I think you're right, but it's really isotropy in combination with independence of x and y. Isotropy means you really are dealing with a function of r where r^2 = x^2 + y^2, and independence means the resulting function p(x,y) must factor as p(x) * p(y). This then forces the exponential form.
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Isn't the key point here about isotropy, similar to the velocities in the ideal gas?
I think you're right, but it's really isotropy in combination with independence of x and y. Isotropy means you really are dealing with a function of r where r^2 = x^2 + y^2, and independence means the resulting function p(x,y) must factor as p(x) * p(y). This then forces the exponential form.