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 !