drift.py

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import gamma

trials = 200
steps = 1000
xs = np.arange(steps)
ds = [1, 10, 100]

fig, axes = plt.subplots(1, 3, sharey=True)

for d, ax in zip(ds, axes):

    ys = np.cumsum(np.random.normal(size=(trials, steps, d)), axis=1)
    dists = np.sqrt(np.sum(ys**2, axis=-1))
    
    mean = np.sqrt(2 * xs) * gamma((d + 1) / 2) / gamma(d / 2)

    ax.plot(xs, dists.T, c='k', alpha=0.02)
    ax.plot(xs, mean, c='r')
    ax.set_xlabel('Time')
    ax.set_ylabel('Distance from origin')
    ax.set_title(f'{d} dimensions')

plt.show()