An array evolves discretely in time following the equation
$$c_{i,j}(t+1)=\frac{p}{4}[c_{i-1,j}(t)+c_{i,j-1}(t)+c_{i+1,j}(t)+c_{i,j+1}(t)]+(1-p)c_{i,j}(t)$$
And the initial condition is $c_{i,j}(0)=\delta_i\delta_j$. I thought the result should have the same symmetry as the matrix
$$\pmatrix{0 & p/4 & 0 \\ p/4 & 1-p & p/4\\0 & p/4 & 0}$$
But apparently it has circular symmetry, at least around the origin. Is there an intuitive way to understand this?
Your evolution equation is morally very similar to a diffusion equation $$ \frac{\partial c(x,y)}{\partial t}=\nabla^2c(x,y)$$ So it is not surpring that it shows similar behavior, namely any initial condition will spread in time to a Gaussian distribution with circular symmetry.