import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns


sns.set(style='white', palette='colorblind', context='talk')
cpal = sns.color_palette('colorblind')


import numpy as np

def random_walk_on_sphere(n, T, epsilon=0.05):
    # Initialize the position on the sphere
    position = np.random.randn(n)
    position /= np.linalg.norm(position)

    # Path of the random walk
    path = [position.copy()]

    for _ in range(T):
        # Generate a small random perturbation
        perturbation = np.random.randn(n) * epsilon
        # Move to the new position
        new_position = position + perturbation
        # Normalize to stay on the sphere
        new_position /= np.linalg.norm(new_position)
        position = new_position

        # Store the position
        path.append(position.copy())

    return np.array(path)

# Example of a random walk on a 3-dimensional sphere with 100 steps
n = 20
T = 100
path = random_walk_on_sphere(n, T)


sns.heatmap(np.corrcoef(path))