Can I match my particle density to my displacement map? I have a plane which uses a noise texture as a displacement map.

I want to put some grass on the plane as particles and match the particle density to the displacement map but I can't find a way to link/save the texture. What is the correct process for this?
You can add a Mapping node, to better control the location/scale/rotation of your texture.

Noise texture always uses the same formula to calculate the noise. 
So every instance of the noise texture looks the same.

Add a Mapping node, so you can precisely control the output.