Geometry of Set of States of Random Process

Question

Consider the following random process in the unit circle. Starting at $(0,0)$, move up, down, left, right a distance $1/2$ with equal probability.

At the next step there are three possibilities: stay where you are with probability $1/2$, go a distance $1/4$ in a direction orthogonal to the previous step with probability $1/4$ (*).

Keep iterating, at step $k$ stay where you are with probability $1/2$, go a distance $1/2^k$ in a direction orthogonal to the previous step with probability $1/4$.

Can we describe the set of points this process can visit?

It seems to have a fractal nature, and I think the closest it gets to the circle is $(2/3,1/3)$.

(*) In context this probability might not be $1/4$ and there are some absorbing states. The idea is to encode products of non commuting orthogonal projections $p, 1-p, q, 1-q$ as points in the unit circle. $p$ would be up, $1-p$ down and similar for $q$ and $1-q$.

Answer 1

Here is a FragM script that renders the fractal (and unit circle) using distance estimation:

#version 330 core

#include "MathUtils.frag"
#include "Progressive2D.frag"

uniform int Iterations; slider[0,10,100]

vec3 color(vec2 q)
{
  q *= 1.05;
  float dq = length(vec4(dFdx(q), dFdy(q)));
  vec2 p = q;
  p *= 2.0;
  float dp = length(vec4(dFdx(p), dFdy(p)));
  p = abs(p);
  if (p.x > p.y)
  {
    p.xy = p.yx;
  }
  p.y -= 1.0; // first step has no stay-put option
  vec2 dir = vec2(1.0, 0.0);
  float dist = 0.5;
  for (int i = 0; i < Iterations; ++i)
  {
    float d = dot(p, dir) / dist;
    if (d < -2.0/3.0)
    {
      p += dist * dir;
      dir.xy = dir.yx;
    }
    if (d > 2.0/3.0)
    {
      p -= dist * dir;
      dir.xy = dir.yx;
    }
    dist *= 0.5;
  }
  return vec3(tanh(clamp(min(length(p) / dp, abs(length(q) - 1.0) / dq), 0.0, 4.0)));
}

Here is what it looks like, within the unit circle:

There is some ambiguity about "orthogonal to previous step", perhaps, I chose to interpret it as "orthogonal to the last step that actually moved some distance".

Answer 2

Well, a bit of simulation shows there isn't anything particular amazing going on: