The random walks start at $(x,y)=(0,0)$ on an infinite chessboard which covers the whole upper plane. Let's say $(0,0)$ is white.
Random walk 1: At every step, I always go up one square, and either one square to the left or to the right with probability $1/2$ for each. (so the displacement occurs only on white squares)
After many random walks 1 stopped at step $t$, the final values of $x$ (position of the end of the paths) are stored and their standard deviation $\sigma_1$ is computed
Random walk 2: On each square of the chessboard, a normal number following $N(0,1)$ is placed randomly. An algorithm computes for every connected path of $t$ steps on white squares the sum of the Gaussian numbers encountered and chooses the path whose corresponding sum is minimal. The end values of $x$ are stored and their standard deviation $\sigma_2$ is computed, for many random walks of length $t$ (the same amount of samples than for random walk $1$).
Should $\sigma_1$ and $\sigma_2$ differ ?
Context: I am asking this question because I was tasked to design an algorithm which picks the path minimizing the sum of Gaussian numbers, and wanted to check my results with a simpler problem which I think is equivalent. I programmed the two algorithms, and get different values for the $\sigma_i$, although I can't see how a significant difference can be justified.
Actually, for the random walk $1$, I didn't need to even code anything, I could solve it using combinatorics too. A random path is basically picked uniformly. All paths have the same length, but there are more paths leading to an end position close to the center. Computing the probabilities I can infer $\sigma_1$ easily.
In random walk $2$, the setting seems to be the same... there can only be one path minimizing the sum, and each path has the same length, so every path has the same probability to be chosen. Using the combinatorics argument, $\sigma_2$ should be the same as $\sigma_1$
So, are my maths wrong or is my algorithm faulty ?