How many paths, starting at the origin, taking N steps either up, down, left or right, end up at a particular place (x, y)? Ignoring the fact that x+y has to have the same parity as N, something that goes like e- k(x2 + y2), right? Just central-limit-type reasoning ought to suffice to see that. It's a nice plain isotropic Gaussian sitting at the origin... so there are about as many paths to (x, y) as there are to (x, y) rotated by some smallish angle θ, yeah? Doesn't this mean there ought to be a combinatorial function --- or, like, "approximate" function or something that "rotates" paths by angles less than 90o, even though the paths themselves are made of chunky up/down/left/right steps?
And yet I can't for the life of me hack up anything that seems to work. Somewhat annoying.
I tried staring at the fact that surely you can pretend |N>, |E>, |W>, |S> are states in a Hilbert space, and take the discrete rotation |N> → |E> → |S> → |W> → |N> and make a continous version of it by saying
|N> → f(t)|N> + f(t-1)|E> + f(t-2)|S> + f(t-3)|W>
f(t) = (1+eπit/2+eπit+e-πit/2)/4
but this doesn't seem to do anything nice when I make a big tensor product and ask what happens when some typical state, like, |NENSENSNWNWNWWE> gets rotated; there doesn't seem to be any tendency of the norm of its displacement to get preserved, even in expectation.