Simulating and Visualizing a Random Walk in Python

Let’s start with the simplest case — moving left or right randomly.

import numpy as np
import matplotlib.pyplot as plt

# Number of steps
n_steps = 500

# Randomly choose steps: +1 (right) or -1 (left)
steps = np.random.choice([-1, 1], size=n_steps)

# Compute position after each step
position = np.cumsum(steps)

# Plot the walk
plt.figure(figsize=(10, 5))
plt.plot(position, lw=2, color='royalblue')
plt.axhline(0, color='black', lw=1, linestyle='--')
plt.title("1D Random Walk Simulation", fontsize=16)
plt.xlabel("Step Number")
plt.ylabel("Position")
plt.grid(True, alpha=0.3)
plt.show()

import numpy as np
import matplotlib.pyplot as plt

n_steps = 1000

# Random angles for each step (in radians)
angles = np.random.uniform(0, 2 * np.pi, n_steps)
x_steps = np.cos(angles)
y_steps = np.sin(angles)

# Compute positions
x = np.cumsum(x_steps)
y = np.cumsum(y_steps)

# Plot the path
plt.figure(figsize=(6, 6))
plt.plot(x, y, lw=1.5, color='royalblue')
plt.scatter(0, 0, color='green', label='Start', zorder=3)
plt.scatter(x[-1], y[-1], color='red', label='End', zorder=3)
plt.title("2D Random Walk", fontsize=16)
plt.xlabel("X position")
plt.ylabel("Y position")
plt.legend()
plt.axis("equal")
plt.grid(alpha=0.3)
plt.show()

import numpy as np
import matplotlib.pyplot as plt

n_walkers = 30
n_steps = 200

plt.figure(figsize=(8, 6))

for i in range(n_walkers):
    steps = np.random.choice([-1, 1], size=n_steps)
    position = np.cumsum(steps)
    plt.plot(position, alpha=0.5)

plt.title("Ensemble of 1D Random Walks")
plt.xlabel("Step Number")
plt.ylabel("Position")
plt.grid(alpha=0.3)
plt.show()

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from IPython.display import HTML

# Parameters
n_steps = 300
step_size = 1

# Initialize positions
x = np.zeros(n_steps)
y = np.zeros(n_steps)

# Random directions for steps (n_steps-1 increments)
angles = np.random.uniform(0, 2*np.pi, n_steps-1)
x[1:] = np.cumsum(np.cos(angles) * step_size)
y[1:] = np.cumsum(np.sin(angles) * step_size)

# Create figure
fig, ax = plt.subplots(figsize=(6,6))
line, = ax.plot(x[:1], y[:1], lw=2, color='royalblue')  # first point
point, = ax.plot([x[0]], [y[0]], 'ro', markersize=6)    # first point as sequence
ax.set_xlim(x.min()-2, x.max()+2)
ax.set_ylim(y.min()-2, y.max()+2)
ax.set_title("Animated 2D Random Walk")
ax.set_aspect('equal')
ax.grid(True, alpha=0.3)

# Update function
def update(frame):
    line.set_data(x[:frame+1], y[:frame+1])
    point.set_data([x[frame]], [y[frame]])  # <-- must be a sequence
    return line, point

# Create animation
ani = FuncAnimation(fig, update, frames=n_steps, interval=30, blit=True)

# Display in Jupyter notebook
HTML(ani.to_jshtml())

import numpy as np
import plotly.graph_objects as go

n_steps = 500
angles = np.random.uniform(0, 2 * np.pi, n_steps)
x = np.cumsum(np.cos(angles))
y = np.cumsum(np.sin(angles))

fig = go.Figure()

fig.add_trace(go.Scatter(
    x=x, y=y,
    mode='lines+markers',
    line=dict(color='royalblue'),
    marker=dict(size=4, color='red'),
    name='Random Walk'
))

fig.update_layout(
    title="Interactive 2D Random Walk",
    xaxis_title="X position",
    yaxis_title="Y position",
    width=600, height=600,
    showlegend=False,
    template='plotly_white'
)
fig.update_xaxes(scaleanchor="y", scaleratio=1)
fig.show()