I would like to generate random functions from an $m$-sphere $S^m$ to an $n$-sphere $S^n$ that are not too wild, some kind of generalization of random Gaussian fields.
More precisely, I want $f(x)$, for any $x\in S^m$, to be a random point on $S^n$ with uniform probability distribution, and a property that $f(x)$ and $f(y)$ are probably close to each other, whenever $x$ and $y$ are close. (The analogy in Gaussian processes is that the covariance decays via some given function)
One question I have in mind is to see, how often a random function is homotopically trivial.
Is there a way to define such maps algorithmically and/or a reference to a rigorous definition of such sphere-valued fields?