Markov Chain — Random Walker & Stationary Convergence (2D)

Markov Chain (2D)

Walker speed 1.0 hops/s

How often the walker samples its next state from the current row of P. Drag a node to move it; tap a node to teleport the walker there and restart the histogram.

Transition matrix P

Rows are auto-normalised so they always sum to 1.

Live Readouts

Hops taken

Current state

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|empirical − stationary| (L1 error)

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How this differs from the 3D version

The 3D companion advances a full probability vector π(t+1) = π(t)·P every frame and shows it as growing node discs. This 2D view instead runs one concrete walker: at each hop it samples the next state by drawing a uniform random number r and walking along the current row's cumulative sums until r falls inside a bucket — exactly how a Markov chain is actually simulated Monte-Carlo style.

cum = 0
for j in states:
  cum += P[i][j]
  if r <= cum: next = j; break

Every hop increments a visit counter for the landed state. The bar chart's filled bars are this empirical frequency (visits[j] / total hops). The thin gold tick on each bar is the theoretical stationary distribution π*, found independently by power-iterating the matrix from a uniform start: v ← v·P, repeated ~500 times, which converges to the left eigenvector of P for eigenvalue 1 for any ergodic chain. Watch the filled bars drift toward the gold ticks as hops accumulate — that convergence is the law of large numbers acting on a single sample path, as opposed to the 3D page's exact deterministic propagation of the whole distribution.