Diffusion CFTP.py

From Werner KRAUTH

Contents 1 Context 2 Python program 3 Output 4 Further information 5 References

Context

This page is part of my 2024 Beg Rohu Lectures on "The second Markov chain revolution" at the Summer School "Concepts and Methods of Statistical Physics" (3 - 15 June 2024).

In the example of a particle diffusing on a path graph with five sites, with moves from configuration i to [i-1, i, i] being proposed, we consider the formulation of a Markov chain in terms of random maps, but run from time t=-infinity up to time t=0.

At time t=-infinity, as shown, the pebble starts on configuration 1, so that, by time t=0, it must sample the stationary distribution (the uniform distribution on the five sites). The below program constructs just as many set of arrows as needed to conclude on where it finishes it infinitely long journey.

Python program

import random
import matplotlib.pyplot as plt

N = 5
pos = []
for stat in range(100000):
   all_arrows = {}
   time_tot = 0
   while True:
      time_tot -= 1
      arrows = [random.choice([-1, 0, 1]) for i in range(N)]
      if arrows[0] == -1: arrows[0] = 0
      if arrows[N - 1] == 1: arrows[N - 1] = 0
      all_arrows[time_tot] = arrows
      positions=set(range(0, N))
      for t in range(time_tot, 0):
         positions = set([b + all_arrows[t][b] for b in positions])
      if len(positions) == 1: break
   a = positions.pop()
   pos.append(a)
plt.title('Backward coupling: 1-d with walls: position at t=0')
plt.hist(pos, bins=N, range=(-0.5, N - 0.5), density=True)
plt.savefig('backward_position_t0.png')
plt.show()

Output

Here is output of the above Python program. The histogram is absolutely flat, without any corrections. But this is normal, given that the simulation has run, for each of the realizations of the random map, an infinite number of iterations.

Further information

References

This program illustrates the coupling-from-the-past approach to Markov-chain sampling introduced by Propp and Wilson, in 1997.

The reference is: