Approximating a correlated random walk on a 2D grid

random walk

I have been working on modeling the movement of ants, and a number of studies suggest the use of correlated random walks. These are biased random walks, where the direction of the next step is dependent on the direction of the previous step. The way this is usually modeled is the agent (ant) can take steps of length $r$, and the direction of next step $\theta$ is chosen relative to the direction of the previous step. This turning angle is drawn from a linear normal distribution of mean $0$. Higher standard deviation results in more tortuous paths, as shown in figures below.

Fig 1

Fig 2

I would like to approximate this kind of a random walk on a 2D grid, where the walker can only take $90^\circ$ or $180^\circ$ turns. In the simplest case, this would be something like the following figure.

enter image description here

What should be the relation between the standard deviation in the original random walk and the way the probabilities of moving towards each direction are calculated during a step of the random walk on a 2D grid such that the final walks in both the cases resemble each other? Simply assigning a higher probability to the 'front' direction and lower probabilities to the other directions does simulate a walk where the agent prefers moving 'ahead'. However, in my opinion (which might be wrong), these probabilities need to have a dependence on time steps, and that is something I'm struggling with and would like some help with.

The figures have been taken from here.

Edit: The reference mentioned also states that for standard deviations greater than $5$, we essentially have a random walk with no correlation.

Best Answer

I think one of the easier ways of doing this is calculating the individual probabilities of every angle using the probability density formula for Gaussian distribution. In the grid, the random walker can take steps only in 90$^{\circ}$ and 180$^\circ$. Diagonal steps ($45^\circ$, $135^\circ$, etc) can be added as well. The probability of the random walker going in one of the directions can be calculated from $$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp\left[\left(-\frac12\right)\left(\frac{x-\mu}{\sigma}\right)^2\right]$$

where $x$ is the angle. This is one of the solutions that I see of approximating a correlated random walk on a 2D grid.

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