Random Walk in Python

Learn how to use Python to make a Random Walk

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A random walk can be thought of as a random process in which a token or a marker is randomly moved around some space, that is, a space with a metric used to compute distance. It is more commonly conceptualized in one dimension ($\mathbb{Z}$), two dimensions ($\mathbb{Z}^2$) or three dimensions ($\mathbb{Z}^3$) in Cartesian space, where $\mathbb{Z}$ represents the set of integers. In the visualizations below, we will be using scatter plots as well as a colorscale to denote the time sequence of the walk.

Random Walk in 1D¶

The jitter in the data points along the x and y axes are meant to illuminate where the points are being drawn and what the tendency of the random walk is.

In [1]:

import plotly.graph_objects as go
import numpy as np
np.random.seed(1)

l = 100
steps = np.random.choice([-1, 1], size=l) + 0.05 * np.random.randn(l) # l steps
position = np.cumsum(steps) # integrate the position by summing steps values
y = 0.05 * np.random.randn(l)

fig = go.Figure(data=go.Scatter(
    x=position,
    y=y,
    mode='markers',
    name='Random Walk in 1D',
    marker=dict(
        color=np.arange(l),
        size=7,
        colorscale='Reds',
        showscale=True,
    )
))

fig.update_layout(yaxis_range=[-1, 1])
fig.show()

Random Walk in 2D¶

In [2]:

import plotly.graph_objects as go
import numpy as np

l = 1000
x_steps = np.random.choice([-1, 1], size=l) + 0.2 * np.random.randn(l) # l steps
y_steps = np.random.choice([-1, 1], size=l) + 0.2 * np.random.randn(l) # l steps
x_position = np.cumsum(x_steps) # integrate the position by summing steps values
y_position = np.cumsum(y_steps) # integrate the position by summing steps values

fig = go.Figure(data=go.Scatter(
    x=x_position,
    y=y_position,
    mode='markers',
    name='Random Walk',
    marker=dict(
        color=np.arange(l),
        size=8,
        colorscale='Greens',
        showscale=True
    )
))

fig.show()

Random walk and diffusion¶

In the two following charts we show the link between random walks and diffusion. We compute a large number N of random walks representing for examples molecules in a small drop of chemical. While all trajectories start at 0, after some time the spatial distribution of points is a Gaussian distribution. Also, the average distance to the origin grows as $\sqrt(t)$.

In [3]:

import plotly.graph_objects as go
import numpy as np

l = 1000
N = 10000
steps = np.random.choice([-1, 1], size=(N, l)) + 0.05 * np.random.standard_normal((N, l)) # l steps
position = np.cumsum(steps, axis=1) # integrate all positions by summing steps values along time axis