2D Random Walk

I am trying to simulate a 2D random walk.

But I keep getting errors

I dont know what I did wrong?

This is my code:

% Define the number of steps for two different runs
N1 = 1000;
N2 = 200;

% Initialize the position vectors for the first run
X1 = zeros(1, N1);
Y1 = zeros(1, N1);

% Perform the random walk for the first run
for i = 2:N1
  % Generate a random direction
  direction = rand(1, 2)*2*pi;
  
  % Update the position
  X1(i) = X1(i-1) + cos(direction);
  Y1(i) = Y1(i-1) + sin(direction);
end

% Initialize the position vectors for the second run
X2 = zeros(1, N2);
Y2 = zeros(1, N2);

% Perform the random walk for the second run
for i = 2:N2
  % Generate a random direction
  direction = rand(1, 2)*2*pi;
  
  % Update the position
  X2(i) = X2(i-1) + cos(direction);
  Y2(i) = Y2(i-1) + sin(direction);
end

% Plot the results for 1000 steps
figure;
subplot(2, 1, 1);
plot(X1, Y1, 'LineWidth', 2);
hold on;
plot(X2, Y2, 'LineWidth', 2);
title('2D Random Walk - Trajectories for 1000 Steps');
xlabel('X-axis');
ylabel('Y-axis');
legend('Run 1', 'Run 2');
grid on;

% Plot the histograms for 1000 steps
subplot(2, 1, 2);
histogram(X1, 'FaceColor', 'blue', 'EdgeColor', 'black', 'BinWidth', 0.5);
hold on;
histogram(Y1, 'FaceColor', 'red', 'EdgeColor', 'black', 'BinWidth', 0.5);
histogram(X2, 'FaceColor', 'cyan', 'EdgeColor', 'black', 'BinWidth', 0.5);
histogram(Y2, 'FaceColor', 'magenta', 'EdgeColor', 'black', 'BinWidth', 0.5);
title('Histograms for X and Y - 1000 Steps');
xlabel('Value');
ylabel('Frequency');
legend('X - Run 1', 'Y - Run 1', 'X - Run 2', 'Y - Run 2');
grid on;

% Repeat for 200 steps
% Initialize the position vectors for the third run
X3 = zeros(1, N2);
Y3 = zeros(1, N2);

% Perform the random walk for the third run
for i = 2:N2
  % Generate a random direction
  direction = rand(1, 2)*2*pi;
  
  % Update the position
  X3(i) = X3(i-1) + cos(direction);
  Y3(i) = Y3(i-1) + sin(direction);
end

% Plot the results for 200 steps
figure;
subplot(2, 1, 1);
plot(X3, Y3, 'LineWidth', 2);
title('2D Random Walk - Trajectory for 200 Steps');
xlabel('X-axis');
ylabel('Y-axis');
grid on;

% Plot the histograms for 200 steps
subplot(2, 1, 2);
histogram(X3, 'FaceColor', 'blue', 'EdgeColor', 'black', 'BinWidth', 0.5);
hold on;
histogram(Y3, 'FaceColor', 'red', 'EdgeColor', 'black', 'BinWidth', 0.5);
title('Histograms for X and Y - 200 Steps');
xlabel('Value');
ylabel('Frequency');
legend('X', 'Y');
grid on;

[Torsten]

What are the possible positions after one random step starting from (x,y) ?

Is it (x+1,y+1), (x+1,y-1), (x-1,y+1) and (x-1,y-1) ?

If this is the case, why do you use cos and sin functions to compute the next position ?

Or are you allowed to reach all points on the circle with radius 1 around (x,y) with equal probability ?

Answer 1

[Voss]

rand(1,2) produces a 1x2 vector:

rand(1, 2)

ans = 1×2

0.6280 0.2123

Seems like you want a scalar instead:

rand(1, 1) % or just rand()

ans = 0.5676

Making that change allows the code to run:

% Define the number of steps for two different runs
N1 = 1000;
N2 = 200;

% Initialize the position vectors for the first run
X1 = zeros(1, N1);
Y1 = zeros(1, N1);

% Perform the random walk for the first run
for i = 2:N1
  % Generate a random direction
  direction = rand()*2*pi;
  
  % Update the position
  X1(i) = X1(i-1) + cos(direction);
  Y1(i) = Y1(i-1) + sin(direction);
end

% Initialize the position vectors for the second run
X2 = zeros(1, N2);
Y2 = zeros(1, N2);

% Perform the random walk for the second run
for i = 2:N2
  % Generate a random direction
  direction = rand()*2*pi;
  
  % Update the position
  X2(i) = X2(i-1) + cos(direction);
  Y2(i) = Y2(i-1) + sin(direction);
end

% Plot the results for 1000 steps
figure;
subplot(2, 1, 1);
plot(X1, Y1, 'LineWidth', 2);
hold on;
plot(X2, Y2, 'LineWidth', 2);
title('2D Random Walk - Trajectories for 1000 Steps');
xlabel('X-axis');
ylabel('Y-axis');
legend('Run 1', 'Run 2');
grid on;

% Plot the histograms for 1000 steps
subplot(2, 1, 2);
histogram(X1, 'FaceColor', 'blue', 'EdgeColor', 'black', 'BinWidth', 0.5);
hold on;
histogram(Y1, 'FaceColor', 'red', 'EdgeColor', 'black', 'BinWidth', 0.5);
histogram(X2, 'FaceColor', 'cyan', 'EdgeColor', 'black', 'BinWidth', 0.5);
histogram(Y2, 'FaceColor', 'magenta', 'EdgeColor', 'black', 'BinWidth', 0.5);
title('Histograms for X and Y - 1000 Steps');
xlabel('Value');
ylabel('Frequency');
legend('X - Run 1', 'Y - Run 1', 'X - Run 2', 'Y - Run 2');
grid on;

% Repeat for 200 steps
% Initialize the position vectors for the third run
X3 = zeros(1, N2);
Y3 = zeros(1, N2);

% Perform the random walk for the third run
for i = 2:N2
  % Generate a random direction
  direction = rand()*2*pi;
  
  % Update the position
  X3(i) = X3(i-1) + cos(direction);
  Y3(i) = Y3(i-1) + sin(direction);
end

% Plot the results for 200 steps
figure;
subplot(2, 1, 1);
plot(X3, Y3, 'LineWidth', 2);
title('2D Random Walk - Trajectory for 200 Steps');
xlabel('X-axis');
ylabel('Y-axis');
grid on;

% Plot the histograms for 200 steps
subplot(2, 1, 2);
histogram(X3, 'FaceColor', 'blue', 'EdgeColor', 'black', 'BinWidth', 0.5);
hold on;
histogram(Y3, 'FaceColor', 'red', 'EdgeColor', 'black', 'BinWidth', 0.5);
title('Histograms for X and Y - 200 Steps');
xlabel('Value');
ylabel('Frequency');
legend('X', 'Y');
grid on;

Answer 2

[Walter Roberson]

for i = 2:N1
  % Generate a random direction
  direction = rand(1, 2)*2*pi;
  
  % Update the position
  X1(i) = X1(i-1) + cos(direction);
  Y1(i) = Y1(i-1) + sin(direction);
end

This will fail for the reasons already identified by @Voss

I would like to suggest

for i = 2:N1
  % Generate a random direction
  direction = randi(4,1,1)/2
  
  % Update the position
  X1(i) = X1(i-1) + cospi(direction);
  Y1(i) = Y1(i-1) + sinpi(direction);
end

This will have only select from 4 positions, rather than from infinite positions.