Top to random eigenvalues.py

From Werner KRAUTH

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).

Python program

import numpy as np
import itertools
import scipy.linalg as la

def factorial(n):
    return 1 if n == 0 else (0 if n == 0 else factorial(n - 1) * n)

for N in [2, 3, 4, 5, 6, 7]:
    FacN = factorial(N)
    ConfCopy = [0] * N
    print(N, 'N', FacN)
#
#   Setup of transition matrix
#
    P   = np.zeros((FacN, FacN))
    Conf = [k for k in range(N)]
    ConfList = list(itertools.permutations(Conf))
    for Conf in ConfList:
        i = ConfList.index(tuple(Conf))
        ConfCopy[:] = Conf
        a = ConfCopy.pop(0)
        for k in range(N):
            TargetConf = ConfCopy[0:k] + [a] + ConfCopy[k:N - 1]
            j = ConfList.index(tuple(TargetConf))
            P[i][j] = 1.0 / float(N)
    eigvals, eigvecsl, eigvecsr = la.eig(P, left=True)
    eigvals.sort()
    stats = [0] * (N+1)
    for a in eigvals:
        index =  int(N * a.real + 0.5)
        stats[index] += 1
    print(stats)