# How to plot a Random walk in 2D ?

Hello!

I'm trying to write a code on a random walk in 2d. This random walk is biased: the probability to choose the next step (vertex) depends on the number of time the walker has already chosen this vertex.

``````
Next_step = function(i,j,Grille){

l = nrow(Grille)
prob = rep(NA,4)

prob[1] = Grille[i+1,j]/(Grille[i,j-1]+Grille[i,j+1]+Grille[i-1,j]+Grille[i+1,j])
prob[2] = Grille[i,j+1]/(Grille[i,j-1]+Grille[i,j+1]+Grille[i-1,j]+Grille[i+1,j])
prob[3] = Grille[i,j-1]/(Grille[i,j-1]+Grille[i,j+1]+Grille[i-1,j]+Grille[i+1,j])
prob[4] = Grille[i-1,j]/(Grille[i,j-1]+Grille[i,j+1]+Grille[i-1,j]+Grille[i+1,j])
next_step = sample(1:4, 1,replace = TRUE, prob = prob)

if (next_step == 1) { # en bas
i = i+1
} else if (next_step == 2) {  # à droite
j = j+1
} else if (next_step == 3) { # à gauche
j= j-1
} else if (next_step == 4) { # en haut
i = i-1
}
return(c(i,j))
}

RW_2d = function(nRow,nCol,pas){

#nRow = 6
#nCol = 6

Grille = matrix(1, nrow = nRow+2, ncol = nCol+2)
Grille[c(1,nRow+2),] = 0
Grille[,c(1,nRow+2)]=0

axe_x = rep(NA,pas)
axe_y = rep(NA,pas)
d = rep(NA,2)

axe_x[1] = sample(1:nRow,1)
axe_y[1] = sample(1:nCol,1)
Grille[axe_x[1],axe_y[1]] = Grille[axe_x[1],axe_y[1]]+1
for (i in 2:pas){
d[1:2]= Next_step(axe_x[i-1],axe_y[i-1],Grille)
axe_x[i]=d[1]
axe_y[i]=d[2]
Grille[d[1],d[2]] = Grille[d[1],d[2]]+1
}
plot(axe_x,axe_y,type = 'l')
return(Grille)
}

``````

I want to plot this with colors like in this image but I don't know how?

Does anyone have an idea?

Moreover, how can I represent the number of passages by each vertex with "heat map" or another function?

Thank you !

For plotting, the best is to have the function `RW_2d()` return the data, rather than plotting inside the function. So I changed the last two rows from

``````plot(axe_x,axe_y,type = 'l')
return(Grille)
``````

To just

``````return(list(path = data.frame(step = seq_along(axe_x), x = axe_x, y = axe_y),
Grille = Grille))
``````

So now I can run the function:

``````res <- RW_2d(50,50,250)
``````

This is a list containing, first, a data.frame with the path followed. We can use it to plot. I couldn't figure out how to plot the colors properly with base graphics (though I'm sure it's possible), but it's easy enough using `{ggplot2}` (needs to be installed):

``````res_path <- res\$path
ggplot(res_path) +
theme_classic() +
geom_path(aes(x = x, y = y, color = step)) +
scale_color_steps(low = "blue",
high = "red")
``````

And for the grid:

``````image(res\$Grille)
``````

Or a potentially better-looking representation:

``````pheatmap::pheatmap(res\$Grille,
cluster_rows = FALSE,
cluster_cols = FALSE)
``````

(requires the package `{pheatmap}`)

thank you this is very helpful !

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