface waters.
Finally, the pressure at the deepest trenches is still somewhat below the level where high-pressure ice would form. Were the trenches deeper we would actually get ice formation down there up until the level where the pressure was insufficient. This is fairly temperature-independent (the upper curve between blue and green is fairly horizontal), so it would mostly depend on depth. 
(Note that in the uppermost atmosphere where the density is low enough molecules actually do show some sorting since they do not collide much, and the hottest molecules go on long parabolic arcs.)
A: Frozen water is less dense than the liquid water, so when icicles are formed, they are moving up, so the ice always builds up at the top of a body of water. If the water reservoir is deep, then, because of thermal capacity of a large volume of water it does not get frozen completely from top to bottom.
The ocean water does not get frozen to the bottom because of many underwater streams moving warm water from one part of the world to the other. Gulfstream is one example of such streams.
A: The water down there does not freeze because of freezing point dpression,
plus the effect of the pressure (In order to freeze water needs to expand).
Since it is not frozen it keeps mixing up with the water coming from above.
