# Interpreting a graphed covariance function

I'm looking through a slide deck (slide 9) about Gaussian Processes, and I came to a slide that describes one example of a covariance function: Matérn $$\frac{3}{2}$$ Covariance.

$$C(x_1,x_2) = (1+\sqrt{6}\frac{|x_1-x_2|}{\ell})*\exp(\sqrt{6}\frac{|x_1-x_2|}{\ell})$$

where
$$\ell>0$$ is the "correlation length parameter" and
$$\sigma^2>0$$ the variance parameter (though this isn't in the formula which confuses me).

Then they show a graph like this: You can see that changing $$\ell$$ changes the shape of the function. However, my understanding is that covariance requires two random variables/vectors as inputs. So, what is $$x$$ (on the x axis) of the graph referring to in this case?

The horizontal-axis label there is wrong, it should be $$r$$, the "radius". The "radius" is essentially the distance between two multidimensional points $$\bf {x_1}$$ and $$\bf {x_2}$$, and this is usually written as:
$$k(x_1, x_2) = f(r) = f(||x_1 - x_2||_p)$$