Stopping circle.py
From Werner KRAUTH
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| ==Context== | ==Context== | ||
| This page is part of my [[BegRohu_Lectures_2024|2024 Beg Rohu Lectures]] on "The second Markov chain revolution" at the [https://www.ipht.fr/Meetings/BegRohu2024/index.html Summer School] "Concepts and Methods of Statistical Physics" (3 - 15 June 2024). | This page is part of my [[BegRohu_Lectures_2024|2024 Beg Rohu Lectures]] on "The second Markov chain revolution" at the [https://www.ipht.fr/Meetings/BegRohu2024/index.html Summer School] "Concepts and Methods of Statistical Physics" (3 - 15 June 2024). | ||
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| + | We continue our exploration of Markov-chain Monte Carlo algorithms that sample perfectly from a given distribution. In Lecture 2 and xxx, we have encountered beautiful such algorithms for card shuffling and for particle diffusion on a path graph, and in Lecture 5, we will encounter more general perfect-sampling algorithms for hard spheres and for the Ising model. All this is nice, but isn't there a simple perfect-sampling algorithm that one can actually understand, say, for a single particle on a one-dimensional graph?? Here it is, for the a particle on an N-cycle (a path graph of N+1 vertices 0, 1, 2, ..., N, with periodic boundary conditions, where we identify vertex N with vertex 0: Start at vertex 0: with probability 1/N: output this vertex. Otherwise, perform a random walk, moving with probability 1/2 to the left and with probability 1/2 to the right. Keep track of the not-visited vertices. Output the final non-visited vertex. The vertex that are output are equally distributed. | ||
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| ==Python program== | ==Python program== | ||
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| + | What was explained above is more easily expressed in terms of a tiny Python program: | ||
| import random | import random | ||
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| ==Output== | ==Output== | ||
| - | Here is output of the above Python program | + | Here is output of the above Python program for N=6. We do 100,000 trials, and see that the six vertices are output with equal probability. |
| site = 0 probability = 0.1673 | site = 0 probability = 0.1673 | ||
Revision as of 17:48, 25 June 2024
Contents |
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).
We continue our exploration of Markov-chain Monte Carlo algorithms that sample perfectly from a given distribution. In Lecture 2 and xxx, we have encountered beautiful such algorithms for card shuffling and for particle diffusion on a path graph, and in Lecture 5, we will encounter more general perfect-sampling algorithms for hard spheres and for the Ising model. All this is nice, but isn't there a simple perfect-sampling algorithm that one can actually understand, say, for a single particle on a one-dimensional graph?? Here it is, for the a particle on an N-cycle (a path graph of N+1 vertices 0, 1, 2, ..., N, with periodic boundary conditions, where we identify vertex N with vertex 0: Start at vertex 0: with probability 1/N: output this vertex. Otherwise, perform a random walk, moving with probability 1/2 to the left and with probability 1/2 to the right. Keep track of the not-visited vertices. Output the final non-visited vertex. The vertex that are output are equally distributed.
Python program
What was explained above is more easily expressed in terms of a tiny Python program:
import random
N_trials = 100000
N = 6
data = [0] * N
for iter in range(N_trials):
x = 0
if random.uniform(0.0, 1.0) < 1.0 / N:
data[x] += 1.0 / N_trials
else:
NotVisited = set([k for k in range(N)])
NotVisited.discard(x)
while len(NotVisited) > 0:
sigma = random.choice([-1, 1])
x = (x + sigma) % N
NotVisited.discard(x)
data[x] += 1.0 / N_trials
print('stopping samples')
for k in range(N):
print('site = ', k,' probability = ', data[k])
Output
Here is output of the above Python program for N=6. We do 100,000 trials, and see that the six vertices are output with equal probability.
site = 0 probability = 0.1673 site = 1 probability = 0.1671 site = 2 probability = 0.1645 site = 3 probability = 0.1657 site = 4 probability = 0.1656 site = 5 probability = 0.1696
Further Information
- What works like a charm for the random walk on the cycle fails for all other graphs except the complete graph (see Lovász and Winkler (1993).
- The coupling from the path approach discussed in Lecture 2 is much more robust.
References
- Lovász, L., Winkler, P., On the last new vertex visited by a random walk, J. Graph Theory 17, 593 (1993)
- Levin, D. A., Peres, Y. & Wilmer, E. L. Markov Chains and Mixing Times (American Mathematical Society, 2008)
