Simulating a 2D Random Walk

How can I turn my 1D Random Walk code into a 2D Random Walk.

Heres my code for the 1D Random walk:

N = 100; % Number of steps
single_trajectory = simulateRandomWalk(N);
figure
plot(0:N, single_trajectory, '-o')
title('1D Random Walk - Single Trajectory')
xlabel('Steps')
ylabel('Position')
P_values = [50, 500, 5000, 50000, 500000];
for p_index = 1:length(P_values)
    P = P_values(p_index);
    final_positions = zeros(1, P);
    
    
    for i = 1:P
        single_trajectory = simulateRandomWalk(N);
        
        final_positions(i) = single_trajectory(end);
        
    end
    
    
    % Plot histogram of final positions
    figure
    
    histogram(final_positions, 'Normalization', 'probability')
    
    title(['Histogram for P = ' num2str(P)])
    
    xlabel('Final Position')
    
    ylabel('Probability')
    
end
function trajectory = simulateRandomWalk(N)
% Initialize position
position = 0;
% Initialize trajectory array
trajectory = zeros(1, N+1);
trajectory(1) = position;
% Simulate random walk
for step = 1:N
    
    % Generate a random number to decide left or right movement
    move = randi([0, 1])*2 - 1; % -1 for left, 1 for right
    
    
    % Update position
    position = position + move;
    
    
    % Store the current position in the trajectory array
    
    trajectory(step+1) = position;
    
end
end

[Walter Roberson]

Already answered at https://www.mathworks.com/matlabcentral/answers/2057329-simulating-a-2d-random-walk#answer_1367059

Answer 1

[Nipun]

Hi William,

I understand that you have one dimensional random walk code and intend to modify the given snippet to yield a two dimensional random walk simulation.

Based on the function "simulateRandomWalk", I assume that for two dimensional random walk, the walker is free to move left, right, up, down

A similar question was answered here: Simulating a 2D Random Walk: - MATLAB Answers - MATLAB Central (mathworks.com)

I will start by changing the "simulateRandomWalk" code to include position changes in Y-coordinate.

function trajectory = simulateRandomWalk(N, position)
%% the starting X and Y coordinates are in position structure
% position has properties: X, Y
% Initialize trajectory array
%% the trajectory is a structure with X and Y coordinates
trajectory.X = zeros(1, N+1);
trajectory.Y = zeros(1,N+1);
trajectory.X(1) = position.X;
trajectory.Y(1) = position.Y;
% Simulate random walk
for step = 1:N
    
    %% Generate a random number to decide left, right, up, down
    % generate a random number between 1 and 4
    % 1 = left, 2 = right, 3 = up, 4 = down
    pos = randi([1,4]);
    
    switch pos
        case 1
            % left
            position.X = position.X - 1;
        case 2
            % right
            position.X = position.X + 1;
        case 3
            % up
            position.Y = position.Y + 1;
        case 4
            % down
            position.Y = position.Y - 1;
    end
    
    %% Store the current position in the trajectory array
    % update both coordinates
    
    trajectory.X(step+1) = position.X;
    trajectory.Y(step+1) = position.Y;
    
end

Now, to plot the random walk, we can use "plot" function, similar to the one dimensional random walk.

% starting coordinates
start.X = 0;
start.Y = 0;
steps = 1000;
t = simulateRandomWalk(steps, start);
%% plot
% Indicate starting position with a circle
plot(0,0,'-ro');
hold on;
plot(t.X,t.Y,'-b');
title("Random walk with " + num2str(steps) + " steps");

Here's an output of the code:

Similarly, use the histogram function to map a histogram of "t.X" and "t.Y"

hold off;
histogram(t.X);
histogram(t.Y);

Refer to the following links for more information on MATLAB plot and histogram function. Hope this helps.

1. MATLAB line plot: https://www.mathworks.com/help/matlab/ref/plot.html
2. MATLAB histogram : https://www.mathworks.com/help/matlab/ref/matlab.graphics.chart.primitive.histogram.html

Regards,

Nipun